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Computational Finance (PGP) Area: Finance Instructor: Michael Carter Term: 6 Objective: This hands-on course aims to develop tools and techniques to implement and analyze the core models of modern finance, as applied in asset pricing and risk management. Financial models will be implemented in Excel, supplemented where appropriate by Visual Basic for Applications (VBA). (It is important to appreciate that it is primarily a course in computation, not a course in programming. Excel and VBA are the vehicles, not the objective.) The course will both enhance understanding of the theory and provide relevant tools for practitioners. The aim is not to produce programmers, but to enable managers to become informed users of the numbers produced for them by others. Outline: Fixed income assets 1. Basic bond pricing 2. Interest rate swaps 3. Estimating the term structure of interest rates Derivatives: lattice methods 4. Binomial and trinomial trees 5. Implied trees 6. American options Derivatives: simulation 7. Random number generation 8. Variance reduction 9. American options 10. Low-discrepancy sequences 11. Non-Gaussian processes Derivatives: Exotic options 12. Asian options 13. Barrier options 14. Basket and spread options 15. Variance swaps Interest rate derivatives 16. Black’s model for bond options, caps and swaptions 17. The Black-Derman-Toy model
Transcript
Page 1: Pg p 12 Handbook

Computational Finance (PGP)

Area: Finance Instructor: Michael Carter Term: 6 Objective: This hands-on course aims to develop tools and techniques to implement and analyze the core models of modern finance, as applied in asset pricing and risk management. Financial models will be implemented in Excel, supplemented where appropriate by Visual Basic for Applications (VBA). (It is important to appreciate that it is primarily a course in computation, not a course in programming. Excel and VBA are the vehicles, not the objective.) The course will both enhance understanding of the theory and provide relevant tools for practitioners. The aim is not to produce programmers, but to enable managers to become informed users of the numbers produced for them by others. Outline:

Fixed income assets 1. Basic bond pricing 2. Interest rate swaps 3. Estimating the term structure of interest rates

Derivatives: lattice methods 4. Binomial and trinomial trees 5. Implied trees 6. American options

Derivatives: simulation 7. Random number generation 8. Variance reduction 9. American options 10. Low-discrepancy sequences 11. Non-Gaussian processes

Derivatives: Exotic options 12. Asian options 13. Barrier options 14. Basket and spread options 15. Variance swaps

Interest rate derivatives 16. Black’s model for bond options, caps and swaptions 17. The Black-Derman-Toy model

Page 2: Pg p 12 Handbook

Pedagogy: Course meetings will combine lecture and practical work. Students must bring their laptop computer to each session. Sessions required: 25 Assessment: Assignments (2) 50% Final exam 40% Homework 10% Prerequisites: Core courses only. Restriction on class size: 30 Relationship with other courses: There is potential overlap with (a) Futures, Options and Risk Management and (b) Fixed Income Securities. However, the computational focus of this course differentiates it from other courses. Bibliography: There is no specific text for the course. Course materials are provided online. Additional student expenses: Nil.

Page 3: Pg p 12 Handbook

PV Present value of an annuity

NPV Net present value of periodic cash flows

FV Future value of an annuity

RATE Rate of return of an annuity

IRR Internal rate of return of periodic c

PRICE Price of a coupon bond

PRICEDISC Price of a discount bond

TBILLPRICE Price of T-bill (special case of PRICEDISC)

YIELD Yield of coupon bond

YIELDDISC Yield of discount bond

TBILLYIELD Yield of T-bill

ACCRINT Accrued interest

COUPNUM Number of coupons remaining

COUPNCD Next coupon date

COUPPCD Previous coupon date

COUPDAYS Number of days in current coupon

COUPDAYBS Number of days between previous settlement

COUPDAYSNC Number of days between settlemencoupon

DURATION Duration of a coupon bond MDURATION Modified duration

EFFECT Effective annual interest rate TBILLEQ Bond equivalent yield of a T-bill

Useful financial functions in Excel

Page 4: Pg p 12 Handbook

Formula auditing toolbar

The Formula Auditing Toolbar enables you to trace graphically the relationships between cells. It also allows you to monitor cell contents by placing them in a Watch Window.

To display the formula auditing toolbar

View > Toolbars > Formula auditing

To trace a cell's precedents

1. Select a cell containing a formula 2. Click on the Trace Precedents button 3. Click on the Trace Precedents button again to display the previous level of precedents. 4. Remove tracer arrows one level at a time by clicking Remove Precedent Arrows

To trace a cell's dependents

1. Select a cell containing a formula 2. Click on the Trace Dependents button 3. Click on the Trace Dependents button again to display the previous level of dependents. 4. Remove tracer arrows one level at a time by clicking Remove Dependent Arrows

To select the cell at the other end of an arrow

Double click the arrow

To remove all tracer arrows

Click the Remove All Arrows button.

To display all the relationships in a worksheet

1. In an empty cell, type = 2. Then click the Select All button and evaluate the cell with Ctrl-Enter 3. Click the Trace Precedents button twice.

To display a formula in a cell

Select the cell and press F2

Page 5: Pg p 12 Handbook

To display all formulae

Click Ctrl-~

To add a cell to the watch window

1. Open the Watch Window by clicking on the Watch Window button in the Formula Auditing Toolbar. 2. Select the cells you want to monitor. 3. Click on the Add Watch button in the Watch Window.

Page 6: Pg p 12 Handbook

Basic bond pricingIn principal, pricing a risk-free bond is deceptively simple - the price or value of a bond is the present value of the futurecash flows, discounted at the prevailing rate of interest, which is known as the yield.

P = ât=1

T

C1

1 + r

t

+ R1

1 + r

T

where P is the price, C is the coupon, R is the redemption value (principal) and T is the term. Alternatively, the yield of abond is the internal rate of return at which the discounted value is equal to market price. Bonds are known as fixed income

assets, because the timing and magnitude of the future cash flows are fixed. Their value however varies inversely with theyield. Bonds of similar risk and term will attract similar yields.

In practice, bond pricing is more complicated because

æ coupons are paid more frequently than annually, typically every six months.

æ a price is required between coupon periods necessitating discounting for fractional periods.

æ interest rates (yields) may be expected to change during the term of the bond.

The first complication is dealt with by treating the coupon period (e.g. 6 months) as the discounting period. If there are mcoupons per year,

P = ât=1

m T C

m

1

1 +r

m

t

+ R1

1 +r

m

m T

Treatment of fractional periods is a matter of market convention. In particular, various markets employ different day

count conventions for calculating the fraction of the coupon period which as elapsed on a given day. Similar conventionsare employed for pricing zero coupon bonds. However, zero coupon bonds issued with a maturity less than one year(notes) are priced with yet another convention. Computation of bond prices and yields requires being familiar with theprevailing conventions.

Changing interest rates (the yield curve) can be accommodated by discounting each cash flow at the appropriate spot rate.Credit risk can be incorporated in a simple way by discounting at a higher rate than the yield on risk-free bonds. Thisdifference, known as the spread, depends upon the credit rating of the issuer. More sophisticated measures employ creditrisk models to allow for the possibility of default and ratings changes during a given horizon. Sophisticated measures willalso account directly for the options embedded in many bonds, as for example in a callable bond.

Page 7: Pg p 12 Handbook

Day count conventions

Coupons  Day count Basis SettlementCorporate bondsGermany +2India Actual/ 365 3Japan 2UK 2 Actual/Actual 1 +7US 2 30/360 0 +2

Government bondsGermany 1 Actual/365 3 +2India 2 30/360 0 +1 Basis Day countJapan 2 Actual/365 3 0 US (NASD) 30/360UK 2 Actual/Actual 1 +1 1 Actual/actualUS 2 Actual/Actual 1 +1 2 Actual/360US ‐ municipal 30/360 0 3 Actual/365

4 European 30/360Government billsGermany Actual/360 2 +2India Actual/365 3 +2US Actual/Actual 1 +1

Money market (LIBOR)Germany Actual/360 2India Actual/365 3Japan Actual/360 2UK Actual/365 3US Actual/360 2

Eurolibor 30/360 0

Basis is an Excel parameterencoding the date count.

From RBI FAQs

Bond market: The day count convention followed is 30/360, which means that irrespective of the actual number of days in a month,  the number of days in a month is taken as 30 and the number of days in a year is taken as 360.

Money market: The day count convention followed is actual/365, which means that the actual number of days in a month is taken for number of days(numerator) whereas the number of days in a year is taken as 365 days.  Hence, in the case of Treasury bills, which are essentially money market instruments, money market  convention is followed.

Day counts are market conventions, which are subject to change and exceptions.The information collected here has been assembled from a variety of sources. It is tentative and provided for educational purposes only. The information needs to be verified before being used for commercial purposes

Page 8: Pg p 12 Handbook

Duration and convexity

Duration and SensitivityAssuming annual coupons, the price of a coupon bond is the discounted value of cash flows

P = ât=1

T

C1

1 + r

t

+ R1

1 + r

T

where P is the full or dirty price, C is the annual coupon, R is the redemption value and T is the term. This

can be rewritten as

P = ât=1

T

C H1 + rL-t+ R H1 + rL-T

Differentiating with respect to the yield gives

¶ P

¶r= â

t=1

T

-t C H1 + rL-t-1- T R H1 + rL-T-1

which can be written as

(1)¶ P

¶r= -

1

1 + rât=1

T

C1

1 + r

t

t + R1

1 + r

T

T

The (Macauley) duration of the bond is

Dur = ât=1

T C I 1

1+rMt

Pt +

R I 1

1+rMT

PT

so that

(2)P � Dur = ât=1

T

C1

1 + r

t

t + R1

1 + r

T

T

which is precisely the term inside the brackets in equation (1). Substituting equation (2) into equation (1)

gives

¶ P

¶r= -

1

1 + rDur � P

Page 9: Pg p 12 Handbook

With m coupons per year, this becomes (see below)

¶ P

¶r= -

1

1 +r

m

Dur � P

To simplify, we call the product on the left modified duration. That is, defining

MDur =

1

1 +r

m

Dur

we have

dP

dr= -MDur � P

For small changes in interest rate, we have

DP

Dr» -MDur � P

or

DP

P» -MDur � Dr

A one percentage point increase in yield will lead to (approx.) MDur fall in price.

Practitioners often express duration (that is, interest-rate sensitivity) in terms of the dollar value of a basis

point (DV01) or more generally price value of a basis point (PV01). This is defined as

PV01 = MDur � P � 0.01 � 0.01

Note that, strictly speaking, it is the invoice or dirty price that should be used for P in this calculation.

à Multiannual coupons

If there are m coupons per year, the price of a bond is

P = ât=1

m T C

m

1

1 +r

m

t

+ R1

1 +r

m

m T

= ât=1

m T C

m1 +

r

m

-t

+ R 1 +

r

m

-m T

Differentiating with respect to the yield

¶ P

¶r= - â

t=1

m T

tC

m1 +

r

m

-t-1 1

m- m T R 1 +

r

m

-m T-1 1

m

= -

1

I1 +r

mM

ât=1

m T t

m

C

m

1

1 +r

m

t

+ T R1

1 +r

m

m T

2 Duration.nb

Page 10: Pg p 12 Handbook

= -

1

I1 +r

mM

ât=1

m T t

m

C

m

1

1+r

m

t

P+ T

R1

1+r

m

m T

PP

= -

1

I1 +r

mM

Dur � P

= mDur � P

A closed formula for durationInverting the previous equation, the duration of a bond is

(3)Dur = -

I1 +r

mM

P

¶ P

¶r

where

P = ât=1

m T C

m

1

1 +r

m

t

+ R1

1 +r

m

m T

By summing the geometric series, the price of the bond can be written in closed form as

P =

C

r1 -

1

I1 +r

mMm T

+

1

I1 +r

mMm T

R

Differentiating this expression and substituting in (2), we obtain a closed formula for the duration of a

bond

(4)Dur =

1 +r

m

r-

T I C

R- r M + I1 +

r

mM

J C

RJI1 +

r

mMm T

- 1N + r N

When the bond is at par, C � R = r , and this simplifies to

Dur =

1 +r

m

r1 -

1

I1 +r

mMm T

The limit of duration for long term bondsAs T goes to infinity, the second term in equation (3) goes to zero. Therefore, the duration of a long-term

bond converges to

Duration.nb 3

Page 11: Pg p 12 Handbook

limT ® ¥

Dur =

1 +r

m

r

For example, with a yield of 5%, the duration of a biannual converges to 1+

5 %

2

5 %= 20.5

ConvexityDuration is related to the first derivative of bond price with respect to yield. Convexity is a measure of the

second derivative, normalised by bond price.

C =

â2P

âr2

P

Though it is possible to derive a formula for convexity, by differentiating the above formula for â P � â r,

we would need to incorporate the complications date count conventions for mid-coupon bonds. Alterna-

tively, we can estimate convexity accurately by numerical differentiation

C =

PHr + â rL - 2 PHrL + PHr - â rL

P â r2

where âr is a small change in interest rate (e.g. 0.0001 for 1 basis point).

Alternatively, we can compute convexity from the first derivative of duration (this is useful if we have a

formula for duration, as in Excel). From above

P C =

â2 P

â r2=

â I âP

ârM

â r

But

â P

â r= - P D

where D is modified duration. Substituting and using the product rule

P C =

=

â H-P DL

â r

= -Dâ P

â r- P

â D

â r

= P D2- P

â D

â r

so that

C = D2-

â D

â r

4 Duration.nb

Page 12: Pg p 12 Handbook

â D � â r can itself be calculated by numerical differentiation.

Duration.nb 5

Page 13: Pg p 12 Handbook

Numerical differentiationThe derivative of a function f HxL is

f ' HxL = lim„xØ0

f Hx + „ xL - f HxL„ x

An obvious method to approximate the derivative is to compute

∑ f

∑ xº

f Hx + „ xL - f HxL„ x

for small „ x. This is known as the forward difference. A better alternative (though more costly to compute) is

∑ f

∑ xº

f Hx + „ xL - f Hx - „ xL2 „ x

which is known as the central difference.

Using central differences, the second derivative can be estimated by

∑2 f

∑ x2º

f ' Ix + 12„ xM - f ' Ix - 1

2„ xM

„ x=

f Hx+„xL- f Hx L„x

-f HxL- f Hx-„xL

„x

„ x=

f Hx+„xL- f Hx L„x

-f HxL- f Hx-„xL

„x

„ x

=f Hx + „ xL - 2 f Hx L + f Hx - „ xL

H„ xL2

Numerical Recipes (Press et al., 2007: 229) discuss the numerical issues in computing numerical derivatives. Inparticular, choice of „x can be crucial.

Page 14: Pg p 12 Handbook

Interest rates swapsThe present value of the floating side of a swap (assuming a notional principal of one) is

PVfloating = âi=1

n

∆i DTi ri-1,i

where ∆i is the discount factor at the end of period i, DTi is the elapsed time adjusted for day count, ri-1,i is

forward rate fixed at the end of period i-1 and payable at the end of period i, and n is the number of floatingrate payments. The forward rate is

ri-1,i =

∆i-1

∆i- 1

DTi

Substituting

PVfloating = âi=1

n

∆i DTi

∆i-1

∆i- 1

DTi

= âi=1

n

∆i

∆i-1

∆i

- 1

= âi=1

n

H∆i-1 - ∆iL = H∆0 - ∆1L + H∆1 - ∆2L + º + H∆n-1 - ∆nL

= 1 - ∆n

since ∆0 = 1. That is

(1)PVfloating = 1 - ∆n

Let N denote the number of fixed payments, ∆ j the discount factor applicable to the jth fixed payment, and DT j

the time over which the jth fixed payment is accrued. The present value of the fixed side (assuming a notionalprincipal of one) and annual fixed payments

(2)PVfixed = âj=1

N

∆ j DT j sN = sN QN

where

QN = âj=1

N

∆ j DT j = QN-1 + ∆N DTN

Note that SN DT j is the “dollar” amount of the fixed payment at time T j.

Consequently, the net present value of a swap is

(3)NPV = HsN QN - H1 - ∆N L L ´ Principal

At fair value (NPV = 0)

1 - ∆N = sN QN

= sN HQN-1 + ∆N DTN L

Page 15: Pg p 12 Handbook

= sN QN-1 + sN ∆N DTN

Assuming previous discount factors ∆1 º∆N-1 are determined, so is QN-1, and we can solve for ∆N

(4)∆N =

1 - sN QN-1

1 + sN DTN

where

QN-1 = âj=1

N-1

∆ j DT j = QN-2 + ∆N-1 DTN-1

This provides a general bootstrapping procedure from inferring discount factors from swap rates.

Generalising, consider a forward swap starting in period t + 1 and ending in period T . The present value of thefloating side is

PVfloating = âi=t+1

T

∆i DTi

∆i-1

∆i- 1

DTi

= âi=t+1

T

∆i

∆i-1

∆i

- 1

= âi=t+1

T

H∆i-1 - ∆iL = H∆0 - ∆1L + H∆1 - ∆2L + º + H∆n-1 - ∆nL

= ∆t - ∆T

The present value of the fixed side (assuming a notional principal of one) is

PVfixed = âj=t+1

T

∆ j DT j sN = sN âj=1

T

∆ j DT j - âj=1

t

∆ j DT j = sN HQT - QtL

where the sum is taken over all fixed payments between times t + 1 and T . Equating fixed and floating sides,the forward swap rate (the fixed rate of a swap starting in t+1 and ending in period T) is given by

(5)sN =

∆t - ∆T

QT - Qt

2 InterestRateSwaps.nb

Page 16: Pg p 12 Handbook

BootstrappingWith annual compounding, the price of a unit par bond with n years remaining is given by

c P1 + c P2 + … + c Pi-1 + H1 + cL Pt = 1

where c is the coupon (yield) and Pi is the discount factor (price of a t-year zero-coupon bond). This can be solved succes-

sively to give the prices of zero-coupon bonds to match a given yield curve.

Pt =

1 - c Úi=1t-1 Pi

1 + c

For semi-annual coupons, the analogous equations are

c

2P 1

2

+

c

2P1 + … +

c

2P

t-1

2

+ 1 +

c

2Pt = 1

and

Pt =

1 -c

i=1

2

t-1

2 Pi

1 +c

2

Page 17: Pg p 12 Handbook

Estimating the term structure

The basic bond pricing equation is

(1)P = âi=1

n C �m

H1 + sti �mLm ti+

R

H1 + stn �mLm tn

where

P = price Hfull or dirty priceLC = annual coupon

R = redemption payment HprincipalLm = frequency of coupons

n = number of remaining coupons

This can be written in terms of the discount factors

P =

C

mâi=1

n

∆ti + ∆tn R

where

∆ti =

1

1 + sti �m

m ti

The spot rates or discount factors also determine the forward rates. Let rti denote the forward (short) rate

1 +

s

m

m ti

= 1 +

s

m

m ti-1

1 +

rti

m

so that

1 +

rti

m=

I1 +s

mM

m ti

I1 +s

mM

m ti-1=

∆ti-1

∆ti

rti = m∆ti-1 - ∆ti

∆ti

= mD∆ti

∆ti

If there is an active market in zero-coupon bonds, these can be used to give immediate market estimates of

the discount rate at various terms. However, such instruments are traded only in the U.K. and U.S.

treasury markets. Moreover, even in these markets, they are usually disregarded because of restricted

maturities, limited liquidity and tax complications.

In principle, discount factors H∆ti L can be inferred from the prices of coupon bonds inverting (1). In turn,

these can be used to infer the spot rate (zti ) and forward rate Hrti L curves. The inversion process is known

as bootstrapping.

In practice, estimation of the spot rate curve is complicated by two basic problems:

Page 18: Pg p 12 Handbook

æ Bonds of the same maturity may be selling at different yields, due to market imperfections,

limited liquidity, tax etc.

æ There may be no data on bonds of other maturities.

These problems are tackled (with varying degrees of success) by statistical estimation and interpolation.

The basic approach is to assume a specific functional form for the forward rate or discount function, and

then adjust the parameters until the best fit is obtained. Simple polynomial functions such as

(2)f HtL = Α0 + Α1 t + Α2 t2+ Α3 t3

have been found not to be very suitable, since they imply that rates go to plus or minus infinity as t ® ¥.

Two basic generalizations are found - exponential functions and polynomial or exponential splines.

à Parsimonious functional forms

The most straightforward generalization of (2) is to substitute an exponential for each power of t, fitting a

model of the form

f HtL = Α0 + Α1 ã-k1 t

+ Α2 ã-k2 t

+ Α3 ã-k3 t

+ …

This is the exponential yield model adopted by J.P Morgan.

The most popular model of this form is due to Nelson and Siegel (1987). They observe that the second

order exponential model is the general solution to a second-order differential equation (assuming real

unequal roots)

f HtL = Β0 + Β1 ã-

t

Τ1 + Β2 ã-

t

Τ2

where Τ1, Τ2 are the rates of decay. Finding that this is overparameterized, they adopt the general solution

for the case of equal roots

(3)f HtL = Β0 + Β1 ã-

t

Τ + Β2

t

Τ

ã-

t

Τ

The short rate is Β0 + Β1, while the long rate is lim t® ¥ f HtL = Β0. Β1 can be interpreted as the weight

attached to the short term component, and Β2 as the weight of the medium term. Τ determines the rate of

decay.

The spot rate, the average of the forward rates, can be obtained by integrating this equation, giving

(4)sHtL = à0

t 1

tf HtL â t = Β0 + H Β1 + Β2L

Τ

tJ1 - ã

-t

Τ N - Β2 ã-

t

Τ

Given values for the parameters Β0, Β1, Β3 and Τ, bonds can be valued using the continuous analogue of

(1)

(5)P = âi=1

n

ã-sHtL t

C

m+ ã

-sHtnL tn R

This is the model adopted by the National Stock Exchange of India for estimating its published spot rate

series.

Svennson (1994) extended this specification by adding an additional term for greater flexibility,

specifically

2 EstimatingTermStructure.nb

Page 19: Pg p 12 Handbook

Svennson (1994) extended this specification by adding an additional term for greater flexibility,

specifically

f HtL = Β0 + Β1 ã-

t

Τ1 + Β2

t

Τ1

ã-

t

Τ1 + Β3

t

Τ2

ã-

t

Τ2

The corresponding spot rate curve is

sHtL = Β0 + H Β1 + Β2LΤ1

tK1 - ã

-t

Τ1 O - Β2 ã-

t

Τ1 + Β3

Τ2

tK1 - ã

-t

Τ2 O - Β3 ã-

t

Τ2

This is the model used by the Deutsche Bundesbank for estimating its published spot rate series.

� Example: National Stock Exchange of India

Estimating the Nelson-Siegel model for bonds traded on 26 June 2004 yields the following parameter

estimates

Β0 = 0.0727, Β1 = -0.0231, Β2 = -0.0210, Τ = 2.8601

5 10 15 20

2

3

4

5

6

7

8

Spot

Forward

� Example: Deutsche Bundesbank

For the 15 September 2004, the Deutsche Bundesbank estimated the following parameters for the Svenn-

son model:

Β0 = 5.4596, Β1 = -3.53042, Β2 = -0.37788, Β3 = -0.98812, Τ1 = 2.70411, Τ2 = 2.53479

These parameters imply the following spot rates.

EstimatingTermStructure.nb 3

Page 20: Pg p 12 Handbook

1 2.303112 2.643363 2.946344 3.212465 3.444256 3.645147 3.818868 3.969059 4.0990410 4.21179

The spot and forward curves are illustrated in the following graph.

