Zvi WienerContTimeFin - 1 slide 1 Financial Engineering Zvi Wiener [email protected] tel: 02-588-3049

Zvi Wiener ContTimeFin - 1 slide 1

Financial Engineering

Zvi [email protected]

tel: 02-588-3049


Main Books

Shimko D. Finance in Continuous Time, A Primer. Kolb Publishing Company, 1992, ISBN 1-878975-07-2 Wilmott P., S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 1996, ISBN 0-521-49789-2


Useful Books

Duffie D., Dynamic Asset Pricing Theory. Duffie D., Security Markets, Stochastic Models. Neftci S., An Introduction to the Mathematics of Financial Derivatives. Steele M., Invitation to Stochastic Differential Equations and Financial Applications. Karatzas I., and S. Shreve, BM and Stochastic Calculus.


Primary Asset Valuation

Discrete-time random walk:

W(t+1) = W(t) + e(t+1);

W(0) = W0;

e~i.i.d. N(0,1)



N(0,1)



Discrete-time random walk refinement:

W(t+) = W(t) + e(t+);

W(0) = W0;

e~i.i.d. N(0, )

This process has the same expected drift and variance over n periods as the initial process had in one period.


Primary Asset ValuationN(0,1)

t = 0 0.25 0.5 0.75 1


Primary Asset ValuationSet dt

W(t+ dt) = W(t) + e(t+ dt);

W(0) = W0;

e~i.i.d. N(0, dt)

Define dW(t) = W(t+dt) - W(t) white noise,

(dt)a = 0 for any a > 1



Needs["Statistics`NormalDistribution`"]

nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]];

tt=NestList[ (#+nor[0, 0.1])&, 0, 300];

ListPlot[tt,PlotJoined->True,PlotLabel->"Random Walk"];



50 100 150 200 250 300

-1

-0.5

0.5

Random Walk


Main Properties

1. E[dW(t)] = 0

2. E[dW(t) dt] = E[dW(t)] dt = 0

3. E[dW(t)2] = dt


Main Properties (cont.)

4. Var[dW(t)2] = E[dW(t)4] - E2[dW(t)2] =

3 dt2 - dt2 = 0

5. E[(dW(t)dt)2] = E [dW(t)2] (dt)2 = 0

6. Var[dW(t)dt] =

E[(dW(t)dt)4] - E2[dW(t)dt] = 0


E[dW(t)] = 0

By definition the mean of the normally distributed variable is zero.


E[dW(t) dt] = E[dW(t)] dt = 0

The expectation of the product of a random variable (dW) and a constant (dt) equals the constant times the expected value of the random variable.


E[dW(t)2] = dt

For any distribution with zero mean the expected value of the squared random variable is the same as its variance.


Var[dW(t)2] =

E[dW(t)4] - E2[dW(t)2] =

3 dt2 - dt2 = 0

The fourth central moment of the standard normal distribution is 3, and (dt)2 = 0.


E[(dW(t)dt)2] = E [dW(t)2] (dt)2 = 0

Follows from properties 2 and 3.


Var[dW(t)dt] =

E[(dW(t)dt)4] - E2[dW(t)dt] = 0

Follows from properties 2 and 5.


Important Property

if

Var[f(dW)] = 0then

E[f(dW)] = f(dW)


Multiplication Rules

Rule 1. (dW(t))2 = dt

Rule 2. (dW(t)) dt = 0

Rule 3. dt2 = 0


W(t) is called a standard Wiener process,

or a Brownian motion.

t

dWWtW

WW

0

0

0

)()(

)0(


Major Properties of W

1. W(t) is continuous in t.

2. W(t) is nowhere differentiable.

3. W(t) is a process of unbounded variation.

4. W(t) is a process of bounded quadratic variation.


Major Properties of W

5. The conditional distribution of W(u) given W(t), for u > t, is normal with mean W(t) and variance (u-t).

6. The variance of a forecast W(u) increases indefinitely as u .


Brownian Motion - BM

The standard BM is useful since many general stochastic processes can be written in terms of W.

X(t+1) = X(t) + (X(t),t) + (X(t),t) e(t+1)

X(0) = X0, e~i.i.d. N(0,1)

generalized drift heteroscedastisity (changing variance)



Choose a shorter time interval :

X(t+) = X(t) + (X(t),t) + (X(t),t) e(t+ )

X(0) = X0, e~i.i.d. N(0, )

generalized drift heteroscedastisity (changing variance)



As we let dt we see that

dX(t) = (X(t),t) dt + (X(t),t) dW(t)

X(0) = X0

generalized univariate Wiener process, (diffusion).

dX = (X,t) dt + (X,t) dW, X(0) = X0


Interpretation

How can we interpret the fact that

dX = dt + dW

the random variable dX has local mean dt and local variance 2dt.

A discrete analogy is X = + z.


Arithmetic BM dX = dt + dW

Let (X,t) = , and (X,t) = two constants.

Then X follows an arithmetic Brownian

Motion with drift and volatility .

This is an appropriate specification for a

process that grows at a linear rate and exhibits

an increasing uncertainty.



1. X may be positive or negative.

2. If u > t, then Xu is a future value of the process

relative to time t. The distribution of Xu given Xt

is normal with mean

Xt+ (u-t) and standard deviation (u-t)1/2.

3. The variance of a forecast Xu tends to infinity

as u does (for fixed t and Xt).



time

X



time

X


Is appropriate for variables that can be positive and negative, have normally distributed forecast errors, and have forecast variance increasing linearly in time.

Example: net cash flows. Is inappropriate for stock price.



Geometric BM dX = Xdt + XdW

Let (X,t) = X, and (X,t) = X.

