Asia-Pacific Financial Markets

, Volume 11, Issue 1, pp 55–77

Understanding the Implied Volatility Surface for Options on a Diversified Index

Article

DOI: 10.1007/s10690-005-4249-4

Cite this article as:
Heath, D. & Platen, E. Asia-Pacific Finan Markets (2004) 11: 55. doi:10.1007/s10690-005-4249-4

Abstract

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts.

Key Words

index derivatives minimal market model random scaling growth optimal portfolio fair pricing binary options 

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, School of Finance and EconomicsUniversity of Technology SydneyBroadwayAustralia

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