Document Type : Research Article
Authors
1 Department of Applied Mathematics, University of Mazandaran
2 Department of Computer Science, University of Mazandaran
Abstract
In this article, we discuss the numerical implementation of the Multilevel Monte-Carlo (MLMC) scheme for option pricing within the Heston asset model. The Heston model is a stochastic volatility model that captures the dynamics of the underlying asset price and its volatility. The MLMC method is a variance reduction technique that exploits the difference between two consecutive levels of discretization to estimate the expected value of a quantity of interest. We begin by providing an overview of the MLMC method, followed by an introduction to the weak methods used to approximate the Heston model. Weak methods are numerical schemes that preserve the distributional properties of the solution, rather than its pathwise behavior. Subsequently, we present the results of some numerical experiments conducted to evaluate the performance of the approach. Two different cases are surveyed.
Keywords
Journal on Financial Mathematics 6 (2015), no. 1, 22–52.
[2] , Discretising the heston model: an analysis of the weak convergence rate, IMA Journal of
Numerical Analysis 37 (2017), no. 4, 1930–1960.
[3] Denis Belomestny and Tigran Nagapetyan, Multilevel path simulation for weak approximation schemes with application to L´evy-driven SDEs, Bernoulli 23 (2017), no. 2, 927–950.
[4] Mark Broadie and Ozg ¨ ur Kaya, ¨ Exact simulation of stochastic volatility and other affine
jump diffusion processes, Operations research 54 (2006), no. 2, 217–231.
[5] Kristian Debrabant, Azadeh Ghasemifard, and Nicky C Mattsson, Weak antithetic mlmc
estimation of sdes with the milstein scheme for low-dimensional wiener processes, Applied
Mathematics Letters 91 (2019), 22–27.
[6] Azadeh Ghasemifard and Mohammad Taghi Jahandideh, Weak multilevel path simulation
for jump-diffusion assets, Mathematical Researches 7 (2021), no. 2, 353–370.
[7] Azadeh Ghasemifard and Mahdieh Tahmasebi, Multilevel path simulation to jump-diffusion
process with superlinear drift, Applied Numerical Mathematics 144 (2019), 176–189.
[8] Michael B Giles, Multilevel monte carlo path simulation, Operations research 56 (2008), no. 3,
607–617.
[9] Michael B Giles and Lukasz Szpruch, Antithetic multilevel monte carlo estimation for multidimensional sdes, Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer, 2013, pp. 367–
384.
[10] Paul Glasserman, Monte carlo methods in financial engineering, vol. 53, Springer, 2004.
[11] Peter E Kloeden and Eckhard Platen, Stochastic differential equations, Springer, 1992.
[12] L Szpruch and A Neuenkirch, First order strong approximations of scalar sdes with values
in a domain, Numerische Mathematik 128 (2014), no. 1, 103–136.
[13] Chao Zheng, Weak convergence rate of a time-discrete scheme for the heston stochastic
volatility model, SIAM Journal on Numerical Analysis 55 (2017), no. 3, 1243–1263.
[14] , Multilevel monte carlo simulation for the heston stochastic volatility model, Advances in
Computational Mathematics 49 (2023), no. 6, 81.
)