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Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model

Pages: 354 - 363, Issue Special, September 2007

 Abstract        Full Text (184.5K)              

Lorella Fatone - Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, Via Campi 213/b, 41100 Modena, Italy (email)
Francesca Mariani - Dipartimento di Scienze Sociali "D. Serrani", Università Politecnica delle Marche, Piazza Martelli 8, 60121, Italy (email)
Maria Cristina Recchioni - Dipartimento di Scienze Sociali "D. Serrani", Università Politecnica delle Marche, Piazza Martelli 8, 60121, Italy (email)
Francesco Zirilli - Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy (email)

Abstract: This paper presents a numerical method to price European options on realized variance. A European realized variance option is an option where payoff depends on the time of maturity, on the observed variance of the log-returns of the stock prices in a preassigned sequence of time values $t_i$, $i$ = 0, 1, . . . ,$N$. The realized variance is the variance observed in the sample of the log-returns considered, so that the value at maturity of the realized variance option depends on the discrete sample of the log-returns of the stock prices observed at the preassigned dates t$_i$, $i$= 0, 1, . . . ,$N$. The method proposed to approximate the price of these options is based on the idea of approximating the discrete sum that gives the realized variance with an integral, using as model of the dynamics of the log-return of the stock price the Heston stochastic volatility model. In this way the price of a realized variance option is approximated with the price of an integrated stochastic variance option where payoff depends on the time of maturity and on the integrated stochastic variance. The integrated stochastic variance option is priced with the method of discounted expectations. We derive an integral representation formula for the price of this last kind of options. This integral formula reduces to a one dimensional Fourier integral in the case of the most commonly traded options that have a simple payoff function. The method has been validated on some test problems. The numerical experiments show that the approach suggested in this paper gives satisfactory approximations of the prices of the realized variance options (relative error 10−$^2$, 10-$^3$). This approach also allows substantial savings of computational time when compared with the Monte Carlo method used to evaluate with approximately the same accuracy. The website http://www.econ.univpm.it/recchioni/finance/w4 contains auxiliary material that can help in the understanding of this paper and makes available to the interested users the codes that implement the numerical method proposed here to price realized variance options. The use of these codes on a computing grid has been made user friendly developing a dedicated application using the software Symphony (that is, a Service Oriented Architecture (SOAM) software of Platform Computing Toronto, Canada). The website mentioned above makes this Symphony application available to the users.

Keywords:  Stochastic volatility models, Fokker Planck equation, variance options.
Mathematics Subject Classification:  Primary: 58F15, 58F17; Secondary: 53C35.

Received: September 2006;      Revised: March 2007;      Published: September 2007.