1

Basic Numerical Procedure

2

Content

1 Binomial Trees

2 Using the binomial tree for options on indices,

currencies, and futures contracts

3 Binomial model for a dividend-paying stock

4 Alternative procedures for constructing trees

5 Time-dependent parameters

6 Monte Carlo simulation

7 Variance reduction procedures

8 Finite difference methods

3

Binomial Trees

In each small interval of time （ Δt ） the stock price is assumed to move up by a proportional amount u or to move down by a proportional amount d

Su

Sd

S

p

1 – p

4

Risk-Neutral Valuation

1. Assume that the expected return from all traded assets is the risk-free interest rate.

2. Value payoffs from the derivative by calculating their expected values and discounting at the risk-free interest rate.

5

Determination of p, u, and d

Mean: e(r-q)t = pu + (1– p )d

Variance:2t = pu2 + (1– p )d 2 – e2(r-q)t

A third condition often imposed is u = 1/ d

6

A solution to the equations, when terms of higher order than t are ignored, is

where

)( tqr

t

t

ea

ed

eu

du

dap

7

Tree of Asset Prices

At time iΔt ：

S0u 2

S0u 4

S0d 2

S0d 4

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3

i0,1,...,j ,duSS j-ij0i

8

Working Backward through the Tree

Example : American put optionS0 = 50; K = 50; r =10%; = 40%; T = 5 months = 0.4167;

t = 1 month = 0.0833

The parameters imply ： u = 1.1224; d = 0.8909;

a = 1.0084; p = 0.5073

9

Example (continued)89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

G

F

ED C

B A

39.690.89091.122450duSS 31310A 14.6435.36-50,0)-max(f TG SK 2.665.45)e0.49270(0.5073efp-1pff 0.0833-0.1t-r

duE 9.9014.64)e0.49275.45(0.5073f 0.0833-0.1A

4.496.96)e0.49272.16(0.5073f 0.0833-0.1D

10

Example (continued)

In practice, a smaller value of Δt, and many more nodes, would be used. DerivaGem shows ：

steps 5 30 50 100 500

f0 4.49 4.263 4.272 4.278 4.283

11

Expressing the Approach Algebraically

j1,i1j1,it-rj-ij

0ji, fp-1pfe ,duS-Kmaxf

12

Estimating Delta and Other Greek Letters

delta （ Δ ）： at time Δt

dS-uS

f-f

S

f

00

1,01,1

S0u 2

S0u 4

S0d 2

S0d 4

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3

13

gamma （ Γ ）： at time 2Δt

)dS-u0.5(S

dS-S

f-f-

S-uS

f-f

S

f2

02

0

200

2,02,1

02

0

2,12,2

2

2

S0u 2

S0u 4

S0d 2

S0d 4

S0

S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3

14

theta （ Θ ）：

t2

f-f

t

f 0,02,1

S0u 2

S0u 4

S0d 2

S0d 4

S0 S0u

S0d S0 S0

S0u 2

S0d 2

S0u 3

S0u

S0d

S0d 3 f

2

1rS 22 rSσ

15

Vega （ ν ）：

Rho （ ρ ）：

ff -*

r

ff

-*

16

Example

89.070.00

79.350.00

70.70 70.700.00 0.00

62.99 62.990.64 0.00

56.12 56.12 56.122.16 1.30 0.00

50.00 50.00 50.004.49 3.77 2.66

44.55 44.55 44.556.96 6.38 5.45

39.69 39.6910.36 10.31

35.36 35.3614.64 14.64

31.5018.50

28.0721.93

G

F

ED C

B A

-0.4144.55-56.12

6.96-2.16

dS-uS

f-f

00

1,01,1

0.0311.65

(-0.64)-(-0.24)

39.69)-0.5(62.9939.69-5010.36-3.77

-50-62.99

3.77-0.64

)dS-u0.5(S

-2

02

0

21

daycalendar per -0.012or

yearper -4.30.08332

4.49-3.77

t2

f-f 0,02,1

17

Using the binomial tree for options on indices, currencies, and futures contracts

As with Black-Scholes ： For options on stock indices, q equals the

dividend yield on the index For options on a foreign currency, q equal

s the foreign risk-free rate For options on futures contracts ： q = r

18

Example

19

Example

20

Binomial model for a dividend-paying stock

Known Dividend Yield ： before ： after ：

Several known dividend yields ：

i0,1,...,j ,duSS j-ij0ji,

i0,1,...,j ,d)u-(1SS j-ij0ji,

j-iji0ji, d)u-(1SS

21

Known Dollar Dividend ：

i k≦ ： i=k+1 ： i=k+2 ：

i0,1,...,j ,duSS j-ij0ji,

i0,1,...,j D,-duSS j-ij0ji,

1-i0,1,...,jfor

D)d-du(S andD)u -du(SS j-1-ij0

j-1-ij0ji,

nodes. 1mkn rather tha 2)m(k are there

mki when and nodes, 1in rather tha 2i are there

22

Simplify the problem

The stock price has two components ： a part that is uncertain and a part that is the present value of all future dividends during the life of the option.

