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In this work we present an alternative methodology to the standard Quantum Accelerated Monte Carlo (QAMC) applied to derivatives pricing. Our pipeline benefits from the combination of a new encoding protocol, referred to as the direct encoding, and an amplitude estimation algorithm, the modified Real Quantum Amplitude Estimation (mRQAE) algorithm. On the one hand, the direct encoding prepares a quantum state which contains the information about the sign of the expected payoff. On the other hand, the mRQAE is able to read all the information contained in the quantum state. Although the procedure we describe is different from the standard one, the main building blocks are almost the same. Thus, all the extensive research that has been performed is still applicable. Moreover, we experimentally compare the performance of the proposed methodology against the standard QAMC employing a quantum emulator and show that we retain the speedups. [ABSTRACT FROM AUTHOR]

Since the early 2000s, fuzzy mathematics has fostered a stream of research on the financial valuation of assets incorporating optionality. This paper makes two contributions to this field. First, it conducts a bibliographical analysis of contributions from fuzzy set theory to option pricing, focusing on fuzzy-random option pricing (FROP) and its applications in binomial and trinomial lattice approaches. Second, it extends the FROP to yield curve modeling within a binomial framework. The bibliographical analysis followed the PRISMA guidelines and was conducted via the SCOPUS and WoS databases. We present a structured review of papers on FROP in discrete time (FROPDT), identifying the principal papers and outlets. The findings reveal that this focus has been applied to price options on stocks, stock indices, and real options. However, the exploration of its application to the term structure of interest-sensitive interest rate assets is very rare. To address this gap, we develop a fuzzy-random extension of the Ho–Lee term structure model, applying it to the European interbank market and price caplet options. [ABSTRACT FROM AUTHOR]

The standard model, which determines option pricing, is the well-known Black-Scholes formula. Heston in addition to Cox-Ingersoll-Ross which is called CIR, respectively, implemented the models of stochastic volatility and interest rate to the standard option pricing model. The cost of transaction, which the Black-Scholes method overlooked, is another crucial consideration that must be made when trading a service or production. It is acknowledged that by employing the log-normal stock diffusion hypothesis with constant volatility, the Black-Scholes model for option pricing departs from reality. The standard log-normal stock price distribution used in the Black-Scholes model is insufficient to account for the leaps that regularly emerge in the discontinuous swings of stock prices. A jump-diffusion model, which combines a jump process and a diffusion process is a type of mixed model in the Black-Scholes model belief. Merton developed a jump model as a modification of jump models to better describe purchasing and selling behavior. In this study, the Heston-Cox-Ingersoll-Ross (HCIR) model with transaction costs is solved using the alternating direction implicit (ADI) approach and the Monte Carlo simulation assuming the underlying asset adheres to the jump-diffusion case, then the outcomes are compared to the analytical solution. In addition, the consistency of the numerical method is proven for the model. [ABSTRACT FROM AUTHOR]

A forward start option is an exotic option with delayed activation and reduced premium that allows investors to avail market opportunities in the future. Moreover, mean-reversion explores the investment strategies. This study examines the pricing of forward start options based on the uncertain mean-reverting stock model by applying Liu’s uncertain finance. Call and put options pricing formulas are derived. The study depicts that the pricing patterns in uncertain environments are consistent with the common characteristics of financial markets.

In order to solve the problem of the large deviation between the classical option pricing model and the actual price data, based on the BS option pricing model, the Fast Fourier Transform (FFT) combined with the Transformer's multi-head attention mechanism of deep learning algorithm was used to conductthe empirical research on the 300ETF options and Shanghai gold options data. The model was quadratically trained by the improved Transformer algorithm on the residual values between the option pricing model based on the FFT algorithm and the option price data of the actual financial market. The results show that compared with other algorithms (BS model, FFT-BS model) and other hybrid algorithm models (FFT-BS+ARIMA model, FFT-BS+LSTM model), the proposed model has a good performance in the statistical indexes such as R2, MSE, NRMSE and MAE, and the hybrid algorithm model achieves a better performance for different volatilitiesand different varieties of options. The study innovatively applies the improved Transformer algorithm to option pricing, which compensates for the shortcomings of the classical option pricing model and provides a more accurate option pricing model.

