Jump-Diffusion

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. ...

Optimal mean-variance portfolio selection under regime-switching-induced stock price shocks ArXiv ID: 2507.19824 “View on arXiv” Authors: Xiaomin Shi, Zuo Quan Xu Abstract In this paper, we investigate mean-variance (MV) portfolio selection problems with jumps in a regime-switching financial model. The novelty of our approach lies in allowing not only the market parameters – such as the interest rate, appreciation rate, volatility, and jump intensity – to depend on the market regime, but also in permitting stock prices to experience jumps when the market regime switches, in addition to the usual micro-level jumps. This modeling choice is motivated by empirical observations that stock prices often exhibit sharp declines when the market shifts from a bullish'' to a bearish’’ regime, and vice versa. By employing the completion-of-squares technique, we derive the optimal portfolio strategy and the efficient frontier, both of which are characterized by three systems of multi-dimensional ordinary differential equations (ODEs). Among these, two systems are linear, while the first one is an $\ell$-dimensional, fully coupled, and highly nonlinear Riccati equation. In the absence of regime-switching-induced stock price shocks, these systems reduce to simple linear ODEs. Thus, the introduction of regime-switching-induced stock price shocks adds significant complexity and challenges to our model. Additionally, we explore the MV problem under a no-shorting constraint. In this case, the corresponding Riccati equation becomes a $2\ell$-dimensional, fully coupled, nonlinear ODE, for which we establish solvability. The solution is then used to explicitly express the optimal portfolio and the efficient frontier. ...

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. ...

On the optimal design of a new class of proportional portfolio insurance strategies in a jump-diffusion framework ArXiv ID: 2407.21148 “View on arXiv” Authors: Unknown Abstract In this paper, we investigate an optimal investment problem associated with proportional portfolio insurance (PPI) strategies in the presence of jumps in the underlying dynamics. PPI strategies enable investors to mitigate downside risk while still retaining the potential for upside gains. This is achieved by maintaining an exposure to risky assets proportional to the difference between the portfolio value and the present value of the guaranteed amount. While PPI strategies are known to be free of downside risk in diffusion modeling frameworks with continuous trading, see e.g., Cont and Tankov (2009), real market applications exhibit a significant non-negligible risk, known as gap risk, which increases with the multiplier value. The goal of this paper is to determine the optimal PPI strategy in a setting where gap risk may occur, due to downward jumps in the asset price dynamics. We consider a loss-averse agent who aims at maximizing the expected utility of the terminal wealth exceeding a minimum guarantee. Technically, we model agent’s preferences with an S-shaped utility functions to accommodate the possibility that gap risk occurs, and address the optimization problem via a generalization of the martingale approach that turns to be valid under market incompleteness in a jump-diffusion framework. ...

Deep learning for quadratic hedging in incomplete jump market ArXiv ID: 2407.13688 “View on arXiv” Authors: Unknown Abstract We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based upon a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feedforward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black-Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle. ...

Mean-variance portfolio selection in jump-diffusion model under no-shorting constraint: A viscosity solution approach ArXiv ID: 2406.03709 “View on arXiv” Authors: Unknown Abstract This paper concerns a continuous time mean-variance (MV) portfolio selection problem in a jump-diffusion financial model with no-shorting trading constraint. The problem is reduced to two subproblems: solving a stochastic linear-quadratic (LQ) control problem under control constraint, and finding a maximal point of a real function. Based on a two-dimensional fully coupled ordinary differential equation (ODE), we construct an explicit viscosity solution to the Hamilton-Jacobi-Bellman equation of the constrained LQ problem. Together with the Meyer-Itô formula and a verification procedure, we obtain the optimal feedback controls of the constrained LQ problem and the original MV problem, which corrects the flawed results in some existing literatures. In addition, closed-form efficient portfolio and efficient frontier are derived. In the end, we present several examples where the two-dimensional ODE is decoupled. ...

A semi-Lagrangian $ε$-monotone Fourier method for continuous withdrawal GMWBs under jump-diffusion with stochastic interest rate ArXiv ID: 2310.00606 “View on arXiv” Authors: Unknown Abstract We develop an efficient pricing approach for guaranteed minimum withdrawal benefits (GMWBs) with continuous withdrawals under a realistic modeling setting with jump-diffusions and stochastic interest rate. Utilizing an impulse stochastic control framework, we formulate the no-arbitrage GMWB pricing problem as a time-dependent Hamilton-Jacobi-Bellman (HJB) Quasi-Variational Inequality (QVI) having three spatial dimensions with cross derivative terms. Through a novel numerical approach built upon a combination of a semi-Lagrangian method and the Green’s function of an associated linear partial integro-differential equation, we develop an $ε$-monotone Fourier pricing method, where $ε> 0$ is a monotonicity tolerance. Together with a provable strong comparison result for the HJB-QVI, we mathematically demonstrate convergence of the proposed scheme to the viscosity solution of the HJB-QVI as $ε\to 0$. We present a comprehensive study of the impact of simultaneously considering jumps in the sub-account process and stochastic interest rate on the no-arbitrage prices and fair insurance fees of GMWBs, as well as on the holder’s optimal withdrawal behaviors. ...

D-TIPO: Deep time-inconsistent portfolio optimization with stocks and options ArXiv ID: 2308.10556 “View on arXiv” Authors: Unknown Abstract In this paper, we propose a machine learning algorithm for time-inconsistent portfolio optimization. The proposed algorithm builds upon neural network based trading schemes, in which the asset allocation at each time point is determined by a a neural network. The loss function is given by an empirical version of the objective function of the portfolio optimization problem. Moreover, various trading constraints are naturally fulfilled by choosing appropriate activation functions in the output layers of the neural networks. Besides this, our main contribution is to add options to the portfolio of risky assets and a risk-free bond and using additional neural networks to determine the amount allocated into the options as well as their strike prices. We consider objective functions more in line with the rational preference of an investor than the classical mean-variance, apply realistic trading constraints and model the assets with a correlated jump-diffusion SDE. With an incomplete market and a more involved objective function, we show that it is beneficial to add options to the portfolio. Moreover, it is shown that adding options leads to a more constant stock allocation with less demand for drastic re-allocations. ...

Monte Carlo simulation for Barndorff-Nielsen and Shephard model under change of measure ArXiv ID: 2306.05750 “View on arXiv” Authors: Unknown Abstract The Barndorff-Nielsen and Shephard model is a representative jump-type stochastic volatility model. Still, no method exists to compute option prices numerically for the non-martingale case with infinite active jumps. We develop two simulation methods for such a case under change of measure and conduct some numerical experiments. Keywords: Barndorff-Nielsen and Shephard Model, Stochastic Volatility, Jump Diffusion, Option Pricing, Monte Carlo Simulation, Options ...