Step 1 ： A tree can be structured in the usual way to model .

Step 2 ： By adding to the stock price at each nodes, the present value of future dividends, the tree can be converted into model S.

*S

ti when , De-SS

ti when , SSt)i-r(-*

*

ti when , DeduSS

ti when , duSSt)i-r(-j-ij*

0ji,

j-ij*0ji,

23

Example

24

Control Variate Technique

1. Using the same tree to calculate both the value of the American option （ ） and the value of the European option （ ） .

2. Calculating the Black-Scholes price of the European option （ ） .

3. This gives the estimate of the value of the American option as

EBSA -fff

Af

Ef

BSf

25

Example

B-S model ： ∴

4.08)(-)(e 12T

BS -dNS-dNKf 0-r

4.32Ef A 4.49f

4.254.32-4.084.49P0

26

Alternative procedures for constructing trees

Instead of setting u = 1/d we can set each of the 2 probabilities to 0.5 and

ttqr

ttqr

ed

eu

)2/(

)2/(

2

2

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Example

0.9703

ed

1.0098

eu

and 0.5pset We

0.25t , 0.75T

0.04

0.10r , 0.06r

0.795K , 0.79S

]0.250.04-.250.0016/2)0-0.1-[(0.06

]0.250.04.250.0016/2)0-0.1-[(0.06

f

0

28

Trinomial Trees

6

1

212

3

2

6

1

212

/1

2

2

2

2

3

rt

p

p

rt

p

udeu

d

m

u

t

S S

Sd

Su

pu

pm

pd

29

Adaptive mesh model（ Figlewski and Gao,1999 ）

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Time-dependent parameters

V/Nt)(t then

steps, timeN of totala is thereIf

step th time theof end theis t

tree theof life theis T , T(T)V Define

maturity t afor y volatilit theis (t) that Suppos

: timeoffuction a make To 2

of valueforward theis (t)

rateinterest forward theis (t)

eset We

: timeoffuction a )(or and make To 1

2

2

t(t)]-(t)[

i

i

qg

f

a

rqr

ii

i

gf

f

31

Monte Carlo simulation

When used to value an option, Monte Carlo simulation uses the risk-neutral valuation result. It involves the following steps:

1. Simulate a random path for S in a risk neutral world.2. Calculate the payoff from the derivative.3. Repeat steps 1 and 2 to get many sample values of

the payoff from the derivative in a risk neutral world.4. Calculate the mean of the sample payoffs to get an

estimate of the expected payoff.5. Discount this expected payoff at risk-free rate to get

an estimate of the value of the derivative.

32

Monte Carlo simulation (continued)

In a risk neutral world the process for a stock price is

We can simulate a path by choosing time steps of length Δt and using the discrete version of this

where ε is a random sample from (0,1)

dS S dt S dz

ttSttStSttS )()(ˆ)(-)(

33

Monte Carlo simulation (continued)

T T 2

ˆ)0(ln)T(ln

thenconstant, are and ˆ if

)()(or

2

ˆ)(ln)(ln

is thisof version discrete The

2

ˆln

.n rather tha ln estimate toaccurate more isIt

2

2/ˆ

2

2

2

SS

etSttS

tttSttS

dzdtSd

SS

tt

34

Derivatives Dependent on More than One Market Variable

When a derivative depends on several underlying variables we can simulate paths for each of them in a risk-neutral world to calculate the values for the derivative ：

ttSttmttt iiiiiii )()(ˆ)()(

35

Generating the Random Samples from Normal Distributions

How to get two correlated samples ε1 and ε2 from univariate standard normal distributions x1 and x2 ？

n.correlatio oft coefficien theis where

-1xx

x

2212

11

36

Cholesky decomposition

ijfor ,

1 where

x

j

1kijjkik

i

1k

2ik

i

1kkiki

37

Number of Trials Denote the mean by μ and the standard deviation

by ω. The standard error of the estimate is

where M is the number of trials. A 95% confidence interval for the price f of the

derivative is

To double the accuracy of a simulation, we must quadruple the number of trials.

M

Mf

M

1.961.96-

38

Applications Advantage ： 1. It tends to be numerically more efficient

（ increases linearly ） than other procedures （ increases exponentially ） when there are more stochastic variables.

