Lévy Processes

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. ...

Analysing Models for Volatility Clustering with Subordinated Processes: VGSA and Beyond ArXiv ID: 2507.17431 “View on arXiv” Authors: Sourojyoti Barick, Sudip Ratan Chandra Abstract This paper explores a comprehensive class of time-changed stochastic processes constructed by subordinating Brownian motion with Levy processes, where the subordination is further governed by stochastic arrival mechanisms such as the Cox Ingersoll Ross (CIR) and Chan Karolyi Longstaff Sanders (CKLS) processes. These models extend classical jump frameworks like the Variance Gamma (VG) and CGMY processes, allowing for more flexible modeling of market features such as jump clustering, heavy tails, and volatility persistence. We first revisit the theory of Levy subordinators and establish strong consistency results for the VG process under Gamma subordination. Building on this, we prove asymptotic normality for both the VG and VGSA (VG with stochastic arrival) processes when the arrival process follows CIR or CKLS dynamics. The analysis is then extended to the more general CGMY process under stochastic arrival, for which we derive analogous consistency and limit theorems under positivity and regularity conditions on the arrival process. A simulation study accompanies the theoretical work, confirming our results through Monte Carlo experiments, with visualizations and normality testing (via Shapiro-Wilk statistics) that show approximate Gaussian behavior even for processes driven by heavy-tailed jumps. This work provides a rigorous and unified probabilistic framework for analyzing subordinated models with stochastic time changes, with applications to financial modeling and inference under uncertainty. ...

American options valuation in time-dependent jump-diffusion models via integral equations and characteristic functions ArXiv ID: 2506.18210 “View on arXiv” Authors: Andrey Itkin Abstract Despite significant advancements in machine learning for derivative pricing, the efficient and accurate valuation of American options remains a persistent challenge due to complex exercise boundaries, near-expiry behavior, and intricate contractual features. This paper extends a semi-analytical approach for pricing American options in time-inhomogeneous models, including pure diffusions, jump-diffusions, and Levy processes. Building on prior work, we derive and solve Volterra integral equations of the second kind to determine the exercise boundary explicitly, offering a computationally superior alternative to traditional finite-difference and Monte Carlo methods. We address key open problems: (1) extending the decomposition method, i.e. splitting the American option price into its European counterpart and an early exercise premium, to general jump-diffusion and Levy models; (2) handling cases where closed-form transition densities are unavailable by leveraging characteristic functions via, e.g., the COS method; and (3) generalizing the framework to multidimensional diffusions. Numerical examples demonstrate the method’s efficiency and robustness. Our results underscore the advantages of the integral equation approach for large-scale industrial applications, while resolving some limitations of existing techniques. ...

Numerical analysis on locally risk-minimizing strategies for Barndorff-Nielsen and Shephard models ArXiv ID: 2505.00255 “View on arXiv” Authors: Takuji Arai Abstract We develop a numerical method for locally risk-minimizing (LRM) strategies for Barndorff-Nielsen and Shephard (BNS) models. Arai et al. (2017) derived a mathematical expression for LRM strategies in BNS models using Malliavin calculus for Lévy processes and presented some numerical results only for the case where the asset price process is a martingale. Subsequently, Arai and Imai (2024) developed the first Monte Carlo (MC) method available for non-martingale BNS models with infinite active jumps. Here, we modify the expression obtained by Arai et al. (2017) into a numerically tractable form, and, using the MC method developed by Arai and Imai (2024), propose a numerical method of LRM strategies available for non-martingale BNS models with infinite active jumps. In the final part of this paper, we will conduct some numerical experiments. ...

Efficient evaluation of joint pdf of a Lévy process, its extremum, and hitting time of the extremum ArXiv ID: 2312.05222 “View on arXiv” Authors: Unknown Abstract For Lévy processes with exponentially decaying tails of the Lévy density, we derive integral representations for the joint cpdf $V$ of $(X_T, \bar X_T,τ_T)$ (the process, its supremum evaluated at $T<+\infty$, and the first time at which $X$ attains its supremum). The first representation is a Riemann-Stieltjes integral in terms of the (cumulative) probability distribution of the supremum process and joint probability distribution function of the process and its supremum process. The integral is evaluated using a combination an analog of the trapezoid rule. The second representation is amenable to more accurate albeit slower calculations. We calculate explicitly the Laplace-Fourier transform of $V$ w.r.t. all arguments, apply the inverse transforms, and reduce the problem to evaluation of the sum of 5D integrals. The integrals can be evaluated using the summation by parts in the infinite trapezoid rule and simplified trapezoid rule; the inverse Laplace transforms can be calculated using the Gaver-Wynn-Rho algorithm. Under additional conditions on the domain of analyticity of the characteristic exponent, the speed of calculations is greatly increased using the conformal deformation technique. For processes of infinite variation, the program in Matlab running on a Mac with moderate characteristics achieves the precision better than E-05 in a fraction of a second; the precision better than E-10 is achievable in dozens of seconds. As the order of the process (the analog of the Blumenthal-Getoor index) decreases, the CPU time increases, and the best accuracy achievable with double precision arithmetic decreases. ...

Simulation of a Lévy process, its extremum, and hitting time of the extremum via characteristic functions ArXiv ID: 2312.03929 “View on arXiv” Authors: Unknown Abstract We suggest a general framework for simulation of the triplet $(X_T,\bar X_ T,τ_T)$ (Lévy process, its extremum, and hitting time of the extremum), and, separately, $X_T,\bar X_ T$ and pairs $(X_T,\bar X_ T)$, $(\bar X_ T,τ_T)$, $(\bar X_ T-X_T,τ_T)$, via characteristic functions and conditional characteristic functions. The conformal deformations technique allows one to evaluate probability distributions, joint probability distributions and conditional probability distributions accurately and fast. For simulations in the far tails of the distribution, we precalculate and store the values of the (conditional) characteristic functions on multi-grids on appropriate surfaces in $C^n$, and use these values to calculate the quantiles in the tails. For simulation in the central part of a distribution, we precalculate the values of the cumulative distribution at points of a non-uniform (multi-)grid, and use interpolation to calculate quantiles. ...

Fourier Neural Network Approximation of Transition Densities in Finance ArXiv ID: 2309.03966 “View on arXiv” Authors: Unknown Abstract This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet’s capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet’s accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including Lévy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process. ...