Analysis of Financial Time Series, Third Edition

6.6 Black–Scholes Pricing Formulas

Black and Scholes (1973) successfully solved their differential equation in Eq. (6.17) to obtain exact formulas for the price of European call-and-put options. In what follows, we derive these formulas using what is called risk-neutral valuation in finance.

6.6.1 Risk-Neutral World

The drift parameter μ drops out from the Black–Scholes differential equation. In finance, this means the equation is independent of risk preferences. In other words, risk preferences cannot affect the solution of the equation. A nice consequence of this property is that one can assume that investors are risk neutral. In a risk-neutral world, we have the following results:

• The expected return on all securities is the risk-free interest rate r.
• The present value of any cash flow can be obtained by discounting its expected value at the risk-free rate.

6.6.2 Formulas

The expected value of a European call option at maturity in a risk-neutral world is

where E* denotes expected value in a risk-neutral world. The price of the call option at time t is

(6.18)

Yet in a risk-neutral world, we have μ = r, and by Eq. (6.10), ln(PT) is normally distributed as

Let g(PT) be the probability ...