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Wiener Processes and Itô’s LemmaChapter 121

Stochastic Processes A stochastic process describes the way a variable evolvesover time that is at least in part random. i.e., temperatureand IBM stock price. A stochastic process is defined by a probability law forthe evolution of a variable xt over time t. For given times,we can calculate the probability that the correspondingvalues x1,x2, x3,etc., lie in some specified range.2

Categorization of Stochastic ProcessesDiscrete time; discrete variableRandom walk: xt xt 1 ε tif ε t can only take on discrete values Discrete time; continuous variable xt a bxt 1 ε tε t is a normally distributed random variable with zeromean. Continuous time; discrete variable Continuous time; continuous variable3

Modeling Stock Prices We can use any of the four types of stochasticprocesses to model stock prices The continuous time, continuous variable processproves to be the most useful for the purposes ofvaluing derivatives4

Markov Processes (See pages 259-60) In a Markov process future movements in avariable depend only on where we are, not thehistory of how we got where we are. We assume that stock prices follow Markovprocesses.5

Weak-Form Market Efficiency This asserts that it is impossible to produceconsistently superior returns with a trading rulebased on the past history of stock prices. In otherwords technical analysis does not work. A Markov process for stock prices is consistentwith weak-form market efficiency6

Example of a Discrete TimeContinuous Variable Model A stock price is currently at 40 At the end of 1 year it is considered that it will havea normal probability distribution of with mean 40and standard deviation 107

Questions What is the probability distribution of thestock price at the end of 2 years? ½ years? ¼ years? t years?Taking limits we have defined a continuousvariable, continuous time process8

Variances & Standard DeviationsIn Markov processes changes in successive periodsof time are independent This means that variances are additive Standard deviations are not additive 9

Variances & Standard Deviations(continued) In our example it is correct to say that thevariance is 100 per year. It is strictly speaking not correct to say that thestandard deviation is 10 per year.10

A Wiener Process (See pages 261-63)We consider a variable z whose value changescontinuouslyDefine φ(µ,v) as a normal distribution with mean µand variance vThe change in a small interval of time t is zThe variable follows a Wiener process if z ε t where ε is ϕ (0,1)The values of z for any 2 different (non-overlapping)periods of time are independent11

Properties of a Wiener Processz (T ) z (0) n εi 1i tMean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is T 12

Taking Limits . . .dz ε dtWhat does an expression involving dz and dtmean? It should be interpreted as meaning that thecorresponding expression involving z and t istrue in the limit as t tends to zero In this respect, stochastic calculus is analogous toordinary calculus 13

Generalized Wiener Processes(See page 263-65) A Wiener process has a drift rate (i.e. averagechange per unit time) of 0 and a variance rate of 1 In a generalized Wiener process the drift rate andthe variance rate can be set equal to any chosenconstants14

Generalized Wiener Processes(continued)The variable x follows a generalized Wiener processwith a drift rate of a and a variance rate of b2 ifdx adt bdzor: x(t) x0 at bz(t)15

Generalized Wiener Processes(continued) x a t bε tMean change in x in time T is aT Variance of change in x in time T is b2T Standard deviation of change in x in time T is b T 16

The Example Revisited A stock price starts at 40 and has a probabilitydistribution of φ(40,100) at the end of the yearIf we assume the stochastic process is Markov with nodrift then the process isdS 10dzIf the stock price were expected to grow by 8 on averageduring the year, so that the year-end distribution isf(48,100), the process would bedS 8dt 10dz17

Whyb t?(1)It’s the only way to make the variance of(xT-x0)depend on T and not on the number ofsteps.1.Divide time up into n discrete periods oflength t, n T/ t. In each period thevariable x either moves up or down by anamount h with the probabilities of p and qrespectively. 18

Whyb t?(2)2.The distribution for the future values of x:E( x) (p-q) hE[( x)2] p( h)2 q(- h)2So, the variance of x is:E[( x)2]-[E( x)]2 [1-(p-q)2]( h)23. Since the successive steps of the random walk areindependent, the cumulated change(xT-x0)is a binomialrandom walk with mean:n(p-q) h T(p-q) h/ tand variance:n[1-(p-q)2]( h)2 T [1-(p-q)2]( h)2 / t19

Why b t?(3)When let t go to zero, we would like the mean andvariance of (xT-x0) to remain unchanged, and to beindependent of the particular choice of p,q, h and t. The only way to get it is to set: h b taa11 t ], q t ][1 [1 p 22bbaap q t 2 hbb20

Why b t?(4)When t goes to zero, the binomialdistribution converges to a normaldistribution, with meanand variancea ht 2 h atb ta 2 b 2 tt[1 ( ) t] b 2tb t21

Sample path(a 0.2 per year,b2 1.0 per year) Taking a time interval of one month, thencalculating a trajectory for xt using theequation:xt xt 1 0.01667 0.2887ε tA trend of 0.2 per year implies a trend of0.0167 per month. A variance of 1.0 per yearimplies a variance of 0.0833 per month, sothat the standard deviation in monthly termsis 0.2887.See Investment under uncertainty, p66

Forecast using generalized Brownian Motion Given the value of x(t)for Dec. 1974, X1974 ,the forecasted value of x for a time T monthsbeyond Dec. 1974 is given by:xˆ0.01667 x1974 1974 TTSee Investment under uncertainty, p67 In the long run, the trend is the dominantdeterminant of Brownian Motion, whereasin the short run, the volatility of the processdominates.

