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Fuzzy random option pricing in continuous time (FROPCT) has emerged as an active research field over the past two decades; thus, there is a need for a comprehensive review that provides a broad perspective on the literature and identifies research gaps. In this regard, we conducted a structure review of the literature by using the WoS and SCOPUS databases while following the PRISMA criteria. With this review, we outline the primary research streams, publication outlets, and notable authors in this domain. Furthermore, the literature review revealed a lack of advancements for the equilibrium models of the yield curve. This finding serves as a primary motivation for the second contribution of this paper, which involves an extension of Vasicek's yield curve equilibrium model. Specifically, we introduce the existence of fuzzy uncertainty in the parameters governing interest rate movements, including the speed of reversion, equilibrium short-term interest rate, and volatility. By incorporating fuzzy uncertainty, we enhance the model's ability to capture the complexities of real-world interest rate dynamics. Moreover, this paper presents an empirical application of the proposed extension to the term structure of fixed-income public bonds in European Union. The empirical analysis suggests the suitability of the proposed extension of Vasicek's model for practical applications. [ABSTRACT FROM AUTHOR]

This paper studies the efficient methods for option pricing under multivariate rough volatility models. The characteristic functions of the asset log-price, which play important role in the option pricing under the multivariate rough volatility models, are determined by a system of parametric nonlinear fractional Riccati equations. This paper obtains the results on the existence, uniqueness and regularity of the solutions to the parametric nonlinear fractional Riccati equations, proposes a high-order scheme to solve the system and proves the high-order convergence. The option pricing problem is solved by the Fourier-cosine formula with the fast approximation of the characteristic functions. Numerical examples are carried out to confirm the theoretical results and show efficiency of the methods. [ABSTRACT FROM AUTHOR]

Based on multiquadric quasi-interpolation, this study presents a meshless numerical method for time fractional Black–Scholes (B–S) model. The method is highly accurate and flexible. The stability and convergence of the proposed scheme are discussed in detail. Numerical results show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Proper pricing of options in the financial derivative market is crucial. For many options, it is often impossible to obtain analytical solutions to the Black–Scholes (BS) equation. Hence an accurate and fast numerical method is very beneficial for option pricing. In this paper, we use the Physics-Informed Neural Networks (PINNs) method recently developed by Raissi et al. (J Comput Phys 378:686–707, 2019) to solve the BS equation. Many experiments have been carried out for solving various option pricing models. Compared with traditional numerical methods, the PINNs based method is simple in implementation, but with comparable accuracy and computational speed, which illustrates a promising potential of deep neural networks for solving more complicated BS equations. [ABSTRACT FROM AUTHOR]

The celebrated Heston's stochastic volatility (SV) model for the valuation of European options provides closed form solutions that are given in terms of characteristic functions. However, the numerical calibration of this five-parameter model, which is based on market option data, often remains a daunting task. In this paper, we provide a theoretical solution to the long-standing 'open problem' of characterizing the class of risk neutral distributions (RNDs), if any, that satisfy Heston's SV for option valuation. We prove that the class of scale parameter distributions with mean being the forward spot price satisfies Heston's solution. Thus, we show that any member of this class could be used for the direct risk neutral valuation of option prices under Heston's stochastic volatility model. In fact, we also show that any RND with mean being the forward spot price that satisfies Heston's option valuation solution must also be a member of the scale family of distributions in that mean. As particular examples, we show that under a certain re-parametrization, the one-parameter versions of the log-normal (i.e., Black–Scholes), gamma, and Weibull distributions, along with their respective inverses, are all members of this class and thus, provide explicit RNDs for direct option pricing under Heston's SV model. We demonstrate the applicability and suitability of these explicit RNDs via exact calculations and Monte Carlo simulations, using already published index data and a calibrated Heston's model (S&P500, ODAX), as well as an illustration based on recent option market data (AMD). [ABSTRACT FROM AUTHOR]

