Wiener Processes and Itô’s Lemma

A Wiener Process We consider a variable z whose value changes continuously Define f(m,v) as a normal distribution with mean m and variance v The change in a small interval of time Dt is Dz The variable follows a Wiener process if The values of Dz for any 2 different (non-overlapping) periods of time are independent

Properties of a Wiener Process Mean of Δz=0 Standard deviation of Δz= Variance of Δz= Δt z follows a Markov process. Mean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is

Wiener process

Generalized Wiener Processes (See page 284-86) A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1 In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants

Generalized Wiener Processes (continued) Mean change in x per unit time is a Variance of change in x per unit time is b2 term add noise or variability to the path followed by x.

Generalized Winner Process

Generalized Wiener Processes (continued) Mean of [x (T ) – x (0)] is aT Variance of [x (T ) – x (0)] is b2T Standard deviation of [x(T ) – x (0)] b*sqrt(T)

Taking Limits . . . What does an expression involving dz and dt mean? It should be interpreted as meaning that the corresponding expression involving Dz and Dt is true in the limit as Dt tends to zero In this respect, stochastic calculus is analogous to ordinary calculus Generalized Winner Process: dx=adt+bdz

Generalized Wiener Processes Mean of [x (T ) – x (0)] is aT Variance of [x (T ) – x (0)] is b2T Standard deviation of [x(T ) – x (0)] b*sqrt(T)

Itô Process In an Itô process the drift rate and the variance rate are functions of time dx=a(x,t) dt+b(x,t) dz The discrete time equivalent is only true in the limit as Dt tends to zero Ito process is also Markov.

An Ito Process for Stock Prices (See pages 286-89) It is reasonable to assume that the percentage return in a short period of time is random, but its variability is the same regardless of the stock price. Thus we have . where m is the expected return s is the volatility. The discrete time equivalent is The process is known as geometric Brownian motion

Monte Carlo Simulation Fro the process of stock price, we have We can sample random paths for the stock price by sampling values for e Suppose m= 0.15, s= 0.30, and Dt = 1 week (=1/52 years), then

Monte Carlo Simulation – One Path Excel code : NORMINV(RAND(),0,1)

Monte Carlo Simulation Monte Carlo methods use statistics gathered from random sampling to model and simulate a complicated distribution. “If “the future” is the random selection of one sequence of events from a probability space, our focus should revolve around modeling the distribution more so than identifying the ultimate future path.”

Itô’s Lemma If we know the stochastic process followed by x, Itô’s lemma tells us the stochastic process followed by some function G (x, t ) Since a derivative is a function of the price of the underlying and time, Itô’s lemma plays an important part in the analysis of derivative securities

Itô’s Lemma-formal argument

Itô’s Lemma-formal argument

Itô’s Lemma-formal argument

Application of Ito’s Lemma to a Stock Price Process We argued that under a reasonable assumption: Both S and G are affected by dz - the noise in the system.

Examples