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

${\frac {dS}{S}}=\mu dt+\sigma dW(t)$

where $W(t)$ is a standard Weiner stochastic process.

Loosely speaking, this means that the return $\Delta S/S$ of the stock over a very small time interval $\Delta t$ can be viewed as a normal random variable with mean $\mu \Delta t$ and variance $\sigma ^{2}(\Delta t)^{2}$. 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 $K$ and time to maturity $T$ is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price $K$ at time $T$. In other words, a European call on the stock S is a contract that provides a single pay-off of

${\hbox{max}}(S(T)-K,0)$

at time $T$. Let $\Phi (t)$ denote the fair value of this contract at time $t<T$. In deriving a formula for $\Phi (t)$, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of $S$ and the call option, one can completely eliminate risk associated to movements in the stock price $S$. 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:

${\frac {\partial \Phi }{\partial t}}+rS{\frac {\partial \Phi }{\partial S}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}\Phi }{\partial S^{2}}}=r\Phi$

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

$\Phi (T)={\hbox{max}}(S(T)-K,0)$

is given by:

$\Phi (t)=S(t)N(d_{1})-Ke^{-r(T-t)}N(d_{2})$

Here $N(x)$ is the cumulative normal distribution function,

$d_{1}={\frac {\log(S(t)/K)+(r+{\frac {1}{2}}\sigma ^{2})(T-t)}{\sigma {\sqrt {T-t}}}}$

and $d_{2}=d_{1}-\sigma {\sqrt {T}}$

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