Zvi WienerContTimeFin - 5 slide 1 Financial Engineering Continuous Time Finance Zvi Wiener...

Zvi Wiener ContTimeFin - 5 slide 1

Financial Engineering

Continuous Time Finance

Zvi [email protected]

tel: 02-588-3049


Futures Contracts Mark to market Convergence property Spot-futures parity Cost-of-carry Martingale Risk-neutral Measure Forwards and Futures Girsanov’s Theorem and its counterpart Feynman-Kac Formula Stochastic optimization The Maximum Principle


Futures Markets

Futures and forward contracts are similar to options in that they specify purchase or sale of some underlying security at some future date.

However a future contract means an obligation of both sides.

It is a commitment rather than an investment.


Basics of Futures Contracts

Delivery of a commodity at a specified place, price, quantity and quality.

Example: no. 2 hard winter wheat or no. 1 soft red wheat delivered at an approved warehouse by December 31, 1997.


Basics of Futures Contracts

Long position – commits to purchase the commodity.

Short position – commits to deliver.

At maturity: Profit to long = Spot pr. at maturity – Original futures pr.

Profit to short = Original futures pr. –Spot pr. at maturity

it is a zero sum game


Futures Markets

The initial investment is zero however some margin is required.

The later cash flow is mark-to-market for a future contract and is concentrated in one point for the forward contract.

Futures are standardized and not specify the counterside.


Futures Markets Currencies

– all major currencies, including cross rate Agricultural

– corn, wheat, meat, coffee, sugar, lumber, rice Metals and Energy

– copper, gold, silver, oil, gas, aluminum Interest Rates Futures

– eurodollars, T-bonds, LIBOR, Municipal, Fed funds Equity Futures

– S&P 500, NYSE index, OTC, FT-SE, Toronto


Mechanism of Trading

Long Shortcommodity

money

ClearinghouseLong Short


Marking to Market

Example: initial margin on corn is 10%.

1 contract is for 5,000 bushels,

price of one bushel is 2.2775,

so you have to post the initial margin =

$1,138.75 = 0.1*2.2755*5000

If the futures price goes from 2.2775 to 2.2975

the clearinghouse credits the margin account of the long position for 5000 bushels x 2 cents or $100 per contract.


Marking to Market

Your balance

time

Maint.margin

margin call

Initialmargin


Marking to Market and Margin

The current futures price for silver delivered in five days is $5.10 (per ounce).

One futures contract is for 5,000 ounces


Marking to Market and Margin

Day Futures Price

0 (today) $5.10

1 $5.20

2 $5.25

3 $5.18

4 $5.18

5 $5.21


Marking to Market and MarginDay Futures P&L/oz. Margin

1 $5.20 5.20-5.10= 0.10 500

2 $5.25 5.25-5.20= 0.05 250

3 $5.18 5.18-5.25=-0.07 -350

4 $5.18 5.18-5.18= 0.00 0

5 $5.21 5.21-5.18= 0.03 150

Total: $550

Compare the total to forward: (5.21–5.10)5000


Convergence Property

The futures price and the spot price must converge at maturity.

Otherwise there will be an arbitrage based on actual delivery.

Sometimes delivery is costly!


Futures Markets

Cash delivery: sometimes is allowed, sometimes is the only way to deliver.

The question of quality is resolved with a conversion factor. The cheapest to deliver option.


Futures Markets

The S&P 500 futures calls for delivery of $500 times the value of the index. If at maturity the index is at 475, then $500x475=$237,500 cash is the delivery value.

If the contract was written on the futures price 470 (some time ago), who will pay money?

Short side will pay to the long side.


Futures Markets Strategies

Hedging and Speculation – efficient tool for hedging and speculation. A significant leverage effect.


Basis Risk and Hedging

The basis is the difference between the futures price and the spot price. (At maturity it approaches zero).

This risk is important if the futures position is not held till maturity and is liquidated in advance.

Spread position is when an investor is long a futures with one ttm and short with another.


Spot-Futures Parity Theorem

Create a riskless position involving a futures contract and the spot position.

Buy one stock for S and take a short futures position in it.

The only difference is from dividends.

Thus F + D – S is riskless.

The amount of money invested is S.


Spot-Futures Parity Theorem

Create a riskless position involving a futures contract and the spot position.

Buy one stock for S and take a short futures position in it.

The only difference is from dividends.

Thus F + D – S is riskless.

The amount of money invested is S.

