The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

CHAPTER 6

Understanding the Black-Scholes Equation

The Black-Scholes equation has many remarkable features. The important ones are outlined and discussed in this chapter.

6.1. BLACK-SCHOLES EQUATION: A TYPE OF BACKWARD KOLMOGOROV EQUATION

First, the final term in the Black-Scholes equation is the discounting due to the time value of cash; if we wrote the formula in terms of the function C′ (S, t) as

which is the option value in maturity dollars, then the last term would disappear as

It is immediately evident that it is a type of backward Kolmogorov equation—but with the “wrong” drift term for the stock price process because the drift of stock includes the risk premium in reality. We have already solved this problem for drift and volatility as arbitrary functions of time but not functions of stock price. So, for this case, from section 4.2 Analytic Formula for the Expected Payoff of a European Option, we can immediately write the Black-Scholes option pricing formulae as

where the expectation value is understood to be the risk-neutral expectation value, and volatilities and rates are ...