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

Journal Article


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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. Finance 10(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.

Publication Date


  • 2015

Citation


  • He, X. & Zhu , S. (2015). Pricing European options with stochastic volatility under the minimal entropy martingale measure. European Journal of Applied Mathematics, 27 (2), 233-247.

Scopus Eid


  • 2-s2.0-84959181009

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6329&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5301

Number Of Pages


  • 14

Start Page


  • 233

End Page


  • 247

Volume


  • 27

Issue


  • 2

Place Of Publication


  • United Kingdom

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. Finance 10(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.

Publication Date


  • 2015

Citation


  • He, X. & Zhu , S. (2015). Pricing European options with stochastic volatility under the minimal entropy martingale measure. European Journal of Applied Mathematics, 27 (2), 233-247.

Scopus Eid


  • 2-s2.0-84959181009

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6329&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5301

Number Of Pages


  • 14

Start Page


  • 233

End Page


  • 247

Volume


  • 27

Issue


  • 2

Place Of Publication


  • United Kingdom