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1、Wiener Processes and Its LemmaChapter 121Types of Stochastic ProcessesDiscrete time;discrete variableDiscrete time;continuous variableContinuous time;discrete variableContinuous time;continuous variable2Modeling Stock PricesWe can use any of the four types of stochastic processes to model stock pric
2、esThe continuous time,continuous variable process proves to be the most useful for the purposes of valuing derivatives3Markov Processes(See pages 259-60)In a Markov process future movements in a variable depend only on where we are,not the history of how we got where we areWe assume that stock price
3、s follow Markov processes4Weak-Form Market EfficiencyThis asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices.In other words technical analysis does not work.A Markov process for stock prices is consistent with weak-for
4、m market efficiency5Example of a Discrete Time Continuous Variable ModelA stock price is currently at$40At the end of 1 year it is considered that it will have a normal probability distribution of with mean$40 and standard deviation$106QuestionsWhat is the probability distribution of the stock price
5、 at the end of 2 years?years?years?Dt years?Taking limits we have defined a continuous variable,continuous time process7Variances&Standard DeviationsIn Markov processes changes in successive periods of time are independentThis means that variances are additiveStandard deviations are not additive8Var
6、iances&Standard Deviations(continued)In our example it is correct to say that the variance is 100 per year.It is strictly speaking not correct to say that the standard deviation is 10 per year.9A Wiener Process(See pages 261-63)We consider a variable z whose value changes continuously Define f(m,v)a
7、s a normal distribution with mean m and variance vThe 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 independent10(0,1)is where fDDtzProperties of a Wiener ProcessMean of z(T)z(0)is 0Varian
8、ce of z(T)z(0)is TStandard deviation of z(T)z(0)is11TTaking 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 zeroIn this respect,stochastic calculus is analogous to or
9、dinary calculus12Generalized Wiener Processes(See page 263-65)A Wiener process has a drift rate(i.e.average change per unit time)of 0 and a variance rate of 1In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants13Generalized Wiener Processes(co
10、ntinued)The variable x follows a generalized Wiener process with a drift rate of a and a variance rate of b2 if dx=a dt+b dz 14Generalized Wiener Processes(continued)Mean change in x in time T is aTVariance of change in x in time T is b2TStandard deviation of change in x in time T is 15tbtaxDDD b TT
11、he Example RevisitedA stock price starts at 40 and has a probability distribution of f(40,100)at the end of the yearIf we assume the stochastic process is Markov with no drift then the process is dS =10dz If the stock price were expected to grow by$8 on average during the year,so that the year-end d
12、istribution is f(48,100),the process would be dS =8dt +10dz16 It Process(See pages 265)In an It process the drift rate and the variance rate are functions of time dx=a(x,t)dt+b(x,t)dzThe discrete time equivalent is only true in the limit as Dt tends to zero17ttxbttxaxDDD),(),(Why a Generalized Wiene
13、r Process Is Not Appropriate for StocksFor a stock price we can conjecture that its expected percentage change in a short period of time remains constant,not its expected absolute change in a short period of timeWe can also conjecture that our uncertainty as to the size of future stock price movemen
14、ts is proportional to the level of the stock price18An Ito Process for Stock Prices(See pages 269-71)where m is the expected return s is the volatility.The discrete time equivalent is19 dzSdtSdSsm tStSSDsDmDMonte Carlo SimulationWe can sample random paths for the stock price by sampling values for S
15、uppose m=0.15,s=0.30,and Dt=1 week(=1/52 years),then20SSS0416.000288.0 DMonte Carlo Simulation One Path(See Table 12.1,page 268)21Its Lemma(See pages 269-270)If we know the stochastic process followed by x,Its lemma tells us the stochastic process followed by some function G(x,t)Since a derivative i
16、s a function of the price of the underlying and time,Its lemma plays an important part in the analysis of derivative securities22Taylor Series ExpansionA Taylors series expansion of G(x,t)gives23DDDDDDD2222222ttGtxtxGxxGttGxxGG Ignoring Terms of Higher Order Than D Dt24txxxGttGxxGGttGxxGGDDDDDDDDD order of is whichcomponent a has because becomes this calculus stochastic In have wecalculusordinary In222Substituting for D Dx25tbxGttGxxGGttbtaxdztxbdttxadxDDDDDDDD2222 than order higher of terms ign