5 10 15 20

2

3

4

5

6

à Spline

A cubic spline comprises a sequence of cubic functions between chosen points called knots, the coeffi-

cients of the cubic functions chosen so that their first and second derivatives match at each of the knots.

This makes the resulting spline smooth.

Author Instrument Estimation Knots

McCullough Discount LS n

Fisher Forward NLLS n �3

Waggoner Forward NLLS n �3

à Evaluation

A recent comprehensive review by Ioannides (2003) found that the parsimonious functional forms out-

performed corresponding spline methods, with the Svennson specification preferred over that of Nelson

and Siegel. However, we note that the Bank of England recently drew the opposite conclusion, switching

from Svensson's method to a spline method (Anderson and Sleath, 1999).

4 EstimatingTermStructure.nb

Page 21: Pg p 12 Handbook

'Implementation of Nelson-Siegel method for estimating forward rate curve ' Michael Carter, 2004 Function Getformula(ThisCell) Getformula = ThisCell.Formula End Function ' Discount function Function df(t As Double, b0 As Double, b1 As Double, b2 As Double, tau As Double) As Double df = Exp(-t * (b0 + (b1 + b2) * (1 - Exp(-t / tau)) * (tau / t) - b2 * Exp(-t / tau))) End Function 'Bond price function Function Pr(t As Double, C As Double, n As Integer, b0 As Double, b1 As Double, b2 As Double, tau As Double) As Double Dim i As Integer Dim P As Double P = 0 For i = 1 To n P = P + df(t + (i - 1) / 2, b0, b1, b2, tau) * (100 * C / 2) Next i Pr = P + df(t + (n - 1) / 2, b0, b1, b2, tau) * 100 End Function

Page 22: Pg p 12 Handbook

BIS Papers No 25 xi

Table 1

The term structure of interest rates - estimation details

Central bank Estimation method

Minimised error

Shortest maturity in estimation

Adjustments for tax

distortions

Relevant maturity spectrum

Belgium Svensson or Nelson-Siegel

Weighted prices Treasury certificates: > few days

Bonds: > one year

No Couple of days to 16 years

Canada Merrill Lynch Exponential Spline

Weighted prices Bills: 1 to 12 months

Bonds: > 12 months

Effectively by excluding bonds

3 months to 30 years

Finland Nelson-Siegel Weighted prices ≥ 1 day No 1 to 12 years

France Svensson or Nelson-Siegel

Weighted prices Treasury bills: all Treasury

Notes: : ≥ 1 month

Bonds: : ≥ 1 year

No Up to 10 years

Germany Svensson Yields > 3 months No 1 to 10 years

Italy Nelson-Siegel Weighted prices Money market rates: O/N and Libor rates from 1 to 12 months

Bonds: > 1 year

No Up to 30 years

Up to 10 years (before February 2002)

Japan Smoothing splines

Prices ≥ 1 day Effectively by price adjustments for bills

1 to 10 years

Norway Svensson Yields Money market rates: > 30 days

Bonds: > 2 years

No Up to 10 years

Spain Svensson

Nelson-Siegel (before 1995)

Weighted prices

Prices

≥ 1 day

≥ 1 day

Yes

No

Up to 10 years

Up to 10 years

Sweden Smoothing splines and Svensson

Yields ≥ 1 day No Up to 10 years

Switzerland Svensson Yields Money market rates: ≥ 1 day

Bonds: ≥ 1 year

No 1 to 30 years

Page 23: Pg p 12 Handbook

xii BIS Papers No 25

Table 1 cont

The term structure of interest rates - estimation details

Central bank Estimation method

Minimised error

Shortest maturity in estimation

Adjustments for tax

distortions

Relevant maturity spectrum

United Kingdom1

VRP (government nominal)

VRP (government real/implied inflation)

VRP (bank liability curve)

Yields

Yields

Yields

1 week (GC repo yield)

1.4 years

1 week

No

No

No

Up to around 30 years

Up to around 30 years

Up to around 30 years

United States Smoothing splines (two curves)

Bills: weighted prices

Bonds: prices

≥ 30 days

No

No

Up to 1 year

1 to 10 years

1 The United Kingdom used the Svensson method between January 1982 and April 1998.

3. Zero-coupon yield curves available from the BIS

Table 2 provides an overview of the term structure information available from the BIS Data Bank. Most central banks estimate term structures at a daily frequency. With the exception of the United Kingdom, central banks which use Nelson and Siegel-related models report estimated parameters to the BIS Data Bank. Moreover, Germany and Switzerland provide both estimated parameters and spot rates from the estimated term structures. Canada, the United States and Japan, which use the smoothing splines approach, provide a selection of spot rates. With the exception of France, Italy and Spain, the central banks report their data in percentage notation. Specific information on the retrieval of term structure of interest rates data from the BIS Data Bank can be obtained from BIS Data Bank Services.

Page 24: Pg p 12 Handbook

U.S. Treasury - Treasury Yield Curve Methodology

Treasury Yield Curve Methodology

This description was revised and updated on February 9, 2006.

The Treasury’s yield curve is derived using a quasi-cubic hermite spline function. Our inputs are the COB bid yields for the on-the-run securities. Because the on-the-run securities typically trade close to par, those securities are designated as the knot points in the quasi-cubic hermite spline algorithm and the resulting yield curve is considered a par curve. However, Treasury reserves the option to input additional bid yields if there is no on-the-run security available for a given maturity range that we deem necessary for deriving a good fit for the quasi-cubic hermite spline curve. In particular, we are currently using inputs that are not on-the-run securities. These are two composite rates in the 20-year range reflecting market yields available in that time tranche. Previously, a rolled-down 10-year note with a remaining maturity nearest to 7 years was also used as an additional input. That input was discontinued on May 26, 2005.

More specifically, the current inputs are the most recently auctioned 4-, 13- and 26-week bills, plus the most recently auctioned 2-, 3-, 5-, and 10-year notes and the most recently auctioned 30-year bond, plus the off-the-runs in the 20-year maturity range. The quotes for these securities are obtained at or near the 3:30 PM close each trading day. The long-term composite inputs are the arithmetic averages of the bid yields on bonds with 18 - 22 years remaining to maturity; and those with 20 years and over remaining to maturity, each inputted at their average maturity. The inputs for the three bills are their bond equivalent yields.

To reduce volatility in the 1-year CMT rate, and due to the fact that there is no on-the-run issue between 6-months and 2-years, Treasury uses an additional input to insure that the 1-year rate is consistent with on-the-run yields on either side of it’s maturity range. Thus, Treasury interpolates between the secondary bond equivalent yield on the most recently auctioned 26-week bill and the secondary market yield on the most recently auctioned 2-year note and inputs the resulting yield as an additional knot point for the derivation of the daily Treasury Yield Curve. The result of this step is that the 1-year CMT is generally the same as the interpolated rate. Treasury has used this interpolated methodology since August 6, 2004.

Treasury does not provide the computer formulation of our quasi-cubic hermite spline yield curve derivation program. However, we have found that most researchers have been able to reasonably match our results using alternative cubic spline formulas.

Treasury reviews its yield curve derivation methodology on a regular basis and reserves the right to modify, adjust or improve the methodology at its option. If Treasury determines that the methodology needs to be changed or updated, Treasury will revise the above description to reflect such changes.

Yield curve rates are normally available at Treasury’s interest rate web sites as early as 5:00 PM and usually no later than 6:00 PM each trading day.

Office of Debt Management Department of the Treasury

Daily Treasury Yield Curve Rates

Daily Treasury Long-Term Rates

Daily Treasury Real Yield Curve Rates

Daily Treasury Real Long-Term Rates

Weekly Aa Corporate Bond Index

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Page 25: Pg p 12 Handbook

The binomial modelMichael CarterA derivative is an asset the value of which depends upon another underlying asset. Consider the simplestpossible scenario, in which the underlying has two possible future states "up" and "down". The value of thederivative in these two states is Vu and Vd respectively.

Underlying

S

u S

d S

Derivative

V

Vu

Vd

The current value of the derivative is enforced by the possibility of arbitrage between the derivative and theunderlying asset. Consider a portfolio comprising x shares and short one option.

Portfolio

x S - V

x u S - Vu

x d S - Vd

=

By choosing x appropriately, we can make the portfolio risk-free. That is, choosing x so that

x u S - Vu = x d S - Vd

Page 26: Pg p 12 Handbook

we have

x S =Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

ü Exercise

Suppose S = 100, u = 1.05, d = 0.95, Vu = 5 and Vd = 0. Calculate the risk-free hedge. Show that it isrisk-free by comparing the value of the portfolio in the two states.

ü

ü

Substituting for x S, the value of the portfolio at time T in either state is

u x S - Vu = u J Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

N - Vu

=u Vu - u Vd - u Vu + d VuÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

u - d

=d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

u - d

The value of the portfolio at time 0 is

x S - V = d Hu x S - VuL = d ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

y{zzwhere d is the discount factor. Let R = 1 ê d. Solving for V

V = x S - d ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

y{zz=

Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

-1ÅÅÅÅÅÅR

ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

y{zz=

1ÅÅÅÅÅÅR

ikjj R Vu - R Vd - d Vu + u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

y{zz=

1ÅÅÅÅÅÅR

ikjj R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

Vu +u - RÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

Vdy{zz

Letting

p =R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

and 1 - p = 1 -R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

=u - d - R + dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

u - d=

u - RÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

we obtain the fundamental option valuation equation

V =1ÅÅÅÅÅÅR

Hp Vu + H1 - pL VdLThe value of the option at time 0 is the discounted expected value of the payoff, where the expectation is takenwith respect to the synthetic or risk-neutral probabilities (defined above) and discounted at the risk-free rate.

2 BinomialModel.nb

Page 27: Pg p 12 Handbook

This value is enforced by arbitrage. To see this, suppose that option is selling at a premium above its true value.

V >1ÅÅÅÅÅÅR

Hp Vu + H1 - pL VdLAn arbitrageur can sell n options and buy n x shares, borrowing the net cost n Hx S - V L. At time T , the portfoliois worth nHx u S - VuL in the "up" state and (equally) nHx d S - VdL in the "down" state. Repaying the loan plusinterest of R n Hx S - V L, the arbitrageur makes a risk-free profit of

profit = payoff - loan= n Hx u S - VuL - R n Hx S - V L

= n ikjj d Vu - u VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

y{zz - R n J Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

- V N= n R JV -

1ÅÅÅÅÅÅR

Hp Vu + H1 - pL VdLNConversely, if the option is selling at a discount, a risk-free profit can be made by reversing this transaction,buying options and selling shares.

ü Exercise

Suppose S = 100, u = 1.05, d = 0.95, Vu = 5, Vd = 0 and R = 1.01. Calculate the true value of the option.Suppose that the option is priced at 3.10. Find a profitable arbitrage.

ü

ü

ü Remarks

æ R is the risk-free total return for the period T . It is given either by R = 1 + r T or R = ‰r T where r is therisk-free (spot) rate for the period T . It is common to use continuous compounding in option evaluation,although discrete compounding is convenient (and appropriate) for the binomial model.

æ The risk-neutral probabilities p and 1 - p are those probabilities at which the expected growth rate of theunderlying asset is equal to the risk-free rate, that is

p u S + H1 - pL d S = R S

Solving for p,

p Hu - d L S + d S = R Sp Hu - d L S = HR - dL S

p =R - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

In the language of probability, p makes the discounted asset price a martingale.

BinomialModel.nb 3

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ü Exercise

What condition is required to ensure the existence of this equivalent martingale measure (probability)?

æ The current asset price S will depend upon the real probabilities q. The expected rate of return

m =q u S + H1 - qL d S - SÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

S= q u + H1 - qL d - 1

must be sufficient to induce investors to hold the asset.

æ The hedge ratio x is equal to delta of the option, the sensitivity of the option price to changes in the price of theunderlying

x =Vu - VdÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHu - dL S =

D VÅÅÅÅÅÅÅÅÅÅÅÅÅD S

æ For a vanilla call option at maturity with a strike price of K

Vu = max Hu S - K, 0L and Vd = max Hd S - K, 0LFor a vanilla put option at maturity with a strike price of K

Vu = max HK - u S, 0L and Vd = max HK - d S, 0LFor a vanilla European option prior to maturity, Vu and Vd are the discounted expected values of the option inthe "up" and "down" states respectively.

For a vanilla American option prior to maturity, Vu and Vd are the maximum of the intrinsic values and dis-counted expected values of the option in the "up" and "down" states respectively.

4 BinomialModel.nb

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The Black-Scholes formula for stock indices, currencies and futuresMichael CarterThe standard Black-Scholes formula is

c = S0 NHd1L - K ‰-r T NHd2Lwhere

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnHS0 ê KL + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

This can be rewritten as

c = S0 NHd1L - K ‰-r T NHd2L= ‰-r T HS0 „r T NHd1L - K NHd2LL= ‰-r T HF0 N Hd1L - K N Hd2LL

where F0 = S0 ‰r T is the expected forward price of S determined at time 0 under the risk-neutral distribution. Astraightforward proof is given in the appendix.

à Continuous dividend

If the underlying assets pays a continuous dividend yield at the rate q, its forward price is

F0 = S0 ‰Hr-qL T

and therefore the call option value is

c = ‰-r T HS0 ‰-Hr-qL T NHd1L - K NHd2LL = S0 ‰- q T NHd1L - K ‰-r T NHd2Lwith

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnHS0 ê KL + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

à Foreign currency

The forward price of a foreign currency is given by

F0 = S0 ‰Hr-r f L T

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which is known as covered interest parity. Therefore, the value of a foreign currency option is

c = ‰-r T HS0 ‰Hr- f f L T NHd1L - K NHd2LL = S0 ‰- r f T NHd1L - K ‰-r T NHd2Lwith

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnHS0 ê KL + Hr - r f + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

In effect, the foreign currency is a dividend yield q = r f .

à Future

The value of a call option on a future is given directly by

c = ‰-r T HF0 NHd1L - K NHd2LLwith

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

à Generalized Black-Scholes formula

All these cases can be subsumed in a generalized Black-Scholes formula

c = S0 ‰Hb-rL T NHd1L - K ‰-r T NHd2Lwhere

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnHS0 ê KL + Hb + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

where b is the cost-of-carry of holding the underlying security, with

b = r non-dividend paying stock

b = r - q stock with dividend yield q

b = r f currency option

b = 0 futures options

Put-call parity gives

p + S0 ‰Hb-rL T = c + K ‰- r T

so that

2 GeneralizedBlackScholes.nb

Page 31: Pg p 12 Handbook

p = H S0 ‰Hb-rL T NHd1L - K ‰-r T NHd2LL + K ‰- r T - S0 ‰Hb-rL T

= K ‰-r T KH1 - NHd2LL - S0 ‰Hb-rL T H1 - NHd1L L= K ‰-r T NH-d1L - S0 ‰Hb-rL T NH-d1L

Traditionally, the Black-Scholes model is implemented in dividend yield form

c = S0 ‰- q T NHd1L - K ‰-r T NHd2L

p = K ‰-r T NH-d2L - S0 ‰- q T NH-d1L

d1 =lnHS0 ê KL + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

with the specific cases being obtained with the following substitutions

q = 0 non-dividend paying stock

q = q stock with dividend yield q

q = r f currency option

q = r futures options

Note that even if the dividend yield is not constant, the formulae still hold with q equal to the average annualizeddividend yield during the life of the option.

à Appendix

THEOREM. If S is lognormally distributed and the standard deviation of ln S is s then

PrHS > KL = NHd2Land

EHS » S > KL = EHSL NHd1Lwhere

d1 =lnHEHSL ê KL + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s

d2 =lnHEHSL ê KL - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s

Consequently

(1)E@maxHS - K, 0LD = EHSL NHd1L - K NHd2LProof:

PrHS > KL = ProbHln S > ln KL = NHd2L

GeneralizedBlackScholes.nb 3

Page 32: Pg p 12 Handbook

For the second part, see Hull (2003: 262-263).

Recognising that (under Black-Scholes assumptions) EHST L = F0 = S0 ‰r T and s = s è!!!!T , the Black-Scholes

formula for a call option

c = ‰-r T HF0 NHd1L - K NHd2LL = ‰-r T HS0 ‰r T NHd1L - K NHd2LL = S0 NHd1L - K ‰-r T NHd2Lis immediate.

4 GeneralizedBlackScholes.nb

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Implementation of Black-Scholes option pricing Michael Carter, 2004

Option Explicit ' ************************************************************ ' Option values Function BSCall(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double Dim d2 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) d2 = d1 - sigma * Sqr(T) BSCall = S * Exp(-q * T) * Application.NormSDist(d1) - K * Exp(-r * T) * Application.NormSDist(d2) End Function Function BSPut(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double Dim d2 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) d2 = d1 - sigma * Sqr(T) BSPut = K * Exp(-r * T) * Application.NormSDist(-d2) - S * Exp(-q * T) * Application.NormSDist(-d1) End Function ' ************************************************************ ' The Greeks Function BSCallDelta(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) BSCallDelta = Exp(-q * T) * Application.NormSDist(d1) End Function Function BSPutDelta(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) BSPutDelta = Exp(-q * T) * (Application.NormSDist(d1) - 1) End Function

Page 34: Pg p 12 Handbook

Function BSCallGamma(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double Dim d1 As Double d1 = (Log(S / K) + (r - q + sigma * sigma / 2) * T) / (sigma * Sqr(T)) Debug.Print d1 BSCallGamma = Exp(-q * T) * Application.NormDist(d1, 0, 1, False) / (S * sigma * Sqr(T)) End Function Function BSPutGamma(S As Double, K As Double, r As Double, q As Double, sigma As Double, T As Double) As Double BSPutGamma = BSCallGamma(S, K, r, q, sigma, T) End Function

Page 35: Pg p 12 Handbook

Dealing with dividendsMichael Carter

European optionsThe Black-Scholes formula is readily adapted to continuous dividends yields (see The Black-Scholes formula forstock indices, currencies and futures).

The price of a dividend paying stock typically falls when the stock goes ex-dividend. A common approach todealing with discrete dividends is to subtract the present value of the dividends from the current stock pricebefore applying the Black-Scholes formula (Hull 2003: 253). For example, if dividends d1, d2, …, dn areanticipated at times t1, t2, …, tn, the present value of the dividends is

D = ‚i=1

n

‰r ti di

and the option is valued as

cHS - D, K, r, s, tL or pHS - D, K, r, s, tLwhere c and p are the Black-Scholes formulae for call and put options respectively.

This is problematic, not the least because historical volatility measures refer to the stock price including divi-dends (Fischling 2002).

Bos and Vandemark (2002) propose a simple modification that closely matches numerical results. Instead ofsubtracting the full present value of future dividends from the current stock price, they propose apportioningeach dividend between the current price and the strike price in proportion to the relative time. Specifically, ifdividends d1, d2, …, dn are anticipated at times t1, t2, …, tn, they compute "near" and "far" components

Dn = ‚i=1

n T - tiÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅT

‰r ti di and D f = ‚i=1

n tiÅÅÅÅÅÅÅT

‰r ti di

The option is valued as

cHS - Dn, K + D f , r, s, tL or pHS - Dn, K + D f , r, s, tLwhere c and p are the Black-Scholes formulae for call and put options respectively.

American optionsDealing with dividends for American options is more complicated, since dividends are closely interwined withthe incentives for early exercise. This is discussed in the complementary note American options.

Page 36: Pg p 12 Handbook

The binomial modelIn a risk-neutral world, the total return from the stock must be r. If dividends provide a continuous yield of q, theexpected growth rate in the stock price must be r - q. The risk-neutral process for the stock price therefore is

„ S = Hr - qL S „ t + s S „ z

The can be approximated in the simple binomial model by adjusting the risk-neutral probabilities, so that

p u S0 + H1 - pL d S0 = S0 ‰Hr-qL Dt

or

p =‰Hr-qL Dt - d

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu - d

With this amendment, the binomial model can be used to value European and American options on indices,currencies and futures.

Discrete proportional dividends are also straightforward to incorporate into the binomial model. Whenever thestock pays a proportional dividend, the stock price tree must be adjusted downwards when the stock goesex-dividend (Hull 2003: 402).

Discrete cash dividends are more difficult, since the adjusted tree becomes non-recombining for nodes afterdividend date. This leads to an impractical increase in the number of nodes. We can finesse this problem in ananalogous way to the treatment of cash dividends with the Black-Scholes formula.

Assume that the stock price S has two components - a risky component S* with volatility s* and the dividendstream ‰- r t D. Develop a binomial tree to represent the stochastic part S* with

S0* = S0 - ‰- r t D, p =

‰r Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

u - d, u = ‰ s* Dt, d = ‰ -s* Dt

Then add back the present value of the dividends to each node (prior to the ex-dividend date) to obtain a bino-mial tree representation of S, which can then be used to value contingent claims in the usual way.

This procedure could be enhanced by apportioning the dividends between current price and strike price accord-ing to the procedure of Bos and Vandemark discussed above.

2 Dividends.nb

Page 37: Pg p 12 Handbook

Hedging strategiesMichael Carter

à Preliminaries

IntroductionConsider a derivative (or portfolio of derivatives) on a single underlying asset. Its value depends upon thecurrent asset price S and its volatility s, the risk-free interest rate r, and the time to maturity t. That is,V = f HS, r, s, tL. (It also depends upon constants like the strike price K.) Taking a Taylor series expansion, thechange in value over a small time period can be approximated by

(1)dV º

∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S

dS +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑r

dr +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑s

ds +∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t

dt +1ÅÅÅÅÅ2

∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 dS2

+ other second order terms+ higher order terms

The partial derivatives in this expansion are known collectively as "the Greeks". They measure the sensitivity ofa portfolio to changes in the underlying parameters. Specifically

D =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S

Delta measures the sensitivity of the portfolio value to changes in the price of the underlying

r =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑r

Rho measures the sensitivity of the portfolio value to changes in the interest rate

v =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑s

Vega measures the sensitivity

of the portfolio value to changes in the volatility of the underlying

Q =∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t

Theta measures the sensitivity of the portfolio value to the passage of time

G =∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 =

∑ DÅÅÅÅÅÅÅÅÅÅÅ∑S

Gamma measures the sensitivity of delta to changes in the price of the underlying,

or the curvature of the S - V curve.

Substituting in (1), the change in value of the portfolio can be approximated by

(2)dV º DdS + rdr + v ds + Qdt +1ÅÅÅÅÅ2

GdS2

Because differentiation is a linear operator, the hedge parameters of a portfolio are equal to a weighted averageof the hedge parameters of its components. In particular, the hedge parameters of a short position are the nega-tive of the hedge parameters of a long position. Consequently, (2) applies equally to a portfolio as to an individ-ual asset. The sensitivity of a portfolio to the risk factors (S, r, s) can be altered by changing the compositionof the portfolio. It can be reduced by adding assets with offsetting parameters.