Copyright of Electric Power Automation Equipment / Dianli Zidonghua Shebei is the property of Electric Power Automation Equipment Press and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We present a method for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, which is based on a system of integral equations that relates terminal density and option prices. Using a discretization of the terminal density, we write these integral equations as a system of linear equations. We show that the kernel matrix of this system is, in general, ill-conditioned, so that it cannot be solved for the discretized density using a naive approach. Instead, we construct a sparse model for the kernel matrix using singular value decomposition (SVD), which allows us not only to systematically improve the condition number of the kernel matrix, but also determines the computational effort and accuracy of our method. In order to allow for the treatment of realistic inputs that may contain arbitrage, we reformulate the system of linear equations as an optimization problem, in which the SVD-transformed density minimizes the error between the input prices and the arbitrage-free prices generated by our method. To further stabilize the method in the presence of noisy input prices or arbitrage, we apply an L 1 -regularization to the SVD-transformed density. Our approach, which is inspired by recent progress in theoretical physics, offers a flexible and efficient framework for the arbitrage-free interpolation of plain-vanilla option prices and implied volatilities, without the need to explicitly specify a stochastic process, expansion basis functions or any other kind of model. We demonstrate the capabilities of our method in a number of artificial and realistic test cases. [ABSTRACT FROM AUTHOR]

In the paper, we prove probabilistic analogs of the universal approximation theorems and link continuous random variables of a certain type and monotonic feedforward artificial neural networks with one-dimensional input, output and one hidden layer. In particular, we show that any continuous infinitely divisible random variable can be successfully approximated with a mix of logistic distributions. Based on the theorems proved in the current paper, we develop a new approach for developing Monte Carlo methods combined with artificial neural networks for pricing options in Lévy models. In contrast to straightforward incorporation of neural networks into Monte Carlo methods, we approximate the cumulative distribution function, but not its inverse. Moreover, we give a clear probabilistic interpretation of the constructed approximator that helps us simulate the Lévy process by using only separate components of our neural network. [ABSTRACT FROM AUTHOR]

In this paper, we present a double exponential jump-diffusion option pricing model with stochastic interest rates, stochastic volatility, and stochastic jump intensity. In addition, Markov regime-switching is introduced to modulate the mean-reverting level of the squared volatility. We obtain the analytical pricing formulae for European options under this model. Finally, we use numerical examples to explore the effects of the regime-switching, stochastic jump intensity and the distribution of jump size on the option price or (and) the implied volatility. [ABSTRACT FROM AUTHOR]

The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most 2 (N − 1) times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility in Gaussian mixtures under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N = 3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima. [ABSTRACT FROM AUTHOR]

Recent studies have demonstrated the efficiency of Variational Autoencoders (VAE) to compress high-dimensional implied volatility surfaces into a low dimensional representation. Although this method can be effectively used for pricing vanilla options, it does not provide any explicit information about the dynamics of the underlying asset. In our work we present an effective way to overcome this problem. We use a Weighted Monte Carlo approach to first generate paths from a simple a priori Brownian dynamics, and then calculate path weights to price options correctly. We develop and successfully train a neural network that is able to assign these weights directly from the latent space. Combining the encoder network of the VAE and this new "weight assigner" module, we are able to build a dynamic pricing framework which cleanses the volatility surface from irrelevant noise fluctuations, and then can price not just vanillas, but also exotic options on this idealized vol surface. This pricing method can provide relative value signals for option traders. Key findings of the paper: We confirmed the success of Variational Autoencoders in compressing volatility surfaces into low dimensional latent spaces in the swaption market. We successfully extracted the physically meaningful information from the latent representation of the market by translating the latent coordinates into weights of Monte Carlo paths. We developed a hybrid approach by combining Deep Learning and Weighted Monte Carlo that can price any type of option consistently with the latent model represented by the fully trained VAE. [ABSTRACT FROM AUTHOR]

A new model which we called time driven model for option pricing is proposed. In this model, the price of underlying asset is driven by different random driving source in different time interval. Explicit option pricing formulas for the European call option are obtained by using the Mellin transform. Moreover, some numerical illustrations are given by computing European call option prices. [ABSTRACT FROM AUTHOR]

In this paper, we develop asymptotic Asian option hedging methods for the Black-Scholes markets with transaction costs. We first construct classical replication strategies and then, using the Leland approach, propose corresponding modifications for the financial markets with proportional transaction costs. Sufficient conditions are found on the transaction costs implying the asymptotic hedging for the constructed strategies. The pricing problem is also considered. Three cases are studied: the case where the option price is the same as for the hedging problem without transaction costs, the case of increasing volatility, and the case where the option price equals the option price of the "buy and hold" strategy for European call options. [ABSTRACT FROM AUTHOR]