Finance-Informed Neural Networks (FINNs), inspired by Physical Information Neural Networks (PINNs) and computational finance, aim to enhance risk assessment and support regulatory decision-making. Despite their promising potential, existing FINNs face significant challenges related to learning efficiency and overall performance. This paper examines the KAFIN framework, based on the Kolmogorov–Arnold representation, which aims to improve financial modeling and analytical calculations. To evaluate the performance of FINNs, this study uses generated data instead of real market data. This choice enables controlled simulations of various financial scenarios while avoiding the complexities and noise inherent in actual market data. By relying on generated data, we are able to isolate and assess the core capabilities of KAFIN under well-defined theoretical conditions, facilitating a clearer analysis of its performance. This study focuses on European option pricing as a case study, using generated input data that simulate a range of market scenarios, including typical regulatory conditions in financial markets. Testing KAFIN under these controlled conditions allows for a rigorous evaluation of its ability to handle the complexities of pricing options across different market assumptions and regulatory constraints. Empirical results demonstrate that KAFIN significantly outperforms the baseline method, improving pricing accuracy by minimizing residual errors and aligning closely with analytical solutions. The architecture of KAFIN effectively captures the inherent complexities of option pricing, integrating financial principles with advanced computational techniques. Performance indicators reveal that KAFIN achieves lower average total losses and reduces the variability in loss components, highlighting its advantage in modeling and analyzing complex financial dependencies while ensuring compliance with regulatory standards. [ABSTRACT FROM AUTHOR]

In this manuscript, we proposed the stability and error analysis for the backward difference operator splitting (BDF-OS) methods to solve the linear complementarity problem (LCP) for pricing the American option under the Black–Scholes framework. The OS schemes have been successfully applied to a variety of Black–Scholes models. It is easy to apply on LCP because the complementarity conditions and the differential equation are segregated and examined separately. We provided an error estimate for these methods and the priori stability estimates for operator splitting strategies based on the BDF1 and BDF2 approaches. We performed numerical experiments and illustrated the order and efficiency of the BDF1 and BDF2 approaches for the test problems to emphasize the convergence behavior of the proposed methods. We have also verified the numerical results with the existing methods in the literature. [ABSTRACT FROM AUTHOR]

In this study, a conditional expectation transform (CET) method was developed for solving high‐dimensional systems arising in vulnerable options pricing. The CET was formulated to recast the pricing of vulnerable options into that of exchange‐traded options under special terminal condition. Then, traditional option pricing methods are deployed to price the exchange‐traded options, and an analytical formula for vulnerable options is obtained. The vulnerable European option, vulnerable Barrier option, and vulnerable floating Lookback option were analyzed as examples to demonstrate the effectiveness and accuracy of the CET method. Numerical examples were provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

This paper explores a numerical method for European and American option pricing under time fractional jump-diffusion model in Caputo scene. The pricing problem for European options is formulated using a time fractional partial integro-differential equation, whereas the pricing of American options is described by a linear complementarity problem. For European option, we present nonuniform discretization along time and the radial basis function (RBF) method for spatial discretization. The stability and convergence analysis of the discrete scheme are carried out in the case of European options. For American option, the operator splitting method is adopted which split linear complementary problem into two simple equations. The numerical results confirm the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The Kennedy model offers a robust framework for modeling forward rates, leveraging Gaussian random fields to accommodate emerging phenomena such as negative rates. In our study, we employ maximum likelihood estimations to determine the parameters of the Kennedy field, utilizing Radon–Nikodym derivatives for enhanced accuracy. We introduce an efficient simulation method for the Kennedy field and develop a Black–Scholes-like analytical pricing formula for diverse financial assets. Additionally, we present a novel parameter estimation algorithm grounded in numerical extreme value optimization, enabling the recalibration of parameters based on observed financial product prices. To validate the efficacy of our approach, we assess its performance using real-world par swap rates in the latter part of this article. [ABSTRACT FROM AUTHOR]

In this paper, we present a generalized adaptive Monte Carlo algorithm using the Diagonal and Off-Diagonal Splitting (DOS) iteration method to solve a system of linear algebraic equations (SLAE). The DOS method is a generalized iterative method with some known iterative methods such as Jacobi, Gauss-Seidel, and Successive Overrelaxation methods as its special cases. Monte Carlo algorithms usually use the Jacobi method to solve SLAE. In this paper, the DOS method is used instead of the Jacobi method which transforms the Monte Carlo algorithm into the generalized Monte Carlo algorithm. we establish theoretical results to justify the convergence of the algorithm. Finally, numerical experiments are discussed to illustrate the accuracy and efficiency of the theoretical results. Furthermore, the generalized algorithm is implemented to price options using the finite difference method. We compare the generalized algorithm with standard numerical and stochastic algorithms to show its efficiency. [ABSTRACT FROM AUTHOR]

Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model. [ABSTRACT FROM AUTHOR]

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In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.