Page 38: Pg p 12 Handbook

The Greeks are not independent. Any derivative (or portfolio of derivatives) V = f HS, r, s, tL must satisfy theBlack-Scholes differential equation

∑ fÅÅÅÅÅÅÅÅÅÅÅ∑ t

+ r S∑ fÅÅÅÅÅÅÅÅÅÅÅ∑S

+1ÅÅÅÅÅ2

s2 S2 ∑2 fÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 = r V

Substituting

∑ PÅÅÅÅÅÅÅÅÅÅÅ∑ t

= Q∑ PÅÅÅÅÅÅÅÅÅÅÅ∑S

= D∑2 PÅÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S2 = G

it follows that the Greeks must satisfy the following relationship

(3)Q + r S D +1ÅÅÅÅÅ2

s2 S2 G = r V

Computing the GreeksThe Greeks of vanilla European options have straightforward formulae, which can be derived from the Black-Sc-holes formula. The generalized Black-Scholes formulae for European options are

c = S ‰-q T NHd1L - K ‰-r T NHd2Lp = K ‰-r T NH-d2L - S ‰-q T NH-d1L

where

d1 =lnHS ê KL + Hr - q + s2 TL ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 =lnHS ê KL + Hr - q - s2 TL ê2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

= d1 - s

The partial derivatives ("the Greeks") are

Call PutDelta ‰-q T NHd1L ‰-q T HNHd1L - 1LGamma

‰-q T N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s

è!!!!T

‰-q T N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s

è!!!!T

Rho K T ‰-r T NHd2L - K T ‰-r T NH-d2LVega ‰-q T S è!!!!T N ' Hd1L ‰-q T S è!!!!T N ' Hd1LTheta -

‰-q T S s N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 è!!!!T

-‰-q T S s N ' Hd1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

2 è!!!!T

+ q ‰-q T S NHd1L - q ‰-q T S NH-d1L-r K ‰-r T NHd2L - r K ‰-r T NH-d2L

2 HedgingStrategies.nb

Page 39: Pg p 12 Handbook

As an example of the derivation, for a call option

G =∑DÅÅÅÅÅÅÅÅÅÅÅ∑S

= ‰-q T N ' Hd1L ∑d1ÅÅÅÅÅÅÅÅÅÅÅÅÅ∑S

= ‰-q T N ' Hd1L 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS s

è!!!!T

Calculating vega from the Black-Scholes formula is an approximation, since the formula is derived under theassumption that volatility is constant. Fortunately, it can be shown that it is a good approximation to the vegacalculated from a stochastic volatility model (Hull 2003: 318).

Some exotic options (e.g. barrier options) have analogous formulae. However, for most exotic options andvanilla options, the Greeks must be estimated by numerical techniques. Since these are the type of options forwhich institutions require such information, this motivates are interest in the accurate computation of optionvalues and sensitivities.

In principle, the Greeks can be estimated by numerical differentiation. For example,

D =cHS1L - cHS0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

S1 - S0and G =

DHS1L - DHS0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅS1 - S0

However, this is not always the most appropriate method, as the small size of the denominator in the limitmagnifies errors in the numerator.

HedgingIn the previous section, we showed the sensitivity of the value of a portfolio of derivatives of a single underlyingto its risk factors can be approximated by

(4)dV º DdS + rdr + v ds + Qdt +1ÅÅÅÅÅ2

GdS2

Hedging is the process of modifying the portfolio to reduce or eliminate the stochastic elements on the right-hand side. Delta-hedging eliminates the first-term on the right-hand side by making the portfolio delta neutral (D= 0). This can be done by taking an offsetting position in the underlying asset, as represented by the tangent tothe portfolio at the current asset price.

HedgingStrategies.nb 3

Page 40: Pg p 12 Handbook

90 100 110 120

2.55

7.510

12.515

17.520

Delta-gamma hedging also eliminates the last term in (4) by making the portfolio gamma neutral (G = 0). Sincethe underlying is gamma neutral, delta-gamma hedging requires the addition of other derivatives to the portfolio.Curvature (Gamma) increases as an option approaches maturity, especially for at-the-money options.

90 100 110 120

2.5

5

7.5

10

12.5

15

17.5

20Increasing curvature approaching maturity - 1, 3 ,6 months

Time

4 HedgingStrategies.nb

Page 41: Pg p 12 Handbook

1 2 3 4 5 6Months to expiry

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Call Gamma over time

Out of the money

At the money

In the money

90 100 110 120

2.55

7.510

12.515

17.520

Hedge option

Recall the fundamental relationship (3)

Q + r S D +1ÅÅÅÅÅ2

s2 S2 G = r V

For a delta-gamma-neutral portfolio, this reduces to

Q = r V

The portfolio earns the risk-free rate.

The closer that hedging option matches the target option, the more robust will be the hedge provided (i.e. thewider the range of parameter variation that will be neutralised). The hedge may be improved by combining twoor more options. For example, combining two options, one with a shorter and one with a longer time to maturity

HedgingStrategies.nb 5

Page 42: Pg p 12 Handbook

would a more accurate match to the gamma of the target option. There is a tradeoff between the robustness ofthe hedge (the frequency of hedge adjustments) and the number of options that must be purchased and managed.The actual performance of a hedge may not reach its theoretical potential (for example, because of model errorsand transaction costs). Consequently, adding too many options to the hedge may give results that are better onpaper than in reality.

A hedge comprising at least two derivatives, in addition to the underlying, can be used to eliminate three termsin equation (3). A hedge comprising three derivatives, in addition to the underlying, can be used to neutralize allfour stochastic terms in equation (3), eliminating all risk to a first-order approximation.

In principle, a hedge can be found by solving a system of linear equations. Suppose there are m potential hedg-ing instruments. Let x1, x2, …, xm denote the amount of hedging instrument j, and let xS denote the amountinvested in the underlying. Then, we seek a solution to the following system of equations.

xS + x1 D1 + x2 D2 + … + xm Dm = Dx1 G1 + x2 G2 + … + xm Gm = Gx1 v1 + x2 v2 + … + xm vm = vx1 r1 + x2 r2 + … + xm rm = r

Provided that the Greeks of the hedging instruments are linearly independent, there will be a unique solution ifm = 3 and multiple solutions if m > 3. However, the solutions may not be economically sensible.

Since T appears explicitly in the formula for vega, options of different maturities will be most effective inhedging against volatility risk. Although interest rate risk can be hedged by options, it may be cost-effective andcertainly more straightforward to hedge interest rate risk by trading bond future contracts, since they are purerho instruments, with no impact on delta, gamma or vega.

ü Example

6 HedgingStrategies.nb

Page 43: Pg p 12 Handbook

Rules of thumbConsider a call option that is at-the-money forward, that is

K = F0 = S0 ‰r T

Then the Black-Scholes formula (assuming no dividend yield) simplifies to

c = S0HNHd1L - NHd2LLwhere

d1 =ln HS0 ê F0L + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=ln HS0 ê S0L - r T + Hr + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=1ÅÅÅÅÅ2

s è!!!!T

d2 = d1 - s è!!!!T = -

1ÅÅÅÅÅ2

s è!!!!T

Therefore the call option value is

c = S0JNJ 1ÅÅÅÅÅ2

s è!!!!T N - NJ-

1ÅÅÅÅÅ2

s è!!!!T NN

Provided that s è!!!!T is small, this can be approximated by

c = S0 ä0.4 s è!!!!T

Since the peak of the standard normal density function is 1 ëè!!!!!!!2 p º 0.4, the area can be approximated by arectangle of height 0.4.

-3 -2 -1 1 2 3

0.1

0.2

0.3

0.4The standard normal distribution

This formula can be inverted to obtain a "rough and ready" estimate of the implied velocity from quotedoption prices, using the average of the two nearest-the-money call options.

s = 2.5c

ÅÅÅÅÅÅÅÅS0

1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅè!!!!T

Page 44: Pg p 12 Handbook

Greeks and the binomial method

Numerical differentiation

Delta measures the sensitivity of the option value to changes in the price of the underlying. It is defined as

D =

¶V HSL

¶S= lim

dS ®0

V HS + dSL - V HSL

dS

An obvious method to evaluate D is to compute

D »

V HS + dSL - V HSL

dS

for small dS. This is known as the forward difference. A better alternative (though more costly to compute) is

D »

V HS + dSL - V HS - dSL

2 dS

which is known as the central difference. The other first-order Greeks (rho, theta and vega) can be estimated similarly.

Gamma is the derivative of delta, or the second derivative of vHSL. Using central differences, gamma can be estimated by

G »

DIS +1

2dSM - DIS -

1

2dSM

dS=

V HS+dSL-V HS LdS

-V HSL-V HS -dSL

dS

dS=

V HS+dSL-V HS LdS

-V HSL-V HS -dSL

dS

dS

=

V HS + dSL - 2 V HS L + V HS - dSL

dS2

Numerical differentiation and the binomial tree

Numerical differentiation is not the best method to be applied to the binomial tree. The problem is illustrated in the followingdiagram.

82.5 87.5 90 92.5 95 97.5

2

4

6

8

10

12

Black Scholes

Page 45: Pg p 12 Handbook

82.5 87.5 90 92.5 95 97.5

2

4

6

8

10

12

Binomial

2 Greeks.nb

Page 46: Pg p 12 Handbook

Implied trees

Michael Carter

� Preliminaries

Implied trees are one practical method to deal with the volatility smile. The objective is to build a tree for the underlying thatreproduces the market prices of traded derivatives. In other words, the resulting tree is calibrated to current market data. Thecalibrated tree can then be used to

æ to compute hedge parameters for traded options

æ to price non-standard derivatives (American options and exotics) on the same underlying.

The starting point is current market prices and quotations for calls and puts of various strikes and maturities. Typically, thesemarket data are converted into a table of implied volatilities. This table is then used to interpolate volatilities and marketprices for arbitrary strikes and maturities.

Binomial tree

� The one-step tree

The value of a one-period call option struck at the initial price S is

(1)VcHSL =

1

Rpu HSu - SL

where pu is the risk-neutral probability of an up-tick, Su = u S is the price of the asset following an up-tick and 1 � R is the

one-period discount factor. Similarly, the value of a one-period put option struck at S is

(2)VpHSL =

1

Rpd HS - SdL

where pd = 1 - pu is the probability of a down-tick, and Sd = d S is the resulting asset price.

In the binomial model, the risk-neutral (arbitrage free) probability pu is that which makes the expected return on the asset

equal to the risk-free rate of return, that is

pu Su + H1 - puL Sd = R S

where R = ãHr-qL Dt is the one-period risk-free rate of capital gains.

Solving for pu

(3)pu =

R - d

u - dor pu =

R S - Sd

Su - Sd

If we add an additional constraint such as u = 1 �d, then we can solve for the binomial tree consistent with market data. For

example, if we have the market price Vc, we obtain from solving equations (1) and (3) the following formula for u, from

which we can derive Su, Sd and pu.

Page 47: Pg p 12 Handbook

u =

∆ S + Vc

∆ R S - Vc

Alternatively, we can use the market price Vp of a put to obtain from (2) and (3)

u =

∆ R S + Vp

∆ S - Vp

Note that ∆ R = 1 if and only if the dividend yield equals zero.

Given u, we can derive

Su =

SH∆ S + VcL

∆ R S - Vc

, Sd =

S H∆ R S - VcL

S ∆ + Vc

, pu =

R S - Sd

Su - Sd

or

Su =

SI∆ R S + VpM

∆ S - Vp

, Sd =

S I∆ S - VpM

∆ R S + Vp

, pu =

R S - Sd

Su - Sd

We can also compute the state contingent prices (discounted probabilities) at the new nodes

Λu = ∆ pu and Λd = ∆ H1 - puL

These will be required in the next step.

We have shown how we can infer a unique one-step binomial tree consistent with given a single market price. Note that theinferred tree will differ depending upon whether we start with a call or a put price, unless the prices are consistent. Inpractice, consistency is usually imposed by starting with an implied volatility for relevant term and strike, and imputingprices for the call or put using a binomial tree with the requisite volatility. This required implied volatility is obtained byinterpolation from the implied volatilities of quoted options.

Assuming that we are dealing with European options, the implied volatility of quoted options is obtained using the Black-Scholes formula. Then, the recommended practice is to use these data to interpolate implied volatilities for the nodes of thetree. The implied prices for options struck at nodes of the tree are then obtained from a binomial model with the requisitevolatility.

� The two - step tree

To complete the second step, we can arbitrarily set the asset price along the middle branch of the tree equal to the startingprice. Given this base line, we can determine the top and bottom nodes of the tree. Suppose we know the value vu of a one-

period call option struck at S = Su, the upper node in the previous step. Then we are in a position analogous to the first step.

From equation (1) we have

vuHSL =

1

Rpu ISup - SM

with

pu =

R - d

u - d=

R S - d S

u S - d S=

R S - Sd

Sup - Sd

Solving for Sup gives

2 ImpliedTrees.nb

Page 48: Pg p 12 Handbook

Solving for Sup gives

(4)Sup =

S2- Sd H∆ S + vuL

S - H∆ Sd + vuL

where in this case Sd = S0.

Similarly, setting the Su = S0, we can determine the bottom node of the tree from the value vd of a one-period put option

struck at the bottom node in the previous step, solving

vd =

1

RH1 - puL HS - SdownL and pu =

R S - Sdown

Su - Sdown

to give

(5)Sdown =

S2- SuH∆ S - vdL

S - H∆ Su - vdL

It remains to supply the supposed data - the values of the one-period options struck at the Su and Sd . We now show how this

can be inferred from market data. Let VcHSu, 2) denote the value of a call option struck at Su and expiring after two periods.

This value can be interpolated from market data. After one period, there are two possibilities. Either the price increased to Su

or decreased to Sd . In the former case, the option is then worth vu. In the latter case it is worthless, as it is bound to expire out

of the money. Consequently

VcHSu, 2L = ∆ H pu vu + H1 - pL 0L = ∆ pu vu

from which we can infer

vu =

VcHSu, 2L

∆ pu

Similarly the value vpHSd , 2) of a two-period put option struck at Sd is

VpHSd , 2L = ∆ H pu 0 + H1 - pL vdL = ∆ H1 - puL vd

which implies

vd =

VpHSd , 2L

∆ H1 - puL

Throughout this note, we will adopt the convention that a capital V refers to a market value, while a small v refers to animputed value.

� The n-step tree

In essence, building an n-step tree involves repeating the preceding two steps again and again. To facilitate this, we need todefine some more precise notation.

� Notation

Let Si, j denote the asset price at the jth node of the ith step. That is, Hi, jL denotes the node reached after j upward and i - j

downward steps. Si,0 is the minimum price attained after i steps while Si,i is the maximum price. Similarly, let Λi, j denote the

state contingent price at the Hi, jLth node. Let pi, j denote the probability of an uptick at the Hi, jLth node with 1 - pi, j the

probability of a downtick. Finally, vi, j denotes the value of at node i of a one-period (maturing at time i + 1) at-the-money

option (struck at Si, j).

ImpliedTrees.nb 3

Page 49: Pg p 12 Handbook

Λ0,0

S0,0 p0,0

v0,0

Λ1,0

S1,0 p1,0

v1,0

Λ1,1

S1,1 p1,1

v1,1

Λ2,0

S2,0 p2,0

v2,0

Λ2,1

S2,1 p2,1

v2,1

Λ2,2

S2,2 p2,2

v2,2

Λ3,0

S3,0 p3,0

v3,0

Λ3,1

S3,1 p3,1

v3,1

Λ3,2

S3,2 p3,2

v3,2

Λ3,3

S3,3 p3,3

v3,3

Λ4,0

S4,0 p4,0

v4,0

Λ4,1

S4,1 p4,1

v4,1

Λ4,2

S4,2 p4,2

v4,2

Λ4,3

S4,3 p4,3

v4,3

Λ4,4

S4,4 p4,4

v4,4

� State contingent prices and probabilities

We have seen in a previous lecture how state prices can be calculated inductively from previous values

Λi,0 = ∆I1 - pi-1,0M Λi-1,0, Λi,i = ∆ pi-1,i-1 Λi-1,i-1

Λi, j = ∆ H I1 - pi-1, jM Λi-1, j + pi-1, j-1 Λi-1, j-1L, 0 < j < i

while the previous probabilities are computed from current asset prices

pi-1, j =

R Si-1, j - Si, j

Si, j+1 - Si, j

The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_. It remains to infer the asset prices from the market data.

� Forward option prices

Let VcISi, j, i + 1M denote the value of a call option struck at Si, j at time t0 and maturing at time Hi + 1L Dt , that is maturing after

i + 1steps in the tree. We can infer this value by interpolation from market data. Note that there is no need to specify thecurrent asset price of the multi-period options - it is always the initial asset price S0,0. Now consider the possibilities for this

option one period prior to maturity. Let vi,kISi, jM denote the value of this option at time i Dt if the asset price reaches Si,k . Then

VcISi, j, i + 1M = âk=0

i

Λi,k vi,kISi,kM

which we can write as

(6)

Out At In

VcISi, j, i + 1M = âk=0

j-1

Λi,k vi,kISi,kM + Λi, j vi, jISi,kM + âk= j+1

i

Λi,k vi,kISi,kM

with

(7)vi,kISi, jM = ∆I pi,k max ISi+1,k+1 - Si, j, 0M + I1 - pi,kM maxISi+1,k - Si, j, 0MM

where ∆ is the one-period discount factor and pi,k is the risk-neutral probability applicable at node Hi, kL.

4 ImpliedTrees.nb

Page 50: Pg p 12 Handbook

where ∆ is the one-period discount factor and pi,k is the risk-neutral probability applicable at node Hi, kL.

For all states k < j, the option must necessarily expire out of the money. Therefore vi,kISi, jM = 0 for all k < j, and the first term

in equation (6) is zero. For all k > j, the option will expire in the money. Consequently, for k > j we can eliminate the max

operator from (7) to give

(8)vi,kISi, jM = ∆I pi,k ISi+1,k+1 - Si, jM + I1 - pi,kM ISi+1,k - Si, jMM = ∆I pi,k Si+1,k+1 + I1 - pi,kM Si+1,kM - ∆ Si, jM

Furthermore, the risk-neutral probability pi,k satisfies the equation

pi,k Si+1,k+1 + I1 - pi,kM Si+1,k = R Si,k

where R = 1 � ∆ equals the one-period risk neutral return. Substituting in (8) gives

vi,kISi, jM = ∆ IR Si,k - Si, jM for k > j

Substituting this in (6) gives

Out At In

VcISi, j, i + 1M = 0 + Λi, j vi, jISi,kM + âk= j+1

i

∆ Λi,k IR Si,k - Si, jM

where vi, j = vi, jISi, jM is the only unknown. (For the at-the-money option vi, j, we can dispense with the argument Si, j, since

there is no ambiguity.) Solving for vi, j= vi, jISi, jM,

(9)vi, j =

VcISi, j, i + 1M - Úk= j+1i

Λi,k ∆ IR Si,k - ∆ Si, jM

Λi, j

Similarly, the value of an Hi + 1L-period put option struck at Si, j is

In At Out

VpISi, j, i + 1M = âk=0

i

Λi,k vi,kISi, jM = âk=0

j-1

Λi,k vi,kISi, jM + Λi, j vi, jISi, jM + âk= j+1

i

Λi,k vi,kISi, jM

where vi,kISi, jM is the value of the option at node Hi, kL. For k < j , the option will expire in the money and

vi,kISi, jM = ∆I pi,k ISi, j - Si+1,k+1M + I1 - pi,kM ISi, j - Si+1,kMM

= ∆ Si, j - ∆I pi,k Si+1,k+1 + I1 - pi,kM Si+1,kM

= ∆ ISi, j - R Si,kM

Conversely, for k > j, the option will expire out of the money with vi,kISi, jM=0. Consequently

VpISi, j, i + 1M = âk=0

j-1

Λi,k ∆I Si, j - R Si,k M + Λi, j vi, j + 0

and

(10)vi, j =

vpISi, j, i + 1M - Úk=0j-1

Λi,k ∆I Si, j - R Si,kM

Λi, j

To summarize, vi, j denotes the implied value at node i, j of an at-the-money call or put option struck at Si, j and expiring after

one-period. The choice of call or put is arbitrary. It is conventional to use (9) to determine vi, j in the top half of the tree and

use (10) determine vi, j in the bottom half of the tree. The choice of call or put along the horizontal is arbitrary.

Note that, once we have determined the asset prices at stage i in the tree, we can then determine state prices and forwardoption prices at the same stage. The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_ ® vi,_.

ImpliedTrees.nb 5

Page 51: Pg p 12 Handbook

Note that, once we have determined the asset prices at stage i in the tree, we can then determine state prices and forwardoption prices at the same stage. The sequence of calculation is Si,_ ® pi-1,_ ® Λi,_ ® vi,_.

� Asset prices

It remains to determine the asset prices in the next stage of the tree, which we compute from the forward asset prices in theprevious stage. If the stage i is even, then we arbitrarily set asset price of the central node equal to the starting price. That is

Si,i�2 = S0,0, i = 2, 4, 6, …

This is known as the centering condition. If the stage i is odd, we determine the values of the two central nodes in identicalfashion to the one-stage tree above. Let m = di �2t, the largest integer less than i �2. m points to the central node when i iseven, and to the lower of the two central nodes when i is odd.

Λi-1, j

Si-1, j pi-1, j

vi-1, j

Λi, j

Si, j pi, j

vi, j

Λi, j+1

Si, j+1 pi, j+1

vi, j+1

Now consider a node above the central node. A one-period call option struck at the preceding (down) node has the valueë

vi-1, j = ∆ pi- j, j ISi, j+1 - Si-1, jM where pi- j, j =

R Si-1, j - Si, j

Si, j+1 - Si, j

in which the only unknown is Si, j+1. Starting at the central node, we can successively solve for all the nodes above the center

Si, j+1 =

∆ R Si-1, j2

- ∆ Si-1, j Si, j - Si, j vi-1, j

∆ R Si-1, j - ∆ Si, j - vi-1, j

, j = :m + 1, m + 2, …, i , i even

m + 2, m + 3, …, i , i odd

Similarly, we can successively compute the prices at all nodes below the central node using the values of one-period putoptions struck at the preceding (up) node.

vi-1, j = ∆ I1 - pi- j, jM ISi-1, j - Si, jM where pi- j, j =

R Si-1, j - Si, j

Si, j+1 - Si, j

giving

Si, j =

∆ R Si-1, j2

- ∆ Si-1, j Si, j+1 + Si, j+1 vi-1, j

∆ R Si-1, j - ∆ Si, j+1 + vi-1, j

, j = m - 1, m - 2, …, 0

� The resulting tree

There is a unique binomial tree fitting a given set of market data (once the central trunk has been specified). The resultingtree is distorted, as the asset prices adapt to fit the volatility at different nodes in the tree.

6 ImpliedTrees.nb

Page 52: Pg p 12 Handbook

An implied binomial tree

The procedure can be adapted to apply to American style options and options with dividends (Chriss 1996). Nothing in theprocedure guarantees that the transition probabilities are nonnegative. If and when "wrong" probabilities arise, the tree mustbe adjusted to eliminate them. The same problem arises with trinomial trees.

Trinomial trees

Trinomial trees give additional degrees of freedom. We can build an asset price tree with regularly spaced nodes, and thenuse market data to determine the transition probabilities. This additional flexibility is especially useful where the market datais noisy, or where it is desired to use the fitted tree to price certain exotic options like barrier options.

Let S0,0 denote the root of the tree, with nodes at step i being labelled Si,-i, Si,-i+1, …, Si,0, …, Si,i. In the context of a given

tree, the value of a call option struck at K and expiring at i DT is (S = S0,0 always)

(11)VcHK, i L = âj=-i

i

Λi, j ISi, j - KM+

= âj= j*

i

Λi, j ISi, j - KM

where j* is such that Si, j*> K and Si, j*

-1 £ K. In particular,

VcISi,i-1, iM = Λi,i ISi,i - Si,i-1M

from which we can deduce

Λi,i =

VcISi,i-1, iM

Si,i - Si,i-1

Similarly

VcISi,i-2, iM = Λi,i ISi,i - Si,i-2M + Λi,i-1 ISi,i-1 - Si,i-2M

so that

Λi,i-1 =

VcISi,i-2, iM - Λi,i ISi,i - Si,i-2M

Si,i-1 - Si,i-2

Proceeding in this fashion, we can iteratively deduce the state contingent prices at each node of the tree from the value ofoptions struck at adjacent nodes. Computationally, it is preferable to use call options to calibrate the top of half of the tree,and then switch to put options for the bottom half of the tree.

