维纳过程和伊藤引理--潘登同学的期权、期货及其他衍生品学习笔记
马尔可夫过程 Markov Process
马尔可夫过程是一种特殊的随机过程 Markov process is a special random process
- 变量的未来运动仅取决于我们当前所处的位置 The future movement of the variable depends only on where we are currently
- 如何到达当前位置与历史过程无关 It is irrelevant with how to get to the current location history process
股票价格的马尔可夫性质与弱型市场有效性是一致的 The Markovian nature of stock prices is consistent with weak-form market efficiency
- 股票的当前价格包含了过去价格的所有信息 The current price of a stock contains all the information about past prices
- 根据过去的股价历史交易规则，无法产生长期的超额回报 Unable to generate long-term excess returns based on past stock price history trading rules
- 技术分析不起作用 technical analysis is invalid
连续时间随机过程 Continuous Time Stochastic Process
- 在马尔可夫过程中，连续时间段的变化是独立的
In a Markov process, successive time periods change independently
- 方差可加/Variance can be added
- 标准差不可加/Standard deviation cannot be added
假设 Suppose
- 股票价格遵循马尔可夫过程
- Stock prices follow a Markov process
- 股票价格当前为 40 美元
- The stock price is currently $40
- 过程是静态的 (即描述过程的参数不会随着时间的推移而改变)
- The process is static (i.e. the parameters describing the process do not change over time)
- 在 1 年结束时，股票价格具有正态概率分布，均值为 \$40，标准差为 \$10 At the end of 1 year, stock prices have a normal probability distribution with mean \$40 and standard deviation \$10
问：2/0.5/0.25/∆𝑡 年末股票价格的概率分布是多少？ What is the probability distribution of stock prices at the end of 2/0.5/0.25/∆𝑡 year? $$ 根据题目可知，一年内变化分布为 \phi(0,100) \ 两年的变化分布为: \phi(0,200) \Rightarrow 两年末股价分布概率为： \phi(40,\sqrt{200})\ 0.5年的变化分布为: \phi(0,50) \Rightarrow 0.5年末股价分布概率为： \phi(40,\sqrt{50})\ 0.25年的变化分布为: \phi(0,25) \Rightarrow 0.25年末股价分布概率为： \phi(40,5)\ \triangle t年的变化分布为: \phi(0,100\triangle t) \Rightarrow 0.25年末股价分布概率为： \phi(40,\sqrt{100 \triangle t})\ $$
维纳过程 Wiener Process
维纳过程是一种均值为0，方差为每年1.0的特殊马尔可夫过程。
变量z具有以下两个性质时称为服从维纳过程：
- 性质1： 在一小段时间区间 $\triangle t$ 内的变化量$\triangle z$为 $$ \triangle z = \epsilon \sqrt{\triangle t}, \epsilon \sim \phi(0,1) $$
- 性质2:在任何两个不相重叠的 t时间区间内，变化量Z之间相互独立。
第一个性质说明Az本身服从正态分布，第二个性质说明变量z 服从马尔可夫过程。 $$ E(\triangle z) = 0 \ SD(\triangle z) = \sqrt{\triangle t} \ Var(\triangle z) = {\triangle t} \ $$
在一段较长时间内的变化$T=N*\triangle t$ $$ z(T) - z(0)=\sum_{i=1}^N\epsilon_i\sqrt{\triangle t} $$
$$ E[z(T)-z(0)]=0\ Var[z(T)-z(0)]=0\ E[z(T)-z(0)]=0\ $$
广义维纳过程 Generalized Wiener Process
- 维纳过程的漂移率 (单位时间内均值的变化) 为 0，方差率 (单位时间内的方差) 为 1 A Wiener process has a drift rate (change in the mean per unit time) of 0 and a variance rate (variance per unit time) of 1
- 在广义维纳过程中，漂移率和方差率可以设置为等于任何选定的常数 In a generalized Wiener process, the drift rate and variance rate can be set equal to any chosen constant $$ dx = adt + bdz \ \triangle x = a \triangle t + b\triangle z\ \triangle x = a \triangle t + b\epsilon \sqrt{\triangle z}\ $$ 可以将其分解为两部分，$adt$项说明变量x的单位时间漂移率为a。$bdz$项说明附加在变量x路径上的噪声或扰动，其幅度为维纳过程的b倍。
