Chapter 13 第十三章
Wiener Processes and Itô’s
Lemma
维纳过程和伊藤引理
1
Stochastic Processes 随机过程
Describes the way in which a variable such as a
stock price, exchange rate or interest rate changes
through time
描述变量(例如股价、汇率、利率)随时间变化的方
式。
Incorporates uncertainties
它包含不确定性
2
Example 1 例子1
Each day a stock price
每天股票价格
increases by $1 with probability 30%
以30%概率上涨1美元
stays the same with probability 50%
以50%的概率保持不变
reduces by $1 with probability 20%
以20%的概率下降1美元
3
Example 2 例子2
Each day a stock price change is drawn from a
normal distribution with mean $0.2 and standard
deviation $1
每天股票价格变动由一个均值为0.2美元,标准差为1
美元的正态分布刻画。
4
5
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 to where
we are
在一个马尔科夫过程里,一个变量未来的路径只依
赖于现在的位置,而不依赖于历史路径。
Is the process followed by the temperature at a certain place
Markov?
某个地方的温度服从马尔科夫过程吗
We assume that stock prices follow Markov processes
我们假设股票价格服从马尔科夫过程
6
Weak-Form Market Efficiency
市场弱式有效
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.
这
表
关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf
明不可能利用基于历史股价的交易规则来获取持续的超额
收益
A Markov process for stock prices is consistent with weak-form
market efficiency
股票价格服从马尔科夫过程与市场弱式有效是一致的。
7
Example 例子
A variable is currently 40
一个变量当期是40
It follows a Markov process
它服从马尔科夫过程
Process is stationary (i.e. the parameters of the process do
not change as we move through time)
该过程是平稳的(即随机过程的参数不随时间变化)
At the end of 1 year the variable will have a normal probability
distribution with mean 40 and standard deviation 10
一年末,该变量服从均值为40,标准差为10的正态分布
8
Questions 问
题
快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题
What is the probability distribution of the
stock price at the end of 2 years?
第二年末该股票的概率分布是什么?
½ years? ½年?
¼ years? ¼年?
Dt years? Dt年?
Taking limits we have defined a continuous
stochastic process
取极限后我们可以定义一个连续的随机过程
9
Variances & Standard Deviations
方差和标准差
In Markov processes changes in successive
periods of time are independent
马尔科夫过程里,变量在相邻时间段里的变化是独
立的
This means that variances are additive
这意味着方差是可加的
Standard deviations are not additive
标准差是不可加的
10
Variances & Standard Deviations
方差和标准差(续)
In our example it is correct to say that the variance
is 100 per year.
在我们的例子里,方差每年为100。
It is strictly speaking not correct to say that the
standard deviation is 10 per year.
严格来说,标准差每年为10是错误的
11
A Wiener Process 维纳过程
Define f(m,v) as a normal distribution with mean
m and variance v
定义f(m,v) 为均值为m,方差为v 的正态分布
A variable z follows a Wiener process if
一个变量z服从维纳过程如果满足如下条件
The change in z in a small interval of time Dt is Dz
在一段很短时间Dt 内,z的变化为Dz
The values of Dz for any 2 different (non-overlapping)
periods of time are independent
Dz的值在任意不同两期(不重叠)是独立的。
(0,1) is where fDD tz
12
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 T
[z (T ) – z (0)]的方差是T
Standard deviation of [z (T ) – z (0)] is
[z (T ) – z (0)]的标准差是
T
T
13
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(即变量每单位时间的平均变
化)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
在广义维纳过程中,漂移率和方差率可以为任
意常数。
14
Generalized Wiener Processes (continued)
广义维纳过程(续)
Mean change in x per unit time is a
单位时间x 的平均变化为a
Variance of change in x per unit time is b2
单位时间x变化的方差为b2
tbtax DDD
15
Taking Limits . . .取极限
What does an expression involving dz and dt mean?
包含dz和dt的表达式如何理解 ?
