Chapter 14: Wiener Processes and Itô's Lemma

The Markov Property

A Markov process is one where only the current value of a variable is relevant for predicting the future — the path history is irrelevant. Stock prices are generally assumed to follow Markov processes, consistent with the weak form of market efficiency.

Continuous-Time Stochastic Processes

Wiener Process (Brownian Motion)

A Wiener process $z(t)$ has two properties:

1. $\Delta z=\epsilon \sqrt{\Delta t}$, where $\epsilon \sim N(0,1)$
2. The values of $\Delta z$ for any two non-overlapping intervals are independent

This means:

• $E[\Delta z]=0$
• $\text{Var}(\Delta z)=\Delta t$
• $\text{Var}(z(T)-z(0))=T$ (variance grows linearly with time)

Generalized Wiener Process

$dx=adt+bdz$

where:

• $a$ = drift rate (expected change per unit time)
• $b$ = variance rate (standard deviation of change per unit time)

$\Delta x=a\cdot \Delta t+b\cdot \epsilon \sqrt{\Delta t}$

Itô Process

Allows $a$ and $b$ to depend on $x$ and $t$:

$dx=a(x,t)dt+b(x,t)dz$

The Process for a Stock Price

Stock prices follow geometric Brownian motion:

$dS=\mu Sdt+\sigma Sdz$

or equivalently:

$\frac{dS}{S}=\mu dt+\sigma dz$

• $\mu$ = expected rate of return (annualized)
• $\sigma$ = volatility (annualized standard deviation of returns)
• In a risk-neutral world, $\mu$ = $r$ (the risk-free rate)

Discrete-time approximation:

$\frac{\Delta S}{S}\sim N(\mu \Delta t,{\sigma }^2\Delta t)$

Itô's Lemma

Itô's lemma is the stochastic calculus equivalent of the chain rule. If $x$ follows an Itô process and $G(x,t)$ is a function of $x$ and $t$:

$dG=\left(\frac{\partial G}{\partial x}a+\frac{\partial G}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}G}{\partial x^2}{b}^2\right)dt+\frac{\partial G}{\partial x}bdz$

Applied to stock prices ($dS=\mu Sdt+\sigma Sdz$):

$dG=\left(\frac{\partial G}{\partial S}\mu S+\frac{\partial G}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}G}{\partial S^2}{\sigma }^2{S}^2\right)dt+\frac{\partial G}{\partial S}\sigma Sdz$

For $G=\ln S$ (a commonly used transformation):

$d(\ln S)=\left(\mu -\frac{{\sigma }^2}{2}\right)dt+\sigma dz$

This implies that $\ln S$ follows a generalized Wiener process with drift $\mu -{\sigma }^2/2$ and variance rate ${\sigma }^2$.

The Lognormal Property

From Itô's lemma applied to $\ln S$, we conclude:

$\ln S_T-\ln S_0\sim N\left[\left(\mu -\frac{{\sigma }^2}{2}\right)T,{\sigma }^{2}T\right]$

or:

$\ln S_T\sim N\left[\ln S_0+\left(\mu -\frac{{\sigma }^2}{2}\right)T,{\sigma }^{2}T\right]$

Expected stock price:

$E[S_T]=S_0\cdot e^{\mu T}$

Expected continuously compounded return (log return):

$E\left[\ln \left(\frac{S_T}{S_0}\right)\right]=\left(\mu -\frac{{\sigma }^2}{2}\right)T$

Note: Expected log return $\ne$ log of expected return.

Fractional Brownian Motion

Fractional Brownian motion (fBm) generalizes standard Brownian motion by allowing for correlation between increments. The Hurst exponent $H$ characterizes the process:

• $H=0.5$: Standard Brownian motion (independent increments)
• $H>0.5$: Positive correlation between increments (persistent/trending)
• $H<0.5$: Negative correlation (mean-reverting)

fBm with $H<0.5$ has been found useful for modeling volatility (rough volatility models).