Numerical Methods

Boundary error control for numerical solution of BSDEs by the convolution-FFT method ArXiv ID: 2512.24714 “View on arXiv” Authors: Xiang Gao, Cody Hyndman Abstract We first review the convolution fast-Fourier-transform (CFFT) approach for the numerical solution of backward stochastic differential equations (BSDEs) introduced in (Hyndman and Oyono Ngou, 2017). We then propose a method for improving the boundary errors obtained when valuing options using this approach. We modify the damping and shifting schemes used in the original formulation, which transforms the target function into a bounded periodic function so that Fourier transforms can be applied successfully. Time-dependent shifting reduces boundary error significantly. We present numerical results for our implementation and provide a detailed error analysis showing the improved accuracy and convergence of the modified convolution method. ...

Almost-Exact Simulation Scheme for Heston-type Models: Bermudan and American Option Pricing ArXiv ID: 2601.00815 “View on arXiv” Authors: Mara Kalicanin Dimitrov, Marko Dimitrov, Anatoliy Malyarenko, Ying Ni Abstract Recently, an Almost-Exact Simulation (AES) scheme was introduced for the Heston stochastic volatility model and tested for European option pricing. This paper extends this scheme for pricing Bermudan and American options under both Heston and double Heston models. The AES improves Monte Carlo simulation efficiency by using the non-central chi-square distribution for the variance process. We derive the AES scheme for the double Heston model and compare the performance of the AES schemes under both models with the Euler scheme. Our numerical experiments validate the effectiveness of the AES scheme in providing accurate option prices with reduced computational time, highlighting its robustness for both models. In particular, the AES achieves higher accuracy and computational efficiency when the number of simulation steps matches the exercise dates for Bermudan options. ...

Numerical valuation of European options under two-asset infinite-activity exponential Lévy models ArXiv ID: 2511.02700 “View on arXiv” Authors: Massimiliano Moda, Karel J. in ’t Hout, Michèle Vanmaele, Fred Espen Benth Abstract We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian theta-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal favourable second-order convergence of our method whenever the exponential Lévy process has finite-variation. ...

Optimal Investment and Consumption in a Stochastic Factor Model ArXiv ID: 2509.09452 “View on arXiv” Authors: Florian Gutekunst, Martin Herdegen, David Hobson Abstract In this article, we study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem, and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Itô diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including – for the first time – the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme. ...

Risk-Neutral Pricing of Random-Expiry Options Using Trinomial Trees ArXiv ID: 2508.17014 “View on arXiv” Authors: Sebastien Bossu, Michael Grabchak Abstract Random-expiry options are nontraditional derivative contracts that may expire early based on a random event. We develop a methodology for pricing these options using a trinomial tree, where the middle path is interpreted as early expiry. We establish that this approach is free of arbitrage, derive its continuous-time limit, and show how it may be implemented numerically in an efficient manner. ...

Revisiting Stochastic Collocation with Exponential Splines for an Arbitrage-Free Interpolation of Option Prices ArXiv ID: 2508.12419 “View on arXiv” Authors: Fabien Le Floc’h Abstract We revisit the stochastic collocation method using the exponential of a quadratic spline. In particular, we look in details whether it is more appropriate to fix the ordinates and optimize the abscissae of an interpolating spline or to fix the abscissae and optimize the parameters of a B-spline representation. ...

Isogeometric Analysis for the Pricing of Financial Derivatives with Nonlinear Models: Convertible Bonds and Options ArXiv ID: 2412.08987 “View on arXiv” Authors: Unknown Abstract Computational efficiency is essential for enhancing the accuracy and practicality of pricing complex financial derivatives. In this paper, we discuss Isogeometric Analysis (IGA) for valuing financial derivatives, modeled by two nonlinear Black-Scholes PDEs: the Leland model for European call with transaction costs and the AFV model for convertible bonds with default options. We compare the solutions of IGA with finite difference methods (FDM) and finite element methods (FEM). In particular, very accurate solutions can be numerically calculated on far less mesh (knots) than FDM or FEM, by using non-uniform knots and weighted cubic NURBS, which in turn reduces the computational time significantly. ...

Efficient inverse $Z$-transform and Wiener-Hopf factorization ArXiv ID: 2404.19290 “View on arXiv” Authors: Unknown Abstract We suggest new closely related methods for numerical inversion of $Z$-transform and Wiener-Hopf factorization of functions on the unit circle, based on sinh-deformations of the contours of integration, corresponding changes of variables and the simplified trapezoid rule. As applications, we consider evaluation of high moments of probability distributions and construction of causal filters. Programs in Matlab running on a Mac with moderate characteristics achieves the precision E-14 in several dozen of microseconds and E-11 in several milliseconds, respectively. ...

Alternative models for FX: pricing double barrier options in regime-switching Lévy models with memory ArXiv ID: 2402.16724 “View on arXiv” Authors: Unknown Abstract This paper is a supplement to our recent paper Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in Lévy models". We introduce the class of regime-switching Lévy models with memory, which take into account the evolution of the stochastic parameters in the past. This generalization of the class of Lévy models modulated by Markov chains is similar in spirit to rough volatility models. It is flexible and suitable for application of the machine-learning tools. We formulate the modification of the numerical method in Alternative models for FX, arbitrage opportunities and efficient pricing of double barrier options in Lévy models", which has the same number of the main time-consuming blocks as the method for Markovian regime-switching models. ...

The Euler Scheme for Fractional Stochastic Delay Differential Equations with Additive Noise ArXiv ID: 2402.08513 “View on arXiv” Authors: Unknown Abstract In this paper we consider the Euler-Maruyama scheme for a class ofstochastic delay differential equations driven by a fractional Brownian motion with index $H\in(0,1)$. We establish the consistency of the scheme and study the rate of convergence of the normalized error process. This is done by checking that the generic rate of convergence of the error process with stepsize $Δ_{“n”}$ is $Δ_{“n”}^{"\min{H+\frac{1"}{“2”},3H,1}}$. It turned out that such a rate is suboptimal when the delay is smooth and $H>1/2$. In this context, and in contrast to the non-delayed framework, we show that a convergence of order $H+1/2$ is achievable. ...