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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]

In this work, we give a thorough analysis of a weak Galerkin finite element technique (WG-FEM) designed specifically for the generalized Black-Scholes equation-based numerical calculation of option price valuation. This novel method offers a strong and adaptable framework for handling the complexity of option pricing models by combining the weak Galerkin method for spatial discretization with the backward-Euler scheme for time discretization. In this study, we aim to establish stability and optimal order error estimates and we obtain these error estimates by a rigorous theoretical study, proving the stability and convergence characteristics of the proposed WG finite element scheme. We perform extensive numerical experiments to validate our theoretical results. These examples are carefully chosen to demonstrate the WG-FEM practical efficacy and dependability in various scenarios and parameter settings. [ABSTRACT FROM AUTHOR]

One of the most discussed problems in the financial world is stock option pricing. The Black-Scholes equation is a parabolic partial differential equation which provides an option pricing model. The present work proposes an approach based on neural networks to solve the Black-Scholes equations. Real-world data from the stock options market were used as the initial boundary to solve the Black-Scholes equation. In particular, times series of call options prices of Brazilian companies Petrobras and Vale were employed. The results indicate that the network can learn to solve the Black-Scholes equation for a specific real-world stock options time series. The experimental results showed that the neural network option pricing based on the Black-Scholes equation solution can reach an option pricing forecasting more accurate than the traditional Black-Scholes analytical solutions. The experimental results making it possible to use this methodology to make short-term call option price forecasts in options markets. [ABSTRACT FROM AUTHOR]

This work introduces a new stochastic volatility model with delay parameters in the volatility process, extending the Barndorff–Nielsen and Shephard model. It establishes an analytical expression for the log price characteristic function, which can be applied to price European options. Empirical analysis on S&P500 European call options shows that adding delay parameters reduces mean squared error. This is the first instance of providing an analytical formula for the log price characteristic function in a stochastic volatility model with multiple delay parameters. We also provide a Monte Carlo scheme that can be used to simulate the model. [ABSTRACT FROM AUTHOR]

Abstract.The Asian-barrier option is an effective hedging instrument that depends on the price correlation between the underlying asset and a predefined threshold over a specified date. This article initially investigates the pricing issues of two types of Asian knock-in options for Liu’s stock model with floating interest rates. Corresponding numerical algorithms are also designed for option evaluation. Subsequently, financial market data are analyzed to estimate the model’s parameters using moment estimation based on the residuals. Uncertain hypothesis test is conducted to assess the effectiveness of the fitted model. Finally, the results are validated through numerical experiments. It is noteworthy that a paradox related to the stochastic differential equation is discussed in Appendix A. [ABSTRACT FROM AUTHOR]

We discuss the joint calibration to SPX and VIX options of an affine Stochastic Volatility model with Jumps in price and Jumps in volatility (the SVJJ model). Conventionally, the SVJJ model assumes exponential jumps in the variance process, leaving the potential benefits of more flexible jump distributions unexplored. The purpose of our study is twofold. First, we show that choosing the gamma distributions for the jumps in variance significantly improves the performance of the joint calibration. However, this improvement comes at the cost of increased computational time. Second, we mitigate this loss of tractability by constructing novel approximations to option prices based on orthogonal polynomial expansions. Unlike the classical method of selecting an explicit reference density, our approach generalizes to all densities with explicit Laplace transform. We apply this methodology to the SVJJ model with gamma jumps and we find that the proposed price expansions achieve the same accuracy as exact transform inversion formulas while requiring only a fraction of the computational time. [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]

The valuation of options holds significant importance within financial markets, and among the various types of options, barrier options stand out. These options exhibit a unique characteristic where the option contract's worthiness is contingent upon the underlying asset price reaching a predetermined level. The focus of this paper is to investigate the pricing of Asian barrier options in an uncertain environment. Specifically, it explores the pricing of these options when the underlying asset price follows the exponential Ornstein–Uhlenbeck Model. Additionally, the paper derives pricing formulas for four types of Asian barrier options (knock-in and knock-out) by utilizing α -paths of uncertain differential equations in this uncertain environment. [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]

