Difference between revisions of "Black-Scholes"

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(The Black-Scholes Formula: Corrected dumb mistake in black scholes diff eqn)
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at time <math>T</math>. Let <math>\Phi(t)</math> denote the fair value of this contract at time <math>t< T</math>. In deriving a formula for <math>\Phi(t)</math>, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of <math>S</math> and the call option, one can completely eliminate risk associated to movements in the stock price <math>S</math>. 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:
 
at time <math>T</math>. Let <math>\Phi(t)</math> denote the fair value of this contract at time <math>t< T</math>. In deriving a formula for <math>\Phi(t)</math>, Black and Scholes' key insight was that by forming a portfolio with the exact right balance of <math>S</math> and the call option, one can completely eliminate risk associated to movements in the stock price <math>S</math>. 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:
  
<math>\frac{\partial\Phi}{\partial t}+\mu S\frac{\partial \Phi}{\partial S}+\frac{1}{2}\sigma^2 S^2 \frac{\partial^2 \Phi}{\partial S^2} = r\Phi
+
<math>\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
 
</math>
 
</math>
  
where <math>r</math> is the continuously compounded risk-free interest rate. The solution to this differential equation, satisfying the boundary condition
+
where <math>r</math> is the continuously compounded risk-free interest rate, and <math>\sigma</math> is the volatility of the stock. The solution to this differential equation, satisfying the boundary condition
  
 
<math>\Phi(T) = \hbox{max}(S(T)-K,0)</math>
 
<math>\Phi(T) = \hbox{max}(S(T)-K,0)</math>

Revision as of 05:19, May 24, 2009

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.