# Black-Scholes

### From Conservapedia

## 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 Δ*S* / *S* of the stock over a very small time interval Δ*t* can be viewed as a normal random variable with mean μΔ*t* and variance σ^{2}(Δ*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

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:

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

Φ(*T*) = max(*S*(*T*) − *K*,0)

is given by:

Φ(*t*) = *S*(*t*)*N*(*d*_{1}) − *K**e*^{ − r(T − t)}*N*(*d*_{2})

Here *N*(*x*) 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 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.