ImpliedTrees.nb 7

Page 53: Pg p 12 Handbook

Proceeding in this fashion, we can iteratively deduce the state contingent prices at each node of the tree from the value ofoptions struck at adjacent nodes. Computationally, it is preferable to use call options to calibrate the top of half of the tree,and then switch to put options for the bottom half of the tree.

Given the state contingent prices, we can then deduce the transition probabilities at each nodes. At the root node, for exam-ple,

Λ1,1 = ∆ p0,0u so that p0,0

u=

Λ1,1

Similarly

p0,0d

=

Λ1,-1

and p0,0m

= 1 - p0,0u

- p0,0d

Furthermore

Λ2,2 = ∆ p1,1u

Λ1,1 so that p1,1u

=

1

Λ2,2

Λ1,1

Next we observe that the risk-neutral transition probabilities must satisfy the equations

p1,1u S2,2 + p1,1

m S2,1 + p1,1d S2,0 � R S1,1

p1,1u

+ p1,1m

+ p1,1d

� 1

where R = ãHr-qL Dt is the risk-neutral drift.

Given p1,1u , we can solve these equations for p1,1

m and p1,1d

p1,1m

=

R S1,1 - S2,0 - p1,1u I S2,2 - S2,0M

S2,1 - S2,0

, p1,1d

= 1 - p1,1u

- p1,1u

Next, we can use p1,1m to deduce the value of p1,0

u from the equation

Λ2,1 = ∆ I p1,1m

Λ1,1 + p1,0u

Λ1,0M

which gives

p1,0u

=

Λ2,1 - ∆ p1,1m

Λ1,1

∆ Λ1,0

Iterating this procedure, we can generate all the transition probabilities in the tree, using the following recursive system ofequations.

(12)

8 ImpliedTrees.nb

Page 54: Pg p 12 Handbook

(12)

pi,iu

=

Λi+1,i+1

∆ Λi,i

pi,ku

=

Λi+1,k+1 - ∆ pi,k+1m

Λi,k+1

∆ Λi,k

, k < i

pi,km

=

R Si,k - Si+1,k-1 - pi,ku I Si+1,k+1 - Si+1,k-1M

Si+1,k - Si+1,k-1

pi,kd

= 1 - pi,ku

- pi,ku

To avoid the accumulation of numerical errors, it is recommended to compute the top half of the tree using (12) and then

compute the bottom half using analogous recursions working from the bottom up, namely

(13)

pi,-id

=

Λi+1,-Hi+1L

∆ Λi,-i

pi,-kd

=

Λi+1,-Hk+1L - ∆ pi,-Hk+1Lm

Λi,-Hk+1L

∆ Λi,-k

, k < i

pi,-km

=

Si+1,-k+1 - R Si,-k - pi,-kd I Si+1,-k+1 - Si+1-,k-1M

Si+1,-k+1 - Si+1,-k

pi,-ku

= 1 - pi,-ku

- pi,-ku

Specifically, in (12) and (13), let k = i, i - 1, …, 0 for successive i = 1, 2, 3, ….

We must e� sure that the transition probabilities are all nonnegative. There are two ways in which negative probabilites canarise

æ the state space is inappropriate

æ the volatility skew is extreme at particular observations

A robust way to avoid the first problem is to build a tree whose volatility is equal to the maximum implied volatility in thedata.

Once a market-consistent tree has been constructed, any contingent claim can be priced in the standard manner. For Euro-pean-style derivatives, we can use the state-contingent prices at maturity; for American-style derivatives, we use backwardinduction.

ImpliedTrees.nb 9

Page 55: Pg p 12 Handbook

Binomial model - implementationMichael CarterThe following is the pseudocode for an efficient implementation of the binomial method for American andEuropean options.

BinomialAmericanCall (S0, K, r, q, s, t, nL

/* initialize parameters */Dt = t/n;u = ‰s è!!!!!!Dt ; d = 1/u;d = ‰-r Dt ;p = ‰Hr-qL Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu-d ; p = 1-p;pu = d p; pd = d H1 - pL; /* pu and pd are discounted probabilities */

/* allocate storage */vectors s[-n,n], v[-n,n];

/* initialize the tree */s[0] = S0;

/* calculate potential asset prices */for j = 1 to n

s[j] = u * s[j-1];s[-j] = d*s[-j+1];

end;

/* calculate option exercise values */for j = -n to n by 2

v[j] = max (s[j] - K,0);end;

/* calculate option values by backward induction */for i = n - 1 to 0 by - 1

for j = -i to i by 2v[j] = max Hpu * v[j+1] + pd * v[j-1], s[j] - K);

end;end;

/* return option value */BinomialAmericanCall = v[0]

The corresponding put option can be evaluated by taking the negative of S and K .

BinomialEuropeanCall (S0, K, r, q, s, t, nL

/* initialize parameters */

Page 56: Pg p 12 Handbook

Dt = t/n;u = ‰s

è!!!!!!!Dt ; d = 1/u;

p = ‰Hr-qL Dt - dÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅu-d ; p = 1- pÅÅÅÅÅÅÅÅÅÅÅp ;

/* calculate maximum asset price and probability*/s = un S0;P = pn;EV = P * maxHs - K, 0L;

/* proceed down final asset prices calculating expected values */for j = n - 1 to 0 by -1

P = j+1ÅÅÅÅÅÅÅÅÅÅn- j P p;s = d2 s;IV = maxHs - K, 0L;if (IV > 0)

EV = EV + P * IV;else

break;endif

end;

/* return option value */BinomialEuropeanCall = ‰-r t EV;

NOTE: We have to be careful that P = pn does not cause an underflow error. As T Ø ¶, p Ø 1ÅÅÅÅ2 andH 1ÅÅÅÅ2 L1050

º 10-316. For example, the Excel VBA data type Double has a minimum value of4.94065645841247 * 10-324, so that n = 1000 is a practical maximum for this algorithm. However, thealgorithm could be modified to exclude very low probability final prices and so increase the number ofbranches.

2 BinomialProgramming.nb

Page 57: Pg p 12 Handbook

Implementation of binomial model for option pricing Michael Carter, 2004

Option Explicit Option Base 0 Enum OptionType callOption = 1 putOption = -1 End Enum Enum OptionStyle American = 1 European = 2 End Enum Function RecursiveBinomialTree(S As Double, K As Double, u As Double, _ d As Double, p As Double, delta As Double, n As Integer) As Double ' A recursive version of the basic option valuation routine ' For illustrative purposes ' Only viable for a small number of stages (e.g. 15) Dim pu As Double, pd As Double pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) If n > 15 Then RecursiveBinomialTree = "Too many steps for recursion" ElseIf n = 0 Then RecursiveBinomialTree = WorksheetFunction.Max(S - K, 0) Else RecursiveBinomialTree = (pu * RecursiveBinomialTree(u * S, K, u, d, p, delta, n - 1) + _ pd * RecursiveBinomialTree(d * S, K, u, d, p, delta, n - 1)) End If End Function Function RecursiveBinomialEurCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) RecursiveBinomialEurCall = RecursiveBinomialTree(S0, K, u, d, p, delta, n) End Function

Page 58: Pg p 12 Handbook

Function BinomialTree(S0 As Double, K As Double, u As Double, d As Double, _ p As Double, delta As Double, n As Integer) As Double ' Efficient implementation of binomial tree for American options ' Allocate storage Dim i As Integer, j As Integer Dim pu As Double, pd As Double Dim S() As Double, V() As Double ReDim S(2 * n) ReDim V(2 * n) ' Initialize parameters pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) ' Initialize the tree S(n) = S0 ' Calculate potential asset prices For j = n + 1 To 2 * n S(j) = u * S(j - 1) Next j For j = n - 1 To 0 Step -1 S(j) = d * S(j + 1) Next j ' Calculate option values at maturity For j = 0 To 2 * n Step 2 V(j) = Application.Max(S(j) - K, 0) Next j ' Calculate option values by backward induction For i = n - 1 To 0 Step -1 For j = n - i To n + i Step 2 V(j) = Application.Max(pu * V(j + 1) + pd * V(j - 1), S(j) - K) Next j Next i ' Return option value BinomialTree = V(n) End Function

Page 59: Pg p 12 Handbook

Function BinomialAmerCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) BinomialAmerCall = BinomialTree(S0, K, u, d, p, delta, n) End Function Function BinomialAmerPut(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) BinomialAmerPut = BinomialTree(-S0, -K, u, d, p, delta, n) End Function Function BinomialEurCall(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, fp As Double, _ pi As Double, S As Double, EV As Double, IV As Double, j As Integer dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u p = (Exp((r - q) * dt) - d) / (u - d) pi = (1 - p) / p ' calculate maximum asset price and probability S = u ^ n * S0 fp = p ^ n EV = fp * WorksheetFunction.Max(S - K, 0)

Page 60: Pg p 12 Handbook

' proceed down final asset prices calculating expected values For j = n - 1 To 0 Step -1 fp = (j + 1) / (n - j) * fp * pi S = d ^ 2 * S IV = WorksheetFunction.Max(S - K, 0) If IV > 0 Then EV = EV + fp * IV Else Exit For End If Next j BinomialEurCall = Exp(-r * T) * EV End Function Function BinomialEurPut(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, fp As Double, _ pi As Double, S As Double, EV As Double, IV As Double, j As Integer dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u p = (Exp((r - q) * dt) - d) / (u - d) pi = p / (1 - p) ' calculate minimum asset price and probability S = d ^ n * S0 fp = (1 - p) ^ n EV = fp * WorksheetFunction.Max(K - S, 0) ' proceed up final asset prices calculating expected values For j = 0 To n - 1 fp = (n - j) / (j + 1) * fp * pi S = u ^ 2 * S IV = WorksheetFunction.Max(K - S, 0) If IV > 0 Then EV = EV + fp * IV Else Exit For End If Next j BinomialEurPut = Exp(-r * T) * EV End Function

Page 61: Pg p 12 Handbook

Function EBinomialTree(S0 As Double, K As Double, u As Double, d As Double, _ p As Double, delta As Double, n As Integer) As Variant 'Extended binomial tree for calculating Greeks n = n + 2 ' extend tree ' Allocate storage Dim A As Variant Dim i As Integer, j As Integer Dim pu As Double, pd As Double Dim S() As Double, V() As Double ReDim S(2 * n) ReDim V(2 * n) ' Initialize parameters pu = delta * p ' pu and pd are discounted prices pd = delta * (1 - p) ' Initialize the tree S(n) = S0 ' Calculate potential asset prices For j = n + 1 To 2 * n S(j) = u * S(j - 1) Next j For j = n - 1 To 0 Step -1 S(j) = d * S(j + 1) Next j ' Calculate option values at maturity For j = 0 To 2 * n Step 2 V(j) = Application.Max(S(j) - K, 0) Next j ' Calculate option values by backward induction For i = n - 1 To 2 Step -1 For j = n - i To n + i Step 2 V(j) = Application.Max(pu * V(j + 1) + pd * V(j - 1), S(j) - K) Next j Next i ' Return option values EBinomialTree = Array(V(n - 2), V(n), V(n + 2)) End Function

Page 62: Pg p 12 Handbook

Function BinomialAmerCallDelta(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(S0, K, u, d, p, delta, n) BinomialAmerCallDelta = (V(2) - V(0)) / ((u ^ 2 - d ^ 2) * S0) End Function Function BinomialAmerPutDelta(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(-S0, -K, u, d, p, delta, n) BinomialAmerPutDelta = (V(2) - V(0)) / ((u ^ 2 - d ^ 2) * S0) End Function Function BinomialAmerCallGamma(S0 As Double, K As Double, r As Double, q As Double, _ sigma As Double, T As Double, n As Integer) As Double Dim dt As Double, u As Double, d As Double, p As Double, delta As Double Dim V As Variant dt = T / n u = Exp(sigma * Sqr(dt)) d = 1 / u delta = Exp(-r * dt) p = (Exp((r - q) * dt) - d) / (u - d) V = EBinomialTree(S0, K, u, d, p, delta, n) BinomialAmerCallGamma = ((V(2) - V(1)) / ((u ^ 2 - 1) * S0) - (V(1) - V(0)) / ((1 - d ^ 2) * S0)) / ((u ^ 2 - d ^ 2) * S0 / 2) End Function

Page 63: Pg p 12 Handbook
Michael
Text Box
Hull (2003) Options, Futures, and Other Derivatives
Page 64: Pg p 12 Handbook
Page 65: Pg p 12 Handbook

Improving the binomial methodMichael Carter

IntroductionAs the number of steps are increased, the binomial method converges to the true value (by the Central

Limit Theorem), but the convergence is slow and awkward. This is illustrated in the following graph for

an American out-of-the-money put option (S = 100, K = 90, r = 5 %, Σ = 30 %, T = 1 �2 ). The

horizontal axis represents the true value as calculated with a 50,000 step tree.

50 100 150 200 250

3.32

3.33

3.34

3.35

3.36

3.37

3.38

Binomial convergence - out-of-the-money put

This pattern repeats indefinitely as the number of steps is increased.

Page 66: Pg p 12 Handbook

500 1000 1500 2000 2500

3.343

3.344

3.345

3.346

3.347

3.348

3.349

Binomial convergence - out-of-the-money put

The next graph illustrates the same option with K = 110.

50 100 150 200 250 300

13.37

13.38

13.39

13.4

13.41

13.42

Binomial convergence - in-the-money put

Clearly, there is a tradeoff between accuracy and efficiency (speed). Various methods are available for

improving the performance of the binomial model. These can be classified into two groups depending on

whether aimed at

æ improving accuracy

æ improving efficiency

Typically, success on one front implies a sacrifice on the other.

2 ImprovingBinomial.nb

Page 67: Pg p 12 Handbook

Improving accuracy

à Successive averages

A popular technique in practice is to average the results of successive integers, n and n + 1.

50 100 150 200 250 300

3.33

3.34

3.35

3.36

3.37

Binomial convergence - successive averages

à Parameterization

In class, we used the simple parameterization

u = ãΣ Dt , d = ã

-Σ Dt , p =

ãHr-qL Dt

- d

u - d

where Dt = t/n. Some improvement in accuracy (at negligble computational cost) can be attained by

modifying the parameterization. Two possibilities are:

u = ãΝDt +Σ Dt , d = ã

ΝDt -Σ Dt , p =

ãHr-qL Dt

- d

u - d

and

u = ãDx, d = ã

-Dx, Dx = Σ2

Dt + Ν2

Dt2 , p =

1

21 + Ν

Dt

Dx

where Ν = r - q -1

2.

Tian (1999) proposed a flexible binomial tree with

u = ãΛ Σ

2Dt +Σ Dt , d = ã

Λ Σ2

Dt -Σ Dt , p =

ãHr-qL Dt

- d

u - d

where Λ is an arbitrarily chosen tilt parameter. Λ can be chosen so as to make one node coincide with the

strike price, thus smoothing convergence.

Another proposal of Tian sounds promising. Convergence of the binomial tree price to the true price

requires that the first and second moments (mean and variance) of the (discrete) binomial distribution of

the nodes match or converge to those of the (continuous) process of the underlying asset, that is

ImprovingBinomial.nb 3

Page 68: Pg p 12 Handbook

Another proposal of Tian sounds promising. Convergence of the binomial tree price to the true price

requires that the first and second moments (mean and variance) of the (discrete) binomial distribution of

the nodes match or converge to those of the (continuous) process of the underlying asset, that is

p u + H1 - pL d = ãr Dt

p u2+ H1 - pL d2

= ãIΣ

2+2 rM Dt

where ãIΣ2

+2 rM Dt is the second moment of asset returns when the log returns have variance Σ2Dt. (See

Chance 2007 for a thorough discussion.) Tian (1993) suggests matching in addition the third moment to

allow for the skew in the underlying distribution, requiring

p u3+ H1 - pL d3

= ã3 IΣ

2+ rM Dt

This gives three equations which can be solved for the three unknowns u, d and p. However, the outcome

is disappointing.

50 100 150 200 250 300

3.32

3.33

3.34

3.35

3.36

3.37

3.38

Binomial convergence - out-of-the-money put

Tian

à Magic numbers

The oscillations arises from the relationship between the strike price and the terminal nodes of the tree.

The graphs reveal that there are particular choices of n that minimize the error in their neighbourhood.

These magic numbers depend upon the precise parameters of the option. By tailoring the size of the tree

to the particular option, we might obtain more accurate results with smaller trees. This becomes especially

important in applying the binomial method to barrier options.

à Binomial-Black-Scholes

Convergence can be significantly enhanced using the Black-Scholes formula to evaluate the penultimate

nodes. This is known as the Binomial-Black-Scholes method.

4 ImprovingBinomial.nb

Page 69: Pg p 12 Handbook

50 100 150 200 250 300

3.335

3.34

3.35

3.355

3.36

3.365

Binomial convergence - out-of-the-money put

Binomial Black-Scholes

à Richardson extrapolation

Richardson extrapolation is a method to improve an approximation that depends on a step size. Applied to

the binomial model, extrapolation attempts to estimate and incorporate the improvement of higher n. For

example, suppose we assume that errors decline inversely with n, so that

Pn1» P +

C

n1

Pn2» P +

C

n2

where P is the (unknown) true value, Pn1 and Pn2

are estimates with step size n1 and n2 respectively, and

C is an unknown constant. Solving for P, we have

P »

n2 Pn2- n1 Pn2

n2 - n1

In particular, when n2 = 2 n1 = n, we have

P »

n Pn -n

2Pn�2

n

2

= 2 Pn - Pn�2

which can be alternatively expressed as

P » 2 Pn - Pn�2 = Pn + HPn - Pn�2L

It is not helpful when applied to the pure binomial model, but is very effective when applied after Black-

Scholes smoothing.

ImprovingBinomial.nb 5

Page 70: Pg p 12 Handbook

50 100 150 200 250 300

3.33

3.34

3.35

3.36

3.37

Binomial convergence - out-of-the-money put

BBS with Richardson Extrapolation

Smoothing and extrapolation also makes a dramatic improvement to the Tian moment-matching tree.

50 100 150 200 250

3.32

3.33

3.34

3.35

3.36

3.37

3.38

Binomial convergence - out-of-the-money put

Tian

In a recent contribution, Widdicks, Andricopoulos, Newton and Duck (2002) have applied extrapolation

to the peaks of the errors, as illustrated in the following diagram from their paper.

6 ImprovingBinomial.nb

Page 71: Pg p 12 Handbook

à Control variable

A simple and effective technique is to use the binomial method to estimate the early exercise premium, as

measured by the difference between the estimated prices of identical American and European options.

This estimate is added to the Black-Scholes value, to give the estimated value of the American option.

P = p + HAb - EbL

where p is the Black-Scholes value, Ab is the binomial estimate of the American option, and Eb is the

binomial estimate of a corresponding European option. This is known as the control variable technique.

Rewriting the previous equation

P = Ab + Hp - EbL

we observe that the effectiveness of this approach depends upon the degree to which the binomial error in

the American option matches that of the European option in sign and magnitude. Chung and Shackleton

(2005) explore this issue, provide a methodology for determining the optimal control, and discuss other

potential control variables.

ImprovingBinomial.nb 7

Page 72: Pg p 12 Handbook

50 100 150 200 250 300

3.335

3.34

3.35

3.355

3.36

3.365

Binomial convergence - out-of-the-money put

Black-Scholes control variable

à Comparison

50 100 150 200 250 300

3.335

3.34

3.35

3.355

3.36

Binomial convergence - out-of-the-money put

Binomial Black-Scholes

BBS with RE

Black-Scholes control variable

8 ImprovingBinomial.nb

Page 73: Pg p 12 Handbook

50 100 150 200 250

3.32

3.33

3.34

3.35

3.36

3.37

3.38

Binomial convergence - out-of-the-money put

Binomial Black-Scholes

BBS with RE

Tian with smoothing and extrapolation

Improving efficiency

à Truncation

Andricopoulos et al. (2004) proposed curtailing the range of nodes that are computed by backward

induction, reporting a near doubling of speed with negligible loss of accuracy.

ImprovingBinomial.nb 9

Page 74: Pg p 12 Handbook

à The diagonal algorithm

Curran (1995) proposed an innovative diagonal algorithm for evaluating binomial trees, which signifi-

cantly reduced the number of nodes that needed to be evaluated. He reported a 10 to 15-fold increase in

speed (with identical accuracy) over the corresponding standard tree. Note that this algorithm achieves a

pure increase in efficiency, returning the same result as the standard method. It is equally applicable to

extrapolation and control variable techniques.

10 ImprovingBinomial.nb

Page 75: Pg p 12 Handbook

The diagonal algorithmMichael CarterCurran (1995) proposed an innovative diagonal algorithm for evaluating binomial trees, which significantlyreduced the number of nodes that needed to be evaluated. He reported a 10 to 15-fold increase in speed (withidentical accuracy) over the corresponding standard tree. Note that this algorithm achieves a pure increase inefficiency, returning the same result as the standard method.It is equally applicable to extrapolation and controlvariable techniques.

The diagonal algorithm depends upon two propositions regarding the evolution of option values in a binary tree.They can be illustrated diagramatically as follows:

Proposition 1If it pays to exercise the option in the next period, it pays to exercise immediately.

exercise á

exercise à

exercise

Proposition 2If it pays to hold the option at some time and asset price, then it pays to hold the option at the same asset price atevery earlier price.

? á à

hold hold à á

?

Proposition 1 applies provided q § r. The intuition is that, on average, the asset price will grow, and thereforethe implicit value will decline. If it is worth exercising in the future, it is worth exercising now. Proposition 2applies irrespective of the dividend yield (provided that u d = 1).

These properties of a binary tree enable two forms of acceleration in the tree.

æ By Proposition 2, once an entire diagonal of no exercise (or hold) nodes has been computed, we can jumpimmediately to the origin, since there are no further exercise nodes. We can evaluate the initial value of theoption by computing the discounted expected value of the implicit values along the no-exercise diagonal in amanner similar to computing expected values of the terminal nodes of a European option.

æ Provided that q § r, we can start evaluation along the diagonal starting immediately below the strike price, sincewe know that all nodes below this diagonal will be exercise nodes (Proposition 1), and therefore their value willbe equal to the implicit value.

Page 76: Pg p 12 Handbook

Proof of Proposition 1: Let S denote the current asset price. Assume that both subsequent nodes are exercisenodes. Then the expected future value is

FV = Hp HK - u SL + H1 - pL HK - d SLL = K - Hp u + H1 - pL dL S

Recall that the risk-neutral probability p is such that

p u + H1 - pL d = ‰Hr-qLDt

Substituting, the expected future value at the subsequent node is

FV = K - ‰Hr-qLDt S

Provided that q § r, the expected future value is less than the current implicit value, that is

FV = K - ‰Hr-qLDt S § K - S

A fortiori, the discounted future value is less than the current implicit value. That is, ‰-rDtHK - SL is more thanyou can expect by waiting. Consequently, the option should be exercised immediately. Note that this is notnecessarily the case if q > r. In this case, expected capital gains are negative. So the option may become morevaluable.

Proposition 2 applies irrespective of the dividend yield. It depends upon the following lemma.

Lemma. Ceteris paribus, the value of an American option increases with time to maturity (Lyuu, Lemma 8.2.1).