2. It can provide a standard error for the estimates.

3. It is an approach that can accommodate complex payoffs and complex stocastic processes.

39

Applications (continued)

An estimate for the hedge parameter is

Sampling through a Tree ：

x

ff *

ˆ-ˆ

40

Variance reduction procedures

Antithetic Variable Techniques ：

standard error of the estimate is

Control Variate Technique ：

2

21 fff

BBAA ffff **

M

41

Variance reduction procedures (continued)

Importance Sampling:

Stratified Sampling:

Moment Matching:

Using Quasi-Random Sequences:

)0.5-

(1-

n

iN

s

mi*i

-

42

Finite difference methods

Define ƒi,j as the value of ƒ at time it when the stock price is jS

ΔT=T/N; ΔS=Smax /M

43

Implicit Finite Difference Method

Forward difference approximation

backward difference approximation

S

ff

S

f jiji

,1,

S

ff

S

f jiji

1,,

44

Implicit Finite Difference Methods(continued)

ƒ2ƒƒƒ

or

ƒƒƒƒƒ

2

ƒƒƒset we

ƒƒ

2

1ƒƒIn

2

,1,1,

2

2

1,,,1,

2

2

1,1,

2

222

SS

SSSS

SS

rS

SS

rSt

jijiji

jijijiji

jiji

45

Implicit Finite Difference Methods(continued)

1, ,

, 1 , , 1 1,

If we also set and S

we obtain:

where t

i j i j

j i j j i j j i j i j

2 2j

2j

f ffS jΔ

t t

a f b f c f f

1 1a (r - q)jΔ - σ j Δt

2 2

b 1 σ

t

t

2

2 2j

j Δt rΔ

1 1 c - (r - q)jΔ - σ j Δt

2 2

46

Implicit Finite Difference Methods(continued)

47

Explicit Finite Difference Methods

:obtain we

point )( at the are they aspoint )1( at the

same thebe toassumed are and If 22

i,j,ji

SfSf

2

1,11,11,

2

2

11,11,

ƒ2ƒƒƒ

2

ƒƒƒ

SS

SS

jijiji

jiji

48

Explicit Finite Difference Methods(continued)

1,1*

,1*

1,1*

, ƒƒƒ

isequation difference The

jijjijjijji cbaf

)t(1

1

)-(1

1

)t(-1

1 where *

Δtjσ2

1(r-q)jΔ

2

1

tr c

Δtjσ1tr

b

Δtjσ2

1(r-q)jΔ

2

1

tra

22*j

22*j

22j

49

Explicit Finite Difference Methods(continued)

50

Difference between implicit and explicit finite difference methods

ƒi +1, j +1

ƒi , j ƒi +1, j

ƒi +1, j –1

ƒi +1, jƒi , j

ƒi , j –1

ƒi , j +1

Implicit Method Explicit Method

51

Change of Variable

ƒZ

ƒ

2

1

Z

ƒ)

2-q-(

ƒ

, ln ZDefine

2

22

2

rrt

S

2j

2j

2j

jijijjijjij

σ2

)-(r-q2

rΔσ 1

σ2

)-(r-q2

ffff

2

2

2

2

2

,11,,1,

Z

t

2-

Z

t-

tZ

t

Z

t

2-

Z

t where

becomes methodimplicit for theequation difference The

52

Change of Variable (continued)

1,1*

,1*

1,1*

, ƒƒƒ

becomes methodexplicit for theequation difference The

jijjijjijjif

)Z

t)

2-(

Z

t(

1

1

)Z

t-(

1

1

)Z

t)

2-(

Z

t(-

1

1 where

2

2

2

2

2*

2*j

2*j

2j

σ2

r-q2tr

σ1tr

σ2

r-q2tr

53

Relation to Trinomial Tree Approaches

The three probabilities sum to unity.

54

Relation to Trinomial Tree Approaches (continued)

2 2

2

2

2

2

2

1- j t is negative when j 13.

We can use change-of-variable approch:

t t - ( - )

Z 2 Zt

-Z

t t ( - )

Z 2 Z

2u

2m

2d

p r - q σ2 2

p 1 σ

p r - q σ2 2

55

Other Finite Difference Methods

Hopscotch method

Crank-Nicolson scheme

Quadratic approximation

56

Summary

We have three different numerical procedures for valuing derivatives when no analytic solution: trees, Monte Carlo simulation, and finite difference methods.

Trees: derivative price are calculated by starting at the end of the tree and working backwards.

Monte Carlo simulation: works forward from the beginning, and becomes relatively more efficient as the number of underlying variables increases.

Finite difference method: similar to tree approaches. The implicit finite difference method is more complicated but has the advantage that does not have to take any special precautions to ensure convergence.