Introduction

The options we have so far considered are sometimes called “vanilla” options to distinguish them from the more exotic options, whose prevalence has increased in recent years. Generally speaking, the value of these options at the exercise time depends not only on the security's price at that time but also on the price path leading to it. In this chapter we introduce three of these exotic-type options – barrier options, Asian options, and lookback options – and show how to use Monte Carlo simulation methods efficently to determine their geometric Brownian motion risk-neutral valuations. In the final section of this chapter we present an explicit formula for the risk-neutral valuation of a “power” call option, whose payoff when exercised is the amount by which a specified power of the security's price at that time exceeds the exercise price.

Barrier Options

To define a European barrier call option with strike price K and exercise time t, a barrier value v is specified; depending on the type of barrier option, the option either becomes alive or is killed when this barrier is crossed. A down-and-in barrier option becomes alive only if the security's price goes below v before time t, whereas a down-and-out barrier option is killed if the security's price goes below v before time t. In both cases, v is a specified value that is less than the initial price s of the security.