Approximate analytical solutions to single-period investments in incomplete market

Abstract

This thesis is focused on studying several aspects of optimal one-period in- vestment strategies in an incomplete market. The well-known Black-Scholes formulae for the European options are only valid for complete market. How- ever, the reality of our financial market is incomplete due to several factors, such as transaction cost, jumps in stock prices, stochastic interest rate and stochastic volatility, etc. In these cases, the argument of perfect hedging, which the Black-Scholes formulae were based on, is no longer held. Opti- mization of expected utility is one of common approaches for pricing options in incomplete markets. In this thesis, we maximize the expected utility on the value of a margin account which allows trading both stock and option. We first consider a specific utility function which allows to the short sale of stocks and options. This is consistent with the behavior of practitioners. We examine how various parameters, such as the drift and volatility of stock, the investor's risk attitude, can affect the person's investment decision and the fair prices of option. We show that, even in an incomplete market, when the difference between the drift of stock price and risk-free rate is small, the fair prices of options are independent of a person's utility function. This is a surprising result for option prices in an incomplete market. Finally, we show that this surprising result also holds for general form of utility function.

Date of Award	3 Oct 2006
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Qiang ZHANG (Supervisor)