Chapter VI - Basic Stochastic Calculus & the Black-Scholes PDE

Basic Stochastic Calculus & the Black-Scholes PDE

Section 1 - Brownian Motion and Integration

Definition 5

A quasi-formal definition of a Brownian Motion (BM)

An ${\mathbb{R}}^n$ - valued stochastic process $\{W_t:t\in [0,+\infty )\}$ on a probability space $(\Omega ,\mathcal{F},P)$ is an n-dimensional Brownian motion when:

• $W_0=0$ (not strictly necessary, but we can always shift the process to make it start at 0)
• the functions $t\to W_t(\omega )$ are continuous for each $\omega \in \Omega$ (i.e., the paths are continuous)
• for each $s,t\in [0,+\infty )$ with $s<t$, the random variable $W_t-W_s$ is dependent of all $W_u$ for $u\in [0,s]$ (i.e., the process has independent increments)
• $W_t-W_s$ has a normal distribution with mean $0$ and covariance matrix $(t-s)I_n$ where $I_n$ is the $n\times n$ identity matrix

Example 16

Assume $n=1$, we can write a BM in discrete time as follows:

$$W_{t+h}=W_t+\sqrt{h}{Z}_t$$

where:

• $h$ = is as small as wanted
• $Z_t$ is a standard normal random variable independent of all $W_u$ for $u\in [0,t]$ with $Z_t$ independent of $Z_s$ for $t\ne s$

Note that for all $t$ and $h$:

$$\begin{array}{rl}E[W_{t+h}-W_t]& =W[\sqrt{h}{Z}_t]=0\\ Var[W_{t+h}-W_t]& =Var[\sqrt{h}{Z}_t]=h\end{array}$$

If $S_t$ is a stock price then a very popular model is: $\ln S_t=\sigma W_t+\mu t$ with $\mu =0.1$ (10% annual), $\sigma =0.254$ (25% annual).

Explanation

But there is a problem with this model, since $W_t$ is a BM, it can take negative values, which means that $S_t$ can also take negative values, which is not possible for a stock price.

Let $S_t$ be the price of stock at $t$.

Then:

$$S_t=S_0+\mu t+\sigma W_t$$

The probability of it being below $0$ is non-zero

$$\begin{array}{rl}P(s_t<0)& =P(S_0+\mu t+\sigma W_t<0)\\ & =P(W_t<\frac{-S_0-\mu t}{\sigma })\\ & =P(Z<\frac{-S_0-\mu t}{\sigma \sqrt{t}})\end{array}$$

Where $Z$ is a standard normal random variable.

The solution? We can model the log of the stock price instead:

$$\ln S_t=\ln S_0+\mu t+\sigma W_t$$

We call this the log normal distribution for $S_t$

$$S_t=S_0{e}^{\mu t+\sigma W_t}$$