Paper
Journal of Derivatives & Hedge Funds (2008) 14, 78–89. doi:10.1057/jdhf.2008.9
Option pricing and hedging bounds in incomplete markets
Practical applications
Black–Scholes option pricing model is widely used within the asset and risk management of traditional and alternative investments. From the investors' point of view, Black–Scholes model provides a unique price of a contingent claim in an ideal, complete and unconstrained market based on the fundamental principles of 'absence of arbitrage opportunities'. The assumptions of the model, however, may not be closely met in practice. I show various approaches to derive closed-form solutions of tight upper and lower bounds, and implications on delta hedging strategies.
Tao Hao1
Correspondence: Tao Hao, Watson Wyatt Worldwide, London, UK. Tel: +44 1737 274503; Fax: + 44 1737 241496; E-mail: tao.hao@watsonwyatt.com, http://www.watsonwyatt.com
1Tao Hao is an economist within a global research team. His role is to develop research across a broad range of areas of strategic interests to Watson Wyatt, including issues concerning corporate pension risk exposure, stochastic financial modelling, asset allocation strategy, derivatives pricing and risk management, dynamic hedging strategy for pension scheme, asset and liability management. Tao holds BSc in Economics from Beijing University, China, and an MSc in Finance and Economics from London School of Economics, UK. He is currently undertaking his PhD study in Finance and Economics at Birkbeck College, University of London.
Received 10 September 2007; Revised 10 September 2007.
Abstract
This paper has reviewed the literature on option pricing in incomplete markets. Tight upper and lower bounds can be derived based on the assumptions of mean and variance of the underlying asset price, not on its entire distribution. The differences between estimated upper or lower bounds and Black–Scholes price are quite small for deep in-the-money options, but can be very significant for deep out-of-the-money options. But at the same time, despite the wide pricing bounds, analysis of the implied hedging bounds suggests that the implications for asset allocation of incomplete markets are fairly limited.
Keywords:
Black–Scholes, option pricing, upper and lower bounds, hedging, incomplete market
