Exponential time integration for fast finite element solutions of some financial engineering problems

Abstract

a b s t r a c tWe consider exponential time integration schemes for fast numerical pricing of European, American, barrier and butterfly options when the stock price follows a dynamics described by a jump-diffusion process. The resulting pricing equation which is in the form of a partial integro-differential equation is approximated in space using finite elements. Our methods require the computation of a single matrix exponential and we demonstrate using a wide range of numerical tests that the combination of exponential integrators and finite element discretisations with quadratic basis functions leads to highly accurate algorithms for cases when the jump magnitude is Gaussian. Comparison with other time-stepping methods are also carried out to illustrate the effectiveness of our methods.

Fast exponential time integration scheme for option pricing with jumps

“…Furthermore, the COS method can be combined with an extrapolation strategy on the Bermudan options for American option pricing. On the other hand, PIDE-based methods are also used for pricing American options recently [25,26]. Therefore, the current work is a stepping-stone to American option pricing and more advanced models like the stochastic volatility with correlated and contemporaneous jumps in return and variance (SVCJ) model [6].…”

Fast approximations of bond option prices under CKLS models

“…The above scheme is known as ETI and has been previously used for pricing options on assets in Tangman et al (2008Tangman et al ( , 2010 and Rambeerich et al (2009Rambeerich et al ( , 2010. However the application to the pricing of interest rate derivatives is new and is used for the first time in this paper for pricing bond options.…”

Numerical pricing of American options under infinite activity Lévy processes

“…The operator splitting technique for the ETI scheme has been considered in Rambeerich et al (2009), where the authors have demonstrated that this numerical algorithm is second-order convergent for various test cases. By using an extra factor ⌳(t) Ն 0, called the Lagrange multiplier, the American option valuation problem becomes ) are given by and respectively.…”

“…In addition, with efficient techniques for matrix function evaluations (Lee, Liu, & Sun, 2010;Schmelzer & Trefethen, 2007), ETI is becoming a promising method for solving systems of equations. Since only spatial discretization is incurred, this scheme combined with quadratic finite Journal of Futures Markets DOI: 10.1002/fut element methods has led to fourth order convergent numerical solutions for American options (Rambeerich, Tangman, Gopaul, & Bhuruth, 2009) under Merton's jump diffusion model (Merton, 1976). This is not the case for the CN method (Wang et al, 2007), as the latter is only second-order accurate in time, unconditionally stable in L 2 norm and is also known for causing undesired zigzag effects in numerical solutions due to lack of the L 0 -stability conditions (Giles & Carter, 2006).…”