Then X follows an geometric Brownian Motion.



This is an appropriate specification for a process

that

grows exponentially at an average rate of

has volatility proportional to the level of the

variable.

It also exhibits an increasing uncertainty.



1. If X(0) > 0, it will always be positive.

2. X has an absorbing barrier at X = 0. Thus if

X hits zero (a zero probability event) it will

remain there forever.



3. The conditional distribution of Xu given Xt is

lognormal. The conditional mean of ln(Xt) is ln(Xt)

+ (u-t) - 0.5 2(u-t) and conditional standard

deviation of ln(Xt) is (u-t)1/2.

ln(Xt) is normally distributed.



4. The conditional expected value of Xu is

Et[Xu] = Xtexp[ (u-t)]

5. The variance of a forecast Xu tends to

infinity as u does (for fixed t and Xt).



time

X



time

X


Is often used to model security values, since the proportional changes in security price are independent and identically normally distributed (sometimes). Example: currency price, stocks. Is inappropriate for dividends, interest rates.



Mean Reverting Process

dX = (-X)dt + XdWOrnstein-Uhlenbeck when = 1

Let (X,t) = (-X), and (X,t) = X, where

0 - speed of adjustment

- long run mean

- volatility



dX = (-X)dt + XdW

This is an appropriate specification for a process

that has a long run value but may be beset by

short-term disturbances.

We assume that , , and are positive for

simplicity.



dX = (-X)dt + XdW

1. If X(0) > 0, it will always be positive.

2. As X approaches zero, the drift is positive

and volatility vanishes.

3. As u becomes infinite, the variance of a

forecast Xu is finite.



dX = (-X)dt + XdW4. If = 0.5, the distribution of Xu given Xt

for u > t is non-central chi-squared, the mean

of the distribution is:

(Xt- ) exp[-(u-t)] +

the variance of the distribution is (CIR 85):

2)(2

)(2)(2

12

tututut eeeX



dX = (-X)dt + XdW

time

X



dX = (-X)dt + XdW

time

X


Mean Reverting Process dX = (-X)dt + XdW

SeedRandom[2]

tt=NestList[

(#+0.3(1-#)+0.1*#*nor[0,0.1])&,

1.01,130];

ListPlot[tt,PlotJoined->True, Axes->False];


Is often used to model economic variables

and do not represent traded assets.

Example: interest rates, volatility.


dX = (-X)dt + XdW


Ito’s lemma

Consider a real valued function f(X): RR.

Taylor series expansion:

)()(6

1

)(2

1)()()(

33

2

oXf

XfXfXfXf

XXX

XXX


Ito’s lemma

If X is a “standard” variable, then 2 is o()

)()()()( oXfXfXf X

dXXfXdf

dXXfXfdXXf

X

X

)()(

)()()(


Ito’s lemma

If X is a stochastic variable (following diffusion) then the term dX2 does NOT vanish.

dtdX

dWdtdWdtdX

dWdtdX

22

22222 )(2)(


Ito’s lemma

2

2

2

2

1

)(2

1)()(

)(2

1)()()(

dXfdXfdf

dXXfdXXfXdf

dXXfdXXfXfdXXf

XXX

XXX

XXX


Ito’s lemma

If f = f(X,t) and dX = dt + dW, then

dWfdtfffdf XtXXX 25.0


Financial Applications A

Suppose that a security with value V guarantees $1dt every instant of time forever. This is the continuous time equivalent of a risk-free perpetuity of $1. If the risk-free interest rate is constant r, what is the (discounted) value of the security?



1. V = V(t), there are NO stochastic variables.

dV = Vtdt

2. The expected capital gain on V is

ECG = E[dV] = Vtdt

3. The expected cash flows to V is ECF = 1 dt

4. The total return on V is

ECG + ECF = (Vt+1)dt



5. Since there is no risk, the total return must

be equal to the risk-free return on V, or rVdt.

(Vt+1) dt = r V dt

6. Divide both sides by dt:

Vt = rV - 1



Vt = rV - 1

DSolve[ V'[t]==r*V[t]-1, V[t], t ]

V(t) = c Exp[r t] + 1/r

given V(0) one can find c


Financial Applications B

Suppose that X follows a geometric Brownian motion with drift and volatility . A security with value V collects Xdt continuously forever. V represents a perpetuity that grows at an average exponential rate of , but whose risks in cash flow variations are considered diversificable. The economy is risk-neutral, and the risk-free interest rate is constant at r. What is the value of this security?


Financial Applications B1. V = V(X), since V is a perpetual claim, its price does not depend on time.

dV = VxdX + 0.5 VxxdX2,

dX = Xdt + XdW,

dX2= 2X2dt

dV = [XVx+0.5 2X2Vxx]dt +XVxdW



2. The expected capital gain:

ECG = E[dV] = [XVx+0.5 2X2Vxx]dt

since E[dW] = 0

3. The Expected cash flow:

ECF = X dt



4. Total return:

TR = ECG + ECF = [XVx+X+0.52X2Vxx]dt

5. But the return must be equal to the risk free return on the same investment V.

rVdt = [XVx+X+0.52X2Vxx]dt

6. Thus the PDE:

rV = XVx+X+0.52X2Vxx



rV = XVx+X+0.52X2Vxx

there are several ways to solve it. One can guess that doubling X will double the price V.

If V is proportional to X, then V = X, Vx= , and Vxx=0, then the equation becomes

r X= X+X

= 1/(r- )

V(X) = X/(r- )

Documents

Zvi WienerContTimeFin - 1 slide 1 Financial Engineering Zvi Wiener [email protected] tel: 02-588-3049