Why a Generalized Wiener Process IsNot Appropriate for Stocks The price of a stock never fall below zero. For a stock price we can conjecture that its expectedpercentage change in a short period of time remainsconstant, not its expected absolute change in a shortperiod of time We can also conjecture that our uncertainty as to thesize of future stock price movements is proportionalto the level of the stock price24

Itô Process (See pages 265) In an Itô process the drift rate and the variancerate are functions of time dx a( x, t)dt b( x, t)dztt00x (t) x0 a ( x, t ) ds b ( x, t ) dz The discrete time equivalent x a( x, t) t b( x, t)ε tis only true in the limit as t tends to zero25

An Ito Process for Stock Prices(See pages 269-71)dS µ S dt σ S dz where µ is the expected return, σ is thevolatility. The discrete time equivalent is S µ S t σ Sε t26

Monte Carlo Simulation We can sample random paths for the stock priceby sampling values for ε Suppose µ 0.15, σ 0.30, and t 1 week( 1/52 years), then S 0.00288S 0.0416Sε27

Monte Carlo Simulation – One Path (See Table12.1, page 268)WeekStock Price atStart of PeriodRandomSample for Change in StockPrice, 05.301.466.704112.00-0.69-2.8928

Itô’s Lemma (See pages 269-270) If we know the stochastic process followedby x, Itô’s lemma tells us the stochasticprocess followed by some function G (x, t ) Since a derivative is a function of the priceof the underlying and time, Itô’s lemmaplays an important part in the analysis ofderivative securities29

Taylor Series Expansion A Taylor’s series expansion of G(x, t) gives G G 2G 2 G x t ½ 2 x x t x 2G 2G 2 x t ? t 2 x t t30

Ignoring Terms of Higher Order Than tIn ordinary calculus we have G G G x t x tIn stochastic calculus this becomes G G 2G2 G x t ? x x t x2because x has a component which is of order t31

Substituting for xSuppose dx a(x ,t)dt b(x ,t)dzso that x a t b ε tThen ignoring terms of higher order than t G G 2G 2 2 G x t ?b ε t2 x t x32

The ε2 t TermSince ε ϕ (0,1) , E(ε ) 0E(ε 2 ) [ E(ε )]2 1E(ε 2 ) 1It follows thatE(ε 2 t) tThe variance of t is proportional to t2 and can be ignored.Hence, G G1 2G 2Gb t x t 2 x t2 x33

Taking LimitsTaking limits: G G 2G 2dG dx dt ?b dt2 x t xSubstituting:dx a dt b dz We obtain: G G 2G 2 GdG a bdtb dz ? 2 t x x x This is Ito'sˆ Lemma34

Application of Ito’s Lemmato a Stock Price ProcessThe stock price process is d S µ S dt σ S d zFor a function G of S and t G G 2G 2 2 G ?dGµS σ S dt σ S dz2 t S S S 35

Examples1. The forward price of a stock for a contractmaturing at time TG S e r (T t )(µ r )G dt σ G dzdG 2. G ln S σ2dG µ 2 dt σ dz 36

Ito’s Lemma for several Ito processes Suppose F F ( x1 , x2 ,., xm , t ) is a function oftime and of the m Ito process x1,x2, ,xm,wheredxi ai ( x1 , x2 ,., xm , t ) dt bi ( x1 , x2 ,., xm , t ) dzi , i 1,2,., mwith E ( dzidz j ) ρijdt Then Ito’s Lemma gives the differential dF as F F1 2 FdF dt dxi dxidx j t2 i j xi x ji xi37

ExamplesSuppose F(x,y) xy, where x and y each follow geometricBrownian motions: dx ax xdt bx xdz x dy a y ydt by ydz ywith E ( dzidz j ) ρijdt . What’s the process followed by F(x,y) and by G logF? dF xdy ydx dxdy ( ax a y ρbxby ) Fdt ( bx dz x by dz y ) F1 2 1 2 dG ax a y bx by dt bx dz x by dz y22 38

Categorization of Stochastic Processes Discrete time; discrete variable Random walk: if can only take on discrete values Discrete time; continuous variable

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