CH12- WIENER PROCESSES AND ITÔ'S LEMMA

OUTLINE .*. The Markov Property Continuous-Time Stochastic Processes The Process For a Stock Price The Parameters Itô's Lemma The Lognormal Property .*.

the value of the variable changes only at certain fixed time point Stochastic Process Continuous Variable Discrete Variable the value of the variable changes only at certain fixed time point .*. only limited values are possible for the variable

12.1 THE MARKOV PROPERTY A Markov process is a particular type of stochastic process . The past history of the variable and the way that the present has emerged from the past are irrelevant. A Markov process for stock prices is consistent with weak-form market efficiency. where only the present value of a variable is relevant for predicting the future.

12.2 CONTINUOUS-TIME STOCHASTIC PROCESSES Suppose $10(now), change in its value during 1 year is f(0,1). What is the probability distribution of the stock price at the end of 2 years? f(0,2) 6 months? f(0,0.5) 3 months? f(0,0.25) Dt years? f(0, Dt) In Markov processes changes in successive periods of time are independent This means that variances are additive. Standard deviations are not additive. N~(μ=0, σ=1) N~(μ=0, σ=2) 5 5

A WIENER PROCESS (1/3) Δz~normal distribution It is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year. A variable z follows a Wiener Process if it has the following two properties: (Property 1.) The change Δz during a small period of time Δt is Wiener Process .*. Δz~normal distribution

A WIENER PROCESS (2/3) (Property 2.) The values of Δz for any two different short intervals of time, Δt, are independent. Mean of Dz is 0 Variance of Dz is Dt Standard deviation of Dz is 7 7

A WIENER PROCESS (3/3) Mean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is Consider the change in the value of z during a relatively long period of time, T. This can be denoted by z(T)–z(0). It can be regarded as the sum of the changes in z in N small time intervals of length Dt, where

EXAMPLE12.1(WIENER PROCESS) Ex：Initially $25 and time is measured in years. Mean：25， Standard deviation :1. At the end of 5 years, what is mean and Standard deviation? Our uncertainty about the value of the variable at a certain time in the future, as measured by its standard deviation, increases as the square root of how far we are looking ahead.

GENERALIZED WIENER PROCESSES(1/3) A Wiener process, dz, that has been developed so far has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1 DR=0 means that the expected value of z at any future time is equal to its current value. VR=1 means that the variance of the change in z in a time interval of length T equals T. Drift rate →DR , variance rate →VR DR=0 , VR=1

GENERALIZED WIENER PROCESSES (2/3) A generalized Wiener process for a variable x can be defined in terms of dz as dx = a dt + b dz DR VR

GENERALIZED WIENER PROCESSES(3/3) In a small time interval Δt, the change Δx in the value of x is given by equations Mean of Δx is Variance of Δx is Standard deviation of Δx is

EXAMPLE 12.2 Follow a generalized Wiener process DR=20 (year) VR=900(year) Initially , the cash position is 50. At the end of 1 year the cash position will have a normal distribution with a mean of ★★ and standard deviation of ●● ANS:★★=70, ●●=30

ITÔ PROCESS dx=a(x,t) dt+b(x,t) dz Itô Process is a generalized Wiener process in which the parameters a and b are functions of the value of the underlying variable x and time t. dx=a(x,t) dt+b(x,t) dz The discrete time equivalent is only true in the limit as Dt tends to zero

12.3 THE PROCESS FOR STOCKS The assumption of constant expected drift rate is inappropriate and needs to be replaced by assumption that the expected reture is constant. This means that in a short interval of time,Δt, the expected increase in S is μSΔt. A stock price does exhibit volatility.

An Ito Process for Stock Prices where m is the expected return and s is the volatility. The discrete time equivalent is

EXAMPLE 12.3 Suppose m= 0.15, s= 0.30, then Consider a time interval of 1 week(0.0192)year, so that Dt =0.0192 ΔS=0.00288 S + 0.0416 S 17 17

MONTE CARLO SIMULATION MCS of a stochastic process is a procedure for sampling random outcome for the process. Suppose m= 0.14, s= 0.2, and Dt = 0.01 then The first time period(S=20 =0.52 ): DS=0.0014*20 +0.02*20*0.52=0.236 The second time period： DS'=0.0014*20.236 +0.02*20.236*1.44=0.611

MONTE CARLO SIMULATION – ONE PATH

12.4 THE PARAMETERS μ、σ We do not have to concern ourselves with the determinants of μin any detail because the value of a derivative dependent on a stock is, in general, independent of μ. We will discuss procedures for estimating σ in Chaper 13 .*.

12.5 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 ) dx=a(x,t)dt+b(x,t)dz Itô's lemma shows that a functions G of x and t follows the process

DERIVATION OF ITÔ'S LEMMA(1/2) IfDx is a small change in x and D G is the resulting small change in G Taylor series

DERIVATION OF ITÔ'S LEMMA(2/2) A Taylor's series expansion of G (x, t) gives 23 23

IGNORING TERMS OF HIGHER ORDER THAN DT

SUBSTITUTING FOR ΔX

THE E2ΔT TERM

APPLICATION OF ITO'S LEMMA TO A STOCK PRICE PROCESS

APPLICATION TO FORWARD CONTRACTS

THE LOGNORMAL PROPERTY We define： 29 29

THE LOGNORMAL PROPERTY The standard deviation of the logarithm of the stock price is 30 30