A barrier option is a kind of path-dependent option whose return depends on whether the price of the underlying asset reaches a certain barrier level. This paper mainly analyzes European barrier option pricing formulas for the uncertain exponential Ornstein–Uhlenbeck model with a floating interest rate. The corresponding numerical algorithms for the knock-in and knock-out option prices are designed. Several numerical examples are given to study the relationship between barrier option prices and parameters. Finally, a real-data example is presented to illustrate the option pricing formulas.

Considering the characteristics of long-range correlations in financial markets, the issue of valuing geometric average Asian options is examined, assuming that the variations of the underlying asset follow the mixed sub-fractional Brownian motion, and the dynamics of short-term interest rate satisfies the mixed sub-fractional Vasicek model. Based on the principle of no arbitrage, the definite solution of PDE of a zero-coupon bond for geometric average Asian options under the circumstance of the mixed sub-fractional is given by the delta hedging technique. The derivation of the explicit pricing formula for geometric average Asian options with fixed strike price is achieved through the utilization of multiple variable substitutions. Furthermore, we perform numerical calculations to analyze the performance of the model.

In this paper, a hybrid method for solving time-fractional Black-Scholes equation is introduced for option pricing. The presented method is based on time and space discretization. A second order finite difference formula is used to time discretization and space discretization is done by a spectral method based on Chelyshkov wavelets and an operational process by defining Chelyshkov wavelets operational matrices. Convergence and error analysis for Chelyshkov wavelets approximation and also for the proposed method are discussed. The method is validated and its accuracy, convergency and efficiency are demonstrated through some cases with given accurate solutions. The method is also utilize for pricing various European options conducted by a time-fractional Black-Scholes model

In this study, we derived pricing formulas for various forward-start style exotic options based on an uncertain stock models with periodic dividends. Specifically, we present valuations for forward-start, Cliquet/Ratchet, and spread options. In addition, we conducted numerical simulations of these formulas and compared them to pricing formulas for the same options based on a dividend-paying stock model driven by standard Brownian motion.

‎This study suggests a novel approach for calibrating European option pricing model by a hybrid model based on the optimized artificial neural network and Black-Scholes model‎. ‎In this model‎, ‎the inputs of the artificial neural network are the Black-Scholes equations with different maturity dates and strike prices‎. ‎The presented calibration process involves training the neural network on historical option prices and adjusting its parameters using the Levenberg-Marquardt optimization algorithm‎. ‎The resulting hybrid model shows superior accuracy and efficiency in option pricing on both in sample and out of sample dataset‎.

This article proposes a new numerical technique for pricing asset-or-nothing options using the Black-Scholes partial differential equation (PDE). We first use the θ−weighted method to discretize the time domain, and then use Haar wavelets to approximate the functions and derivatives with respect to the asset price variable. By using some vector and matrix calculations, we reduce the PDE to a system of linear equations that can be solved at each time step for different asset prices. We perform an error analysis to show the convergence of our technique. We also provide some numerical examples to compare our technique with some existing methods and to demonstrate its efficiency and accuracy.

In this study, we propose to determine option pricing by using Black-Scholes model numerically. The Keller box method, a numerical method with a box-shaped implicit scheme, is chosen to solve the problem of pricing stock options, especially European-put option. This option pricing involves several parameters such as stock price volatility, risk-free interest rate and strike price. The numerical stability of the method is checked using Von Neumann stability before the simulation is conducted. The influence of interest rates, volatility, and strike price on the option price state that the higher the value of the interest rate parameter, the lower the option price value, while the greater the value of stock price volatility and strike price, the higher the option price.

