Finite Difference Methods

Numerical methods for solving PIDEs arising in swing option pricing under a two-factor mean-reverting model with jumps ArXiv ID: 2511.01587 “View on arXiv” Authors: Mustapha Regragui, Karel J. in ’t Hout, Michèle Vanmaele, Fred Espen Benth Abstract This paper concerns the numerical valuation of swing options with discrete action times under a linear two-factor mean-reverting model with jumps. The resulting sequence of two-dimensional partial integro-differential equations (PIDEs) are convection-dominated and possess a nonlocal integral term due to the presence of jumps. Further, the initial function is nonsmooth. We propose various second-order numerical methods that can adequately handle these challenging features. The stability and convergence of these numerical methods are analysed theoretically. By ample numerical experiments, we confirm their second-order convergence behaviour. ...

American option pricing using generalised stochastic hybrid systems ArXiv ID: 2409.07477 “View on arXiv” Authors: Unknown Abstract This paper presents a novel approach to pricing American options using piecewise diffusion Markov processes (PDifMPs), a type of generalised stochastic hybrid system that integrates continuous dynamics with discrete jump processes. Standard models often rely on constant drift and volatility assumptions, which limits their ability to accurately capture the complex and erratic nature of financial markets. By incorporating PDifMPs, our method accounts for sudden market fluctuations, providing a more realistic model of asset price dynamics. We benchmark our approach with the Longstaff-Schwartz algorithm, both in its original form and modified to include PDifMP asset price trajectories. Numerical simulations demonstrate that our PDifMP-based method not only provides a more accurate reflection of market behaviour but also offers practical advantages in terms of computational efficiency. The results suggest that PDifMPs can significantly improve the predictive accuracy of American options pricing by more closely aligning with the stochastic volatility and jumps observed in real financial markets. ...

Option Pricing with Stochastic Volatility, Equity Premium, and Interest Rates ArXiv ID: 2408.15416 “View on arXiv” Authors: Unknown Abstract This paper presents a new model for options pricing. The Black-Scholes-Merton (BSM) model plays an important role in financial options pricing. However, the BSM model assumes that the risk-free interest rate, volatility, and equity premium are constant, which is unrealistic in the real market. To address this, our paper considers the time-varying characteristics of those parameters. Our model integrates elements of the BSM model, the Heston (1993) model for stochastic variance, the Vasicek model (1977) for stochastic interest rates, and the Campbell and Viceira model (1999, 2001) for stochastic equity premium. We derive a linear second-order parabolic PDE and extend our model to encompass fixed-strike Asian options, yielding a new PDE. In the absence of closed-form solutions for any options from our new model, we utilize finite difference methods to approximate prices for European call and up-and-out barrier options, and outline the numerical implementation for fixed-strike Asian call options. ...

Estimation of domain truncation error for a system of PDEs arising in option pricing ArXiv ID: 2401.15570 “View on arXiv” Authors: Unknown Abstract In this paper, a multidimensional system of parabolic partial differential equations arising in European option pricing under a regime-switching market model is studied in details. For solving that numerically, one must truncate the domain and impose an artificial boundary data. By deriving an estimate of the domain truncation error at all the points in the truncated domain, we extend some results in the literature those deal with option pricing equation under constant regime case only. We differ from the existing approach to obtain the error estimate that is sharper in certain region of the domain. Hence, the minimum of proposed and existing gives a strictly sharper estimate. A comprehensive comparison with the existing literature is carried out by considering some numerical examples. Those examples confirm that the improvement in the error estimates is significant. ...

Boundary conditions at infinity for Black-Scholes equations ArXiv ID: 2401.05549 “View on arXiv” Authors: Unknown Abstract We propose a numerical procedure for computing the prices of European options, in which the underlying asset price is a Markovian strict local martingale. If the underlying process is a strict local martingale and the payoff is of linear growth, multiple solutions exist for the corresponding Black-Scholes equations. When numerical schemes such as finite difference methods are applied, a boundary condition at infinity must be specified, which determines a solution among the candidates. The minimal solution, which is considered as the derivative price, is obtained by our boundary condition. The stability of our procedure is supported by the fact that our numerical solution satisfies a discrete maximum principle. In addition, its accuracy is demonstrated through numerical experiments in comparison with the methods proposed in the literature. ...