Venture capital (VC) often involves complex equity contracts with so-called preferential rights affecting the allocation of exit proceeds among different share classes and investors. We structure exit-relevant preferential rights in a two-dimensional framework and develop a contingent claims model that allows for ex-ante valuation of separate shareholdings. The model generates insights on the valuation effects of varying setups in VC financing and indicates considerable mispricing potential of VC investments when applying commonly used heuristics such as the most recent funding round. Applying the model to a sample of ventures indicated an average 'overvaluation' on a per-share basis of 26.7 % , with common stocks and early-stage investments being the most affected. In addition, our analysis provides different implications regarding the effects of preferential right structuring for early and late stage investors. [ABSTRACT FROM AUTHOR]

A numerical method based on the Haar wavelet is introduced in this study for solving the partial differential equation which arises in the pricing of European options. In the first place, and due to the change of variables, the related partial differential equation (PDE) converts into a forward time problem with a spatial domain ranging from 0 to 1. In the following, the Haar wavelet basis is used to approximate the highest derivative order in the equation concerning the spatial variable. Then the lower derivative orders are approximated using the Haar wavelet basis. Finally, by substituting the obtained approximations in the main PDE and doing some computations using the finite differences approach, the problem reduces to a system of linear equations that can be solved to get an approximate solution. The provided examples demonstrate the effectiveness and precision of the method. [ABSTRACT FROM AUTHOR]

This paper proposes a pricing model for equity warrants under the sub-mixed fractional Brownian motion regime with the interest rate following the Merton short rate model. By using the delta hedging strategy, the corresponding partial differential equations for equity warrants are obtained. Moreover, the explicit pricing formula for equity warrants and some numerical results are given.

In this work we rigorously establish mathematical models to obtain the capital valuation adjustment (KVA) as part of the total valuation adjustments (XVAs). For this purpose, we use a semi-replication strategy based on market theory. We formulate single-factor models in terms of expectations and PDEs. For PDEs formulation, we rigorously obtain the existence and uniqueness of the solution, as well as some regularity and qualitative properties of the solution. Moreover, appropriate numerical methods are proposed for solving the corresponding PDEs. Finally, some examples show the numerical results for call and put European options and the corresponding XVA that includes the KVA. • New mathematical models for XVA including KVA are rigorously stated. • A PDE model for risky derivatives taking into account KVA is analyzed. • Numerical methods based on IMEX and LDG methods are proposed. • Numerical examples illustrate the performance of the models and numerical methods. [ABSTRACT FROM AUTHOR]

This study suggests the pricing problems of options dependent on multiple assets, spread, basket, and quanto options when the asset dynamics are described by the uncertain fractional differential equation. The solutions of these option prices are analytically provided and the algorithms related to each one of these derivatives are designed. For the first time, we apply the minimum cover method to estimate the parameters of the uncertain fractional differential equations based on the real data related to the stock prices of some markets. Through the uncertain hypothesis test, we demonstrate that the estimated uncertain fractional differential equations can successfully fit the observed data. We then experimentally show that the α -paths obtained by the estimated uncertain fractional differential equations favorably cover the sample data. Finally, some numerical experiments based on the uncertain fractional differential equation estimated by the minimum cover method are accomplished to confirm the achievement of the presented results. • Parameter estimation of uncertain fractional differential equations (UFDEs) by minimum cover method. • Option pricing written on multiple assets when the asset price dynamics are described by UFDEs. • Valuation of uncertain hypothesis test for UFDEs estimated by minimum cover method. [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.

The option is an important derivative tool in financial market, and after decades of development, the option has emerged in various forms. This paper studies an exotic option with a proactive investment strategy. Compared with the classic option theory, we assume that the option holders continuously trade the underlying assets according to a predetermined investment strategy. Taking the European call option as an example, we first give some assumptions and define the loss function in terms of a logarithmic investment strategy. Then, the specific pricing expression of the exotic option is derived from the Black-Scholes option pricing formula. Next, numerical simulations are presented to visualize the mechanism of the exotic option in 2D and 3D forms by selecting appropriate parameters. The results indicate that the exotic option has a significant price advantage (up to 43.9% under specific parameter settings) over the classic option. Moreover, the empirical results illustrate a perfect fit between 50 ETF (issued by the Shanghai Stock Exchange) and our exotic option. The new proposed exotic option extends the Black-Scholes option theory from a no-trading condition (do not buy or sell underlying assets during the validity period) to a dynamic investment condition and has important practical significance in real life.