S

SDFrf


Spot-Futures Parity Theorem Cost-of-carry relationship

)1()1( drSDrSF ff


Spot-Futures Parity Theorem Cost-of-carry relationship

Tf drSF )1(

For contract maturing in T periods


Relationship for Spreads

)(12

2

1

12

2

1

)1)(()(

)1()(

)1()(

TTf

Tf

Tf

drTFTF

drSTF

drSTF

This is a rough approximation based on anassumption that there is a single source ofrisk and all contracts are perfectly correlated.


Martingale

X - a stochastic time dependent variable.

Et - expectation based on information available at time t.

Xt is a martingale if for any s > t

Et(Xs) = Xt


Martingale

Most financial variables are not martingales because of the drift component (inflation, interest rates, cost of storage, etc.)

However one can change a numeraire so that the new financial variable becomes a martingale.

What can be chosen for an ABM, GBM?


Martingale

dX = dt + dZ ABM

Et(Xs) = Xt+ (s-t)

set Yt = Xt- t

then dYt= dt + dZ - dt = dZ

hence Et(Ys) = Yt


MartingaledX = Xdt + XdZ GBM

What is Et(Xs)?

set Yt = Xte-t

dY = e-t dX - e-tXdt =

e-tXdt + e-t XdZ - e-tXdt =

( - )Ydt + YdZ.

What is Et(Ys)?


Martingale

dY = ( - )Ydt + YdZ

then d(lnY) = ( - - 0.5 2) dt + dZ

lnYt = lnY0 + ( - - 0.5 2) t + Z

if a~N(, ), then E(ea) =exp(+0.52)

lnYt~N(lnY0 + ( - - 0.5 2) t, t)

Then E0(Yt) = Y0exp(( - - 0.5 2)t+0.52t).


Martingale

E0(Yt) = Y0exp(( - - 0.5 2)t+0.52t).

Set =

E0(Yt) = Y0

Et(Ys) = Yt - martingale!

What is the economic meaning of Y?


Equivalent Martingale Measure

Harrison and Kreps

Harrison and Pliska

There exists a risk neutral probability measure.

There exists an equivalent martingale measure.

For a detailed explanation, see Duffie.

Extension to a stochastic volatility, see Grundy,

Wiener.


Forward Contract

)()(exp0 t

T

t

Qt FWdssrE

T

t

Qt

T

t

Qt

t

dssrE

WdssrEF

)(exp

)(exp

if W and r are independent Ft=EtQ(W)


Futures Contract

WEQtt

Mark-to-market procedure equatesthe instantaneous price to zero.


Girsanov’s Theorem

Let dX = (X,t)dt + (X,t)dZ.

If there exist and , such that = - ,

then there exists a new probability measure equivalent to the original one, such that

relative to the new measure the original process X becomes:

dX = (X,t)dt + (X,t)dZ*


by a change of the probability measure (note B*),if there exists a process such that .

Girsanov’s TheoremGirsanov’s Theorem

can be transformed totdZtXdttXdX ),(),(

*tdZtXdttXdX ),(),(


),( tXFchange of variables

tdZtXdttXdX ),(),( can be transformed to

(Girsanov)tdZtXdttXdX ),(),( *1.

tdZtadtdF )(... (Theorem 1)2.

tdZtFadtdF )(... (Theorem 1’)3.


Monotonic change of variables preserves order

x

y

x x y y2 1 2 1

2

y2

x

1

1

y

x


Monotonic change of variables preserves order

x

y1

x

11 PrPr yyxx

1


tdZxdttxdx )(),(

change of variables: x

K

xdx

xy)(

1)(

K

xxd

xxy

x

K

lnˆ1

ˆ1

)(

Constant volatility case: tdBxdttxdx ˆ),(

tdZdtx

x

txdy

2

)('

)(

),(

leads to

Example


Theorem 1. The diffusion process

is transformed by the following change of variables

into a process with a deterministic diffusion parameter

dZtadtFtadF )(2

)( 21

Free parameters:a(t) – defines the resulting diffusion parameterA(t) – defines zero level of the new variable

tdZtxdttxdx ),(),(

dt

tatxF

x

tA

)( ),(

)(),(


Feynman-Kac Formula

0.52fxx + fx + ft - rf + h=0

f(X,T) = g(X)

The solution is given by:

)(),(),( ,,, TTt

T

t

ssttx XgdssXhEtXf

T

t

dXr

st e

),(

, the discount factor


Stochastic Optimization

In many cases financial assets involve decisions. In some cases we should assume that decision makers are rational and try to use an optimal decision, in some cases we assume not rational behavior.


A Time-Homogeneous Problem

Values do not depend on time explicitly.

A financial asset V, which depends on a set of variables X, and time t.

