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
where W(t) is a standard Weiner stochastic process.
Loosely speaking, this means that the return ΔS / S of the stock over a very small time interval Δt can be viewed as a normal random variable with mean μΔt and variance σ2(Δt)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
max(S(T) − K,0)
at time T. Let Φ(t) denote the fair value of this contract at time t < T. In deriving a formula for Φ(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:
Φ(T) = max(S(T) − K,0)
is given by:
Φ(t) = S(t)N(d1) − Ke − r(T − t)N(d2)
Here N(x) is the cumulative normal distribution function,
This is the famous Black-Scholes formula for the price of a European call. Note that all the variables except for σ can be observed in directly in the market at time t. The volatility, σ of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.