Hedging exotic options in incomplete markets.

摘要

This thesis studies risk management of exotic options, which are custom-design options that are not traded in exchanges. We exploit the fact that standard or vanilla options are liquid assets in today's financial markets, and they should be considered as hedging instruments for exotic options.; We first introduce the two most common hedging techniques related to exotic options that are valid in the so-called Black-Scholes market model, and compare them under realistic market assumptions using Monte-Carlo simulations.; We study optimal hedging of exotic options using a combination of a static position in standard options and dynamic trading of the underlying asset in a general semi-martingale model. In incomplete markets, there can be risk factors that are not hedgeable through trading the underlying stock, for example the volatility risk. As standard options are also exposed to these risk factors, they can be utilized as means to hedge these risk factors. In reality, standard options are available with certain strikes, and transaction costs associated with trading these options are higher than those associated with stock and bond trading. Therefore, we allow only static positions in the available options whereas the underlying stock is traded dynamically. The problem is formulated as the maximization of a function, which is itself the value function for a stochastic control problem. It reduces to computing the Fenchel-Legendre transform of the utility indifference price as a function of the number of standard options used to hedge, evaluated at the market price of the standard options. We give conditions guaranteeing differentiability and strict convexity of the indifference price in the hedging quantity, and hence a unique solution to the hedging problem.; Finally, we illustrate the approach within Markovian stochastic volatility models. In this case, the utility indifference price problem is described by a quasilinear partial differential equation. We establish existence and uniqueness of a classical solution, and conclude with a computational example.