The Black-Scholes equation

I do not remember the first time I heard about the Black-Scholes equation, but I finally fulfilled my curiosity. The Black-Scholes equation is a differential equation for the value of a contract that establishes the optional right to buy (call options) or sell stocks (put options). The typical call option is a contract that gives the option to buy the stock at a certain point in the future T for a given strike price k. The value of the stock changes in time so that if I buy a call option written with a strike price k, I am betting that the price of the stock at time T will be higher than k (plus the cost of the contract itself), so that I will be able to execute the transaction recovering the difference. Otherwise, I could let the contract expire, losing all the money I paid for the call option.

The value of the contract is well know in the last day T, when the option can be executed because it is exactly the difference of price of the stock minus the strike price of the contract if the stock price is higher than k, otherwise, the contract's value is ZERO. The big problem is to estimate the fair value of the contract as a function of the stock price S and time t. Under certain assumptions, the value of the contract obeys the Black-Scholes partial differential equation.

The figure below shows a typical solution of the Black-Scholes partial differential equation. The curve in red corresponds to the value of the contract at time T as a function of the stock price, acting as boundary condition. The remaining curves correspond to times farther and farther in the past. The strike price is k=120, the maturity time is T=1, the interest rate is r=0.05 and the volatility is sigma=0.5


This is an example adapted from the book:
Computational Financial Mathematics using Mathematica, by Srdjan Stojanovic.

Now, having a reasonable idea of the value of the contract, I can even buy and sell contracts!.