We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option. [ABSTRACT FROM AUTHOR]

In this paper, we present a new precipitation model based on a multi-factor Ornstein-Uhlenbeck approach of pure-jump type. In this setup, we derive a representation for the related precipitation swap price process and infer its risk-neutral time dynamics. We further deduce a pricing formula for European options written on the precipitation swap and obtain the minimal variance hedging portfolio in the underlying weather market. In the second part of the paper, we provide a precipitation swap price representation under future information modeled by an initially enlarged filtration. We finally derive a formula for the associated information premium and investigate minimal variance hedging of precipitation derivatives under future information. [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 paper, we develop a 'jump diffusion type' financial model based on renewal processes for the discontinuous part of the risk driver, and study its ability to price options and accurately reproduce the corresponding implied volatility surfaces. In this model, the log-returns process displays finite mean and variance, and non-vanishing skewness and excess kurtosis over a long period. The proposed construction is parsimonious, and it allows for a simple and intuitive pricing formula for European vanilla options; furthermore, it offers an efficient Monte Carlo sample path generator for the pricing of exotic options. We illustrate the performance of the proposed framework using observed market data, and we study the features of the best fitting model specifications. [ABSTRACT FROM AUTHOR]

This article deals with an European option pricing via proportional transaction costs in the incomplete environment with and without arbitrage opportunities under two long memory versions of the Heston model. Observing and introducing a traded proxy for the volatility in the modern market, we use the conditional expectation and the delta hedging strategies and present the generalized fractional Ito formula to obtain the option price partial differential equations (PDEs). To solve these PDEs, we apply the finite difference method and employ the K-antithetic variates algorithm based on the Monte-Carlo simulation as a benchmark for this method. Finally, we provide numerical results to illustrate the effectiveness of the proposed model. [ABSTRACT FROM AUTHOR]

This paper proposes a fourth-order numerical scheme for solving multi-asset nonlinear variable coefficient Black–Scholes PDEs. First, using the Newton linearization technique, we linearize the given nonlinear PDE, and obtain a sequence of linear PDEs. Then, we discretize the time derivative by the Crank–Nicolson scheme, and the resultant semi-discrete problem by the central difference scheme on uniform meshes. In order to enhance the order of convergence of the proposed scheme, we use the Richardson extrapolation method, by solving the fully discrete problem on two different meshes. The stability and convergence are studied. To validate the proposed technique, several numerical experiments are carried out. [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.

In this article we will propose a novel, self-financing, dynamic and path dependent portfolio trading strategy which is based on the well known principle "sell high – buy low". Trading strategies are important also for the hedging problem selling/buying an option. The main problem of the writer of an option is how to invest the amount that she has received selling the option therefore the proposed trading strategy play an important role here. We will see that the hedging problem reduces to an optimization one and therefore the portfolio optimization and the hedging problem are closely related. We will also propose a deterministic portfolio selection method (i.e., without making any assumption-guess about the assets) and a notion of a deterministic fair price of an option. [ABSTRACT FROM AUTHOR]

We introduce a Path Shadowing Monte Carlo method, which provides the prediction of future paths, given any generative model. At any given date, it averages future quantities over generated price paths whose past history matches, or 'shadows', the actual (observed) history. We test our approach using paths generated from a maximum entropy model of financial prices, based on a recently proposed multi-scale analogue of the standard skewness and kurtosis called 'Scattering Spectra'. This model promotes the diversity of generated paths while reproducing main statistical properties of financial prices, including stylized facts such as volatility roughness. Our method yields state-of-the-art predictions for future realized volatility and allows one to determine conditional option smiles for the S&P500 that outperform both the most recent low-parametric models and the option market itself. The code is available at (This work is supported by the PRAIRIE 3IA Institute of the French ANR-19-P3IA-0001 program and the ENS-CFM models and data science chair.). [ABSTRACT FROM AUTHOR]

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]