Black-Scholes

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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 is a standard Weiner stochastic process.

Loosely speaking, this means that the return of the stock over a very small time interval can be viewed as a normal random variable with mean and variance . One can make this notion precise by invoking the concepts from the Ito calculus.

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 and time to maturity is a financial contract that gives the holder the option, but not the obligation, to purchase the stock for price at time . In other words, a European call on the stock S is a contract that provides a single pay-off of

at time . Let denote the fair value of this contract at time . In deriving a formula for , Black and Scholes' key insight was that by forming a portfolio with the exact right balance of and the call option, one can completely eliminate risk associated to movements in the stock price . 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:

where is the continuously compounded risk-free interest rate. The solution to this differential equation, satisfying the boundary condition

is given by:

Here is the cumulative normal distribution function,

and

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