$$ E[\triangle x] = a\triangle t \ SD[\triangle x] = b\sqrt{\triangle t} \ Var[\triangle x] = b^2 \triangle t \ $$
在一段较长时间内的变化$T=N*\triangle t$ $$ E[\triangle x] = aT \ SD[\triangle x] = b\sqrt{T} \ Var[\triangle x] = b^2 T \ $$
- 期初的股票价格 40 美元，年底的概率分布为𝜙(40,100)
The stock price at the beginning of the period is \$40, and the probability distribution at the end of the period is 𝜙(40,100)
- 如果我们假设随机过程是没有漂移的马尔科夫过程 If we assume that the random process is a Markov process without drift $$ dS = 10dz $$
- 如果预测股票价格在年内平均增长 8 美元，年终分布为𝜙(48,100)，则随机过程表达为 If the stock price is predicted to increase by an average of \$8 during the year, with a year-end distribution of 𝜙(48,100), the stochastic process is expressed as $$ dS = 8dt + 10dz $$
伊藤过程 Itô Process
- 在 Itô process 中，漂移率和方差率是关于变量𝑥和时间𝑡的函数 In the Itôprocess, the drift rate and variance rate are functions of variables 𝑥 and time t $$ dx = a(x,t)dt + b(x,t)dz $$
对于离散时间过程 For a discrete time process $$ \triangle x = a(x,t)\triangle t + b(x,t)\epsilon \sqrt{\triangle t} $$
为什么广义维纳过程不适用于股票价格
Why the Generalized Wiener Process Doesn't Work for Prices?
- 股票价格 stock price
- 在短时间内的预期百分比变化保持不变(投资者的预期收益率保持不变，股价期望漂移率不一定不变) The expected percentage change over a short period of time remains constant
- 而不是预期的变化量 instead of the expected amount of change
- 对未来股价变动幅度的不确定性与股价水平成正比 Uncertainty about the magnitude of future stock price movements is directly proportional to the stock price level
写出股票价格变动的过程 $$ dS = \mu Sdt + \sigma Sdz \ \frac{dS}{S} = \mu dt + \sigma dz $$
- 其中 𝜇 是预期收益 𝜎 是波动率 where 𝜇 is the expected return and 𝜎 is the volatility
- 风险中性世界里，𝜇等于无风险利率 In a risk-neutral world, 𝜇 equals the risk-free rate
对于离散时间过程 For a discrete time process $$ \triangle S = \mu S\triangle t + \sigma S \epsilon \sqrt{\triangle t}\ \frac{\triangle S}{S} = \mu \triangle t + \sigma \epsilon \sqrt{\triangle t}\sim\phi(\mu\triangle t,\sigma^2\triangle t) $$
- 该过程被称为几何布朗运动 This process is called Geometric Brownian motion
伊藤引理 Itô’s Lemma
当知道变量 x 的随机过程，通过伊藤引理可以推断出函数$G(x,t)$的随机过程 When the random process of the variable 𝑥 is known, the random process of the function 𝐺(𝑥,𝑡) can be deduced by Ito's lemma
- 如果𝑥服从以下伊藤过程 If 𝑥 obeys the following Ito process $$ dx = a(x,t)dt+b(x,t)dz $$
- 其中， a(x,t) 为变量𝑥的漂移率， b(x,t) 为变量𝑥的方差率 Among them, 𝑎(𝑥,𝑡) is the drift rate of variable 𝑥, and 𝑏(𝑥,𝑡) is the variance rate of variable 𝑥
