Local volatility function models under a benchmark approach

Abstract

Without requiring the existence of an equivalent risk-neutral probability measure this paper studies a class of one-factor local volatility function models for stock indices under a benchmark approach. It is assumed that the dynamics for a large diversified index approximates that of the growth optimal portfolio. Fair prices for derivatives when expressed in units of the index are martingales under the real-world probability measure. Different to the classical approach that derives risk-neutral probabilities the paper obtains the transition density for the index with respect to the real-world probability measure. Furthermore, the Dupire formula for the underlying local volatility function is recovered without assuming the existence of an equivalent risk-neutral probability measure. A modification of the constant elasticity of variance model and a version of the minimal market model are discussed as specific examples together with a smoothed local volatility function model that fits a snapshot of S&P500 index options data.

Keywords:

• Local volatility function
• Index derivatives
• Growth optimal portfolio
• Benchmark approach
• Fair pricing
• Dupire formula
• Modified CEV model
• Minimal market model

Acknowledgements

The authors would like to thank the three referees as well as Hardy Hulley and Truc Le for valuable comments on the manuscript. MSCI data was provided by Thomson Financial. Options data was downloaded from http://finance.yahoo.com.