Purpose The primary objective of this research is to develop new algorithms in the framework of deep neural networks for the valuation of options under dynamics driven by stochastic volatility models. We aim to use the Heston model for equity options to demonstrate the accuracy of our approach. Design/methodology/approach Physics-informed neural networks (PINNs) are trained to minimize a loss function that includes terms from the partial differential equation residuals, initial condition and boundary conditions evaluated at selected points in the space-time domain. Speed and accuracy comparisons are carried out against single hidden-layer neural networks, called physics-informed extreme learning machines (PIELMs). American options are formulated as linear complementarity problems, and PINNs are applied in conjunction with penalty methods for the computation of the option prices. Findings For American options under the Heston model, PINNs yield accurate prices. Computed Greeks sensitivities are in close agreement with those reported for mesh-based methods. In contrast to mesh-based penalty methods for American options, PINNs work with smaller values of the penalty term. For the real estate index American option problem, numerical prices obtained using PINNs have comparable accuracies as those obtained by a high-order radial basis functions finite difference scheme. Practical implications There is a lack of reliable pricing models for pricing property derivatives. This work contributes to developing accurate neural network algorithms. Originality/value The development of informed neural networks and informed extreme learning machines for realistic two-factor real-world option contracts constitutes a novel approach in the field of financial derivatives.

ABSTRACT This paper extends the existing two-factor Schwartz model into a comprehensive three-factor framework that incorporates stochastic volatility through the Heston model, enabling more precise pricing of European options on crude oil futures. In this enhanced model, we account for key variables including market volatility and yield changes, which significantly influence futures pricing dynamics. We present an analytical formula for the pricing of futures contracts, facilitating robust calibration against actual market data. To solve the associated partial differential equations for option pricing, we implement the Deep Galerkin Method (DGM), a novel approach leveraging deep neural networks that proves to be efficient and accurate, particularly in high-dimensional settings. The obtained results demonstrate that the DGM method outperforms traditional numerical techniques, offering substantial improvements in both accuracy and computational efficiency.

ABSTRACT In this paper, we propose a regime‐switching (R‐S) Heston‐ α model and consider the analytical pricing of European options. With a constant elasticity of variance specification to reflect the level dependence between the high volatility and the volatile volatility and an R‐S mechanism to capture the impact of changing economic conditions, this model is proved, through an empirical study, to have much better pricing performance than the original Heston model. The non‐affine nature of the newly‐proposed model has however, precluded the use of most existing analytical approaches developed for affine models, and the R‐S mechanism has undoubtedly brought in additional difficulties in deriving analytical option pricing formula. Albeit the inherent mathematical difficulties, we have managed to derive an analytical approximation for the price of European options under such a complicated model, which allows the calibration of the model to be completed at an appropriate speed. Numerical experiments suggest that the newly derived formula has an acceptable degree of accuracy for general parameter settings. Empirical results also confirm the practicability of the newly‐proposed model to real financial markets.

Moment generating functions (MGFs) of the terminal log-asset price and integrated variance of stochastic volatility models conditional on the terminal variance value are required in exact simulation algorithms and Fourier based algorithms for pricing European path dependent options. Broadie and Kaya (Exact simulation of stochastic volatility and other affine jump diffusion processes. Oper. Res., 2006, 54(2), 217–231) initiate the analytic derivation of conditional MGF of integrated variance of the Heston model based on related analytic results for the Bessel bridge. Kang et al. (Exact simulation of the Wishart multidimensional stochastic volatility model. Oper. Res., 2017, 65(5), 1190–1206) and Zeng et al. (Analytical solvability and exact simulation in models with affine stochastic volatility and Lévy jumps. Math. Finance, 2023, 33, 842–890) employ different techniques of measure changes to obtain the conditional MGFs of the Heston model, multidimensional Wishart stochastic volatility model, 4/2-model and Ornstein–Uhlenbeck-driven stochastic volatility model. In this paper, we develop systematic and comprehensive measure change techniques that provide effective derivation procedures for the associated conditional MGFs. We establish an interesting linkage between joint conditional MGFs and their unconditional counterparts. Interestingly, the conditional MGFs under the 4/2-model can be deduced from those under the Heston model via an appropriate measure change. Besides, we employ the partial transform method to derive conditional MGFs that go beyond the Heston-type stochastic volatility models.

We develop a novel deep learning approach for pricing European options in diffusion models, that can efficiently handle high-dimensional problems resulting from Markovian approximations of rough volatility models. The option pricing partial differential equation is reformulated as an energy minimization problem, which is approximated in a time-stepping fashion by deep artificial neural networks. The proposed scheme respects the asymptotic behavior of option prices for large levels of moneyness, and adheres to a priori known bounds for option prices. The accuracy and efficiency of the proposed method is assessed in a series of numerical examples, with particular focus in the lifted Heston model.

