Model of the Behavior of Stock Prices 股票价格行为模型

1. Model of the Behaviorof Stock Prices股票价格行为模型 Chapter 11

2. Introduction简介 Any variable whose value changes over time in an uncertain way is said to follow a stochastic process. 如果某一变量的价值以某种不确定的方式随时间变化，这一变量被称为服从某种随机过程。 Discrete time vs. continuous time A discrete-time stochastic process is one where the value of the variable can change only at certain fixed points in time A continuous-time stochastic process is one where changes can take place at any time.

3. Introduction简介 Discrete time vs. continuous time 离散时间vs连续时间 A discrete-time stochastic process is one where the value of the variable can change only at certain fixed points in time 一个离散时随机过程是指标的变量值只能在某些确定的时间点上变化 A continuous-time stochastic process is one where changes can take place at any time. 连续时间随机过程是指标的变量值可以在任何时刻上变化

4. Introduction简介 Continuous variable vs. discrete variable 连续变量vs离散变量 In a continuous variable process, the underlying variable can take any value within a certain range 在连续变量过程中标的变量可以在某一范围内取任意值 In a discrete-variable process, only certain discrete values are possible. 在离散变量过程中，标的变量只能取得某些离散值

5. Categorization of Stochastic Processes随机过程的分类 Discrete time; discrete variable 离散时间，离散变量 Discrete time; continuous variable 离散时间，连续变量 Continuous time; discrete variable 连续时间，离散变量 Continuous time; continuous variable 连续时间，连续变量

6. Modeling Stock Prices股票价格建模 We can use any of the four types of stochastic processes to model stock prices 我们可以应用四种随机过程的任一种对股票价格进行建模 The continuous time, continuous variable process proves to be the most useful for the purposes of valuing derivatives 连续时间，连续变量过程对于衍生品定价是最有用的模型

7. Markov Processes马尔科夫过程 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 马尔科夫构成中，标的变量的为了预测只依赖于当前值，与变量从过去到现在的演变方式无关 A Markov process is a particular type of stochastic process where only the present value of a variable is relevant for predicting the future. 马尔科夫过程是一个特殊类型的随机过程，只有标的变量的当前值与未来的预测有关。 We assume that stock prices follow Markov processes 我们假定股票价格变动符合马尔科夫过程

8. Weak-Form Market Efficiency弱型市场有效性 Weak-Form Market Efficiency asserts that it is impossible to produce consistently superior returns with a trading rule based on the past history of stock prices. 弱型市场有效性认为基于股票价格的历史数据交易不可能获得高于平均收益率的收益 In other words technical analysis does not work. 换言之，技术分析无效 A Markov process for stock prices is clearly consistent with weak-form market efficiency 股票价格的马尔科夫与弱型市场有效性一致 Efficiency resource有效性来源: competition竞争

9. Example of a Discrete Time Continuous Variable Model离散时间连续变量模型举例 A stock price is currently at $10 and suppose it follows a Markov stochastic process. 股票当前价格为$10，服从马尔科夫随机过程 The change in its value during one year is f(0,1) 变量一年间的变化服从f(0,1) Where f(0,1) is a normal distributionwith mean 0 and standard deviation 1. f(0,1)为均值为0，标准差为1的正态分布 What is the probability distribution of the stock price at the end of 2 years? 股票价格在第二年末的概率分布？

10. Example of a Discrete Time Continuous Variable Model离散时间连续变量模型举例 The change in two years is the sum of two normal distributions. 变量在两年内的变化等于两个正态分布的和 The mean is the sum of means and the variance is the sum of variances 均值为独立变量均值之和，方差为独立变量方差之和 The change in the stock price over two years is therefore f(0,sqrt(2)) 股票价格在两年间的变动符合f(0,sqrt(2))

11. Variances & Standard Deviations 方差&标准差 In Markov processes changes in successive periods of time are independent 马尔科夫过程中，相邻时间区间的变化是独立的 This means that variances are additive 这意味着方差具有可加性 Standard deviations are not additive 标准差不具有可加性

12. Questions 问题 ½ years? ¼ years? Dt years? Taking limits we have defined a continuous variable, continuous time process 取极限，我们定义了连续变量，连续时间过程

13. A Wiener Process维纳过程 The process above is known as a Wiener process 这里变量所服从的过程为维纳过程 It is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1 per year. 维纳过程是马尔科夫过程中变化的期望值为0，方差为1的特殊形式 It is sometimes referred to as Brownian motion. 有时被称为布朗运动 13

14. A Wiener Process维纳过程 Formal definition正式定义 We consider a variable z whose value changes continuously 变量z连续变化 The change in a small interval of time Dt is Dz在Dt内变量变化为Dz The variable follows a Wiener process if The values of Dz for any 2 different (non-overlapping) periods of time are independent, that is, it follows a Markov process. 14

15. A Wiener Process维纳过程 The variable follows a Wiener process if 当变量服从以下两个性质时服从维纳过程 The values of Dz for any 2 different (non-overlapping) periods of time are independent, that is, it follows a Markov process. 对于任意两个不同时间间隔，变化量Dz相互之间独立 15

16. Properties of a Wiener Process维纳过程的性质 Mean of Dz is 0 Dz均值为0 Variance of Dz is Dt Dz的方差为Dt Standard deviation of Dz is Dz的标准差为

17. Properties of a Wiener Process维纳过程的性质 The change of z during a relative long period of time T can be denoted by z (T ) – z (0)= 一段相对较长的时间段T内z的变化可以用z (T ) – z (0)= 表示

18. Properties of a Wiener Process维纳过程的性质 Mean of [z (T ) – z (0)] is 0 [z (T ) – z (0)]的均值为0 Variance of [z (T ) – z (0)] is NΔt=T [z (T ) – z (0)]的方差为NΔt=T Standard deviation of [z (T ) – z (0)] is [z (T ) – z (0)]的标准差为

