Black-Scholes

The Black-Scholes Model

The Black-Scholes model for a stock price assumes that the stock price follows geometric Brownian motion with constant drift and volatility. More precisely, if S(t) the stock price at time t, then

<math> \frac{dS}{S} = \mu dt + \sigma dW(t) </math>

where <math>W(t)</math> is a standard Weiner stochastic process.

Loosely speaking, this means that the return <math> \Delta S/S</math> of the stock over a very small time interval <math>\Delta t</math> can be viewed as a normal random variable with mean <math>\mu \Delta t</math> and variance <math>\sigma^2(\Delta t)^2 </math>. One can make this notion precise by invoking the concepts from the Ito calculus.

The Black-Scholes Formula

The Black-Scholes pricing formula for a European call option can be deduced from the Black-Scholes model for a stock price. A European call option on a stock with strike price <math>K</math> and time to maturity <math>T</math> is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price <math>K</math> at time <math>T</math>. In other words, a European call on the stock S is a contract that provides a single pay-off of

<math>\hbox{max}(S(T)-K,0)</math>

at time <math>T</math>. Let <math>\Phi(t)</math> denote the fair value of this contract at time <math>t< T</math>. In deriving a formula for <math>\Phi(t)</math>, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of <math>S</math> and the call option, one can completely eliminate risk associated to movements in the stock price <math>S</math>. Moreover, the resulting portfolio, being risk-free, has to grow at the risk free rate. These observations implied that the fair price of the call option had to satisfy the differential equation:

<math>\frac{\partial\Phi}{\partial t}+r S\frac{\partial \Phi}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Phi}{\partial S^2} = r\Phi </math>

where <math>r</math> is the continuously compounded risk-free interest rate, and <math>\sigma</math> is the volatility of the stock. The solution to this differential equation, satisfying the boundary condition

<math>\Phi(T) = \hbox{max}(S(T)-K,0)</math>

is given by:

<math>\Phi(t) = S(t)N(d_1) - Ke^{-r(T-t)}N(d_2)</math>

Here <math>N(x)</math> is the cumulative normal distribution function,

<math>d_1 = \frac{\log(S(t)/K)+(r+\frac{1}{2}\sigma^2)(T-t)}{\sigma\sqrt{T-t}}</math>

and <math>d_2 = d_1-\sigma\sqrt{T}</math>

This is the famous Black-Scholes formula for the price of a European call. Note that all the variables except for <math>\sigma</math> can be observed in directly in the market at time <math>t</math>. The volatility, <math>\sigma</math> of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.