1、Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.1,Model of the Behavior of Stock Prices Chapter 10,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.2,C
2、ategorization of Stochastic Processes,Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10
3、.3,Modeling Stock Prices,We can use any of the four types of stochastic processes to model stock prices The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivative securities,Options, Futures, and Other Derivatives, 4th edition 2000 by John C.
4、 Hull Tang Yincai, 2003, Shanghai Normal University,10.4,Markov Processes,In a Markov process future movements in a variable depend only on where we are, not the history of how we got where we are We will assume that stock prices follow Markov processes,Options, Futures, and Other Derivatives, 4th e
5、dition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.5,Weak-Form Market Efficiency,The assertion is 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
6、 Markov process for stock prices is clearly consistent with weak-form market efficiency,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.6,Example of a Discrete Time Continuous Variable Model,A stock price is currently at $40
7、At the end of 1 year it is considered that it will have a probability distribution of f(40,10), where f(m,s) is a normal distribution with mean m and standard deviation s.,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.7,Que
8、stions,What is the probability distribution of the change in stock price over/during 2 years? years? years?Dt years?Taking limits we have defined a continuous variable, continuous time process,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Norma
9、l University,10.8,Variances & Standard Deviations,In Markov processes changes in successive periods of time are independent This means that variances are additive Standard deviations are not additive,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shangha
10、i Normal University,10.9,Variances & 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. (You can say that the STD is 10 per square root of years),Options, Future
11、s, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.10,A Wiener Process (See pages 220-1),We consider a variable z whose value changes continuously The change in a small interval of time Dt is Dz The variable follows a Wiener process if1. ,wher
12、e is a random drawing from (0,1). 2. The values of Dz for any 2 different (non- overlapping) periods of time are independent,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.11,Properties of a Wiener Process,Mean of z (T ) z (
13、0) is 0 Variance of z (T ) z (0) is T Standard deviation of z (T ) z (0) is,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.12,Taking Limits . . .,What does an expression involving dz and dt mean? It should be interpreted as
14、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,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.13,
15、Generalized Wiener Processes (See page 221-4),A Wiener process has a drift rate (ie average change per unit time) of 0 and a variance rate of 1 In a generalized Wiener process the drift rate & the variance rate can be set equal to any chosen constants,Options, Futures, and Other Derivatives, 4th edi
16、tion 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.14,Generalized Wiener Processes (continued),The variable x follows a generalized Wiener process with a drift rate of a & a variance rate of b2 if dx = a dt + b dz,Options, Futures, and Other Derivatives, 4th edition 2000 by J
17、ohn C. Hull Tang Yincai, 2003, Shanghai Normal University,10.15,Generalized Wiener Processes (continued),Mean 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,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull
18、Tang Yincai, 2003, Shanghai Normal University,10.16,The Example Revisited,A stock price starts at 40 & has a probability distribution of f(40,10) at the end of the year If we assume the stochastic process is Markov with no drift then the process is dS = 10dz If the stock price were expected to grow
19、by $8 on average during the year, so that the year-end distribution is f(48,10), the process is dS = 8dt + 10dz,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.17,Ito Process (See pages 224-5),In an Ito process the drift rate
20、 and the variance rate are functions of timedx=a(x,t) dt + b(x,t) dz The discrete time equivalentis only true in the limit as Dt tends to zero,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.18,Why a Generalized Wiener Proces
21、s is not Appropriate for Stocks,For a stock price we can conjecture that its expected proportional change in a short period of time remains constant We can also conjecture that our uncertainty as to the size of future stock price movements is proportional to the level of the stock price,Options, Fut
22、ures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.19,An Ito Process for Stock Prices (See pages 225-6),where m is the expected return, s is the volatility.The discrete time equivalent is,Options, Futures, and Other Derivatives, 4th edition
23、 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.20,Monte Carlo Simulation,We can sample random paths for the stock price by sampling values for e Suppose m= 0.14, s= 0.20, and Dt = 0.01, then,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai
24、, 2003, Shanghai Normal University,10.21,Monte Carlo Simulation One Path (continued. See Table 10.1),Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.22,Itos Lemma (See pages 229-231),If we know the stochastic process followed
25、 by x, Itos lemma tells us the stochastic process followed by some function G (x, t ) Since a derivative security is a function of the price of the underlying & time, Itos lemma plays an important part in the analysis of derivative securities,Options, Futures, and Other Derivatives, 4th edition 2000
26、 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.23,Taylor Series Expansion,A Taylors series expansion of G (x , t ) gives,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.24,Ignoring Terms of Higher Order Tha
27、n Dt,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.25,Substituting for Dx,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.26,The e2Dt Term,Options,
28、Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.27,Taking Limits,This is Itos Lemma.,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.28,Application of Itos Lemma to a Stock Price Process,Options, Futures, and Other Derivatives, 4th edition 2000 by John C. Hull Tang Yincai, 2003, Shanghai Normal University,10.29,Examples,