The authors proved three theorems about the exact solutions of a generalized or interacting Black–Scholes equation that explicitly includes arbitrage bubbles. These arbitrage bubbles can be characterized by an arbitrage number AN. The first theorem states that if AN = 0, then the solution at maturity of the interacting equation is identical to the solution of the free Black–Scholes equation with the same initial interest rate of r. The second theorem states that if AN ≠ 0, then the interacting solution can be expressed in terms of all higher derivatives of the solutions to the free Black–Scholes equation with an initial interest rate of r. The third theorem states that for a given arbitrage number, the interacting solution is a solution to the free Black–Scholes equation but with a variable interest rate of r(τ) = r + (1/τ)AN(τ), where τ = T − t.

In this paper, a rapid and high accurate numerical method for pricing discrete single and double barrier knock‐out call options is presented. With regard to the well‐known Black‐Scholes model, the price of an option in each monitoring date could be calculated by computing a recursive integral formula that is based on the heat equation solution. We have approximated these recursive solutions with the aid of Lagrange interpolation on Jacobi polynomial nodes. After that, an operational matrix, which makes our computation significantly fast, has been derived. In some theorems, the convergence of the presented method has been shown and the rate of convergence has been derived. The most important benefit of this method is that its complexity is very low and does not depend on the number of monitoring dates. The numerical results confirm the accuracy and efficiency of the presented numerical algorithm. [ABSTRACT FROM AUTHOR]

In this paper, the valuation of European option is investigated when the discrete dividends are described by Markov-modulated Merton jump-diffusion process. According to the dividend discount theory, we regard the stock price as the net present value of all future dividends. The regime switching Esscher transform is applied to determine a risk-neutral measure. The closed form solution of European option is obtained under the condition that the dividend payments are announced in advance. Numerical simulations for the European call option prices are provided. [ABSTRACT FROM AUTHOR]

Mathematical difficulty remains in many classical financial problems, especially for a closed-form expression of asset value. The European option evaluation problem based on a regime-switching has been formally modeled since early 2000, for which a recursive algorithm was developed to solve it. The key mathematical difficulty of this problem relies on the expectation I E [ h (Y T) ] , where h is a payoff function and { Y t } t ∈ [ 0 , T ] denotes a geometric Brownian motion with Markovian regime-switching. It is long since attempted to conclude this problem with closed form formulas. Towards the same target, this paper applies some novel techniques to draw explicit formulas for cases with more states for regime-switching (whereas the former deals the cases with two states) for any integrable function h (whereas the former only applies to the payoff of a European option). This paper combines the technique of occupation time of Markov chains and inverse Laplace transform to achieve the density function of geometric Brownian motion with Markovian regime-switching. Extension along this technique creates potential for probabilistic computations in addition to European option pricing. The reflection from the inverse Laplace transform to the expression of a moment-generating function is the core technique developed by this paper, and it is presented in symmetric forms. [ABSTRACT FROM AUTHOR]

Preintegration is an extension of conditional Monte Carlo to quasi-Monte Carlo and randomized quasi-Monte Carlo. Conditioning can reduce but not increase the variance in Monte Carlo. For quasi-Monte Carlo it can bring about improved regularity of the integrand with potentially greatly improved accuracy. We show theoretically that, just as in Monte Carlo, preintegration can reduce but not increase the variance when one uses scrambled net integration. Preintegration is ordinarily done by integrating out one of the input variables to a function. In the common case of a Gaussian integral one can also preintegrate over any linear combination of variables. For continuous functions that are differentiable almost everywhere, we propose to choose the linear combination by the first principal component in an active subspace decomposition. We show that the lead eigenvector in an active subspace decomposition is closely related to the vector that maximizes a computationally intractable criterion using a Sobol' index. A numerical example of Asian option pricing finds that this active subspace preintegration strategy is competitive with preintegrating the first principal component of the Brownian motion, which is known to be very effective. The new method outperforms others on some basket and rainbow options where there is no generally accepted counterpart to the principal components construction. [ABSTRACT FROM AUTHOR]