19. Example Suppose that the value, z, of a variable follows a Wiener process, is initially 25 and the time is measured in years. At the end of one year, the value of the variable is normally distributed with a mean of 25 and a standard deviation of 1.0. At the end of five years, it is normally distributed with a mean of 25 and a standard distribution of

20. 举例 变量z服从维纳过程, 其初始值为25，时间以年为计量. 在一年末，变量值服从正态分布，期望值为25，标准差为1.0.在第五年末，变量值服从正态分布，期望值为25，标准差为

21. Taking Limits . . .取极限。。。 What does an expression involving dz and dt mean?包含dz 和dt的表达式意味着什么？ It should be interpreted as meaning that the corresponding expression involving z and t is true in the limit as t tends to zero dz代表当t趋近为0时， z 的极限形式， dt为当t趋近为0时， t的极限形式 In this respect, stochastic calculus is analogous to ordinary calculus 在这一方面，随机微分与普通微分相对应

22. Generalized Wiener Processes广义维纳过程 A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1 维纳过程的漂移率（单位时间内变量变化的均值）为0，方差率为1 In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants 广义维纳过程中漂移率和方差率可以为任意给定常数

23. Generalized Wiener Processes广义维纳过程(continued) The variable x follows a generalized Wiener process with a drift rate of a and a variance rate ofb2 if dx=adt+bdz 若满足dx=adt+bdz，则变量x服从漂移率为a，方差率为b2的广义维纳过程

24. Generalized Wiener Processes(continued)广义维纳过程 Mean change in x in time Tis aT x的变化的均值为aT Variance of change in x in timeTis b2T x的变化的方差为b2T Standard deviation of change in xin time Tis x的变化的标准差为

25. Example A stock price starts at 40 and 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 by $8 on average during the year, so that the year-end distribution is f(48,10), the process is dS = 8dt + 10dz

26. 举例 股票当前价格为40 ，年末概率分布为f(40,10) 我们假定股票价格遵循无漂移率的马尔科夫随机过程 dS = 10dz 股票价格在该年内增长的期望值为$8，年末概率分布为f(48,10)，则服从以下关系式 dS = 8dt + 10dz

27. Ito Process伊藤过程 In an Ito process the drift rate and the variance rate are functions of timedx=a(x,t)dt+b(x,t)dz 伊藤过程中漂移率和方差率均为时间的函数， dx=a(x,t)dt+b(x,t)dz The discrete time equivalent is only true in the limit as t tends to zero离散时间过程 仅在t趋近于0时成立

28. Why a Generalized Wiener Process is not Appropriate for Stocks Price? For 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 time 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 In other words, the variability of the percentage return in a short period of time, Δt, is the same regardless of the stock price.

29. 为什么股票价格变动不适用广义维纳过程？ 我们假定股票价格的预期变化率在一个短的时间段内保持恒定，而不是预期绝对变化量在短时间段内保持恒定 我们假定未来股票价格变动的标准差与股票价格水平成正比 换句话讲，无论股票价格为多少，在一个较短的时间Δt后，股票价格的百分比收益的波动率为常数

30. An Ito Process for Stock Prices股票价格的伊藤过程 Where m is the expected return m 为预期收益率 s is the volatility of the stock price. s 为股票价格波动率 The discrete time equivalent is 离散形式为

31. Example 11.3 P223 Consider a stock that pays no dividends, has a volatility of 30% per annum, and provides an expected return of 15% per annum with continuous compounding. In this case, μ=0.15, σ=0.30.某无股息股票，波动率为每年30%，连续复利预期收益率为15%，这时μ=0.15, σ=0.30 The process for the stock price is 股票价格过程为

32. Example 11.3 P223 Consider a time interval of one week, or 0.0192 years, and suppose that the initial stock price is $100. Then Δt=0.0192,S=100 and 假定时间间隔为一个星期，即0.0192年，股票价格的初始值为100美元，因此Δt=0.0192,S=100 ΔS=100(0.15*0.0192+0.0416ε) =0.288+4.16ε Showing that the price increase is a random drawing from a normal distribution with mean 0.288 and std 4.16.上式说明股票价格增量服从正态分布，期望值为0.288，标准差为4.16

33. Monte Carlo Simulation蒙特卡罗模拟法 We can sample random paths for the stock price by sampling values for e 通过对e抽样来模拟股票价格的变换路径 Suppose m= 0.14, s= 0.20, and t = 0.01, then

34. Monte Carlo Simulation – One Path

35. Monte Carlo Simulation蒙特卡罗模拟法 Different random samples would lead to different price movements. 不同的随机抽样会产生不同的价格运动 In the limit as t0, a perfect description of the stochastic process is obtained. 在t0的极限状态，可以取得对于布朗运动的完美描述 By repeatedly simulating movements in the stock price, a complete probability distribution of the stock price at the end of the time is obtained.反复模拟股票价格变动，可以得出股票价格的一个完整的概率分布

36. Ito’s Lemma 伊藤引理 If we know the stochastic process followed by x, Ito’s 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 and time, Ito’s lemma plays an important part in the analysis of derivative securities

37. Taylor Series Expansion泰勒级数展开 A Taylor’s series expansion of G(x, t) gives 运用G(x, t) 的泰勒展开式

38. Ignoring Terms of Higher Order Than Dt 忽略比Dt高阶的项 38

39. Substituting for Dx 39

40. The e2Dt Term 40

41. Taking Limits 取极限 41

42. Application of Ito’s Lemmato a Stock Price Process 股票价格遵循伊藤引理的应用 42

43. Examples举例 43