# Black-Scholes

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## 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  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 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  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, and  is the volatility of the stock. 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 at time . The volatility,  of the stock must be estimated using either statistical data, or inferred from the price of options being sold in the market.