By obtaining some new estimates for classical solutions of filtration equations and singular parabolic equations we derive consequences for fully nonlinear parabolic equations of dual filtration type, with nonlinearities that can be both singular and degenerate elliptic and also non-homogeneous. Such equations appear in the theory of option pricing with market impact. [ABSTRACT FROM AUTHOR]

We consider a parsimonious framework of jump-diffusion models for price dynamics with stochastic price volatilities and stochastic jump intensities in continuous time. They account for conditional heteroscedasticity and also incorporate key features appearing in financial time series of price volatilities and jump intensities, such as persistence of contemporaneous jumps (cojumps), mean reversion and feedback effects. More precisely, the stochastic variance and stochastic intensity are jointly modelled by a generalised bivariate shot-noise process sharing common jump arrivals with any non-negative jump-size distributions. This framework covers many classical and important models in the literature. The main contribution of this paper is that, we develop a very efficient scheme for its exact simulation based on perfect decomposition where neither numerical inversion nor acceptance/rejection scheme is required, which means that it is not only accurate but also the efficiency would not be sensitive to the parameter choice. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of this scheme. Our algorithm substantially outperforms the classical discretisation scheme. Moreover, we unbiasedly estimate the prices of discrete-barrier European options to show the applicability and flexibility of our algorithms. [ABSTRACT FROM AUTHOR]

We study the convergence of the binomial, trinomial, and more generally m-nomial tree schemes when evaluating certain European path-independent options in the Black–Scholes setting. To our knowledge, the results here are the first for trinomial trees. Our main result provides formulae for the coefficients of 1 / n and 1 / n in the expansion of the error for digital and standard put and call options. This result is obtained from an Edgeworth series in the form of Kolassa–McCullagh, which we derive from a recently established Edgeworth series in the form of Esseen/Bhattacharya and Rao for triangular arrays of random variables. We apply our result to the most popular trinomial trees and provide numerical illustrations. [ABSTRACT FROM AUTHOR]

A partial case of the Guéant–Pu model of options pricing, which takes into account transaction costs and the impact of operations on the market in the absence of a permanent market impact, is considered. Three recursion operators for the considered equation are found using P. Olver's criterion. An hierarchy of generalized symmetries for the model is constructed using the recursion operators. [ABSTRACT FROM AUTHOR]

Copyright of Acta Universitatis Lodziensis. Folia Oeconomica is the property of Wydawnictwo Uniwersytetu Lodzkiego and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Insurance guaranty schemes have been adopted to compensate policyholders for losses due to the insolvency of insurance companies. We derive explicit pricing formulas for risk-based premiums to focus on insurers' financial stability incorporating sudden changes of insurers' liabilities and asset-liability management (ALM) risk. The pricing formula of insurance guaranty fund is derived under regulatory forbearance. To deal with the insurance guaranty funds' payoff structure, this paper introduces new types of exchange options: early exchange options and barrier exchange options which allow multi-step boundaries. With these options, we can reflect jumps in value of underlying asset. Explicit pricing formulas are derived by using jump models with binomial approach. Also, we present sensitivity analysis on jump sizes, ALM risks and regulatory levels to provide guidance to financial authorities and insurers. [ABSTRACT FROM AUTHOR]

A fast and accurate explicit finite difference scheme for the Black–Scholes (BS) model with no far-field boundary conditions is proposed. The BS equation has been used to model the pricing of European options. The proposed numerical solution algorithm does not require far-field boundary conditions. Instead, the computational domain is progressively reduced one by one as the time iteration increases. A Saul'yev-type scheme for temporal discretization and non-uniform grids for the underlying asset variables are used. Because the scheme is stable, practically sufficiently large time steps can be applied. The main advantages of the proposed method are its speed, simplicity, and efficiency because it uses a stable explicit numerical scheme without using far-field boundary conditions. In particular, the proposed method is suitable for nonlinear boundary profiles such as power options because it does not require far-field boundary conditions. To validate the speed and efficiency of the proposed scheme, standard computational tests are performed. The computational test results confirmed the superior performance of the proposed method. [ABSTRACT FROM AUTHOR]

