"Frontmatter". In: Analysis of Financial Time Series

234 CONTINUOUS-TIME MODELS∂G t∂t∂G t+ rP t + 1 ∂ P t 2 σ 2 Pt2 ∂ 2 G t∂ Pt2= rG t . (6.17)This is the Black–Scholes differential equation for derivative pricing. It can be solvedto obtain the price of a derivative with P t as the underlying variable. The solution soobtained depends on the boundary conditions of the derivative. For a European calloption, the boundary condition isG T = max(P T − K, 0),where T is the expiration time and K is the strike price. For a European put option,the boundary condition becomesG T = max(K − P T , 0).Example 6.4. As a simple example, consider a forward contract on a stockthat pays no dividend. In this case, the value of the contract is given byG t = P t − K exp[−r(T − t)],where K is the delivery price, r is the risk-free interest rate, and T is the expirationtime. For such a function, we have∂G t∂t=−rK exp[−r(T − t)],∂G t∂ P t= 1,∂ 2 G t∂ P 2t= 0.Substituting these quantities into the left-hand side of Eq. (6.17) yields−rK exp[−r(T − t)]+rP t = r{P t − K exp[−r(T − t)]},which equals the right-hand side of Eq. (6.17). Thus, the Black–Scholes differentialequation is indeed satisfied.6.6 BLACK–SCHOLES PRICING FORMULASBlack and Scholes (1973) successfully solve their differential equation in Eq. (6.17)to obtain exact formulas for the price of European call and put options. In whatfollows, we derive these formulas using what is called Risk-Neutral Valuation infinance.6.6.1 Risk-Neutral WorldThe drift parameter µ drops out from the Black–Scholes differential equation. Infinance, this means the equation is independent of risk preferences. In other words,