a Consider a share S traded in the market. Let the current inter- est rate be r and the volatility (roughly speaking, the variance of the price) of the share be o. We suppose that r and o are constants. A European call option V(S,t) on S gives the holder the right to buy the share at a fixed price K (the strike price) at some future time T. If the price of the share at T is greater than K, the option holder can buy the share for K, then sell it for S, making a profit of S K dollars. If the price of the share at T is less than K, the option is worthless, because no one would buy a share for more than it is worth on an exchange. Problem: How much is European call option worth? Black and Scholes showed that if V(S,t) = u(S,T t), then u(s, t) satisfies the partial differential equation 7 7 = at = 1 2 S202, trs ru, as2 as = S K, S >K with u(S,0) This is the famous Black- 0, SK with u(S,0) This is the famous Black- 0, S