An analytical derivation of the conditional moment-generating function (MGF) for a regime-switching nonlinear drift constant elasticity of variance process is established. The proposed model incorporates both regime-switching mechanisms and nonlinear drift components to better capture market phenomena such as volatility smiles and leverage effects. Regime-switching models can match the tendency of financial markets to often change their behavior abruptly and the phenomenon that the new behavior of financial variables often persists for several periods after such a change. Closed-form formulas for the MGF under various conditions, which are then applied for option pricing, are also derived. The efficacy and accuracy of the results are validated through a discrete Markov chain simulation. The results obtained from the proposed formulas completely match with those from MC simulations, while requiring significantly less computational time. [ABSTRACT FROM AUTHOR]

In this study, we derived pricing formulas for various forward-start style exotic options based on an uncertain stock models with periodic dividends. Specifically, we present valuations for forward-start, Cliquet/Ratchet, and spread options. In addition, we conducted numerical simulations of these formulas and compared them to pricing formulas for the same options based on a dividendpaying stock model driven by standard Brownian motion. [ABSTRACT FROM AUTHOR]

We present a unified, market-complete model that integrates both Bachelier and Black–Scholes–Merton frameworks for asset pricing. The model allows for the study, within a unified framework, of asset pricing in a natural world that experiences the possibility of negative security prices or riskless rates. Unlike the classical Black–Scholes–Merton, we show that option pricing in the unified model differs depending on whether the replicating, self-financing portfolio uses riskless bonds or a single riskless bank account. We derive option price formulas and extend our analysis to the term structure of interest rates by deriving the pricing of zero-coupon bonds, forward contracts, and futures contracts. We identify a necessary condition for the unified model to support a perpetual derivative. Discrete binomial pricing under the unified model is also developed. In every scenario analyzed, we show that the unified model simplifies to the standard Black–Scholes–Merton pricing under specific limits and provides pricing in the Bachelier model limit. We note that the Bachelier limit within the unified model allows for positive riskless rates. The unified model prompts us to speculate on the possibility of a mixed multiplicative and additive deflator model for risk-neutral option pricing. [ABSTRACT FROM AUTHOR]

In financial computation, Field Programmable Gate Arrays (FPGAs) have emerged as a transformative technology, particularly in the domain of option pricing. This study presents the impact of Field Programmable Gate Arrays (FPGAs) on computational methods in finance, with an emphasis on option pricing. Our review examined 99 selected studies from an initial pool of 131, revealing how FPGAs substantially enhance both the speed and energy efficiency of various financial models, particularly Black–Scholes and Monte Carlo simulations. Notably, the performance gains—ranging from 270- to 5400-times faster than conventional CPU implementations—are highly dependent on the specific option pricing model employed. These findings illustrate FPGAs' capability to efficiently process complex financial computations while consuming less energy. Despite these benefits, this paper highlights persistent challenges in FPGA design optimization and programming complexity. This study not only emphasises the potential of FPGAs to further innovate financial computing but also outlines the critical areas for future research to overcome existing barriers and fully leverage FPGA technology in future financial applications. [ABSTRACT FROM AUTHOR]

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems. [ABSTRACT FROM AUTHOR]

The paper suggests a hybrid numerical method to price barrier options under Levy processes. As the main ingredient of our approach, we model the values of the Wiener-Hopf factors using artificial neural networks in the exact formula for the solution. The numerical Wiener-Hopf factorization typically reduces the problem to the factorization of the polynomial of exp(iξ), which is interpreted as the characteristic function of the random variable that approximates the Lévy process at the exponentially distributed time moment. We design and train a feedforward neural network with one hidden layer that approximates the coefficients of factors based on the input vector of the factorized polynomial coefficients. We implemented in the software a training data generator and a generalized loss function to factorize a polynomial of arbitrary degree. We demonstrate the performance of our approach using examples of factorization of second-, sixth- and 254th-degree polynomials. It takes a fraction of a second for our trained artificial neural networks to calculate the factors. [ABSTRACT FROM AUTHOR]

We address the stock predictability puzzle, a challenge in the stock market often discussed in behavioral finance. Our approach formulates a statistical model within rational finance, avoiding reliance on behavioral finance assumptions, and integrates stock return predictability into the Black–Scholes option pricing framework. Empirical analysis focuses on the predictability of stock prices by option and spot traders, introducing a forward‐looking measure we term "implied excess predictability." Results show that option traders' predictability of stock returns positively correlates with moneyness, whereas for spot traders, this relationship is inverse. These findings suggest a potential asymmetry in stock price predictability between spot and option traders. Additionally, we demonstrate the importance of incorporating stock return predictability into option pricing formulas, particularly for options with strike prices significantly different from the stock price. Conversely, when moneyness is close to unity, predictability is not integrated into option pricing, indicating equal information among spot and option traders. Comparison of volatility measures reveals the difference between implied and realized variances or variance risk premia as potential predictors of stock returns. [ABSTRACT FROM AUTHOR]

