# American Institute of Mathematical Sciences

2007, 2007(Special): 354-363. doi: 10.3934/proc.2007.2007.354

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

 1 Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, Via Campi 213/b, 41100 Modena, Italy 2 Dipartimento di Scienze Sociali "D. Serrani", Università Politecnica delle Marche, Piazza Martelli 8, 60121, Italy, Italy 3 Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy

Received  September 2006 Revised  March 2007 Published  September 2007

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.
Citation: Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni, Francesco Zirilli. Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model. Conference Publications, 2007, 2007 (Special) : 354-363. doi: 10.3934/proc.2007.2007.354
 [1] Xingchun Wang, Yongjin Wang. Variance-optimal hedging for target volatility options. Journal of Industrial & Management Optimization, 2014, 10 (1) : 207-218. doi: 10.3934/jimo.2014.10.207 [2] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [3] Kais Hamza, Fima C. Klebaner, Olivia Mah. Volatility in options formulae for general stochastic dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 435-446. doi: 10.3934/dcdsb.2014.19.435 [4] Axel Klar, Florian Schneider, Oliver Tse. Approximate models for stochastic dynamic systems with velocities on the sphere and associated Fokker--Planck equations. Kinetic & Related Models, 2014, 7 (3) : 509-529. doi: 10.3934/krm.2014.7.509 [5] Jacinto Marabel Romo. A closed-form solution for outperformance options with stochastic correlation and stochastic volatility. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1185-1209. doi: 10.3934/jimo.2015.11.1185 [6] Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165 [7] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [8] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [9] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [10] Shuang Li, Chuong Luong, Francisca Angkola, Yonghong Wu. Optimal asset portfolio with stochastic volatility under the mean-variance utility with state-dependent risk aversion. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1521-1533. doi: 10.3934/jimo.2016.12.1521 [11] John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371 [12] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 [13] Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681 [14] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 [15] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [16] Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 [17] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 [18] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [19] Hwa-Sung Kim, Bara Kim, Jerim Kim. Catastrophe equity put options under stochastic volatility and catastrophe-dependent jumps. Journal of Industrial & Management Optimization, 2014, 10 (1) : 41-55. doi: 10.3934/jimo.2014.10.41 [20] Ian Knowles, Ajay Mahato. The inverse volatility problem for American options. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020235

Impact Factor: