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Pricing European options with stochastic volatility under the minimal entropy martingale measure

Published online by Cambridge University Press:  20 October 2015

XIN-JIANG HE  and
SONG-PING ZHU
XIN-JIANG HE
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: [email protected], [email protected]
SONG-PING ZHU
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia email: [email protected], [email protected]
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Abstract

In this paper, a closed-form pricing formula in the form of an infinite series for European call options is derived for the Heston stochastic volatility model under a chosen martingale measure. Given that markets with the stochastic volatility are incomplete, there exists a number of equivalent martingale measures and consequently investors face a problem of making a choice of appropriate measure when they price options. The one we adopt here is the so-called minimal entropy martingale measure shown to be related to the expected utility maximization theory (Frittelli 2000 Math. Finance10(1), 39–52) and the financial rationality for choosing this measure will be further illustrated in this paper. A great advantage of our newly-derived pricing formula is that the convergence of the solution in series form can be proved theoretically; such a proof of the convergence is also complemented by some numerical examples to demonstrate the speed of convergence. To further show the validity of our formula, a comparison of prices calculated through the newly derived formula is made with those obtained directly from the Monte Carlo simulation as well as those from solving the PDE (partial differential equation) with the finite difference method.

Keywords

Series expansionMinimal entropy martingale measureExpected utility maximizationConvergence
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 2 , April 2016 , pp. 233 - 247
Copyright
Copyright © Cambridge University Press 2015 

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