Proof of lemma. Suppose otherwise. Sell the more expensive shorter option and buy the one with the longermaturity for a positive cash flow. Let t denote the time at which the shorter option is exercised or expires, and Pt

the value of the longer option at this time (assuming a put for example).

Case 1: Pt > max HK - St, 0L. Sell the longer option.

Case 2: Pt § max HK - St, 0L. In this case, the short option will be exercised. Offset this by exercising thelonger option.

In either case, we have a positive cash flow at time zero, and a nonnegative cash flow at time t.

Proof of Proposition 2. Let Pu and Pd denote the possible values of the option given an asset price of S, and letPu+2 and Pd

+2 denote the possible values of the option two periods later. By assumption, the holding value at time+2 is greater than the exercise value. That is

‰-rDtHp Pu+2 + H1 - pL Pd

+2 L ¥ K - S

By the lemma, the possible values are at least as great as they will be two periods later.

Pu ¥ Pu+2 and Pd ¥ Pd

+2

Therefore, the current holding is at least as great as the current exercise value.

‰-rDtHp Pu + H1 - pL Pd L ¥ ‰-rDtHp Pu+2 + H1 - pL Pd

+2 L ¥ K - S

2 DiagonalAlgorithm.nb

Page 77: Pg p 12 Handbook
Michael
Text Box
Broadie and Detemple, 1996
Page 78: Pg p 12 Handbook

Timing VBA code executionMichael CarterAccurately timing execution speed on a multitasking computer is surprisingly difficult, since the CPU can beregularly interrupted by other processes. It is normal to record different times on repeated runs. So good practicewould be to average (or minimum) over a number of runs. It is also sensible to close other applications whenundertaking timing comparisons.

VBA contains a function Timer() that gives the number of seconds since midnight. By calling Timer() at thebeginning and end of a lengthy computation, it is possible to estimate the time taken by the function as follows:

Dim StartTime, EndTime, ComputationTime As Single

StartTime = TimerDo lengthy computation

EndTime = TimerComputationTime = EndTime − StartTime

More accurate timing can be achieved using the Windows operating system function GetTickCount(), whichreturns the time in milliseconds since the system was started. It is claimed to have a resolution of 10 millisec-onds (approximately). To use this function, it must first be declared as follows:

Declare Function GetTickCount Lib " Kernel32 " HL As LongDim StartTime, EndTime, ComputationTime As Long

StartTime = GetTickCountDo lengthy computation

EndTime = GetTickCountComputationTime = EndTime − StartTime

More information is available in the Microsoft tutorial note How To Use QueryPerformanceCounter to TimeCode.

Page 79: Pg p 12 Handbook

Random number generationMichael CarterA pseudorandom number generator is a deterministic algorithm that generates a series of numbers in a

manner that appears to be random. It can be described by a recursive function

xi = f Hxi-1L

or more generally

xi = f Hxi-1, xi-2, … , xi-kL

f may be linear or nonlinear. The starting value x0 (or values) is called the seed. By default, the seed is often

set from the system clock.

The most common pseudorandom number generator is the linear congruential generator

xi = Ha xi-i + cL mod m

Rescaling the sequence by dividing my m gives a sequence in the ui = xi �m in the unit interval (0,1).

By careful choice of a, c and m, a sequence of period m can be obtained. Then, the resulting sequence of real

numbers will appear to be uniformly distributed on H0, 1L. However, not that inappropriate choice of a, c and

m can give very poor results.

Excel VBA's RND function is a linear congruential generator with a = 1 140 671 485, c = 12820163 and

m = 224, with a period of 224= 16, 777, 216. Starting with the default seed of 327680, the first values in

the sequence are: 0.705548, 0.533424, 0.579519, 0.289562, and 0.301948. (It is not possible to replicate this

generator in VBA or a spreadsheet because the internal representation of long integers is not accessible.)

Press, Teukolsky, Vetterling and Flannery (1992) is a valuable source of reliable random number generators

that have been well-tested, including the so-called "minimal standard" generator of Park and Miller which

uses

a = 75= 16 807 c = 0 m = 231

- 1 = 2 147 483 647

The period of this generator is 231- 2 » 2.1 � 109. For general use, they recommend ran1 which combines

the minimal standard generator with an additional shuffle to eliminate low-order serial correlations. Where a

longer period is required, they describe ran2 which combines two linear congruential generators with

different periods. They also describe some "quick and dirty alternatives" for less demanding applications.

An increasingly popular algorithm is the Mersenne twister. It has a period of 219,937- 1, is fast to compute

and is guaranteed to have equidistribution properties in at least 633 dimensions. An implementation for Excel

is freely available from NT Technologies.

Excel 2003 implemented a new RAND function based on the Wichman and Hill (1982) algorithm, which

combines three linear congruential generators. It is claimed to have a period of 1013. This is retained in Excel

2007 (see Description of the RAND function in Excel 2007 and in Excel 2003).

Page 80: Pg p 12 Handbook

Normal random deviatesPseudo-random values from a distribution other than uniform (e.g. normal) are known as random deviates, to

distinguish them from uniform random numbers. Normal random deviates are almost always generated in a

two-step procedure, first generating normal random numbers and then transforming them in some way to

produce numbers that appear to come from a standard normal distribution. A variety of very clever methods

have been proposed for doing this transformation, of which the most common are the Box-Muller method

and normal inverse transformation.

� Box-Muller transformation

à Draw two random numbers u1 and u2 from U @0, 1D such that w = u12

+ u22

< 1

à Calculate c = -2log HwL

w

à z1 = c u1 and z2 = c u2 are normally distributed (N@0, 1D)

The Box-Muller transformation is very efficient. However, there are circumstances in which it is

inappropriate

à It should not be used with quasi-random number sequences, as it will scramble the low-

discrepancy property of the sequence.

à It should not be used in conjunction with a simple LCG generator (Bratley, Fox, Schrage 1982:

223)

� Inverse normal transformation

Let U be uniformly distributed on (0,1). Let FHxL be the distribution function of a continuous distribution.

Then F is strictly increasing on [0,1] and has an inverse F-1 such that

u = FHxL if and only if x = F-1HuL

Therefore, we can compute a random deviate X with distribution FHxL simply by setting X = F-1HU L.Then

PHX £ xL = PIF-1HU L £ xM = PHU £ FHxLL

But since U is uniform, PHU £ uL = u for 0 < u < 1. Therefore

PHX £ xL = PHY £ FHxLL = FHxL

So the problem of generating random deviates is essentially a numerial problem of inverting the distribution

function.

2 RandomNumberGeneration.nb

Page 81: Pg p 12 Handbook

The cumulative distribution function of the standard normal distribution is

FHxL =

1

2 Π

à-¥

x

ãt2�2

â t

This has no closed form, and so much be computed numerically. Hull (2003) outlines a polynomial approxi-

mation that gives six place decimal accuracy (p. 248).

Similarly, the inverse CDF much be computed numerically. Moro (1995) proposed a rational polynomial

approximation that gives eight place decimal accuracy. It has become very popular in finance applications.

The NORMSINV function in Excel XP has been reported to still give slightly erroneous results (McCullough

and Wilson, 2002).

Multivariate normal distributionGiven two independent standard normal random variables z1 and z2, the following transformation generates

two correlated normal random variables with mean 0, standard deviation 1 and correlation coefficient Ρ.

x1 = z1

x2 = Ρ z1 + 1 - Ρ2 z2

� Exercise

Do it!

RandomNumberGeneration.nb 3

Page 82: Pg p 12 Handbook

Furthermore, if we set

x1 = Σ1 z1 + Μ1

x2 = Σ2K Ρ z1 + 1 - Ρ2 z2O + Μ2

then Hx1, x2L has a bivariate normal distribution with mean HΜ1, Μ2L and covariance matrix

S =Σ1

2Ρ Σ1 Σ2

Ρ Σ1 Σ2 Σ22

, as can be confirmed by direct computation.

The transformation comes from the Cholesky decomposition of S =Σ1

2Ρ Σ1 Σ2

Ρ Σ1 Σ2 Σ22

.

Any positive definite matrix S can be factored into a product of triangular matrices S = H HT . In this case,

H =

Σ1 0

Ρ Σ2 1 - Ρ2

Σ2

with H HT= S

If Z is a vector of independent normal random variables, that is Z ~ N(0,I), then

X = H Z + Μ

is N(Μ,S) where S = H HT . Conversely, if X ~ N(Μ,S), then

Z = H-1HX - ΜL

is N(0,I).

4 RandomNumberGeneration.nb

Page 83: Pg p 12 Handbook

Article ID: 828795 - Last Review: January 16, 2007 - Revision: 5.3

Description of the RAND function in Excel 2007 and in Excel 2003

This article describes the modified algorithm that is used in the random numbergenerator function, RAND, in Microsoft Office Excel 2007 and in Microsoft Office Excel 2003.

The RAND function in earlier versions of Excel used a pseudo-random numbergeneration algorithm whose performance on standard tests of randomness was not sufficient. Although this is likely toaffect only those users who have to make a large number of calls to RAND, such as a million or more, and not to be aconcern for almost every user, the pseudo-random number generation algorithm that is described here was firstimplemented for Excel 2003. It passes the same battery of standard tests.

The battery of tests is named Diehard (see note 1). The algorithm that is implemented in Excel 2003 was developed byB.A. Wichman and I.D. Hill (see note 2 and note 3). This random number generator is also used in the RAT-STATSsoftware package that is provided by the Office of the Inspector General, U.S. Department of Health and HumanServices. It has been shown by Rotz et al (see note 4) to pass the DIEHARD tests and additional tests developed by theNational Institute of Standards and Technology (NIST, formerly National Bureau of Standards).

Notes

The basic idea is that if you take three random numbers on [0,1] and sum them, the fractional part of the sum is itself arandom number on [0,1]. The critical statements in the Fortran code listing from the original Wichman and Hill articleare:

Therefore IX, IY, IZ generate integers between 0 and 30268, 0 and 30306, and 0 and 30322 respectively. These arecombined in the last statement to implement the simple principle that was expressed earlier: if you take three randomnumbers on [0,1] and sum them, the fractional part of the sum is itself a random number on [0,1].

Because RAND produces pseudo-random numbers, if a long sequence of them is produced, eventually the sequence willrepeat itself. Combining random numbers as in the Wichman-Hill procedure guarantees that more than 10^13 numberswill be generated before the repetition begins. Several of the Diehard tests produced unsatisfactory results with earlierversions of RAND because the cycle before numbers started repeating was unacceptably short.

Results in Earlier Versions of Excel

The RAND function in earlier versions of Excel was fine in practice for users who did not require a lengthy sequence ofrandom numbers (such as a million). It failed several standard tests of randomness, making its performance an issuewhen a lengthy sequence of random numbers was needed.

Results in Excel 2003

A simple and effective algorithm has been implemented. The new generator passes all standard tests of randomness.

Conclusions

Power users of RAND who require lengthy sequences of random numbers are better offwith the new generator of Excel 2003. Other users should be undeterred from using RAND in earlier versions of Excel.

SUMMARY

MORE INFORMATION

The tests were developed by Professor George Marsaglia, Department of Statistics, Florida State University andare available at the following Web site: http://i.cs.hku.hk/~diehard (http://i.cs.hku.hk/~diehard)

Wichman, B.A. and I.D. Hill, Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator,Applied Statistics, 31, 188-190, 1982.Wichman, B.A. and I.D. Hill, Building a Random-Number Generator, BYTE, pp. 127-128, March 1987.Rotz, W. and E. Falk, D. Wood, and J. Mulrow, A Comparison of Random Number Generators Used in Business,presented at Joint Statistical Meetings, Atlanta, GA, 2001.

C IX, IY, IZ SHOULD BE SET TO INTEGER VALUES BETWEEN 1 AND 30000 BEFORE FIRST ENTRY IX = MOD(171 * IX, 30269) IY

= MOD(172 * IY, 30307) IZ = MOD(170 * IZ, 30323) RANDOM = AMOD(FLOAT(IX) / 30269.0 + FLOAT(IY) / 30307.0 +

FLOAT(IZ) / 30323.0, 1.0)

Description of the RAND function in Excel 2007 and in Excel 2003 http://support.microsoft.com/kb/828795/en-us

1 of 2 05/03/2009 10:52

Page 84: Pg p 12 Handbook

Help and Support©2009 Microsoft

For more information about an issue that was documented to occur in RAND, click thefollowing article number to view the article in the Microsoft Knowledge Base: 834520 (http://support.microsoft.com

/kb/834520/ ) The RAND function returns negative numbers in Excel 2003

APPLIES TO

Keywords: kbfuncstat kbfunctions kbinfo KB828795

Get Help Now

Contact a support professional by E-mail, Online, or Phone

REFERENCES

Microsoft Office Excel 2007Microsoft Office Excel 2003

Description of the RAND function in Excel 2007 and in Excel 2003 http://support.microsoft.com/kb/828795/en-us

2 of 2 05/03/2009 10:52

Page 85: Pg p 12 Handbook

1/24/2012

1

Numerical Recipes

Neveruse a LCG generator (alone)use a generator with a period < 2 ≈ 2 × 10use a built‐in generator in C or C++

Avoidgenerators designed for cryptographic usegenerators with period > 10

Numerical Recipes

Alwaysuse a generator  that (correctly) combines two unrelated methods

If criticalconfirm results with a different generator

Normal transformation

Don’tuse Box‐Mueller transformation 

Insteaduse Moro transformation or alternative

Excel’s NormSInv now OK

Excel

Set seed? Volatile? QualityRnd() Y N WeakRand() N Y UnknownPRand Y Y &N Good

PRand

‘Every‐day’ RNG from Numerical Recipes

Combines  64‐bit  XorShift with LCG

Period  2 ≈ 2 × 10

NormMInv () – Moro inverse normal CDF

Uniform Normal Seed

Volatile URand() NRand() SetSeed

Non‐volatile URandNonV() NRandNonV() SetSeedNonV()

Comparative efficiency

NormSInv NormMInv

Rand 3.4 – 4.2 3.6

URand 4.2 3.6

NRand 3.6

vbaNRand 4.7

Time required for 1 million simulations (minutes)

2.1 GHz Pentium

Page 86: Pg p 12 Handbook

f f f i * o N r E c A R t o s l M U t A T t o N

THE FULL MoNITETronsforming quosi'rondom numbers into useful Monte Corlosimulotion voriobles con be o bit of o gomble. Boris Moroexploins how fo do it quickly ond occurotely

I I onte Carlo simulat ion is often used forf V I pricing European-slyle derivative secu-ri t ies n' i th complicated pavoffs, especiai lvthose dependent upon severa l Bror t 'n ianmotions.

I t invo lves s imulat ing the path o f theunderlf ing security price a number of timesand calculating the expected option value asan average over all prices obtained at maturi-ry. The path depends on the drift and volatil-ity, n'hich measures the size of normalh' dis-tributed random fl uctuations.

In practice, one first has to calculate pseudo-random numbers in the interval (0,1) - or aunit h1'percube, (0,1)s, for an s-factor model -and then transform them into a set of Gaussiandeviates using the Box-Muller algorithm (seeFlannery, Press and Teukolsky (1992), page289 ) . Un fo r t una te l y , i f t he Mon te Ca r l omethod is based on standard pseudo-randomnumbers, it is very slow because the pricingerror decreases only with 1/{N where N is thenumber of simulation runs.

Recently, Jov, Bo1'le and Tan (1994) haveshown that the efficienry of the method canbe greatly increased if quasi-random rather

than pseudo-random numbers are used (Fox(1986)). These are determinist ic sequencesdefined on a unit hvpercube n'hich have aIon'discrepanc\/, thails, they fill the domainnearl;' uniformll'. Unlike a pseudo-randomsequence of finite length, there can be no clus_ters of points, so u'hen the function is calcu-lated by averaging over such a set, the esti-mate is more accurate.

In a Monte Carlo simulation, such num-bers must be transformed into Gaussian devi-ates so that tl're low discrepancy of the origi-nal sequence is preserved. The Box-Mullermethod and its variants scramble the quasi-random sequence and therefore should not beused for that purpose.

Instead, one has to computex(Y), of the cumulative normalfunction

the inverse,distr ibution

t '1 x

Y(x) = -= te 'd t

^l2n -*

u'hich can be done using Newton,s iteration.in practice, howevet this is too slow to be use_ful. The approximation'derived by Hill and

Relotive errors in the foil of distribution1 . 0

I O-rt0-:

I0- l

l 0 *

l 0 -5

l0 -6

l 0 t

l0-8

l0 -e

l 0 - , 0

l 0 - "

l 0 - , r

l 0 - , ,

I 0 - , 0

2 .6 .104 I . 9 . 1 0 - 8 1 . 0 . 1 0 - r r

( :

Davis (1973) also gives the result correct tomachine precision but is not much faster thanNen'ton's ntethod. Rational approximationsfor x(Y) n'ere deriyed by Odeh and Evans

LT.974) and by Beastey and Springer (1977).The l a t t e r p rog ram (ppND) i J f as t bu tbecomes inaccurate in the tails of the distrib_ution.

We obta ined an approx imat ion to theinverse of the cumulative normal distributionfunction n'hich has a high acoracy for all y inthe inten'al [10-r:, !-Lo-L2]and ii faster thanstandard implementations of the Box-Mullermethod.

We shall use symmetry of the inverse ofthe cumulative normal distribution and con_sider onlv the interval (0.5, 1). For smalldeviations (Y < 0.92) we use the functionderived by Beasley and Springer:

H 2 ^La"y

x(y)=v-s* . _ (1)

1+ I bny2 "

w h e r e Y : Y - 0 . 5 .For Y > 0.92 \^/e approximate x(y) by a

polynomial:

n=8

x (Y )= I . n .nn=0

where: . -

kr in(- ln(1- y)) + k, ; an, bn and cnare suitabh' chosen coefficients and the con_stants k, and lg are such that z = _! whenY = 0.92 andz : lwheny = 1 - !O-r2 .

The rat ional approximation (1) has thelargest absolute error of 3x10-s in the interval[0.5, 0.92]. In the tails of the distribution, ouralgori thm retains this accuracy for up toseven standard deviations. High_order poly_nomial approximations which have 14_digitaccuracy in the interval [10-rs, t-tg-rs; cineas i ly be const ructed but they run muchslower.

The table shoH's the time needed for gen-erating 107 double precision normally dis_tributed numbers on a Sun Sparc 10-30 usingthree programs: GASDEV (Box-Muller algol

11B as implemented by Flanne ry et i ly,PIND (Beasley and Springer) and CNDEVu'hich is based on our algorithm. In the firstset of runs, uniformly distributed pseudo_

RISKVOL 8/NO 2/FEBRUARY

Page 87: Pg p 12 Handbook

. l a M O N T E C A R L O S I M U L A T I O N

randonl numbers from the inten'al (O,1) n'eregenerated by the function DRAND48. In thesecond set of runs (PPND and CNDEV) nor-mal deviates \^'ere produced from uniformlydrarvn numbers from the interval (0,1), ie,rr'ithout function DRAND48. I

Boris Moro is o researcher for TMG FinonciolProducts in Greenwich, Connecticut. He thonks

Professor Phelin Boyle for comments ondsuggesfions

THE CNDEV.C PROGRAM#include <moth.h>

Are you taking enough care?

RISkcareThe Derivative Development & Consultancy Specialists

Experienced independent advice on hedging strategies, investment opportunities, individual & portfolio pricing and ongoing risk analysis.For users, market makers and software developers of derivative products, Riskcare provides

the mathematics, technology, training and business awareness to suit your requirements.

Complex structures need care.Telephone: 44 (0) l7t 251 4748 or Fax: 251 2663

Complete confidentiality guaranteed.

RISK vor. B/No 2/IEBRUARY r ee5

Page 88: Pg p 12 Handbook

Simulating asset pricesMichael CarterAs a computation method, simulation has two major advantages

æ it can easily deal with multiple random factors, for example random interest rates or volatility, oroptions on multiple assets

æ it can easily incorporate more alternative asset processes, such as jumps or non-normal stochasticprocesses

The major disadvantage is that it is computationally very intensive.

Our starting point is the observation that periodic returns are approximately normally distributed around atrend

CNX Nifty 1990-2006

50010001500200025003000

Prices

-0.1-0.05

00.050.1

Returns

-4 -2 2 4

0.05

0.1

0.15

0.2

0.25

CNX Nifty distribution v. normal

That is, we assume

lnJ StÅÅÅÅÅÅÅÅÅÅÅÅÅSt-1

N ~ NIn Dt, sè!!!!!

Dt M

with mean and variance are proportional to Dt. This implies that the distribution of asset prices is lognormal.This process can be simulated by computing

ln St = ln St-1 + n Dt + sè!!!!!

Dt zt

Page 89: Pg p 12 Handbook

This equation distinguishes two components of the periodic return - the steady drift n Dt together with thestochastic variation s è!!!!!

Dt zt. The previous equation can be rewritten as

St = ‰n Dt + sè!!!!!!

Dt z St-1

where zt is a standard normal random variable. Since the exponential function is nonlinear

E@ StD = ‰Hn + s2ê2L t S0 = ‰m t S0

where m = n + s2 ê 2 is the expected return of the asset.

More formally, the fundamental assumption underlying Black-Scholes and most other financial models isthat asset prices follow a particular stochastic process called geometric Brownian motion (GBM)

(1)dS = m S dt + s S dz

or

dSÅÅÅÅÅÅÅÅÅS

= m dt + s dz

This implies that the logarithm of the stock price follow

(2)d ln S = n dt + s dz

where n = m - 1ÅÅÅÅ2 s2. The derivation of (2) from (1) is an application of Ito's lemma (Hull: 2003:232-233,Luenberger 1998:313).There are two methods of simulating stock prices, depending upon whether we take a discrete version of (1)or (2).

A discrete version of (1) is

dS = St - St-1 = m St-1 Dt + s St-1 et è!!!!!

Dt

where et is a standard normal random variable, so that

(3)St = I1 + m Dt + s et è!!!!!

Dt M St-1

This is the approach taken by a Hull (2003: 224).

A discrete version of (2) is

d ln S = lnHStL - lnHSt-1L = n Dt + s et è!!!!!

Dt

where et is a standard normal random variable and n = m - s2 ê 2, so that

lnHStL = lnHSt-1L + n Dt + s et è!!!!!

Dt

or

(4)St = ‰n Dt + s et è!!!!!!

Dt St-1

In practice, the second approach is preferred because it is more accurate (although it is computationallyslower). Whereas (3) is only true in the limit as Dt Ø 0, equation (4) is true for all Dt.

2 SimulatingPrices.nb

Page 90: Pg p 12 Handbook

Monte Carlo simulation

Statistical foundations

The value of a derivative instrument is discounted expected value of the payoff

V0 = EAã-r T V HSt, 0 £ t £ TLE

where the expectation is taken with respect to the risk-neutral distribution. Monte Carlo simulation can be used to estimatethis expected value by averaging over repeated samples from the assumed distribution of the underlying.

V0 = ã- r T E@V D » ã

- r TV1 + V2 + … Vn

n

where Vi is the value of the of the ith realization under the risk-neutral distribution.

The statistical foundations of Monte Carlo simulation rest on the two fundamental theorems of probability. Assume thatHX1, X2, ..., XnL are independent and identically distributed random variables with mean Μ and variance Σ2.

