PRICING AND HEDGING OF EXOTIC OPTIONS THROUGH MONTE CARLO SIMULATION

Pricing and Hedging of Exotic Options through Monte Carlo Simulation

Abstract

This paper attempts to implement Monte Carlo simulations in order to price and hedge exotic options. Many exotic options have no analytic solutions, either because they are too complex or because the volatility specification is wrong. Consequently, numerical solutions are a necessity. We discuss the advantages and the drawbacks of such a pricing approach for the main exotic options. Given the strong assumptions of the Black-Scholes world, we attempt to relax them and, in particular, we focus on stochastic volatility models. After a review of the literature, we analyze via simulations the impact of stochastic volatility on the valuation of Asian and spread options. Next we construct and evaluate a dynamic hedging strategy for an exchange option under discrete rebalancing, stochastic volatility and transaction costs. We study the effect of each of these market imperfections on the hedge performance. Finally, we shortly discuss possible hedging approaches for various exotic options and compare static and dynamic hedging.

Executive Summary

Options that are trivial to price (like binary options) are difficult to hedge. Options that are difficult to price (like Asian options) are trivial to hedge. (Howard Savery, Exotic Options Trader)

Over the last years, the size of the exotic options market has expanded considerably. Today a large variety of such instruments is available to investors and they can be used for multiple purposes. Several factors can provide an explanation for the recent success of these instruments. One possibility is their almost unlimited flexibility in the sense that they can be tailored to the specific needs of any investor. It is why exotic options are also called: "special-purpose options" or "customer-tailored options".

Secondly, these options are playing a significant hedging role and, thus, they meet the hedgers' needs in cost effective ways. Corporations have moved away from buying some form of general protection and they are designing strategies to meet specific exposures at a given point in time. These strategies can be based on exotic options which are usually less expensive and more efficient than standard instruments.

Thirdly, exotic options can be used as attractive investments and trading opportunities. As a result, views on the spot evolution, various preferences on time horizons and

Premium contingency can all be accommodated by exotic patterns. Moreover, it becomes possible to undertake a much leveraged position which would be unattainable in the spot or standard options market.

The main types of exotic options have been priced either numerically or analytically. A major element in the derivation of the prices has been the construction of hedging or replicating portfolios. Thus, these two issues are strongly interrelated and we consider that, for a global view, it is necessary to discuss them both. The approach we adopt for pricing and hedging is based on Monte Carlo simulations and it is implemented in Gauss.

The rest of this paper is organized as follows. In Chapter 1, we shortly justify the choice of Monte Carlo simulations for pricing and hedging purposes...