- 那么𝐺服从以下伊藤过程 Then 𝐺 obey the following Ito process $$ \begin{aligned} dG &= \frac{\partial{G}}{\partial{x}}dx+\frac{1}{2}\frac{\partial^2{G}}{\partial{x^2}}dx^2 + \frac{\partial{G}}{\partial{t}}dt+\frac{1}{2}\frac{\partial^2{G}}{\partial{t^2}}dt^2\ &=\frac{\partial{G}}{\partial{x}}(adt+bdz)+\frac{1}{2}\frac{\partial^2{G}}{\partial{x^2}}(adt+bdz)^2 + \frac{\partial{G}}{\partial{t}}dt+\frac{1}{2}\frac{\partial^2{G}}{\partial{t^2}}dt^2\ &=\frac{\partial{G}}{\partial{x}}(adt+bdz)+\frac{1}{2}\frac{\partial^2{G}}{\partial{x^2}}b^2dt + \frac{\partial{G}}{\partial{t}}dt \quad \text{(仅保留$d_t$和z,且$dz^2=dt$)}\ &=(\frac{\partial{G}}{\partial{x}}a+ \frac{\partial{G}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{G}}{\partial{x^2}}b^2)dt + \frac{\partial{G}}{\partial{x}}bdz \end{aligned} $$
𝐺的漂移率和方差率是什么？ What are the drift and variance rates for 𝐺? $$ \text{漂移率为：} \frac{\partial{G}}{\partial{x}}a+ \frac{\partial{G}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{G}}{\partial{x^2}}b^2 \ \text{方差为: } (\frac{\partial{G}}{\partial{x}})^2b^2 $$
假设股票价格符合以下伊藤过程 Assume that the stock price conforms to the following Ito process $$ dS = \mu Sdt+\sigma Sdz $$
- 𝐺(𝑆,𝑡)服从以下伊藤过程 𝐺(𝑆,𝑡) obeys the following Ito process $$ dG = (\frac{\partial{G}}{\partial{S}}\mu S+ \frac{\partial{G}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{G}}{\partial{S^2}}\sigma^2 S^2)dt + \frac{\partial{G}}{\partial{S}}\sigma Sdz $$
G的波动率与S的波动率满足如下关系 $$ \sigma_G G = \frac{\partial{G}}{\partial{S}}\sigma_S S $$
伊藤引理应用于远期合约
远期合约forward contract
- $S_0$为0时刻的即期价格，$F_0$为0时刻的远期价格， 𝑇为期限 $S_0$ is the spot price at time 0, $F_0$ is the forward price at time 0, and 𝑇 is the time $$ F_0 = S_0e^{rT} $$
- 𝑆为t时刻的即期价格，定义𝐹为t时刻的远期价格 𝑆 is the spot price at time t, define 𝐹 as the forward price at time t $$ F = Se^{r(T-t)} $$
- 根据伊藤引理 $$ \begin{aligned} dF &= (\frac{\partial{F}}{\partial S}\mu S+\frac{\partial{F}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{F}}{\partial S^2}\sigma^2 S^2)dt +\frac{\partial{F}}{\partial S} \sigma S dz\ &= (e^{r(T-t)}\mu S-rSe^{r(T-t)}+0)dt+e^{r(T-t)}\sigma S dz\ &= (\mu -r)Fdt + \sigma Fdz \ \end{aligned} \ \Rightarrow \frac{dF}{F} = (\mu -r) dt + \sigma dz $$
可知远期价格F也服从几何布朗运动，且与S具有相同的波动率，但期望增长率为$\mu-r$
股票价格服从对数正态分布
$$ \begin{cases} G = ln S \ dS = \mu Sdt + \sigma Sdz \end{cases} \ \Rightarrow \begin{aligned} dG &= (\frac{\partial{G}}{\partial{S}}\mu S + \frac{\partial{G}}{\partial{t}}+\frac{1}{2}\frac{\partial^2{G}}{\partial{S^2}}\sigma^2 S^2)dt + \frac{\partial{G}}{\partial{S}}\sigma Sdz\ &=(\frac{1}{S}\mu S + 0 + -\frac{1}{2S^2}\sigma^2 S^2)dt + \frac{1}{S}\sigma Sdz\ &=(\mu -\frac{1}{2}\sigma^2)dt + \sigma dz \end{aligned} $$
说明G服从一个广义维纳过程，其漂移率为常数$\mu -\frac{1}{2}\sigma^2$,方差率为常数$\sigma^2$,因此有 $$ ln S_t-ln S_0 \sim \phi[(\mu -\frac{1}{2}\sigma^2)T,\sigma^2T] \ ln S_t \sim \phi[ln S_0+(\mu -\frac{1}{2}\sigma^2)T,\sigma^2T] \ $$
说明股票价格服从对数正态分布。