Two models with uncertain volatility and the Heston model are considered. For the European type option, Bellman equations are obtained for calculating the range of fair prices in the case of models with uncertain volatility, as well as for calculating the fair price in the case of the Heston model. For comparison, the boundary value problem solution method, the Monte Carlo method and the method based on recurrent formulas on a binary tree are used. An obvious advantage of Bellman equations is their invariance with respect to methods of approximating continuous models. The main conclusion is that models with uncertain volatility are an adequate replacement for the Heston model in cases where it is necessary to calculate the range of fair prices of a European type option.

In this paper we consider the pricing of variable annuities (VAs) with guaranteed minimum withdrawal benefits. Among three models of the equity index, we consider two pricing approaches, the classical risk-neutral approach and the real-world pricing under the benchmark approach, and we examine the associated static and optimal behaviors of both the investor and insurer. The first model considered is the so-called minimal market model, where real-world pricing is achieved using the benchmark approach, introduced by Platen (2001. A minimal financial market model. In Trends in mathematics (pp. 293–301). Birkhäuser). Under this approach, valuing an asset involves determining the minimum-valued replicating portfolio, with reference to the growth optimal portfolio under the real-world probability measure, and it both subsumes classical risk-neutral pricing as a particular case and extends it to situations where risk-neutral pricing is impossible. The second and third models are the Black–Scholes model and the Heston model for the equity index, where the pricing of contracts is performed within the risk-neutral framework. We demonstrate that when the insurer prices and reserves using the minimal market model, the reserves are significantly lower than those under either the Black–Scholes or the Heston model. Furthermore, when the insured employs a dynamic withdrawal strategy based on the minimal market model, the total withdrawal amount exceeds that received when using either the Black–Scholes or the Heston model. We employ a novel approximation to the valuation of the VA having GMWB features, which permits fast valuation and multiple backtests of reserving strategies to be performed that would otherwise be impractical.

Efficiently generating or loading probability distributions on quantum computers is a foundational task in Quantum Monte Carlo methods. While the quantum amplitude estimation algorithm offers a quadratic speed-up over classical Monte Carlo techniques, this advantage can be negated if the probability distribution is not effectively loaded. Moreover, any practical approach must be flexible enough to accommodate a wide range of market models�a requirement unmet by existing methods. In this paper, we propose a novel and efficient approach for loading probability distributions tailored to derivative pricing. The proposed method, the Quantum Binomial Tree, serves as a quantum analog of the classical Binomial Tree model. This approach enables exponential scaling in the number of Monte Carlo paths, while retaining an overall quadratic speed-up compared to classical algorithms. We demonstrate how the Quantum Binomial Tree framework can load prominent financial models�including the local volatility model, and the Heston model onto a quantum computer. Furthermore, this paper includes a detailed implementation for option pricing under time-dependent volatility, as well as numerical results.

A mixed finite element method for pricing American options and Greeks in the Heston model

Foreign exchange option pricing with a three-factor Heston model with regime switching and stochastic interest rate

We show the existence of a stationary measure for a class of multidimensional stochastic Volterra systems of affine type. These processes are, in general, not Markovian, a shortcoming, which hinders their large-time analysis. We circumvent this issue by lifting the system to a measure-valued stochastic evolution equation introduced by Cuchiero and Teichmann [Generalized Feller processes and Markovian lifts of stochastic Volterra processes: The affine case, J. Evol. Equ. 20 (2020), pp. 1301–1348], whence we retrieve the Markov property. Leveraging on the associated generalized Feller property, we extend the Krylov–Bogoliubov theorem to this infinite-dimensional setting and thus establish an approach to the existence of invariant measures. We present concrete examples, including the rough Heston model from Mathematical Finance.

Option pricing is one of the core problems in modern financial mathematics. This paper systematically reviews the mathematical models used in option pricing, including classical models (Black-Scholes model, binomial tree model), modern stochastic models (Heston model, Merton jump-diffusion model), numerical methods (Monte Carlo simulation, finite difference method), and machine learning techniques. Through theoretical analysis and empirical comparisons, the study reveals the mathematical principles, applicability, and limitations of these models. Furthermore, the study discusses model optimization directions in the context of real financial markets, particularly for special cases such as China's A-share market. The research shows that the evolution of mathematical models has always balanced market incompleteness and computational efficiency. Future trends will focus on hybrid models integrating stochastic analysis and data science.

With the incorporation of the conditional on one-step survival technique, estimators for the price and some Greeks of an autocallable are constructed under the Heston model. With some restrictions on the model parameters, these estimators have strong-order-one convergence with regards to the discretization of the variance process. Numerical examples illustrate superior performances of these estimators as well as their property of strong-order-one convergence.

Option pricing for Heston model with tempered fractional Brownian motion

Insurance companies often seek reinsurance to spread risk and protect themselves against catastrophic losses. Consider a bidimensional risk model for an insurer and his reinsurer that divides claims in some specified proportions under the assumption that the model allows a time-dependent structure. This paper introduces a càdlàg process to describe the model's investment return process and firstly proposes a condition for the càdlàg process that can ensure the uniform asymptotic estimates for ruin probabilities when the claim size distribution is heavy-tailed. Especially, many important stochastic processes satisfy the condition, such as the Lévy process, Cox-Ingersoll-Ross model, Vasicek model, Heston model, and Stochastic volatility model. Finally, numerical simulations are provided to demonstrate the accuracy of the asymptotic estimates.