� First fundamental theorem (Law of large numbers)

The sample average converges to the mean, that is

X =

X1 + X2 + ... + Xn

n� Μ

� Variance of independent random variables

Since the variables are independent, their variance is additive, that is

Var@X D = VarB1

nX1F + VarB

1

nX2F + ... + VarB

1

nXnF

=

1

n2Var@X1D +

1

n2Var@X2D + ... +

1

n2Var@XnD

=

1

nVar@X D =

Σ2

n

so that ΣX = Σ � n .

� Second fundamental theorem (Central limit theorem)

Whatever the distribution of X , the sample mean X converges to a normal distribution. That is, for large samples,

X ~ NJΜ, Σ2

nN.

Page 91: Pg p 12 Handbook

Variance reduction

The principal methods of variance reduction are:

� Antithetic variables

A sufficient condition for variance reduction with uniform or normal random variables is that the outcome function ismonotonic.

� Control variables

With Α selected optimally, that is Α*= Cov@Y , CVD �Var@CVD, the ratio of the variances of the control variable estimator to

the standard estimator is

VarAY - Α*ICV - E@CVDME

Var@Y D= 1 - ΡY ,CV

2

æ The effectiveness of the control variable method depends solely upon the degree of correlation betweentarget variable and the control variable (the sign is absorbed into Α).

æ In practice, the population moments are unknown and Α* is estimated from the sample. This introducessome bias in finite samples. This bias can be eliminated by estimating Α on a different random sample.Typically, the bias is so small that this is not worthwhile.

� Importance sampling

Other methods include:

� Moment matching

� Stratified sampling

� Latin hypercube sampling

Comments:

æ Moment matching can also be achieved by using the moments as control variables. Boyle, Broadie andGlasserman, BS show that including using moments as control variables is asymptotically better thanmoment matching.

æ Low-discrepancy (quasi-random) sequences automatically achieve first-moment matching, stratified andLatin hypercube sampling.

Simulating the Greeks

The delta of a derivative is

D =

d V

d S0

With simulation, this can be estimated by the average of the finite-differences over a sample n of replications.

2 MonteCarloSimulation.nb

Page 92: Pg p 12 Handbook

D�

=

1

V�

HS0 + ΕL - V�

HS0L

Ε

If we use different random samples for estimating V�

HS0L and V�

HS0 + ΕL, the variance of D�

becomes very large as Ε becomessmall.

A better estimate of D is generally obtained by using the same random numbers in estimating both V�

HS0L and V�

HS0 + ΕL. The

variance of D� is

Var@D�

D =

VarAV�

HS0LE + VarAV�

HS0 + ΕLE - 2 CovAV�

HS0L, V�

HS0 + ΕLE

Ε2

Provided that V�

HS0L and V�

HS0 + ΕL are positively correlated, the estimate obtained from common random numbers will have alower variance.

� Pathwise derivative

The pathwise derivative decomposes the total derivative into two components.

d V�

d S0

=

d V�

d ST

d ST

d S0

Assuming geometric Brownian motion and the risk-neutral distribution

ST = S0 ãKr-q-

Σ2

2O T + Σ T Z

so that

d ST

d S0

= ãKr-q-

Σ2

2O T + Σ T Z

=

ST

S0

The discounted payoff of a vanilla European call option is ã-r T max HST - K, 0L

d V�

d ST

= ã-r T

d

d S0

max HST - K, 0L =ã

-r T , ST > K

0, ST < K

The function is not differentiable at ST = K, but this is a zero-probability event. Combining the two factors, the pathwiseestimator of delta for a vanilla European call option is

d V�

d S0

=

d V�

d ST

d ST

d S0

= ã-r T

ST

S0

I 8SHTL > K<

where I 8SHTL > K< equals one if SHTL > K and zero otherwise. This estimator can easily be computed by simulation. Aminor modification of this procedure can be used to estimate vega, but it is generally inapplicable to estimating gamma.

MonteCarloSimulation.nb 3

Page 93: Pg p 12 Handbook

The Cholesky decomposition

Michael Carter

� Preliminaries

Multivariate normal distribution

Given two independent standard normal random variables Z1 and Z2, the following transformation generates two correlatednormal random variables with mean 0, standard deviation 1 and correlation coefficient Ρ.

X1 = Z1, X2 = Ρ Z1 + 1 - Ρ2 Z2

Furthermore, if we set

X1 = Μ1 + Σ1 Z1 , X2 = Μ2 + Σ2K Ρ Z1 + 1 - Ρ2 Z2O

then HX1, X2L has a bivariate normal distribution with mean HΜ1, Μ2L and covariance matrix S =Σ1

2Ρ Σ1 Σ2

Ρ Σ1 Σ2 Σ22

, as

can be confirmed by direct computation.

Generalizing, if Z is a n-vector of independent normal random variables (Z ~ N(0,I)) and an L is an n´n square matrix, then

X = Μ + L Z

is N HΜ, SL where S = L LT . Therefore, in order to produce a random sample from a multivariate normal distribution, we

need to find a matrix L with the property that L LT= S. Then

E@XD = Μ + A E@ZD = Μ

EAHX - ΜL HX - ΜLT E = EAL ZHL ZLT E = EAL Z ZT LT E = L EAZ ZT E LT= L I LT

= L LT

The Cholesky decomposition

Any square, symmetric, positive-definite matrix A can be factored into the product of two triangular matrices

A = L LT

For example, the Cholesky decomposition of the matrix2 1

1 2is

2 0

1

2

3

2

since

2 0

1

2

3

2

2 1

2

0 3

2

=2 1

1 2

Now consider the general case for n = 3

Page 94: Pg p 12 Handbook

L LT=

l1,1 0 0

l2,1 l2,2 0

l3,1 l3,2 l3,3

l1,1 l2,1 l3,1

0 l2,2 l3,2

0 0 l3,3

=

l1,12 l1,1 l2,1 l1,1 l3,1

l1,1 l2,1 l2,12

+ l2,22 l2,1 l3,1 + l2,2 l3,2

l1,1 l3,1 l2,1 l3,1 + l2,2 l3,2 l3,12

+ l3,22

+ l3,32

=

a1,1 a2,1 a3,1

a2,1 a2,2 a3,2

a3,1 a3,2 a3,3

Equating the resulting matrices element by element, we can solve the system row by row

l1,12

� a1,1

l1,1 l2,1 � a2,1

l1,1 l3,1 � a3,1

l1,1 l2,1 � a2,1

l2,12

+ l2,22

� a2,2

l2,1 l3,1 + l2,2 l3,2 � a3,2

l1,1 l3,1 � a3,1

l2,1 l3,1 + l2,2 l3,2 � a3,2

l3,12

+ l3,22

+ l3,32

� a3,3

to give

l1,1 = a1,1 ,

l2,1 =

a2,1

l1,1

, l2,2 = a2,2 - l2,1

l3,1 =

a3,1

l1,1

, l3,2 =

a3,2 - l2,1 l3,1

l2,2

, l3,3 = a3,3 - l3,12

- l3,22

The general case is

li,i = ai,i - âj=1

i-1

li, j2

l j,i =

1

li,iai, j - â

k=1

i-1

li,k l j,k , j = i + 1, …, n

Implementation

2 Cholesky.nb

Page 95: Pg p 12 Handbook

Asian options

IntroductionAsian options are popular in currency and commodity markets because

æ they offer a cheaper method of hedging exposure to regular periodic cash flows

æ they are less susceptible to manipulation of the spot market.

There are two classes of Asian options

æ average price options (payoff max IS - K, 0M or max IK - S, 0M)

æ average strike options (payoff max IST - S, 0M or max IS - ST , 0M)

Average strike options are sometimes called floating strike options. Vanmaele et al. (2006) provide transforma-tions between average price and average strike options.

In addition, there are two ways of calculating the average S - arithmetic and geometric.

A =

1

m + 1HS0 + S2 + … + SmL ³ G = HS0 �S1 �… �SnL

1

m+1

Note that by convention, the average includes the price on the first day of averaging. Jensen's inequality statesthat the arithmetic mean is larger than the geometric mean, with equality if and only if the observations areequal.

Unfortunately, most Asian options in practice are based on arithmetic averaging, while precise results arereadily available only for geometric averages. In practice, we use the price of an equivalent geometric averageoption as a lever to price its arithmetic counterpart.

� Upper and lower bounds

Jensen's inequality implies that the value of an arithmetic Asian option is bounded below by its geometriccounterpart.

cGHTL £ cAHTL

We can also develop useful upper bounds. The value of the arithmetic Asian option is

cAHTL = ã-r T EAIST - KM

+E

where expectation is taken with respect to the risk neutral distribution. This implies

cAHTL = ã-r T EAIST - KM

+E

= ã-r T EB

1

m + 1âi=0

m

SHtiL - K+

F

=

1

m + 1ã

-r T EB âi=0

m

HSHtiL - KL+

F

£

1

m + 1ã

-r T EBâi=0

m

HSHtiL - KL+F

=

1

m + 1âi=0

m

ã-r T E@HSHtiL - KL+D

Page 96: Pg p 12 Handbook

£

1

m + 1âi=0

m

cHtiL

The value of an Asian option is less than the value of a portfolio of intermediate options with the same strikeprice. Furthermore, absent dividends, cHtiL £ cHTL and therefore

cGHTL £ cAHTL £

1

m + 1âi=0

m

cHtiL £ cHTL

� Put-call parity

Put-call parity applies to European average price options, whether arithmetic or geometric. To see this, observethat the difference between the terminal values of a put and call option is

cH0L - pH0L = IS - KM+

- IK - SM+

= S - K

Taking risk-neutral expectations

cHTL - pHTL = ã-r T IE@SD - KM

The expectation E@SD can be calculated exactly for both arithmetic and geometric averages. Consequently, itsuffices to compute call values and derive the values of puts from (1).

Geometric average optionsSuppose that St is lognormal with mean Ν t and variance Σ2 t. Then the geometric average

Gm = HS0 �S1 � … � SmL1

m+1

is lognormal with mean

(1)E@GmD = S0 ãKΝ +

2 m+1

m+1

Σ2

6O

T

2

and variance

(2)V @log GmD =

2 m + 1

m + 1

Σ2

6T

See Appendix. Under the risk neutral distribution Ν = r - q - Σ2 �2. Substituting in (1)

E@GmD = S0 ãKr -q -

m+2

m+1

Σ2

6O

T

2

By the fundamental theorem (Hull 2003: 262), the value of a European call option on Gm is

cG = ã-r t E@maxHGm - K, 0LD = ã

-r tHEHGmL NHd1L - K NHd2LL

where NHL denotes the standard normal distribution function and

d1 =

lnI E@GmDK

M +V 2

2T

V, d2 =

lnJ E@GmDK

-V 2

2TN

V

With continuous averaging, m � ¥ and

E@G¥D = S0 ãKr -q -

Σ2

6O

T

2

with

V @log G¥D =

Σ2

3T

2 Asian.nb

Page 97: Pg p 12 Handbook

Making the substitution

q*=

1

2r + q +

Σ2

6

the expected value is

E@G¥D = S0 ãKr -q -

Σ2

6O

T

2 = S0 ãHr - q*L T

which is the same as the expected value of ST assuming the dividend yield q*.

Consequently, the value of European geometric Asian option with continuous averaging is given by Black-Scholes formula with the substitutions (Hull 2003, p. 444)

dividend yield =

1

2r + q +

Σ2

6volatility =

Σ

3

This also provides a useful approximation for large n.

Moments of the geometric mean under the risk neutral distribution

� Discrete Continuous

E@GD S0 ãKr -q -

m+2

m+1

Σ2

6O

T

2 S0 ãKr -q -

Σ2

6O

T

2

V @log GD I 2 m + 1

m+1M Σ

2

6T Σ

2

3T

Arithmetic average optionsThe arithmetic average of lognormal random variables is not lognormal, and its precise distribution has provedintractable. There are three practical approaches to accurate valuation of arithmetic Asian options:

æ analytical approximation

æ simulation using the geometric average option as a control variate

æ a modified binomial method

� Analytical approximation

Although the distribution of the arithmetic average is A is intractable, its moments EAAkE can be readily

calculated. This has spurred a variety of approximation methods.

� Moments of A

Observe that

S1 = S0 R1

S2 = S1 R2 = S0 R1 R2

Sm = S0 R1 R2 … Rn

âi=0

m

Si = S0H1 + R1 + R1 R2 + … + R1 R2 … RmL

where the gross returns Ri are independent and identically distributed. Let a = E [Ri] . Then

EBâi=0

m

SiF = S0 E@1 + R1 + R1 R2 + … + R1 R2 … RnD = S0I1 + a + a2+ … + amM = S0

1 - am+1

1 - a

Under the risk neutral distribution, a = E@RiD = ãHr-qL T�m. Therefore

Asian.nb 3

Page 98: Pg p 12 Handbook

E@AmD =

1

m + 1EBâ

i=0

m

SiF =

S0

m + 1

1 - ãHr-qL THm+1L�m

1 - ãHr-qL T�m

For continuous averaging, we have (Hull 2003: 444)

E@A¥D = limn® ¥

E@AnD = limm® ¥

S0

m + 1

1 - ãHr-qL THm+1L�m

1 - ãHr-qL T�m

=

IãHr-qL T

- 1M

Hr - qL TS0

� An upper bound

We have previous observed that G £ A (Jensen's inequality). This implies that CG £ GA, but also that thedifference in payoffs is

HA - KL+- HG - KL+

=

0, G £ A £ K

A - K, G £ K £ A

A - G, K £ G £ A

In every case we have

HA - KL+- HG - KL+

£ A - G

Taking risk-neutral expectations

CAHTL - CGHTL £ ã-r T HE@AD - E@GDL

and therefore

CGHTL £ CAHTL £ CGHTL + ã-r T HE@AD - E@GDL

Example. For S0 = 50, r = 10 %, q = 0, Σ = 40 %, T = 1 and n = 250, the expected values of the arithmeticand geometric means are 52.59 and 51.86 respectively. The value of geometric average call option is 5.13.Therefore, we can deduce that the value of an arithmetic average option CA is bounded as follows

5.13 £ CA £ 5.13 + ã-0.1 H52.37 - 51.87L = 5.79

� Simple modified geometric

The simplest analytical approximation assumes that arithmetic average is lognormally distributed with mean

E@AnD, but assuming the standard deviation of ln An is Σg= Σ/ 3 . By the fundamental theorem (Hull 2003:

262)

CA = ã-r T E@maxHS - K, 0LD = ã

-r T HE@AnD NHd1L - K NHd2LL

where

d1 =

lnHE@AnD �KL + ΣG2 T �2

ΣG T, d2 = d1 - ΣG T

� Other analytical approximations

Extensions of the previous idea include:

æ Levy: Assume An is lognormally distributed with true mean and variance (Hull 2003: 444)

æ Turnbull and Wakeman: Approximate the actual distribution of An using Edgeworth expansions.

There is evidence that Levy approximation is not generally more accurate (and may be less accurate) than thesimple modified geometic method, and that the Turnbull and Wakeman method is no more accurate unlesshigher (third and fourth) moments are used (James 2003: 215-216).

� Bounds

Earlier, we developed upper and lower bounds for an arithmetic average option

4 Asian.nb

Page 99: Pg p 12 Handbook

CGHTL £ CAHTL £ CGHTL + ã-r T HE@AD - E@GDL

In a recent contribution, Nielsen and Sandmann (2003) develop tighter upper and lower bounds that are easilycalculated. In a numerical analysis of 32 options, they find that the Levy approximation satisfies the bounds inonly eight cases while the Turnbull and Wakeman method satisfies the bounds in only seven cases. Theyconclude "more information is gained from the easily calculated bounds than from the pricing approximations"in their sample. They also show how the bounds can be used to calculate hedge parameters.

Asian.nb 5

Page 100: Pg p 12 Handbook
Michael
Text Box
Hull (2003) Options, Futures, and Other Derivatives
Page 101: Pg p 12 Handbook
Page 102: Pg p 12 Handbook
Page 103: Pg p 12 Handbook

Barrier optionsMichael Carter

à Preliminaries

IntroductionA barrier option is similar to a vanilla call or put, but the final payoff depends on whether or not the underly-ing price achieves a particular level (the barrier) during the life of the option. The vanilla option may eitherknock-in or knock-out when the barrier is crossed. Sometimes, a rebate is paid if the vanilla option lapsesbecause the barrier condition is not met. Barrier options are one of the most commonly traded exoticoptions, since they are less expensive than vanilla calls and puts.

European barrier optionsFollowing Björk (1998), we consider the class of derivatives on a single underlying asset where the payoffFHST L depends on ST (and not St, t < T). Throughout, we consider continuous monitoring. Since a portfoliocomprising a knock-in and a knock-out option with the same barrier and strike will always pay the same as avanilla option (e.g. HS - KL+), we can use the in-out parity condition

Knock - in option + Knock - out option = Vanilla option

to deduce the price of one type from the other. A knock-out option can be either down-and-out or up-and-out, depending upon the relationship between the current asset price S and the barrier H .

85 90 95 100 105

2

4

6

8

10

12

14European barrier call options

Barrier

Strike

Vanilla

Down-out

Down-in

Theorem 1 (Down-and-out) Suppose that V HS, F L denotes the value of a European derivative contractF HST L on S maturing at time T. The value of a down-and-out version of the derivative is

VdoHS0, FL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz

Page 104: Pg p 12 Handbook

where H < S0 is the barrier, n = r - q - s 2 ê2 and

FH HSL = 9 FHSL,0,

for S > Hfor S § H

= FHSL . IHS > LL

FH is F truncated below H .

Proof: Björk 1998: 185-186

ü Proof

ü

Corollary 1 (Down-and-in) Suppose that V HSL denotes the value of a European derivative contract on Smaturing at time T. The value of a down-and-in version of the same option is

VdiHS0, FL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, FHy{zzz

where H < S0 is the barrier, n = r - q - s 2 ê2 and

FH HSL = 9 FHSL,0,

for S < Hfor S ¥ H

= FHSL . IHS < LL

Note that Vdi depends upon both FH and FH .

Proof Risk neutral valuation implies that the value functional V is linear in F. Since F = FH + FH

V HS0, FL = V HS0, FH L + V HS0, FH LBy in-out parity

Vdi HS0, FL = V HS0, FL - Vdo HS0, FL

= V HS0, FH L + V HS0, FH L -i

kjjjjj V HS0 FH L - J H

ÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz y

{zzzzz

= V HS0, FH L + J HÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz

ü Example: Vanilla European call with H £ K

The payout for a vanilla European call is

FHSL = max HS - K, 0LThe truncated payouts FH and FH depend upon whether H § K or H > K. For H § K

FH HSL = FHSL = max HS - K, 0L and FH HSL = 0

2 Barrier.nb

Page 105: Pg p 12 Handbook

H K

FH FH

and therefore

V HS0, FH L = V HS0, FL and V HS0, FH L = 0

By Theorem 1 and Corollary 1

VdoHS0, FL = V HS0, FL - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Fy{zzz = cHS0, KL - J H

ÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

VdiHS0, FL = V HS0, 0L + J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Fy{zzz = J H

ÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

where cHS0, KL is the Black-Scholes formula for a vanilla European call. Note that we can directly use theBlack-Scholes formula for valuing these options. How would you obtain the hedge parameters?

ü Exercise

Show that this is the same as the formula given in Hull (2003:439) for the value of a vanilla down-and-incall, namely

cdi = S0 ‰- q T J HÅÅÅÅÅÅÅÅS0

N2 l

NHyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0

N2 l-2

NIy - s è!!!!T M

where

l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 , y =lnHH2 ê HS0 KLL + Hr - q + s2 ê 2LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnHH2 ê HS0 KLLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ l s è!!!!T

ü Answer

cdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

= J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

‰-q T N HyL - K ‰-r T N Iy - s è!!!!T My

{zzz

= S0 ‰-q T J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2 +2

N HyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

N Iy - s è!!!!T M

Barrier.nb 3

Page 106: Pg p 12 Handbook

where

y =lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

and n = r - q -s2ÅÅÅÅÅÅÅÅÅÅ2

Define

l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =r - q - s2 ê 2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =n

ÅÅÅÅÅÅÅÅÅÅs2 + 1

so that

2 l =2 nÅÅÅÅÅÅÅÅÅÅs2 + 2

Substituting, we derive the formula in Hull.

cdiHS0, K, HL = S0 ‰-q T J HÅÅÅÅÅÅÅÅS0

N2 l

NHyL - K ‰-r T J HÅÅÅÅÅÅÅÅS0

N2 l-2

N Iy - s è!!!!T M

l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2

y =lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K M + s2 l TÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

=lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ s lè!!!!T

Alternatively, we can start with the formula in Hull. Factoring out the power, this can be written as

cdi = J HÅÅÅÅÅÅÅÅS0

N2 l-2

ikjjjj S0 ‰- q T J H

ÅÅÅÅÅÅÅÅS0

N2

NHy1L - K ‰-r T NIy1 - s è!!!!T My

{zzzz

= J HÅÅÅÅÅÅÅÅS0

N2 l-2

ikjjjikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

y{zzz ‰- q T NHy1L - K ‰-r T NIy1 - s

è!!!!T My{zzz

= J HÅÅÅÅÅÅÅÅS0

N2 Hl-1L

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

where cI H2ÅÅÅÅÅÅÅÅS0

, KM is the value of a vanilla call with underlying price H2 ê S0 and strike K

l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2

y =lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ s lè!!!!T

Furthermore

l - 1 =r - q + s2 ê2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 - 1 =r - q + s2 ê2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =r - q - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =n

ÅÅÅÅÅÅÅÅÅÅs2

where n = r - q - s2 ê 2.Therefore, the value of a down-and-in call with barrier H § K is

cdi = J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

with

4 Barrier.nb

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y =lnI H2

ÅÅÅÅÅÅÅÅÅÅÅS0 K MÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ s lè!!!!T =

lnI H2ÅÅÅÅÅÅÅÅÅÅÅS0 K M + s2 l T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T=

lnI H2ÅÅÅÅÅÅÅÅÅÅÅS0 K M + Hr - q + s2 ê 2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T

which is the same as corollary 1.

ü Example: Vanilla European call H > K

The payout for a vanilla European call is

FHSL = max HS - K, 0L

HK

FH FH

For H > K

FH HSL = maxHS - H , 0L + HH - KL . IHS > HLThe down-truncated payout is a portfolio of a standard call and HH - KL cash-or-nothing binary options bothwith strike H , and therefore

V HS, FH L = cHS, HL + HH - KL bHS, HLwhere bHS, HL is the value of an option that pays 1 if ST > H and zero otherwise.

Substituting in Theorem 1

(1)

cdo HS0, K, HL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, FHy{zzz

=

c HS0, HL + HH - KL b HS0, HL - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjjc

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzz + HH - KL b

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzzy{zzz

Similarly

FH HSL = FHSL - FH HSL = max HS - K, 0L - maxHS - H , 0L - HH - KL . IHS > HL

Barrier.nb 5

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The up-truncated payoff is a portfolio comprising a bull spread together with a short position in H - Kbinary options.

V HS, FH L = cHS, KL - cHS, HL - HH - KL bHS, HLSubstituting in Corollary 1

(2)cdi HS0, K, HL =

cHS0, KL - cHS0, HL - HH - KL bHS0, HL + J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjjcik

jjj H2ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzz + HH - KL bi

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzzy{zzz

Note that

cdi HS0, K, HL + cdo HS0, K, HL = cHS0, KL

ü Exercise

Show that the value of a single binary option is

bHS, HL = ‰-r T NHd2Lwhere N is the cumulative distribution function of the standard normal distribution and

d2 =lnHS0 ê HL + nTÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sè!!!!T

, n = r - q - s2 ê 2

Note that this is analogous to the second term of the Black-Scholes formula with H in place of K.

