第13章 维纳过程和伊藤引理

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13.2 连续时间随机过程
Example




A variable is currently 40 It follows a Markov process Process is stationary (i.e. the parameters of the process do not change as we move through time) At the end of 1 year the variable will have a normal probability distribution with mean 40 and standard deviation 10
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13.2.1 标准维纳过程 A Wiener Process


Define f(m,v) as a normal distribution with mean m and variance v A variable z follows a Wiener process if


Dt内的变化
The change in z in a small interval of time Dt is Dz

T 内的变化 Dz Mean of [z (T ) – z (0)] is 0 Variance of [z (T ) – z (0)] is T Standard deviation of [z (T ) – z (0)] is
T
在Excel中的模拟。
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3.5 3 2.5 2 1.5
随机过程(stochastic [sto'kæstɪk] process)的 含义
随机变量 静态 随机向量 随机序列 动态 随机过程


每时每刻都是一个随机变量。按时间顺序排列的 随机变量集。 随机变量服从于某分布，随机过程遵循某种过程。
随机过程空间）
离散时间(discrete time) 连续时间(continuous time)
陕西师范大学 国际商学院
曹培慎
Chapter 13 Wiener Processes and Itô’s Lemma 维纳过程和伊藤引理
Options, Futures, and Other Derivatives, 8th Edition, Copyright © John C. Hull 2012
13.2.2 广义维纳过程 Generalized Wiener Processes


漂移率 方差率 A Wiener process has a drift rate (i.e. average change per unit time) of 0 and a variance rate of 1 In a generalized Wiener process the drift rate and the variance rate can be set equal to any chosen constants
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The Example Revisited



A stock price starts at 40 and has a probability distribution of f(40,100) 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(48,100), the process would be dS = 8dt + 10dz
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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
dx adt bdz
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Generalized Wiener Processes
(continued)
Dx a Dt b Dt

Mean change in x per unit time is a Variance of change in x per unit time is b2
1
0.5 0
0 -0.5
0.2
0.4
0.6
0.8
1
1.2
3
2
1
0 0 -1 0.2 0.4 0.6 0.8 1 1.2
-2
-3
重要的一点
(Dz) 的期望:
2
根据 D(X)=E ( X ) ( EX )
2 2
所以 E ((Dz) ) D(Dz ) ( E ( Dz )) Dt
2 2
例13-1 假定随机变量遵循维纳过程，其初始 值为25，时间以年为单位。在1年末，变量值 服从正态分布，其期望值为25，标准差为1.在 5年末，变量服从正态分布，其期望值为25， 标准差为 5 。 变量在将来某一确定时刻由标准差来定义不 确定性，并且与未来时间长度的平方根成正 比。
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Taking 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 zero In this respect, stochastic calculus is analogous to ordinary calculus
Dx a( x, t )Dt b( x, t ) Dt is true in the limit as Dt tends to zero
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Why a Generalized Wiener Process Is Not Appropriate for Stocks


For a stock price we can conjecture that its expected percentage change in a short period of time remains constant (not its expected actual change) 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
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人们在实际中常遇到具有下述特性的随机过程：在已知它 所处的状态的条件下，它未来的演变不依赖于它以往的演 变。这种已知“现在”的条件下，“将来”与“过去”独 立的特性称为马尔可夫性，具有这种性质的随机过程叫做 马尔可夫过程。 荷花池中一只青蛙的跳跃是马尔可夫过程的一个形象化的 例子。青蛙依照它瞬间或起的念头从一片荷叶上跳到另一 片荷叶上，因为青蛙是没有记忆的，当所处的位置已知时， 它下一步跳往何处和它以往走过的路径无关。如果将荷叶 编号并用X0,X1,X2，…分别表示青蛙最初处的荷叶号码及 第一次、第二次、……跳跃后所处的荷叶号码，那么{Xn， n≥0} 就是马尔可夫过程。
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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.


按变量（状态空间）
离散变量(discrete variable) 连续变量(continuous variable)

按具有的性质


本章将建立关于股票价格的连续变量、连续时间 的随机过程模型。 定价中的一个重要原理：伊藤引理(Ito`s Lemma)
Stochastic Processes
Weak-Form Market Efficiency


This 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-form market efficiency

Each day a stock price

increases by $1 with probability 30% stays the same with probability 50% reduces by $1 with probability 20%