Black-Scholes Model - 2023.2 English

The Continue form of Black-Scholes is:

\[dS_t = \mu S_t dt + \sigma S_t dz\]

where \(S_t\) is the stock price at time \(t\). \(\mu\) is the stock’s expected rate of return. \(\sigma\) is the volatility of the stock price. The random variable \(z\) follows Wiener process, i.e. \(z\) satisfies the following equation.

The change of \(\Delta z\) during a sample period of time \(\Delta t\) is \(\Delta z = \epsilon \sqrt{\Delta t}\), where \(\epsilon\) has a standardized normal distribution \(\phi(0,1)\).

The value of \(\Delta z\) for any two different short intervals of time, \(\Delta t\), are independent.

It follows that \(\Delta z\) has a normal distribution with the mean of \(0\) and the variance of \(\Delta t\).

The Discrete form of Black-Scholes is:

\[\Delta S_t = \mu S_t \Delta t + \sigma S_t \epsilon \sqrt{\Delta t} => \frac{\Delta S_t}{S_t} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t}\]

The left side of the equation is the return provided by the stock in a short period of time, \(\Delta t\). The term \(\mu \Delta t\) is the expected value of this return, and the \(\sigma \epsilon \sqrt{\Delta t}\) is the stochastic component of the return. It is easy to see that \(\frac{\Delta S_t}{S_t} \sim \phi (\mu \Delta t, \sigma^2 \Delta t)\), i.e. the mean of \(\frac{\Delta S_t}{S_t}\) is \(\mu \Delta t\) and the variance is \(\sigma^2 \Delta t\).