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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

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

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 σ2t)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:

\frac{\partial\Phi}{\partial t}+r S\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 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(d1) − Ker(Tt)N(d2)

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 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.

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