In this paper, a hybrid method for solving time-fractional Black-Scholes equation is introduced for option pricing. The presented method is based on time and space discretization. A second order finite difference formula is used to time discretization and space discretization is done by a spectral method based on Chelyshkov wavelets and an op- erational process by defining Chelyshkov wavelets operational matrices. Convergence and error analysis for Chelyshkov wavelets approximation and also for the proposed method are discussed. The method is validated and its accuracy, convergency and efficiency are demonstrated through some cases with given accurate solutions. The method is also utilize for pricing various European options conducted by a time-fractional Black- Scholes model. [ABSTRACT FROM AUTHOR]

The accuracy and efficiency for computing option prices play very important in the financial risk management and hedging for the investors. In this paper, we for the first time develop a fast and accurate numerical method that combines Laplace transform for time variable and compact difference for spatial discretization, for computing option prices governed by the partial integro-differential equation system under the regime-switching jump-diffusion models. We then invert the Laplace transform through the numerical contour integral to recover the option prices. Furthermore, we prove that the method is convergent in the discrete L 2 and L ∞ norms with fourth-order in space and exponential-order with respect to the quadrature nodes for the numerical Laplace inversion. Finally, several numerical examples are reported to illustrate the convergence theory and show the advantages of the method over the benchmarks in regards to the accuracy and efficiency. [ABSTRACT FROM AUTHOR]

We developed a new method to price double barrier options under pure non-Gaussian Lévy processes admitting jumps of unbounded variation. In our approach, we represent the underlying process as a sequence of spectrally positive, negative, and again positive jumps, with spectrally positive jumps corresponding to half of the time moment. This rule is applied to the increments of the Lévy process at exponentially distributed time points. The justification for the convergence of the method in time is carried out using the interpretation of time randomization as the numerical Laplace transform inversion with the Post-Widder formula. The main algorithm consists of the recurrent calculation of the sufficient simple expectations of the intermediate price function depending on the position of the extremum of spectrally positive or negative part of the underying Lévy process at a randomized time moment. It corresponds to a sequence of problems for integro-differential equations on an interval, each of which is solved semi-explicitly by the Wiener–Hopf method. We decompose the characteristic function of a randomized spectrally positive (negative) process as the product of two Wiener–Hopf factors, the first of which corresponds to the exponential distribution and the second is calculated as a ratio. The parameter of the exponential distribution is numerically found as the only root of the factorizing operator symbol using Newton's method. Thus, by combining the sequential application of the Wiener–Hopf operators in explicit form with the characteristic function of our interval, we obtain the solution to each auxiliary problem. The main advantage of the suggested methods is that being very simple for programming it makes it possible to avoid dealing with Wiener–Hopf matrix factorization or solving systems of coupled nontrivial Wiener–Hopf equations that require application of tricky approximate factorization formulas. [ABSTRACT FROM AUTHOR]

This paper values fixed-income (discrete- and continuous-time) European Asian and Australian options. We assume that the term structure of interest rates is modelled by the specification proposed in Moreno et al. (Econ Model 72:140–150, 2018, https://doi.org/10.1016/j.econmod.2018.01.015). We obtain closed-form expressions for the premiums of geometric average options and, for arithmetic average options, premiums are computed by numerical methods. We also perform a sensitivity analysis with respect to different parameters for both (geometric and arithmetic) options. [ABSTRACT FROM AUTHOR]

We present a geometric approach to portfolio theory with a focus on risk-adjusted returns, in particular Jensen's alpha. We find that while the alpha/beta approach has severe limitations, especially in higher dimensions, only minor conceptual modifications (e.g., using orthogonal Sharpe ratios rather than risk-adjusted returns) are needed to identify the efficient trading strategies. We further show that, in a complete market, the so-called market price of risk vector is identical to the growth optimal Kelly vector, albeit expressed in coordinates of a different basis. This implies that a derivative, having an orthogonal Sharpe ratio of zero, has a price given by the minimal martingale measure. [ABSTRACT FROM AUTHOR]

We introduce a novel stochastically correlated two-factor (i.e., bivariate) diffusion process under the square-root format, for which we analytically obtain the corresponding solutions for the conditional moment-generating functions and conditional characteristic functions. Such solutions recover verbatim those of the uncorrelated case which encompasses a range of processes similar to those produced by a bivariate square-root process in which entries are correlated in the standard way, that is, via a constant correlation coefficient. Note that closed-form solutions for the conditional characteristic and moment-generating functions are not available for the latter. We focus on the financial scenario of obtaining closed-form expressions for the exact price of a zero-coupon bond and Asian option prices using a Fourier cosine series method. [ABSTRACT FROM AUTHOR]