Control variable .

0

0 ),(max dsXueEV rs

dZsXdsXdX ),(),(



Sometimes the control variable is a constant, sometimes it is a function of time and state.

The expected cash flow is:

ECF = u(X, )ds

The capital gain is:

CG = dV = VxdX+0.5Vxx(dX)2

The expected capital gain is:

ECG = (Vx+0.52Vxx)dt



The value of V does not depend on time.

The optimally managed total return per unit of

time is given by:

ETR = max(ECF+ECG)=

max[u(X, )+ (X, )Vx +0.52 (X, )Vxx]

It must be equal the risk free return:

rV= max[u(X, )+ (X, )Vx +0.52 (X, )Vxx]


The Maximum Principle

X follows an ABM with parameters and .

An asset pays continuous cash flow at the rate Xdt.

There is no limited liability option.

A manager can influence the growth rate of X.

Suppose that for any one has to pay 2dt to

managers.

What is the optimal strategy?



dZdsdXts

dsXeEV rs

..

max0

20

xxx VVXrV 22 5.0max



xxx VVXrV 22 5.0max

xopt

x

V

V

5.0

20

Note that Shimko assumes that one can not replacea manager, thus opt is constant and hence Vxx=0.



With this assumption we get V=2X opt + C

34

1

2)2(

rr

XV

XCXrrV optopt



Assuming one-time decision we can value thesecurity as a sum of linearly growing perpetuity(ABM) minus a level perpetuity (constant paymentof 2 forever.

rrr

XV

2

2

Optimizing with respect to we obtain:

34

1

rr

XV



Without this assumption we get:

xxx VVXrV 22 5.025.0

A non-linear ODE, must be solved numerically.

What are the appropriate boundary conditions?


Multiple State Variables

Consider a perpetually lived value-maximizing monopolist who produces output at a rate of qdt, but faces a stochastically varying demand.

Assume that the demand is linear p = a - bq, where p is the price of the good, and a, b are given by:

b

a

dZqbadtqbagdb

dZqbadtqbafda

),,(),,(

),,(),,(

dtdZdZ ba



The initial conditions are a(0)=a0, b(0)=b0.

Assume that the cost of production is zero.

The value of the firm is V, such that:

0

0 )(max qdsbqaeEV rs

q



The expected cash flow is:

(a-bq)qdt

The capital gain component is:

dV = Vada+Vbdb+0.5Vaa(da)2+Vabdadb+0.5Vbb(db)2

The expected capital gain is:

ECG=E[dV]=fVa+ gVb+0.52Vaa+Vab+0.52Vbb



The maximum total return is:

max(TR) = max(ECF+ECG) = rV

Therefore

bbabaabaq

VVVgVfVqbqarV 22 5.05.0)(max

The first order condition is:

05.0)(5.02 bbqqqabaaqbqaq VVVVgVfbqa


Multiple State VariablesAssume that

f(a,b,q) = af0

g(a,b,q) = bg0

(a,b,q) = a0

(a,b,q) = b0

The value of the firm is:

bbabaaba VbabVVaVbgVafb

arV 2

02

0020

200

2

5.05.04

)22(4 2000

2000

2

gfrb

aV


Optimal Asset Allocation

Merton 1971.

Utility function: U= r - 0.5A2

Here r is the expected rate of return and - its standard deviation.

A - is the individual’s coefficient of risk aversion.



Denote by - proportion invested in risky assets. Then

fP rrr )1( 222 P



Maximizing utility with respect to , we get:

02

1)1( 22

Pf ArrU

2

A

rr fopt


Dynamic Asset Allocation

How one can apply the Girsanov’s theorem?

Perfect markets, no taxes, costs, restrictions.

The budget equation:

rPdtdP

XdZXdtdX

XdZWdtcWrPXWdW ])1([


Dynamic Asset AllocationThe objective function is to maximize the expected lifetime discounted utility.

XdZWdtcWrPXWdWts

dsscUeEJ s

c

])1([..

))((max0

0,


Problem 4.3

The height of a tree at time t is given by Xt, where Xt follows an ABM. We must decide when to cut the tree.

The tree is worth $1 per unit of height, and if the tree is cut down at time at height Y, then its value today is:

V = e-rY.


Problem 4.3

a. What PDE must the value of the tree satisfy?

b. What are the boundary conditions?

c. Value the tree, assuming that the value is zero

when the tree’s height is -.

d. What is the optimal cutting policy?

Date post:	19-Dec-2015
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Zvi WienerContTimeFin - 5 slide 1 Financial Engineering Continuous Time Finance Zvi Wiener...

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