American Style Stock Options

Abstract

The previous Black-Scholes analysis was based on the premise that an early exercise of options is not allowed. These are so-called European style options. Their possible exercise date is fixed in advance. On the other hand, the fact is that options that are usually traded on the option market can be exercised at any time before the expiry, although most often they are not. Such options are called American options. As seen so far, the problem of pricing European options was solved by solving the associated Black-Scholes partial differential equations, while the problem of implied volatility in its most sophisticated form so far was solved by solving an optimal control problem for the associated Dupire partial differential equation. Both the equations Black-Scholes and Dupire were linear although the optimal control problem was non-linear.