8 - Black–Scholes PDE and formulas

Summary

OUTLINE

• sum-of-squares for asset price

• replicating portfolio

• hedging

• Black–Scholes PDE

• Black–Scholes formulas for a European call and put

Motivation

At this stage we have defined what we mean by a European call or put option on an underlying asset and we have developed a model for the asset price movement. We are ready to address the key question: what is an option worth? More precisely,

can we systematically determine a fair value of the option at t = 0?

The answer, of course, is yes, if we agree upon various assumptions. Although our basic aim is to value an option at time t = 0 with asset price S(0) = S₀, we will look for a function V(S, t) that gives the option value for any asset price S ≥ 0 at any time 0 ≤ t ≤ T. Moreover, we assume that the option may be bought and sold at this value in the market at any time 0 ≤ t ≤ T. In this setting, V(S₀, 0) is the required time-zero option value. We are going to assume that such a function V(S, t) exists and is smooth in both variables, in the sense that derivatives with respect to these variables exist. It was mentioned in Section 7.1 that S(t) is not a smooth function of t − it is jagged, without a well-defined first derivative.