Introduction to Black-Scholes Stock Price Modeling

Wiener Process (Brownian Motion)

• \(W_0=0\)
• \(W_t\) is a continuous function of time \(t\)
• \(\Delta W_{t,s}=W_t-W_s\sim\mathcal{N}(0,t-s)\) for all \(s<t\)
• \(\Delta W_{u,v}\) and \(\Delta W_{t,s}\) are independent for all \(s<t\leq v<u\)

Arithmetic Wiener process

\[X_t=X_0+at+\sigma W_t\] This can be unrealistic if negative values are produced possibly due to \(a=0\), \(X_0\to0\) or large \(\sigma\) (volatility).

Geometric Wiener process

The natural logarithm is an arithmetic Wiener process.

Generalized Wiener Process

\[\mathrm{d}X_t=\mu\mathrm{d}t+\sigma\mathrm{d}W_t\]

Itô Process

\[\mathrm{d}X_t=\mu(X,t)\mathrm{d}t+\sigma(X,t)\mathrm{d}W_t\]

Stock Price Modeling

• SDE \[\mathrm{d}S_t=\mu S_t\mathrm{d}t+\sigma S_t\mathrm{d}W_t\]
• Closed-form solution \[S_t=S_0\exp[(\mu-\sigma^2/2)t+\sigma W_t]\]
• Approximation \[\Delta S_t=\mu S_t\Delta t+\sigma S_t\Delta W_t\]

Itô Lemma

Given Itô process \(X_t\) and \(Y_t\), a function \(G(X_t,Y_t)\) follows the process \[\mathrm{d}G=\frac{\partial G}{\partial x}\mathrm{d}X_t+\frac{\partial G}{\partial y}\mathrm{d}Y_t+\frac{1}{2}\left(\frac{\partial^2 G}{\partial x^2}(\mathrm{d}X_t)^2+2\frac{\partial^2 G}{\partial x\partial y}\mathrm{d}X_t\mathrm{d}Y_t+\frac{\partial^2 G}{\partial y^2}(\mathrm{d}Y_t)^2\right)\]

If \(Y_t=t\), then \(G(X,t)\) follows the process \[\mathrm{d}G=\left(\mu\frac{\partial G}{\partial x}+\frac{\partial G}{\partial t}+(\sigma^2/2)\frac{\partial^2 G}{\partial x^2}\right)\mathrm{d}t+\sigma\frac{\partial G}{\partial x}\mathrm{d}W_t\]