Option pricing under fuzzy environment is a hot research topic nowadays. Traditionally, option pricing were made in the case of fixed interest rate, while the uctuate of interest rate may result in profit loss or bring unexpected risk. Thus, based on credibility theory, a new option pricing model under fuzzy interest rate are constructed in this paper. In fact, almost all fuzzy option pricing uses expected value method. In this paper, a new pricing method, fuzzy term structure and fuzzy affine term structure method, is adopted, and two European call option pricing formulas are obtained, one is that the fuzzy interest rate coefficients are constants, the other is that the fuzzy interest rate drift coefficient is a fuzzy process. [ABSTRACT FROM AUTHOR]

In modern financial market, option is a very effective tool to hedge the risks brought by various uncertainties in real society. Therefore, it is of great significance to select an appropriate stock model to price options. To this aim, the paper presents a general stock model with fuzzy volatility for fuzzy financial market, that is, fuzzy constant elasticity of variance model. The advantage is that the fuzzy volatility of underlying stock is related to its price and can explain volatility smile. In addition, we consider the impact of elasticity coefficient on stock price and then limit the elasticity coefficient to a reasonable range. Subsequently, the European call and European put option pricing formulas are given, separately. Finally, some figures and tables are given to illustrate the impact of parameter changes on option prices. [ABSTRACT FROM AUTHOR]

We present a stochastic local volatility model for derivative contracts on commodity futures. The aim of the model is to recover the prices of derivative claims both on future contracts and on indices on future strategies. Numerical examples for calibration and pricing are provided for the S&P GSCI Crude Oil excess-return index. [ABSTRACT FROM AUTHOR]

Exponential Lévy processes provide a natural and tractable generalization of the classic Black–Scholes–Merton model which account for several stylized features of financial markets, including jumps and kurtosis. In the existing literature, closed-form option pricing formulas are sparse for exponential Lévy models, outside of special cases such as Merton's jump diffusion, and complex numerical techniques are required even to price European options. To bridge the gap, this work provides a comprehensive and unified pricing framework for vanilla and exotic path independent options under the Variance Gamma (VG), Finite Moment Log Stable (FMLS), one-sided Tempered Stable (TS), and Normal Inverse Gaussian (NIG) models. We utilize the Mellin Transform and residue calculus to obtain closed-form series representations for the price of several options, including vanillas (European), digitals, power, and log options. These formulas provide nice theoretical representations, but are also efficient to evaluate in practice, as numerous numerical experiments demonstrate. The closed-form nature of these option pricing formulas makes them ideal for adoption in practical settings, as they do not require complicated pricing methods to achieve high-accuracy prices, and the resulting pricing error is reliably controllable. [ABSTRACT FROM AUTHOR]

When approximating the expectations of a functional of a solution to a stochastic differential equation, the numerical performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and improve the regularity structure of the problem, we consider cases in which analytic smoothing (bias-free mollification) cannot be performed and introduce a novel numerical smoothing approach by combining a root-finding method with a one-dimensional numerical integration with respect to a single well-chosen variable. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e. the Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, focusing on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we demonstrate the advantages of combining numerical smoothing with the ASGQ and QMC methods over these methods without smoothing and the Monte Carlo approach. Finally, our approach is generic and can be applied to solve a broad class of problems, particularly approximating distribution functions, computing financial Greeks, and estimating risk quantities. [ABSTRACT FROM AUTHOR]

The use of pre-sale options for tickets is underutilized in sports. In this study, we show that the use of pre-sale options increases profits, but also increases consumer utility—making both parties better off. Thus, the efficiency gains and profit gains make incorporating options with advanced selling, as opposed to just advanced selling, an optimal strategy. Through this ticket pricing strategy, the organizer can realize a significant increase in profits from a separating equilibrium pricing strategy while simultaneously consumer welfare increases as fans know, with certainty, they have tickets to the game if their team participates. If these options were offered by participants instead of the organizer it also allows the participant to smooth their revenues over time. [ABSTRACT FROM AUTHOR]