While asymmetric mixture models improve option pricing over generic pricing models, mispricing remains due to their inability to capture the effect of economic factors on price levels. This paper uses the hidden truncation normal (HTN) distribution introduced by Arnold et al. (1993) and the NGARCH model of Engle and Ng (J Finance, 48:1749–1778, 1993) to price options. Compared to the Black–Scholes model, the HTN -NGARCH option pricing model has extra parameters linked to economic dynamics and with economic interpretations. The model integrates some stylized facts underlying option prices such as a time-varying price of risk, non-normal innovations, asymmetry, and kurtosis. The model can be estimated by maximum likelihood. With an application to market data, we show that the HTN -NGARCH model accurately prices index options and captures adequately the smirk of implied volatility. [ABSTRACT FROM AUTHOR]

The recently introduced deep parametric PDE method combines the efficiency of deep learning for high-dimensional problems with the reliability of classical PDE models. The accuracy of the deep parametric PDE method is determined by the best-approximation property of neural networks. We provide (to the best of our knowledge) the first approximation results, which feature a dimension-independent rate of convergence for deep neural networks with a hyperbolic tangent as the activation function. Numerical results confirm that the deep parametric PDE method performs well in high-dimensional settings by presenting in a risk management problem of high interest for the financial industry. [ABSTRACT FROM AUTHOR]

We present a new term-structure model for commodity futures prices based on Trolle and Schwartz (2009), which we extend by incorporating seasonal stochastic volatility represented with two different sinusoidal expressions. We obtain a quasi-analytical representation of the characteristic function of the futures log-prices and closed-form expressions for standard European options' prices using the fast Fourier transform algorithm. We price plain vanilla options on the Henry Hub natural gas futures contracts, using our model and extant models. We obtain higher accuracy levels with our model than with the extant models. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the convergence rates of an implicit-explicit (IMEX) difference scheme for solving a two-dimensional partial integro-differential equation (PIDE) with an inexact boundary which arises in option pricing with stochastic intensity jumps. First, the IMEX scheme is proposed to solve the two-dimensional PIDE and its inexact boundary governed by a one-dimensional PIDE. Then the second-order convergence rates of the IMEX scheme for the main PIDE are proved for both time and space based on the second-order convergence analysis in the discrete H 1 -norm of the IMEX scheme for the boundary PIDE. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This study introduces an adapted Fourier-cosine series (COS) method that focuses on numerically solving characteristic functions linked to interest rate processes. The adaptation extends to encompass models within the affine jump-diffusion niche to assess the impact of different probability distributions on path-dependent option prices, with emphasis on the influence of stochastic volatility models on skewness and kurtosis. This study leverages the COS method, modified to numerically address characteristic functions linked to interest rate processes, to calculate the price of path-dependent derivatives. It investigates diverse models within the affine jump-diffusion framework, encompassing elements such as stochastic volatility, jumps, and correlated Brownian motion. An innovative approach is introduced, wherein the characteristic function is generated from the integral of the interest rate, as opposed to the interest rate itself. The research generated notable findings, highlighting the adaptability and effectiveness of the modified COS method. This significantly expands the range of applicable models for those with analytically unsolved characteristic functions. Remarkably, even in cases with analytically solvable characteristic functions, an unexpectedly low number of terms can accurately priced options. This study introduces original contributions by adapting the COS method to address the characteristic functions associated with interest rate processes. The distinct approach of generating the characteristic function from the interest rate integral, rather than the interest rate itself, is a substantial original contribution. The application of Kibble's bivariate gamma probability distribution to correlate interest rates and volatility jump sizes further enhances the originality of this research. [ABSTRACT FROM AUTHOR]

We develop a binary tree pricing model with underlying asset price dynamics following Itô–McKean skew Brownian motion. Our work was motivated by the Corns–Satchell, continuous-time, option pricing model. However, the Corns–Satchell market model is incomplete, while our discrete-time market model is defined in the natural world, extended to the risk-neutral world under the no-arbitrage condition where derivatives are priced under uniquely determined risk-neutral probabilities, and is complete. The skewness introduced in the natural world is preserved in the risk-neutral world. Furthermore, we show that the model preserves skewness under the continuous-time limit. We provide empirical applications of our model to the valuation of European put and call options on exchange-traded funds tracking the S&P Global 1200 index. [ABSTRACT FROM AUTHOR]