It should be interpreted as meaning that the corresponding
expression involving Dz and Dt is true in the limit as Dt
tends to zero
可以解释为当Dt趋向0时,包含Dz 和Dt的相应表达式是正确的
In this respect, stochastic calculus is analogous to ordinary
calculus
随机微积分可以类比于普通微积分
16
Taking Limits . . .取极限
17
The Example Revisited 回到上面例子
A stock price starts at 40 and has a probability distribution of
f(40,100) at the end of the year
股票价格开始为40美元,年末股票价格服从f(40,100) 的正态
分布
If we assume the stochastic process is Markov with no drift
then the process is
如果我们假设该过程为没有漂移率的马尔科夫过程,那么
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
如果股票价格在一年内预期涨8美元,那么在年末股票价格的
分布为f(48,100),那么
dS = 8dt + 10dz
18
Generalized Wiener Processes (continued)
广义维纳过程(续)
19
Itô Process 伊藤过程
In an Itô process the drift rate and the variance
rate are functions of time
伊藤过程的漂移率和方差率是时间的函数
dx=a(x,t) dt+b(x,t) dz
The discrete time equivalent is true in the limit as
Dt tends to zero
当Dt 趋于0时,离散时间相应表达式是正确的
ttxbttxax DDD ),(),(
20
Why a Generalized Wiener Process Is Not
Appropriate for Stocks
为什么广义维纳过程对股票不合适
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)
对于股票价格,我们可以猜测在短期内股票价格百分比变化的期望
保持不变(不是实际价格变化的期望)
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
我们同样可以猜测我们对未来股票价格变动的不确定性与股票价格
水平成比例
21
An Ito Process for Stock Prices
股票价格的伊藤过程
where m is the expected return s is the volatility.
这里m是期望收益, s是波动率
The discrete time equivalent is
离散时间相应表达式为
The process is known as geometric Brownian
motion
这个过程称作为几何布朗运动
dzSdtSdS sm
tStSS DsDmD
Interest Rates 利率
What would be a reasonable stochastic process to
assume for the short-term interest rate?
应该假设短期利率服从怎样一个合理的随机过
程呢?
22
23
Monte Carlo Simulation
蒙特卡洛模拟
We can sample random paths for the stock price
by sampling values for
我们可以通过对值的随机取样来为股票价格的路径
随机取样
Suppose m= 0.15, s= 0.30, and Dt = 1 week (=1/52
or 0.192 years), then
假设m= 0.15, s= 0.30, Dt = 1 周( =1/52 年)
D
SSS
ε ..S..ΔS
04160002880
or
0192030001920150
..
24
Monte Carlo Simulation – Sampling one Path
蒙特卡洛模拟---对一条路径取样
Week
Stock Price at
Start of Period
Random
Sample for
Change in Stock
Price, DS
0 100.00 0.52 2.45
1 102.45 1.44 6.43
2 108.88 −0.86 −3.58
3 105.30 1.46 6.70
4 112.00 −0.69 −2.89
Correlated Processes 相关过程
Suppose dz1 and dz2 are Wiener processes with
correlation r
假设dz1 和dz2 维纳过程,它们的相关性为 r
25
r
DD
DD
is ncorrelatio w hereondistributi
normal standard bivariate a from
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22
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26
Itô’s Lemma 伊藤引理
If we know the stochastic process followed by x,
Itô’s lemma tells us the stochastic process
followed by some function G (x, t )
如果我们知道x服从的随机过程,那么伊藤引
理能告诉我们G (x, t )所服从的随机过程
Since a derivative is a function of the price of
the underlying asset and time, Itô’s lemma
plays an important part in the analysis of
derivatives
因为衍生品价格是标的资产价格和时间的函
数,伊藤引理在
分析
定性数据统计分析pdf销售业绩分析模板建筑结构震害分析销售进度分析表京东商城竞争战略分析
衍生品中扮演重要的角色。
Taylor Series Expansion
泰勒展开
A Taylor’s series expansion of G(x, t) gives
G(x, t)的泰勒展开得到
27
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Ignoring Terms of Higher Order Than Dt
忽略Dt的高阶项
28
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29
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30
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Taking Limits 取极限
31
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Application of Ito’s Lemma to a Stock Price
Process 将伊藤引理应用于股票随机过程
32
dzS
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22
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Examples 例子
33
1. The forward price of a stock for a contract
maturing at time T
Examples 例子
34
2. The log of a stock price
lnG S