Multi-asset option pricing using a parallel Fourier-based technique

The deterministic sparse grid method, also known as Smolyak’s method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about randomized versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the $D$-dimensional unit cube $[0,1)^D$. Let $d,s \in {\mathbb {N}}$ be such that $D=d\cdot s$. The $s$-dimensional building blocks of the sparse grid quadratures are based on stratified sampling for $s=1$ and on scrambled $(0,m,s)$-nets for $s\ge 2$. The spaces of integrands and the error criterion we consider are Haar wavelet spaces with parameter $\alpha $ and the randomized error (i.e., the worst case root mean square error), respectively. We prove sharp (i.e., matching) upper and lower bounds for the convergence rates of the $N$th minimal errors for all possible combinations of the parameters $d$ and $s$. Our upper error bounds still hold if we consider as spaces of integrands Sobolev spaces of mixed dominated smoothness with smoothness parameters $1/2< \alpha < 1$ instead of Haar wavelet spaces.

The large misalignment angle error model of strapdown inertial navigation system (SINS) is a kind of nonlinear Gauss error model. The Gauss Hermite method can approach Gauss integral with any precision, and achieve high precision filtering effect. The number of Gauss Hermite integral points is exponentially related to the dimension of nonlinear system. In the process of real-time filtering, a large number of integral points of multidimensional system may lead to dimension disaster. According to the initial alignment error model of SINS with large misalignment angle, this paper presents a strong tracking reduced dimension sparse grid Gauss Hermite filtering method, which reduces the nonlinear integration dimension through separation of the linear and nonlinear terms, further reduces the number of Gauss hermit filtering integration points by sparse grid method, and ensures the stability of the system by strong tracking method.

Option pricing, a fundamental problem in finance, often requires solving non-linear partial differential equations (PDEs). When dealing with multi-asset options, such as rainbow options, these PDEs become high-dimensional, leading to challenges posed by the curse of dimensionality. While deep learning-based PDE solvers have recently emerged as scalable solutions to this high-dimensional problem, their empirical and quantitative accuracy remains not well-understood, hindering their real-world applicability. In this study, we aimed to offer actionable insights into the utility of Deep PDE solvers for practical option pricing implementation. Through comparative experiments, we assessed the empirical performance of these solvers in high-dimensional contexts. Our investigation identified three primary sources of errors in Deep PDE solvers: (i) errors inherent in the specifications of the target option and underlying assets, (ii) errors originating from the asset model simulation methods, and (iii) errors stemming from the neural network training. Through ablation studies, we evaluated the individual impact of each error source. Our results indicate that the Deep BSDE method (DBSDE) is superior in performance and exhibits robustness against variations in option specifications. In contrast, some other methods are overly sensitive to option specifications, such as time to expiration. We also find that the performance of these methods improves inversely proportional to the square root of batch size and the number of time steps. This observation can aid in estimating computational resources for achieving desired accuracies with Deep PDE solvers.

This paper develops a time‐space double radial basis function (TSDRBF) method to solve PDEs arising from multiasset option pricing. By TSDRBF discretization for the high‐dimensional PDEs, a linear system (LS) is obtained. After solving the LS, multi‐asset options of European and American style are restored, with number N of space discretization, number of time nodes, and number M of time RBF discretization. Numerical examples confirm that the TSDRBF method has exponentially convergent rates for the number N and number M, while the convergence is linear for the number . Compared with finite difference, TSDRBF avoids the difficulty of space discretization, and the convergence is greatly improved.

Multiphase pumps are used as an important tool for natural gas hydrate extraction owing to their excellent gas–liquid mixing and transport properties. This paper proposes an adaptive response surface-based integrated optimization design method. A model pump is designed based on the axial flow pump design theory. The model pump is numerically simulated and analyzed to obtain its performance parameters. Then the structural and performance parameters of the pump are parameterized to establish a closed-loop input–output system. Based on this closed-loop system, a sensitivity analysis is performed on the structural parameters of the impeller and guide vane, and the parameters that affect the performance of the gas–liquid hybrid pump the most are derived. The Sparse Grid method was introduced to design the experiment and construct the approximate model. The structural parameters of the impeller and guide vane are used as design variables to optimize the pressure increment and efficiency of the pump. After optimization, the pressure increment of the multiphase pump was increased by 10.78 KPa and the efficiency was increased by 0.89% compared to the original model. Finally, we validate the accuracy of the optimized model with tests.

Black-Scholes equation of dual currency exchange option pricing is a typical multi-asset option pricing model. It is important to research numerical solution of this equation. In this paper, accelerated additive operator splitting (AOS) differential algorithm is used to transform the multi-dimensional Black-Scholes equation into equivalent one-dimensional equations. Then ’explicit-implicit’ and ’implicit-explicit’ schemes will be constructed. These schemes proved to be stable and convergent unconditionally and they have second-order accuracy. Finally, the numerical example shows the effectiveness of the accelerated AOS difference algorithm. It illustrates that accelerated AOS algorithm effectively avoids the computational complexity of the high-dimensional Black-scholes equation, and dramatically improves the computational speed. Therefore, it is applicable for multi-asset option pricing.