ü

ü Exercise

Show that equations (1) and (2) are equivalent to the formulae given in Hull (2003: 439), namely

cdo = S0 ‰-q T NHx1L - K ‰-r T NHx2L - S0 ‰-q T J HÅÅÅÅÅÅÅÅS0

N2 l

NHy1L + J HÅÅÅÅÅÅÅÅS0

N2 l-2

K ‰-r T NHy2L

cdi = c - cdi

where

x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T=

lnHS0 ê HLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ l s è!!!!T , x2 = x1 - s

è!!!!T

y1 =lnHH ê S0L + Hr - q + s2 ê2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T=

lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T+ l s

è!!!!T , y2 = y1 - s è!!!!T

6 Barrier.nb

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ü Answer

cdoHS0, K, HL = cHS0, HL + HH - KL bHS0, HL - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjjci

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzz + HH - KL bi

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzzy{zzz

= S0 ‰-q T NHx1L - H ‰-r T NHx2L + HH - KL ‰-r T NHx2L -

J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

‰-q T NHy1L - H ‰-r T NHy2L + HH - KL ‰-r T NHy2Ly{zzz

= S0 ‰-q T NHx1L - K ‰-r T NHx2L - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

‰-q T NHy1L - K ‰-r T NHy2Ly{zzz

= S0 ‰-q T NHx1L - K ‰-r T NHx2L - J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2 +2

S0 ‰-q T NHy1L + J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

K ‰-r T NHy2L

where

x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T, x2 = x1 - s

è!!!!T , n = r - q -s2ÅÅÅÅÅÅÅÅÅÅ2

y1 =lnHH ê S0L + Hr - q + s2 ê2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T, y2 = y1 - s

è!!!!T

Define

l =r - q + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =r - q - s2 ê 2 + s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 =n

ÅÅÅÅÅÅÅÅÅÅs2 + 1

Substituting and simplifying

cdoHS0, K, HL = S0 ‰-q T NHx1L - K ‰-r T NHx2L - S0 ‰-q T J HÅÅÅÅÅÅÅÅS0

N2 l

NHy1L + J HÅÅÅÅÅÅÅÅS0

N2 l-2

K ‰-r T NHy2L

where

x1 =lnHS0 ê HL + Hr - q + s2 ê 2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T=

lnHS0 ê HLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

+ l s è!!!!T , x2 = x1 - s

è!!!!T

y1 =lnHH ê S0L + Hr - q + s2 ê2L T

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T=

lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅs

è!!!!T+ l s

è!!!!T , y2 = y1 - s è!!!!T

ü

Analogous results for up options HH > S0L follow.

Theorem 2 (Up-and-out) Suppose that V HS, F L denotes the value of a European derivative contract F HST Lon S maturing at time T. The value of an up-and-out version of the derivative is

VuoHS0, FL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz

where H > S0 is the barrier, n = r - q - s 2 ê2 and

FH HSL = 9 FHSL,0,

for S < Hfor S ¥ H

= FHSL . IHS < HL

Barrier.nb 7

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Corollary 2 (Up-and-in) Suppose that V HSL denotes the value of a European derivative contract on Smaturing at time T. The value of an up-and-in version of the same option is

VuiHS0, FL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, FHy{zzz

where H > S0 is the barrier, n = r - q - s 2 ê2,

FH HSL = FHSL . IHS > HL and FH HSL = FHSL . IHS < HL

ü Example Vanilla European call

When H § K, an up-and-in call is equivalent to a vanilla whereas the value of an up-and-out call is alwayszero. Therefore, we need only consider explicitly the case H > K.

Recall that the up-truncated payoff of a vanilla European call when H > K is a portfolio comprising a bullspread together with a short position in H - K binary options.

V HS, FH L = cHS, KL - cHS, HL - HH - KL bHS, HLSubstituting the payoff in Theorem 2

(3)

cuo HS0, K, HL = V HS0, FH L - J HÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz

= cHS0, KL - cHS0, HL - HH - KL bHS0, HL -

J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjj ci

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz - ci

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzz - HH - KL bi

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzzy{zzz

The down-truncated payoff of a vanilla European call when H > K is a portfolio comprising a standard calland HH - KL cash-or-nothing binary option both with strike H .

V HS, FH L = cHS, HL + HH - KL bHS, HLSubstituting the payoff in Corollary 2

(4)

cui HS0, K, HL = V HS0, FH L + J HÅÅÅÅÅÅÅÅS

N2 nÅÅÅÅÅÅÅÅs2

Vikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS

, FHy{zzz

= c HS0, HL + HH - KL b HS0, HL +

J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjj c

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz - c

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzz - HH - KL b

ikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Hy{zzzy{zzz

ü Exercise

Show that equations (3) and (4) are equivalent to the formula on p. 440 of Hull (2003).

ü

8 Barrier.nb

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ü

ü Discrete monitoring

The above formulae assume continuous monitoring of the asset price. Many real-world contracts assumemonitoring at discrete intervals, e.g. daily at a specific time. Broadie, Glasserman and Kou (1997) suggestan adjustment to the barrier to account for discrete monitoring. Specifically, the barrier H is adjusted asfollows

H ö9 H ‰+ b s è!!!!!!!!!!Têm , H > S0 HupL

H ‰- b s è!!!!!!!!!!Têm , H < S0 HdownL

where b = 0.5826 and m is the monitoring frequency.

ü Summary

The following table summarizes the valuation formulae for European style barrier call options.

Ñ H £ K H > K

cdi I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 cI H2ÅÅÅÅÅÅÅÅS0

, KM cHS0, KL - cHS0, HL - HH - KL bHS0, HL +

I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 IcI H2ÅÅÅÅÅÅÅÅS0

, HM + HH - KL bI H2ÅÅÅÅÅÅÅÅS0

, HMM

cdo cHS0, KL - I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 cI H2ÅÅÅÅÅÅÅÅS0

, KM cHS0, HL + HH - KL bHS0, HL -

I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 IcI H2ÅÅÅÅÅÅÅÅS0

, HM + HH - KL bI H2ÅÅÅÅÅÅÅÅS0

, HMMcui cHS0, KL cHS0, HL + HH - KL bHS0, HL +

I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 I cI H2ÅÅÅÅÅÅÅÅS0

, KM - cI H2ÅÅÅÅÅÅÅÅS0

, HM - HH - KL bI H2ÅÅÅÅÅÅÅÅS0

, HMMcuo 0 cHS0, KL - cHS0, HL - HH - KL bHS0, HL -

I HÅÅÅÅÅÅÅS0M

2 nÅÅÅÅÅÅÅÅs2 I cI H2ÅÅÅÅÅÅÅÅS0

, KM - cI H2ÅÅÅÅÅÅÅÅS0

, HM - HH - KL bI H2ÅÅÅÅÅÅÅÅS0

, HMM

ü Rebate

For knock-out options, it is customary for the rebate to be paid immediately that the barrier is hit. Thiscomplicates the valuation, since the rebate is paid at a random time.

The discounted expected value of the rebate R is

Vrebate = RikjjjjJ

HÅÅÅÅÅÅÅÅS0

Na+b

NHh zL + J HÅÅÅÅÅÅÅÅS0

Na-b

NIh z - 2 h b s è!!!!T M y

{zzzz

where

a = l - 1, b =è!!!!!!!!!!!!!!!!!!!!!!!

n2 + 2 r s2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s2 , z =lnHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sè!!!!T

+ b s è!!!!T , n = r - q - s2 ê 2

and h = 1 if the barrier is approached from above HS0 > HL and h = -1 if the barrier is approached frombelow HS0 < HL.

ü Proof

Barrier.nb 9

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ü

Standard American barrier optionsWe showed above that, for H § K, the value of a down-and-in vanilla European option is related to thevalue of the underlying non-barrier option by the formula

cdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

Fortunately, the same relationship applies to knock-in American options (Haug 2001, Dai and Kwok 2004).That is, where H § K,

(5)CdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

Cikjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz

where Cdi is the value of a knock-in option entitling the holder to a standard American option of strike K ifand when the asset price S falls below the barrier H , and C is the value of the underlying American option.More generally

(6)CdiHS0, K, HL = J HÅÅÅÅÅÅÅÅS0

N2 nÅÅÅÅÅÅÅÅs2

ikjjjCi

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzz - ci

kjjj H2

ÅÅÅÅÅÅÅÅÅÅÅS0

, Ky{zzzy{zzz + cdiHS, K, HL

provided that

(7)H § max ikjjK,

rÅÅÅÅÅq

Ky{zz

When H § K, cdiHS, K, HL matches the second term inside the brackets.

Analogous (though different formulae) can be given when (7) is not satisfied (Dai and Kwok 2004). Thismeans that we can easily value a knock-in American option by applying our best numerical techniques tocorresponding standard options.

Unfortunately (for computation), in-out parity does not hold for American barrier options. Consequently, wecannot use (6) to compute the value of a knock-out option. Indeed,

CdiHS, K, HL + CdoHS, K, HL > CHS, KLTo see this, consider a portfolio comprising a down-and-in plus a down-and-out with the same strike, barrierand maturity. We show that this portfolio dominates a corresponding nonbarrier call. Suppose that theportfolio holder adopts an exercise policy for the down-and-out identical to that for the nonbarrier option(though this is suboptimal for the down-and-out). The exercise payoff of the portfolio is always higher thanthat of the nonbarrier call because the portfolio has an additional option. Both the portfolio and the optionhave the same payoff at maturity because one of the options will be knocked-out. In all scenarios, theportfolio is worth at least as much as the option and possibly more.

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Numerical methodsThe preceding analysis provides exact formulae for standard European barrier options, plus an efficientapproach for dealing with American knock-in options. However, this does not exhaust the variety of barriersoptions traded in over-the-counter markets. In particular, where

æ the barrier is non-constant (and non-exponential)

æ early exercise is allowed (American knock-out options)

æ it is desired to allow for non-lognormality (volatility skew)

it is necessary to resort to numerical methods. Barrier options pose a special problem for numerical methods,which is illustrated by the binomial method.

à Binomial trees

We have previously examined the convergence of the basic binomial method. The following diagramdepicts the errors for a one-year vanilla European call option (S = 95, K = 100, r = 10 %, q = 0,s = 25 %).

50 100 150 200 250 300

-0.04

-0.02

0.02

0.04

Performance deteriorates dramatically when applied to Barrier options. For a down-and-out version of theprevious option with the barrier at 90, we find

Barrier.nb 11

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100 200 300 400

0.25

0.5

0.75

1

1.25

1.5

1.75

2Binomialerror in down-and-out call

The reason is that, when the barrier lies between the nodes of the tree, the effective barrier is different to thetrue barrier. Performance can be significantly improved if we ensure that the barrier lies on or just above arow of nodes. This requires a judicious choice of the number of steps n.

Recall that in the basic binomial model

u = ‰s è!!!!!!!!!Tên , d = ‰-s

è!!!!!!!!!Tên

Barrier H will be precisely aligned with a row of nodes in a tree of size n when

S0 ‰k s è!!!!!!!!!Tên = H

lnS0ÅÅÅÅÅÅÅÅH

= k s $%%%%%%%TÅÅÅÅÅÅn

for some k = ≤1, ≤2, …, ≤n.

n = ikjj k s

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅlnHS0 ê HL

y{zz

2 T , k = ≤1, ≤2,

Barrier H will lie on or just above a row of nodes when n is the largest integer which is smaller than theright-hand side. For the option depicted above, these magic numbers are 21, 85, 192, 342 and 534.

At these magic numbers, the binomial method delivers similar accuracy to that obtained for vanilla options.

12 Barrier.nb

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n Error21 0.045089785 0.0228345192 0.00895174342 0.00144772534 0.00352852

100 200 300 400

0.05

0.1

0.15

0.2

0.25

Binomialerror in down-and-out call

Derman, Kani, Ergener and Bardhan (1995) propose an enhancd binomial method to allow for nonconstantbarriers.

à Trinomial trees

We have just seen how the accuracy of binomial method can be preserved by careful selection of the treesize n to align the barrier with the nodes in the tree. Since the trinomial tree has an additional degree offreedom, this alignment can achieved for any n through judicious choice of the "stretch" parameter l. Thestandard parameterization of the trinomial tree is

u = ‰l s è!!!!!!

Dt

d =1ÅÅÅÅÅu

n = r - q -1ÅÅÅÅÅ2

s2

pu =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l2 +

è!!!!!Dt

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l s

n

pd =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l2 -

è!!!!!Dt

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 l s

n

pm = 1 - pu - pd = 1 -1

ÅÅÅÅÅÅÅÅl2

Barrier.nb 13

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Consider an up-and-out barrier option. To find the appropriate value of l, start with l = 1. Let m denote themaximum number of consecutive up moves allowed before breaching the barrier. That is

S0 um < H but S0 um+1 ¥ H

Given l = 1, we have

S0 ‰m s è!!!!!!Dt < H ï m <logHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sè!!!!!

Dt

Define

m* =logHH ê S0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sè!!!!!

Dt

and then choose l > 1 such that l m = m* which implies that

S0 ‰l m s è!!!!!!Dt = H

With this choice of l, the tree is precisely aligned with the barrier after m upward steps.

For example, suppose S0 = 95, H = 125, with s = 25 % and T = 1. For a 5-step tree, Dt = 1 ê 5,m* = 2.45464 and m = 2, so that l = m* ê m = 1.22732. For this choice of l, the tree coincides with thebarrier after precisely 2 steps.

The following graph depicts the convergence of a trinomial tree in valuing a 6-month European up-and-outcall option (S = 100, K = 90, H = 125, r = 5 %, q = 2 %, and s = 30 %L. The red path shows the errorof a trinomial tree with l =

è!!!!!!!!!3 ê 2 against n, while the blue line depicts the error a trinomial tree in which lis chosen optimally for each n so as to align a row of nodes with the barrier.

100 200 300 400

-0.2

0.2

0.4

Trinomial error in up-and-out call

The following graph displays the percentage error.

14 Barrier.nb

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100 200 300 400

-10

10

20

Trinomial percentage error in up-and-out call

Barrier.nb 15

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Michael
Text Box
Hull (2003) Options, Futures, and Other Derivatives
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Spread options� Preliminaries

Examples� Commodity markets

The soybean crush spread traded on the CBOT is

CS =

48

2000SM

+

11

100SO

- SB

where SM is the price is the futures price of soy meal in dollars per ton, SO is the futures price of soy oil indollars per 100 pounds, and St is the futures price of soybean in dollars per bushel. The payoff of a call optionon the soybean crush is

CSC = HCS - KL+

� Energy markets

The payoff of the 1:1:0 gasoline crack spread call is

CSGC = H42 UG - CO - KL+

where UG is the price of unleaded gasoline ($ / gallon) and CO is the price of crude oil ($ / barrel). The payoff

of the 3:2:1 crack spread call is

CSHC = 422

3UG +

1

3HO - CO - K

+

where HO is the price of heating oil. Crack spread options are traded on NYMEX.

A spark spread is a proxy for cost of converting a specific fuel (usually natural gas) into electricity.

SS = E - Heff NG

where E and NG are the futures prices of electricity and natural gas, and Heff is the heat rate.

� Interest rate markets

The TED spread measures the difference between 3-month Tbills and 3-month LIBOR. The MOB spreadmeasures the difference in yield between municipal and treasury bonds.

Exchange optionsA spread option with a zero strike is known as an option to exchange

payoff = H S1 - S2 L+

for which we have an exact formula (known as the Magrabe formula).

c = ã-q1 T S1 NHd1L - ã

-q2 T S2 NHd2L

where

d1 =

LnHS1 �S2L + Iq2 - q1 + Σ2 �2M T

Σ T

Page 122: Pg p 12 Handbook

d2 = d1 - Σ T

Σ = Σ12

- 2 Ρ Σ1 Σ1 + Σ22

Note that the value is independent of the risk-free rate. Alternatively, in terms of futures prices, we have

c = ã-r T HF1 NHd1L - F2 NHd2L L

where

d1 =

LnHF1 � F2L + Σ2 T �2

Σ T

d2 = d1 - Σ T

Σ = Σ12

- 2 Ρ Σ1 Σ1 + Σ22

Approximations� The Kirk approximation

The payoff of a spread option is

payoff = H S1 - S2 - KL+

The popular Kirk approximation values the spread option as an exchange between F1 and F2 + K, treating

F2 + K as lognormal with volatilty J F2

F2+KN Σ2.

c = ã-r T H F1 NHd1L - HF2 + KL NHd2L L

where

Fi = ãHr-qiL T Si, i = 1, 2

Σ = Σ12

- 2 b Ρ Σ1 Σ1 + b 2Σ2

2 , b =

F2

F2 + K

d1 =

LnJ F1

F2+KN +

1

2 T

Σ T

d2 = d1 - Σ T

Clearly, the Kirk approximation reduces to the Margrabe formula when K = 0, and will be most useful whenK << S2.

� Bjerksund and Stensland

The payoff of the spread can be written as

CHTL = H S1 - S2 - KL+= H S1 - S2 - KL × I HS1HTL ³ S2HTL + KL

where IH L is the indicator function, taking the value one when its argument is true, and zero otherwise. Bjerk-sund and Stensland consider the related derivative with the payoff

cHTL = H S1 - S2 - KL+= H S1 - S2 - KL × I S1HTL ³

a HS2HTLLb

EAHS2HTLLbE

where a = F2 + K, b = F2 � HF2 + KL and F2 is the forward price F2 = S2 ãHr-q2L T . This has two implications:

2 SpreadOptions.nb

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æ They can compute the exact value of the related derivative, which they propose as a superior alternativeto the Kirk approximation.

æ The related derivative can be used as a control variable in simulating the value of a spread option.

The exact value is given by the following formula.

c = ã-r T H F1 NHd1L - F2 NHd2L - K NHd3L L

where

Fi = ãHr-qiL T Si, i = 1, 2

Σ = Σ12

- 2 b Ρ Σ1 Σ1 + b 2Σ2

2

d1 =

LnJ F1

aN + I 1

2Σ1

2- b Ρ Σ1 Σ2 +

1

2b2

Σ22M T

Σ T

d2 =

1

Σ TLn

F1

a+ -

1

2Σ1

2+ Ρ Σ1 Σ2 +

1

2b2

Σ22

- b Σ22 T

d2 = d1 -

IΣ12- H1 + bL Ρ Σ1 Σ2 + b Σ22M

Σ

T

d3 = d1 -

IΣ12- b Ρ Σ1 Σ2M

Σ

T

d3 =

LnJ F1

aN + I-

1

2Σ1

2+

1

2b2

Σ22M T

Σ T

GreeksFor the BjS

c = ã-r T H F1 NHd1L - F2 NHd2L - K NHd3L L

First note

¶di

¶ F1

=

1

F1 Σ T

¶c

¶ F1

= ã-r T K NHd1L + HF1 ΦHd1L - F2 ΦHd2L - K ΦHd3LL

¶di

¶ F1

O

= ã-r T NHd1L +

F1 Φ Hd1L - F2 Φ1Hd2L - K Φ Hd3L

F1 Σ T

and therefore

¶c

¶S1

=

¶c

¶ F1

¶ F1

¶S1

= ã-q T NHd1L +

F1 Φ Hd1L - F2 Φ1Hd2L - K Φ Hd3L

F1 Σ T

Computing the derivative with respect to F2 is a a little more difficult, since the adjusted volatility dependsupon b, which is a function of F2.

¶b

¶ F2

=

b2 K

F22

SpreadOptions.nb 3

Page 124: Pg p 12 Handbook

¶d0

¶ F2

=

1

2 T Σ3

Σ2 HΡ Σ1 - b Σ2LK b2

F22LogB

F1

F2 + KF - T Σ

2-

2 Σ2

F2 + K

¶d1

¶ F2

=

1

2 T Σ3

Σ2 HΡ Σ1 - b Σ2LK b2

F222 LogB

F1

F2 + KF - T Σ

2-

2 Σ2

F2 + K

¶d2

¶ F2

=

¶d1

¶ F2

-

b2 K

F22

H1 - bL I1 - Ρ2M Σ12

Σ22

Σ3

T

4 SpreadOptions.nb

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Volatility and Variance swaps

IntroductionA volatility swap is a forward contract on realized volatility. Its payoff is

payoff = HΣR - KvolL ´ N

where ΣR is realised volatility (annualized), Kvol is the specified delivery price, and N is the notional amountof the swap in "dollars" per annualized volatility point. A variance swap is a forward contract on realizedvariance, with payoff

payoff = IΣR2

- KvarM ´ N

where N is expressed in "dollars" per annualized volatility point squared.

For example, a 3-month volatility swap on the S&P 500 struck at 20% with a notional $100,000 would pay(25 - 20) × 100000 = $500000 if the realised volatility was 25%.

A volatility or variance swap is a position on realized volatility. It is related to, but distinct from, options onimplied volatility (VIX) which are traded on the CBOE.

� Questions

æ how to hedge volatility.

æ how to replicate (hence price and hedge) a variance swap?

æ how is the VIX computed.

Hedging volatilityIn the Black-Scholes framework, vega of a vanilla call or put is

Vega =

¶V

¶ Σ

= ã-q T S N ' Hd1L

where

d1 =

LnHS �KL + Ir - q - Σ2 T �2M

Σ T

For this purpose, it is convenient to measure variance vega, which (by the chain rule) is a multiple of vega.

VarVega =

¶V

¶ Σ

¶ Σ

¶ Σ2

=

1

2 Σ

´Vega

The following graph shows vega on traded options on the Nifty on 5 January 2010, allowing for the volatilitysmile. We observe that:

æ (Var)vega is sharply peaked around the strike of an option.

æ We also observe that (var)vega increases with the strike.

Page 126: Pg p 12 Handbook

4000 5000 6000 7000

500

1000

1500

Vega of a one-month options on the Nifty at selected strikes

This suggests that vega can be hedged by a weighted portfolio of traded options. This is illustrated in thefollowing diagram, where the red line is the (var)vega of a portfolio of options weighted inversely to strikesquared.

Out[6]=

4000 5000 6000 7000

500

1000

1500

In fact, a portfolio of options at all strikes, weighted inversely proportional to strike, will give an exposure tovariance which is independent of asset price. This is precisely what is needed to trade pure variance. It is easyto show that the terminal payoff of such a portfolio comprising puts K £ K* and calls K > K* is

(1)

payoff = à0

K* 1

K2HK - ST L+

â K + àK*

¥ 1

K2HK - ST L+

â K

=

ST - K*

K*

- logST

K*

where log is natural log, and K* is a boundary strike.

Valuing a variance swapFor an asset following geometric Brownian motion, it can be shown that expected (realized) variance overperiod @0, TD is

(2)E@VarD =

2

THr - q L T - EBlog

ST

S0

F

Inverting equation (1) shows that a log contract is equivalent to a forward together with a portfolio of vanillacalls and puts.

2 VarSwaps.nb

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logST

K*

=

ST - K*

K*

forward

- à0

K* 1

K2HK - ST L+

â K put options

- àsK*

¥ 1

K2HST - KL+

â K call options

Substituting and setting F = E@ST D = S0 ãHr-qL T

(3)

E@VarD =

2

THr - qL T - log

F

S0

- logK*

F-

F - K*

K*

+ à0

K* 1

K2ã

r T PHKL â K + àK*

¥ 1

K2ã

r T CHKL â K

=

2

T- log

F

K*

-

F - K*

K*

- ãr T à

0

K* 1

K2PHKL â K - ã

r T àK*

¥ 1

K2CHKL â K

=

2

0

K* 1

K2ã

r T PHKL â K + àK*

¥ 1

K2ã

r T CHKL â K -

1

T

F - K*

K*

2

where we have used the fact that

logF

K*

»

F - K*

K*

To value a variance swap, we need to approximate this with traded options.

