An IMEX-Scheme for Pricing Options under Stochastic Volatility Models with Jumps

Abstract

Abstract. Partial integro-differential equation (PIDE) formulations are often preferable for pricing options under models with stochastic volatility and jumps, especially for American-style option contracts. We consider the pricing of options under such models, namely the Bates model and the so-called stochastic volatility with contemporaneous jumps (SVCJ) model. The nonlocality of the jump terms in these models leads to matrices with full matrix blocks. Standard discretization methods are not viable directly since they would require the inversion of such a matrix. Instead, we adopt a two-step implicit-explicit (IMEX) time discretization scheme, the IMEX-CNAB scheme, where the jump term is treated explicitly using the second-order Adams-Bashforth (AB) method, while the rest is treated implicitly using the Crank-Nicolson (CN) method. The resulting linear systems can then be solved directly by employing LU decomposition. Alternatively, the systems can be iterated under a scalable algebraic multigrid (AMG) method. For pricing American options, LU decomposition is employed with an operator splitting method for the early exercise constraint or, alternatively, a projected AMG method can be used to solve the resulting linear complementarity problems. We price European and American options in numerical experiments, which demonstrate the good efficiency of the proposed methods. It is well known that fitting empirically observed option prices into the Black-Scholes model typically implies a volatility distribution with a smile-like shape. This volatility smile becomes more pronounced near the maturity date. The usual modifications of the BlackScholes model to explain such implied volatility patterns include models with jumps and/or stochastic volatility. The underlying asset price can be completely modeled by an infinite activity model such as the Carr-Geman-Madan-Yor (CGMY) model [8]. Over long time intervals the component behaving like the geometric Brownian motion becomes the dominant part in the model. For options with long maturities, the stochastic volatility models, for example the Heston model [24], are often regarded as more appropriate. For options with short maturities, however, jumps become increasingly important as a purely geometric Brownian motion driven process would require extremely high levels of volatility to explain the pronounced volatility smile pattern. Well-known jump-diffusion models in the literature include the Merton [34]

Adaptive finite differences and IMEX time-stepping to price options under Bates model

“…This method is only first-order accurate in time. The IMEX-midpoint [18], [17] and IMEX-CNAB [28], [30] methods are similar, but secondorder accurate methods in time. In this paper, we employ the IMEX-CNAB method.…”

“…Here we employ a direct solver based on LU decomposition. Recently this approach was shown to be efficient and convenient for these problems in [30]. For American options an LCP with the same operator needs to be solved at each time step.…”

“…For American options an LCP with the same operator needs to be solved at each time step. The LCP can be approximated and solved using various methods including an operator splitting method [12], [15], [30], penalty methods [39], [9], and multigrid methods [7], [22], [26], [14], [35]. Here we use the operator splitting method to approximate the solutions of LCPs as it is accurate and easy to implement.…”

Efficient Hedging in Bates Model Using High-Order Compact Finite Differences

“…The implicit-explicit scheme is based on the approaches in Düring and Fournié [3] and Salmi et al [5]. The scheme is fourth order accurate in space and second order accurate in time.…”

LSV Models with Stochastic Interest Rates and Correlated Jumps

“…These models as applied to pricing American options were intensively studied in Salmi et al (2014), for basket options in Shirava and Takahashi (2013).…”