To address the critical challenges faced by the elderly population in the field of long-term care, this paper proposes a novel insurance product that flexibly combines a variable annuity with a guaranteed lifelong withdrawal benefit (GLWB) rider and long-term care (LTC) insurance. Specifically, the proposed insurance product incorporates a conversion period during which the policyholder can flexibly decide whether to convert the standard annuity contract into a combo product that integrates LTC benefits. Using the Fourier cosine (COS) method combined with Markov chain approximation techniques, we develop a unified framework to price the GLWB annuity under general stochastic volatility models. By improving the algorithm through cubic spline interpolation, we achieve efficient valuation of this contract. The stochastic volatility models considered in this study include the Heston model, 3/2 model, 4/2 model, Hull-White model, and Scott model. The impact of annuity model parameters, conversion timing, and the policyholder's gender on the GLWB value is analyzed.

Modeling the volatility of crude oil prices is essential because it gives substantial influence to the oil producing countries. Nigeria, the biggest oil producer in Africa and a major participant in the world oil market, has significant economic difficulties changes in oil prices. This study uses 14 years of crude oil price data (2010–2023) to assess and compare the forecasting effectiveness of the Heston stochastic volatility model and GARCH-type models (GARCH, EGARCH, IGARCH, TGARCH, and FIGARCH). According to the analysis, GARCH-type models with Student’s t-distribution perform better than models with typical innovation. With a log-likelihood value of 12022.3, an AIC of -4.7012, a mean error (ME) of 0.0254, and a root mean square error (RMSE) of 0.0534, the EGARCH model outperformed the others. Nonetheless, the Heston model outperformed all GARCH-type models in terms of forecast accuracy, achieving the smallest error (0.000564) and successfully capturing fat-tail characteristics in daily return distributions. The study indicates that the Heston model offers a better fit and more accurate forecast than GARCH-type models using data from January to December 2023 for out-of-sample forecasting. These results provide stakeholders and policymakers with important information for controlling the volatility of Nigeria’s crude oil market.

Since the emergence of the Black–Scholes model (BSM) in the early 1970s, models for the pricing of financial options have been developed and evolved with mathematical tools that provide greater efficiency and accuracy in the valuation of these assets. In this research, we have used the generalized conformable derivatives associated with seven obtained conformable models with a closed-form solution that is similar to the traditional Black and Scholes. In addition, an empirical analysis was carried out to test the models with Mexican options contracts listed in 2023. Six foreign options were also tested, in particular three London options and three US options. With this sample, in addition to applying the seven generalized conformable models, we compared the results with the Heston model. We obtained much better results with the conformable models. Similarly, we decided to apply the seven conformable models to the data of the Morales et al. article, and we again determined that the conformable models greatly outperform the approximation of the Black, Scholes (BS), and Merton model with time-varying parameters and the basic Khalil conformable equation. In addition to the base sample, it was decided to test the strength of the seven generalized conformable models on 10 stock options that were out-sampled. In addition to the MSE results, for the sample of six options whose shares were traded in the London and New York stock markets, we tested the positivity and stability of the results. We plotted the values of the option contracts obtained by applying each of the seven generalized conformable models, the values of the contracts obtained by applying the traditional Heston model, and the market value of the contracts.

This paper presents a comprehensive simulation study on estimating parameters for the popular Heston stochastic volatility model. Leveraging high-frequency data, we explore, in a data-science type exercise, various spot-volatility estimation and sampling techniques, improving existing methods to enhance parameter accuracy. Through extensive simulations, we report difficulties in generating correct parameter estimates for realistic parameter settings where the volatility dynamic does not satisfy the Feller condition. This study contributes valuable insights into the practical implementation of the Heston model and its applicability to high-frequency data. We find that the scheme of Azencott et al. (2020) with uniform kernel weighting provides reliable and efficient parameter estimates. It is advised to also apply a Jackknife estimation to corroborate the findings.

This work incorporates a stochastic liquidity risk, stochastic volatility, and Caputo-type fractional derivatives into European option pricing and establishes two novel space-time fractional hybrid models to capture the nonlinearity and non-stationarity of price evolution processes. The combination neural network algorithm with the defined nonlinear test solutions is designed to solve the fractional derivative models with the initial condition, Dirichlet and Robin boundary conditions. When the influence of a liquidity risk is removed, the studied models are reduced to the space-time fractional Heston models, and the pricing results are compared with the analytical formula of the classical Heston model. In the presence of a liquidity risk, the pricing models under the Caputo and Caputo-Fabrizio fractional derivatives are tested based on the market data. The applications and comparison results prove that the dynamical models demonstrated in the work have small prediction errors and can highly fit market data. The designed combination neural network can effectively handle the mixed problems of the high-dimensional fractional derivative equations and derive the optimal approximations under a stochastic liquidity risk and volatility.