Deep learning has drawn great attention in the financial field due to its powerful ability in nonlinear fitting, especially in the studies of asset pricing. In this paper, we proposed a long short-term memory option pricing model with realized skewness by fully considering the asymmetry of asset return in emerging markets. It was applied to price the ETF50 options of China. In order to emphasize the improvement of this model, a comparison with a parametric method, such as Black-Scholes (BS), and machine learning methods, such as support vector machine (SVM), random forests and recurrent neural network (RNN), was conducted. Moreover, we also took the characteristic of heavy tail into consideration and studied the effect of realized kurtosis on pricing to prove the robustness of the skewness. The empirical results indicate that realized skewness significantly improves the pricing performance of LSTM among moneyness states except for in-the-money call options. Specifically, the LSTM model with realized skewness outperforms the classical method and other machine learning methods in all metrics. [ABSTRACT FROM AUTHOR]

Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferential equations (PIDEs) on state spaces of possibly high dimension d. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump Lévy processes. We prove for such PIDEs arising from a class of jump-diffusions on ℝ d , that for any suitable measure μ d on ℝ d , there exist constants C , , > 0 such that for every ∈ (0 , 1 ] and for every d ∈ ℕ the DNN L 2 (μ d) -expression error of viscosity solutions of the PIDE is of size with DNN size bounded by C d − . In particular, the constant C > 0 is independent of d ∈ ℕ and of ∈ (0 , 1 ] and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to suitable Markovian jump-diffusion processes. As a consequence of the employed techniques, we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD. [ABSTRACT FROM AUTHOR]

Generally, it is well known that the constant elasticity of variance (CEV) model fails to capture the empirical results verifying that the implied volatility of equity options displays smile and skew curves at the same time. In this study, to overcome the limitation of the CEV model, we introduce a new model, which is a generalization of the CEV model, and show that it can capture the smile and skew effects of implied volatility. Using an asymptotic analysis for two small parameters that determine the volatility shape, we obtain approximated solutions for option prices in the extended model. In addition, we demonstrate the stability of the solution for the expansion of the option price. Furthermore, we show the convergence rate of the solutions in Monte‐Carlo simulation and compare our model with the CEV, Heston, and other extended stochastic volatility models to verify its flexibility and efficiency compared with these other models when fitting option data from the S&P 500 index. [ABSTRACT FROM AUTHOR]

This paper investigates the pricing of European-style lookback options when the price dynamics of the underlying risky asset are assumed to follow a Markov-modulated Geometric Brownian motion; that is, the appreciation rate and the volatility of the underlying risky asset depend on states of the economy described by a continuous-time Markov chain process. We derive an exact, explicit and closed-form solution for European-style lookback options in a two-state regime switching model. [ABSTRACT FROM AUTHOR]

Abstract The Backward Stochastic Differential Equation (BSDE) is an important tool for pricing and hedging. Highly accurate pricing for low computation time becomes interesting for minimizing monetary loss. Therefore, we explore the opportunity of parallelizing high-order multistep schemes in option pricing. In the multistep scheme the computations at each space grid point are independent and this fact motivates us to select massively parallel GPU computing using CUDA. In our investigations we identify performance bottlenecks and apply appropriate optimization techniques to reduce the computation time in a uniform space domain. Runtime experiments manifest optimistic speedups for the parallel implementation on a single GPU, NVIDIA GeForce 1070 Ti.

The present work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.

This paper studies the method of constructing high order recombined multinomial tree based on fast Fourier transform (FFT), and applies multinomial tree option pricing under the Lévy process. First, the Lévy option pricing model and Fourier transform are introduced. Then, the network model based on FFT (Markov chain) is presented. After that, a method of constructing a recombined multinomial tree based on FFT is given. It is proved that the discrete random variables corresponding to the multinomial tree converge to the Lévy distributed continuous random variable. Next, we obtain the European option pricing formula of FFT multinomial tree pricing, and apply the reverse iteration method to the American option pricing. Finally, under the Jump-diffuse process, the difference between the computational accuracy and computational efficiency of the Semi-analytical solution of European Option and Merton European Call Option which are priced under FFT is compared. The results show that the method of constructing a high-order recombined multinomial tree based on FFT has very high calculation precision and calculation speed, which can solve the problem of traditional risk-neutral multinomial tree construction and it is a promising pricing method for derivative products.