A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

34 Pages Posted: 2 Jan 2004

See all articles by Vicky Henderson

Vicky Henderson

University of Warwick

David G. Hobson

University of Bath - School of Mathematical Sciences

Sam Howison

University of Oxford

Tino Kluge

University of Oxford - Nomura Centre for Mathematical Finance

Date Written: October 3, 2003

Abstract

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk.

As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same.

As a concrete example, we specialise to a variant of the Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Choice of "q" is shown to influence the level of the implied volatility smile for options of varying maturity.

Keywords: stochastic volatility, pricing measure, market price of volatility risk, Heston model

JEL Classification: D52, G13

Suggested Citation

Henderson, Vicky and Hobson, David G. and Howison, Sam and Kluge, Tino, A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation (October 3, 2003). Available at SSRN: https://ssrn.com/abstract=459594 or http://dx.doi.org/10.2139/ssrn.459594

Vicky Henderson (Contact Author)

University of Warwick ( email )

Gibbet Hill Rd.
Coventry, West Midlands CV4 8UW
United Kingdom
44 (0)2476 574811 (Phone)

David G. Hobson

University of Bath - School of Mathematical Sciences ( email )

Bath, BA2 7AY
United Kingdom

Sam Howison

University of Oxford ( email )

Mathematical Institute
andrew Wiles Building, Woodstock Road
Oxford, OX2 6GG
United Kingdom

Tino Kluge

University of Oxford - Nomura Centre for Mathematical Finance ( email )

OCIAM Mathematical Institute
24-29 St Giles
Oxford OX1 3LB, Ox1 3LB
United Kingdom

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