In the previous chapter the solution of linear equation systems was discussed. Often there are even optimal order methods, so that an equation system of n equations can be solved in O(n) operations. This is optimal in the sense that any method which produces an output of n numbers has at least O(n) complexity, even if the method is as simple as write the vector zero. Of course, one may be able to tune given methods or find new ones with a smaller constant in the complexity, but in general a lower limit is reached. Nevertheless, it may be desirable to solve a given PDE even faster. In this book we will present two ways to do this; we introduce parallel algorithms in chapter 5. This is a general approach to reducing the complexity by using more than just one processor. Another way to reduce the complexity is to change the discretisation and to use fewer degrees of freedom. This can be done for functions which are smooth or regular enough by a higher order or by a sparse grid method, see also chapter 2.5. In cases of non-smooth functions, adaptive mesh refinement can be used to concentrate the computational effort to regions where it is needed most. We will cover the topic of adaptive mesh refinement in this chapter.KeywordsHash TableMesh RefinementUnstructured MeshSparse GridError IndicatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A 6M digital twin for modeling and simulation in subsurface reservoirs

This study quantifies the hazard associated with extreme sea levels due to eastern Adriatic meteotsunamis—long waves generated by traveling atmospheric disturbances—and assesses the sensitivity of the ocean response to the disturbances responsible for those events. In this spirit, a surrogate model of meteotsunami maximum elevation based on generalized polynomial chaos expansion (gPCE) methods, is implemented. The approach relies on the definition of a synthetic pressure disturbance—depending on six different stochastic parameters known to be important for meteotsunami generation, which is used as forcing to produce series of meteotsunami simulations defined with sparse grid methods (up to 10 689 used in this study). The surrogate model and the sensitivity study are then obtained with a pseudo-spectral approximation (PSA) method based on the chosen meteotsunami simulations. This study mainly presents the developed methodology and discusses the feasibility of implementing such gPCE-based surrogate models to assess the hazard and to study the sensitivity of meteorologically driven extreme sea levels.

Multiphase pumps play an important role in the exploitation of natural gas hydrate. Compared with ordinary pumps, they can handle fluids with higher gas volume fraction (GVF). Therefore, it is important to improve the performance of the pump under high GVF. A model pump is designed based on the design theory of axial flow pump and centrifugal pump inducer. The hydraulic performance of the model pump is verified by numerical simulation and experiment. The Sparse Grid method is applied to the design of experiment (DOE), and three different adaptive refined response surface methods (RSM) are applied to the build the approximate model. Refinement points and verification points are used to improve and verify the precision of the response surface, respectively. The model with high precision and high computational efficiency is obtained through comparison and analysis. The multi-objective optimization of the optimal response surface model is carried out by MOGA (Multi-Objective Genetic Algorithm) method. The pressure increment of the optimized model is increased by 38 kPa. The efficiency is significantly improved under large mass flow conditions. The hydraulic performance of the optimized model is compared with that of the basic model. And the reasons that affect the performance of the multiphase pump are analyzed.

The stationary monochromatic radiative transfer equation (RTE) is a partial differential transport equation stated on a five-dimensional phase space, the Cartesian product of physical and angular domain. We solve the RTE with a Galerkin FEM in physical space and collocation in angle, corresponding to a discrete ordinates method (DOM). To reduce the complexity of the problem and to avoid the “curse of dimension”, we adapt the sparse grid combination technique to the solution space of the RTE and show that we obtain a sparse DOM which uses essentially only as many degrees of freedom as required for a purely spatial transport problem. For smooth solutions, the convergence rates deteriorate only by a logarithmic factor. We compare the sparse DOM to the standard full DOM and a sparse tensor product approach developed earlier with Galerkin FEM in physical space and a spectral method in angle. Numerical experiments confirm our findings.

In this article, the authors develop a new multi-asset option pricing model where underlying dynamics are assumed to follow the normal tempered stable (NTS) process and its correlation structure evolves over time. The model is constructed by extending the constant correlation term in the previous NTS framework to the stochastic correlation making use of the Ornstein-Uhlenbeck process. We then derive its closed-form solution under the risk-neutral measure and apply it to an empirical study of a quanto option. The in-sample tests and calibration practices justify our model specification reflecting the empirical stylized facts on asset returns such as heavy tails, skewness, kurtosis, and stochastic correlation. Building on the empirical results, we can also identify the presence of stochastic correlation in the risk-neutral world.

In this paper, a modification of the original global radial basis functions-based differential quadrature (RBF-DQ) method is set forth and analyzed. The improved RBF-DQ method is applicable to the numerical approximation of solutions of a wide range of partial differential equations with mixed derivative terms. However, it appears to be considerably faster than the original method. In support of this contention, the multi-asset option pricing problems under exponential Levy framework have been solved numerically by using the proposed method and compared with results obtained via the original RBF-DQ method. For accuracy achieved versus work expended, the improved method performs better.

The evaluation of option premium is a very delicate issue arising from the assumptions made under a financial market model, and pricing of a wide range of options is generally feasible only when numerical methods are involved. This paper is based on our recent research on numerical pricing of path-dependent multi-asset options and extends these results also to the case of Asian options with fixed strike. First, we recall the three-dimensional backward parabolic PDE describing the evolution of European-style Asian option contracts on two assets, whose payoff depends on the difference of the strike price and the average value of the basket of two underlying assets during the life of the option. Further, a suitable transformation of variables respecting this complex form of a payoff function reduces the problem to a two-dimensional equation belonging to the class of convection-diffusion problems and the discontinuous Galerkin (DG) method is applied to it in order to utilize its solving potentials. The whole procedure is accompanied with theoretical results and differences to the floating strike case are discussed. Finally, reference numerical experiments on real market data illustrate comprehensive empirical findings on Asian options.

This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak’s sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.