� First method: numerical integration

Approximate (3) by

(4)E@VarD =

2

K £ K*

DK

K2ã

r T PHKL + âK > K*

DK

K2ã

r T CHKL -

1

T

F - K*

K*

2

This is the basis of the method employed to calculate the VIX.

� Second method: replicating the log contract

Rewriting (2)

E@VarD =

2

THr - q L T - EBlog

ST

S0

F

=

2

THr - q L T -

E@ST D - K*

K*

+

E@ST D - K*

K*

- EBlogK*

S0

+ logST

K*

F

=

2

THr - q L T -

S0 ãHr-qL T

- K*

K*

- logK*

S0

+ E@ f HST LD

where

f HST L =

2

TBST - K*

K*

- logST

K*

f HST L is a payoff function which can be approximated by a portfolio traded vanilla calls and puts, as shown inthe workbook VarSwaps.xlsm.

VarSwaps.nb 3

Page 128: Pg p 12 Handbook

4500 5000 5500 6000 6500

0.05

0.10

0.15

0.20

The payoff function f HST L

To approximate this, consider first the segment around K*.

Out[25]=

5400 5500 5600 5700

0.001

0.002

0.003

0.004

0.005

A segment of the payoff function f HST L

Computing the VIXThe CBOE VIX is an estimate of the 30-day expected volatility of the S&P 500 index, computed as a weightedaverage estimated volatility for near- and next-term options. Specifically

VIX = 100 ´

T2 - T30

T2 - T1

T1 Σ12

+

T30 - T1

T2 - T1

T2 Σ22

where Σ12 and Σ2

2 are the estimated variance of near- and next-term options respectively, T1 and T2 are their

respective times to maturity and T30 = 30 �365.

Each variance computed by evaluating (4) using all quoted options with active bid price. In the evaluation

æ times to maturity are computed in minutes (although expressed on an annual basis).

æ the forward price F is computed by put-call parity applyed to the nearest-to-the-money strike

F = KNTM + ãr T HCHKL - PHKL

where KNTM is the strike at which the absolute difference between call and put prices is mininized.

æ the boundary K* is selected as the strike immediately below the forward price F .

In computing India-VIX, the NSE uses the same methodology, with the following modifications.

æ the risk-free rate is MIBOR 30 or 90 days. (It appears that the rate is not interpolated to match time tomaturity.)

4 VarSwaps.nb

Page 129: Pg p 12 Handbook

æ the forward price F is set at the appropriate futures closing price.

æ cubic splines are used to compute mid-price where the bid-ask spread is greater than 30%.

VarSwaps.nb 5

Page 130: Pg p 12 Handbook

Interest rate derivativesMichael CarterInterest rate derivatives are financial assets whose payoff depends in some way on the level of interest rates.Examples include

æ Bond options

æ Bond futures options

æ Callable and putable bonds

æ Mortgages

æ Mortgate-backed securities

æ Interest rate caps and floors

æ Swaps

æ Swapoptions

Interest rate derivatives are more difficult to value than stock or foreign exchange derivatives, because

æ The behavior of interest rates is more complicated. It is necessary to develop of model describingthe behavior of the entire zero-coupon yield curve.

æ The volatilities at different point of the yield curve are different.

æ Interest rates are used for discounting as well as defining the payoff of the derivative.

Modeling the term structure - discrete timeWe can represent the term structure of interest rates in three equivalent ways:

æ discount function (prices of zero coupon bonds)

æ spot rates (yield to maturity of zero coupon bonds)

æ forward rates

Let R1, R2, …, RT denote the spot rates of interest appropriate to 1, 2, …, T years (we assume annualcompounding for simplicity). Then the current prices of zero coupon bonds are

(1)PH0, 1L =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + R1

, PH0, 2L =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1 + R2L2 , …, PH0, TL =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅH1 + RT LT

Conversely

(2)RT = J 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL N

1ÅÅÅÅÅT

- 1

Page 131: Pg p 12 Handbook

Forward rates are rates agreed now for a loan in the future. For example, f3,5 denotes the rate agreed now fora 2-year loan beginning in 3 years. Absence of arbitrage requires that

H1 + R2L2 = H1 + R1L H1 + f1,2Land more generally

H1 + RT LT = H1 + RtLt H1 + ft,T LT-t

Substituting from (1), we have

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 2L =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L H1 + f1,2L and

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, tL H1 + ft,T L

so that

f1,2 =PH0, 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 2L - 1 and ft,T =

PH0, tLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL - 1

Note also that

H1 + R3L3 = H1 + R2L2 H1 + f2,3L = H1 + R1L H1 + f1,2L H1 + f2,3Lso that

H1 + RT LT = H1 + R1L H1 + f1,2L H1 + f2,3L … H1 + fT-1,T LIn a discrete model, the short rate r is the one period forward rate. That is

r0 = f0,1 =PH0, 0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L - 1 =

1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, 1L - 1 = R1, r1 = f1,2, r2 = f2,3, …, rT-1 = fT-1,T

and therefore

(3)H1 + RT LT = H1 + r0L H1 + r1L H1 + r2L … H1 + rT-1L

Modeling the term structure - continuous timeAs in the Black-Scholes model, theoretical work is almost always exposited in continuous time. As withdiscrete compounding, we can represent the term structure of interest rates in three equivalent ways:

æ discount function (prices of zero coupon bonds)

æ spot rates (yield to maturity of zero coupon bonds)

æ forward rates

Each of these familiar interest rate concepts has a continuous time counterpart.

2 InterestRateDerivatives.nb

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Discrete Continuous

Discount function PHt, TL = J 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + RHt, TL N

T-tPHt, TL = ‰-RHt,TL HT-tL

Spot rate RHt, TL = PHt, TL 1ÅÅÅÅÅÅÅÅÅÅT-t - 1 RHt, TL = -ln PHt, TLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

T - t

Forward rate f Ht, T , T + 1L =PHt, T + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

PHt, TL - 1 f Ht, TL = -∑

ÅÅÅÅÅÅÅÅÅÅÅ∑T

ln PHt, TL

To derive the expressions for the forward rate, note that one period forward rates f Ht, t, t + 1L must satisfy

(4)

H1 + RHt, TLLT-t = H1 + RHt, t + 1LLäH1 + f Ht, t + 1, t + 2LL ä… äH1 + f Ht, T - 1, TLL =

‰i=1

T-t

H1 + f Ht, t + i - 1, t + iLL

where f Ht, t, t + 1L = RHt, t + 1L. In particular, this implies that

(5)PHt, T + 1L =PHt, TL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + f Ht, T , T + 1L

which yields

f Ht, T , T + 1L =PHt, TL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPHt, T + 1L - 1

For small time periods, (5) requires

PHt, T + DTL = ‰- f Ht,T ,T+1L DT PHt, TLor

ln PHt, T + DTL = - f Ht, T , T + D TL DT ä ln PHt, TL

f Ht, T , T + D TL = -ln PHt, T + D TL - ln PHt, TL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD T

Letting D T Ø 0, the instantaneous forward rate is

(6)f Ht, TL = limD T Ø 0

-ln PHt, T + D TL - ln PHt, TL

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅD T

= -∑

ÅÅÅÅÅÅÅÅÅÅÅ∑T

ln PHt, TL

From (1)

PHt, TL = ‰i=1

T-t 1ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ1 + f Ht, t + i - 1, t + iL

With continuous discounting, this becomes

PHt, TL = ‰i=1

T-t

‰- f Ht,t+i-1,t+iLä1 = expi

kjjjjj ‚

i=1

T-t

- f Ht, t + i - 1, t + iLä1y

{zzzzz

In continuous time, this becomes

PHt, TL = ‰ -ŸtT f Ht,tL „t

InterestRateDerivatives.nb 3

Page 133: Pg p 12 Handbook

This can be derived more formally by integrating (3)

‡t

Tln PHt, tL „ t = ln PHt, TL - ln PHt, tL = -‡

t

Tf Ht, tL „ t

But since PHt, tL = 1

ln PHt, TL = -‡t

Tf Ht, tL „ t ï PHt, TL = ‰ -Ÿt

T f Ht,tL „t

The price of the discount bond is the final cashflow discounted by instantaneous forward rates.

It follows that the spot rate is the continuous average of forward rates

RHt, TL =1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅT - t

‡t

Tf Ht, tL „ t

Further

f Ht, TL = -∑

ÅÅÅÅÅÅÅÅÅÅÅ∑T

ln PHt, TL =∑

ÅÅÅÅÅÅÅÅÅÅÅ∑T

HT - tL RHt, TL = RHt, TL + HT - tL ∑ÅÅÅÅÅÅÅÅÅÅÅ∑T

RHt, TL

The short rate r is the rate on instantaneous borrowing or lending, i.e. RHt, tL = f Ht, tL . A sum of $1invested at the short rate at time zero and continuously rolled over is called a money market account. Itsvalue at time t is

pt = ‰Ÿ0t rs ds

Interest rate derivatives as portfolios of bond optionsSeveral popular interest rate derivatives can be valued as portfolios of European options on discount bonds.Therefore, it suffices to develop formulae or algorithms for discount bonds. This is especially useful incalibrating interest-rate models to market data.

à Coupon bonds

Assuming that there is only one factor of uncertainty, so that all rates are positively related to r0, we canvalue options on coupon bonds by treating them as a portfolio of options on zero-coupon bonds. Consider acall option maturing at time T on a coupon bond maturing at time tn > T , where n is the number of couponpayments remaining at time T . The price of the bond at time T will be

B = ‚i=1

n

ci PrT HT , tiL

and the payoff of a call option maturing at time T will be

maxikjjjjj‚

i=1

n

ci PrT HT , tiL - K, 0y{zzzzz

4 InterestRateDerivatives.nb

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where rT is the short rate at time T . Note that cn includes the principal repayment. Note that B depends uponthe short rate at time T . Let rK denote the short-rate at time T at which the price of the coupon bond is equalto the strike price. That is

‚i=1

n

ci PrK HT , tiL = K

The option will be exercised provided rT < rK and not exercised if rT ¥ rK . Let Ki denote the value at timeT of a zero-coupon bond paying $1 at time ti given rT = rK

Ki = PrK HT , tiLso that

‚i=1

n

ci Ki = ‚i=1

n

ci PrK HT , tiL = K

Therefore the payoff of the coupon bond option can be written as

maxikjjjjj‚

i=1

n

ci PrT HT , tiL - K, 0y{zzzzz = max

ikjjjjj‚

i=1

n

ci PrT HT , tiL - ‚i=1

n

ci Ki, 0y{zzzzz

= maxikjjjjj‚

i=1

n

ciHPrT HT , tiL - KiL, 0y{zzzzz

= ‚i=1

n

ci maxHPrT HT , tiL - Ki, 0L

Consequently, the option can be valued as a portfolio of options on discount bonds.

à Swaptions

A swaption is an option on an interest rate swap, and can therefore be regarded as an option to exchange acoupon bond for a floating rate bond. At the start of the swap, the value of the floating rate bond equals theprincipal amount of the swap. A European swaption can therefore be regarded as an exchange a couponbond for the principal amount of the swap. If the swaption gives the holder the right to pay fixed and receivefloating (a payer swaption), it is a put option on a fixed rate bond with strike price equal to the notionalprincipal. If the swaption gives the holder the right to pay floating and receive fixed (a receiver swaption), itis a call option on the fixed rate bond.

à Interest rate caps and floors

An interest rate cap is a portfolio of interest rate options. Each component option on a forward rate isknown as a caplet.

We show that an interest rate cap can be valued as a portfolio of European put options on discount bonds.Assuming annual tenor, the payoff from each caplet discounted to the beginning of the period is

payoff =P max Hr - cap, 0LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

1 + r

where is P is the principal and cap is the cap rate. This is equivalent to

InterestRateDerivatives.nb 5

Page 135: Pg p 12 Handbook

payoff = P max J r - capÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

1 + r, 0N

= P max J 1 + r - H1 + capLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

1 + r, 0N

= max JP -PH1 + capLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

1 + r, 0N

The second term in the brackets, PH1 + capL ê H1 + rL, is the value at the beginning of the period of a discountbond that pays P(1+cap) at the end of the period. Therefore, the payoff of the t-period caplet is equal to thepayoff of a put option on a discount bond with a face value of PH1 + capL and a maturity of t + 1. The optionhas a strike price of P and maturity of t. Consequently, the cap can be regarded as a portfolio of Europeanput options on zero-coupon bonds.

Analogously, an interest rate floor can be valued as a portfolio of European call options on discount bonds.A collar is a combination of a long position in a cap and a short poisiton in a floor.

There is a put-call parity relationship between the prices of caps and floors with the same strike place RK ,namely

cap price + floor price = value of swap

where the swap is an agreement to receive floating and pay fixed RK , with no exchange of payments on thefirst reset date.

6 InterestRateDerivatives.nb

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Interest rate derivatives: Standard market modelsMichael CarterThe three most popular over-the-counter interest rate derivatives are bond options, interest rate caps and floorsand swap options. Many traders use a Black-Scholes type formula known as Black's model for valuing thesederivatives. It is based on the following key result.

THEOREM. If S is lognormally distributed and the standard deviation of ln S is s then

PrHS > KL = NHd2Land

EHS » S > KL = EHSL NHd1Lwhere

d1 =lnHEHSL ê KL + s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s, d2 =

lnHEHSL ê KL - s2 ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s= d1 - s

Consequently

(1)E@maxHS - K, 0LD = EHSL NHd1L - K NHd2LProof: Hull (2003: 262-263)

Recognising that (under Black-Scholes assumptions) EHST L = S0 ‰r T and s = s è!!!!T , the Black-Scholes formula

for a call option

c = ‰-r T E@maxHS - K, 0LD = ‰-r T HS0 ‰r T NHd1L - K NHd2LL = S0 NHd1L - K ‰-r T NHd2Lis immediate.

Consider a European call option on a variable S. Let

T = time to maturity of the optionF = forward price of S with maturity TF0 = value of F at time 0K = strike price of optionP H0, TL = price at time 0 of a discount bond paying $1 at time T .

Assume

æ S is lognormally distributed with the standard deviation of ln S equal to s è!!!!T

æ EHST L = F0

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Then the expected payoff of the option at time T is

E@maxHST - K, 0LD = F0 NHd1L - K NHd2LThe present value of the option is

(2)c = PH0, TL E@maxHST - K, 0LD = PH0, TL HF0 NHd1L - K NHd2LLwhere

d1 =lnHF0 ê KL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 =lnHF0 ê KL - s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

= d1 - s è!!!!T

Similarly the value of a corresponding put option is

(3)p = PH0, TL HK NH-d2L - F0 NH-d1LLThis is usually referred to as Black's model since the formulas are similar to those in the model suggested byFisher Black for commodity futures. Note that it is not necessary to make any assumption about the stochasticprocess followed by V of F, only that VT is lognormal at time T . Although the volatility parameter s is usuallyreferred to as the volatility of F or the forward volatility of V , its only role is determine the standard deviation ofln S at time T .

à Bond options

In addition to trading in the OTC market, bond options are frequently embedded in standard bonds in order tomake them more attractive to the issuer or potential purchasers. Examples include callable bonds, puttable fondsand conventional mortages with early payment privileges. Most OTC bond options and some embedded bondoptions are European.

In using Black's model to value European bond options, it is assumed that the bond price ST at the maturity ofthe option is lognormal with the standard deviation of ln ST = s

è!!!!T . The forward price of the bond is

F0 =S0 - CÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅPH0, TL

where S0 is the current price and C is the present value of the coupons that will be paid during the life of theoption. The value of a call option is given by (2) and the value of a put option by (3).

The spot and forward prices in this formula are cash ("dirty") prices, and so the strike price should also be thecash price. In practice, the strike price is often the quoted ("clean") price applicable when the option is exercised,in which case K should be set equal to the strike price plus accrued interest at the expiration date of the option.

The volatility parameter s is the volatility of the forward price F. Volatilities are usually quoted in terms ofyield volatilities. Practitioners typically use bond duration to convert yield volatilities sy to price volatilities by

s = D y0 sy.

where D is modified duration and y0 is the forward yield.

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The Black model assumes that volatility (variance) of the underlying increases linearly with time to maturity.However, the price of a bond as it approaches maturity must be equal to the principal plus coupon ("pull to pareffect"). Therefore, the volatility of a bond over its life first increases then decreases to zero. Pricing of Euro-pean bond options using the Black model should therefore be limited to options of short maturity compared tothe maturity of the bond. A rule of thumb used by some traders is that the time to maturity of the option shouldbe no more than one-fifth of the maturity of the underlying bond.

à Interest rate cap

Recall that an interest rate cap is a portfolio of interest rate options. The caplet corresponding to the rate rk

observed at time tk provides a payoff at time tk+1 of

L Dt maxHrk - cap, 0Lwhere is L is the notional principal. This is effectively a call option on rk with the payoff made at time tk+1. Inusing Black's model to value caps and floors, it is assumed that the interest rate rk is lognormal with volatilitysk. Applying equation (1), the expected payoff at time tk+1 is

E@L Dt maxHrk - cap, 0LD = L Dt @ fk NHd1L - cap NHd2LDwhere fk is the forward rate between time tk and tk+1, and

d1 =lnH fk ê capL + sk

2 tk ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sk è!!!!tk

, d2 = d1 - s è!!!!tk

The present value of the caplet is

vcaplet = PH0, t + 1L L Dt @ fk NHd1L - cap NHd2LDThe value of an interest rate cap is the sum of the values of its constituent caplets, each valued according theprevious equation. Similarly, the value of an interest rate floor is the sum of its constituent floorlets, each ofwhich is analagous to a put option

vfloorlet = PH0, t + 1L L Dt @ floor NH-d2L - fk NH-d1L DWhere the same volatility is used to price each caplet, these are known as flat volatilities. In contrast, spotvolatilities are specific to each caplet. While many traders like to use spot volatilities, cap prices are usuallyquoted in the market in terms of flat implied volatilities.

à Swaption

The payoff from a payer swaption in each period is

L Dt maxHrT - rS , 0Lwhere L is the notional principal of the swap, rT is the actual swap rate at the maturity of the option, and rS is thefixed rate payable under the option. To apply Black's model to value the swaption, it is assumed that rT islognormal with volatility (of log rT ) equal to s. The expected value of the cash flow received at time Ti is

Black'sModel.nb 3

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v = PH0, Ti L L Dt @r0 NHd1L - rS NHd2LDwhere r0 is the forward swap rate and

d1 =lnHr0 ê rSL + s2 T ê 2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

s è!!!!T

, d2 = d1 - s è!!!!T

The total value of the swaption is

vpayer = ‚i=1

m n

PH0, Ti L L Dt @r0 NHd1L - rS NHd2LDwhere m = 1 ê Dt is the compounding frequency, n is the length of option (in years), and Ti = T + i ê m. Define

A =1

ÅÅÅÅÅÅÅm

‚i=1

m n

PH0, Ti LThe value of a payer swaption is

vpayer = A L @r0 NHd1L - rS NHd2LDSimilarly, the payoff of a receiver swaption in each period is

L Dt maxHrS - rT , 0Lwhich is analogous to a put option on rT . The value of the receiver swaption

vreceiver = A L @rS NH-d2L - r0 NH-d1L DBrokers provide tables of implied volatilities for European swap options.

à Conclusion

Practitioners typically use an adaptation of Black's model for valuing the most popular interest rate derivatives -bond options, interest rate caps and swaptions. Each model is based on the assumption that a key variable islognormal. These assumptions are inconsistent with one another. For example, if future bond prices are lognor-mal, future zero rates and swap rates are not. Similarly, if future zero rates are lognormal, future bond prices andswap rates are not. However, the great popularity of this model for pricing both caps and swaptions indicatesthat this inconsistency is not significant economically.

4 Black'sModel.nb

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Determining price volatility from yield volatility

First consider an option maturing at t on a discount bond which matures at T . The initial price of the bond is

S0 = ã-rT´T

At maturity of the option, the price of the bond is

St = ãrt´1 S0 = ã

-rT-t´HT-tL

where rT-t is a random variable.

Differentiating

¶St

¶rT-t

= -HT - tL ã-rT-t´HT-tL

= -HT - tL St

so that for small changes

D St

St

= -HT - tL D rT-t = -HT - tL´ rT-t ´

D rT-t

rT-t

and so the price volatility is

VolD St

St

= HT - tL´ rT-t ´VolD rT-t

rT-t

The first term T - t is the duration of the discount bond at the maturity of the option.

Analogously, for a coupon bond, we have

VolD St

St

= D´ rT-t ´VolD rT-t

rT-t

where D is the modified duration of the bond at maturity of the option.

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The Black-Derman-Toy modelThe single factor model of Black-Derman-Toy stands out as a very attractive tool for pricinginterest rate options in India. This model is lattice based, incorporates mean reversion, assumeslevel independent volatility and is calibrated through an exogenously specified yield curve.

Jayant Varma, 1996

Michael CarterThe fundamental assumption underlying Black-Scholes and most other financial models is that asset pricesfollow a particular stochastic process called geometric Brownian motion (GBM)

(1)dS = m S dt + s S dz

or

dSÅÅÅÅÅÅÅÅÅS

= m dt + s dz

This implies that the logarithm of the stock price follows

(2)d ln S = n dt + s dz

where n = m - 1ÅÅÅÅ2 s2. The derivation of (2) from (1) is an application of Ito's lemma (Hull: 2003:232-233,Luenberger 1998:313). This implies that the asset price is lognormal, that is

lnHStL = lnHSt-1L + n Dt + s et è!!!!!

Dt

or

St = ‰n Dt + s et è!!!!!!

Dt St-1

The basic binomial model sets n = 0, so that

St = u St-1 or St = d St-1

with

u = ‰ s è!!!!!!Dt and d = ‰- s è!!!!!!Dt

The Black-Derman-Toy model assumes that short rates follow the stochastic process

d ln r = nHtL dt + sHtL dz

which implies that the short rate is lognormal

lnHrtL = lnHrt-1L + nHtL Dt + sHtL et è!!!!!

Dt

or

rt = ‰nHtL Dt + sHtL et è!!!!!!Dt rt-1

This can be approximated by a binomial model

Page 142: Pg p 12 Handbook

rt = u rt-1 or rt = d rt-1

with

u = ‰ nt Dt + stè!!!!!!Dt and d = ‰nt Dt - st

è!!!!!!Dt

where nt and st are chosen to match the actual term structure of interest rates and volatilities.

The single asset and interest rate models are compared in the following table.

Lognormal lnHStL = lnHSt-1L + n Dt + s et è!!!!!

Dt lnHrtL = lnHrt-1L + nHtL Dt + sHtL et è!!!!!

Dtor or

or St = ‰n Dt + s et è!!!!!!Dt St-1 rt = ‰nHtL Dt + sHtL et è!!!!!!Dt rt-1

∞ ∞

Binomial St = u St-1 or St = d St-1 rt = u rt-1 or rt = d rt-1

with with

u = ‰ s è!!!!!!Dt d = ‰- s è!!!!!!Dt u = ‰ nt Dt + stè!!!!!!Dt d = ‰nt Dt - st

è!!!!!!Dt

2 BlackDermanToy.nb

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Kolmogorov�Smirnov One�Sided Test

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Michael
Text Box
Source: Peter M. Lee, Department of Mathematics, University of York

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