Multi-asset Options Research Articles

An Efficient Fourth-Order Numerical Scheme for Nonlinear Multi-asset Option Pricing Problems

An Efficient Fourth-Order Numerical Scheme for Nonlinear Multi-asset Option Pricing Problems

Three-Layer Artificial Neural Network for Pricing Multi-Asset European Option

This paper studies an artificial neural network (ANN) for multi-asset European options. Firstly, a simple three-layer ANN-3 is established with undetermined weights and bias. Secondly, the time–space discrete PDE of the multi-asset option is given and the corresponding discrete data are fed into the ANN-3. Then, using least squares error as the objective function, the weights and bias of ANN-3 are trained well. Numerical examples are carried out to confirm the stability, accuracy and efficiency. Experiments show the ANN’s relative error is about 0.8%. This method can be extended into multi-layer ANN-q(q>3) and extended into American options.

Pricing multi-asset options with tempered stable distributions

We derive methods for risk-neutral pricing of multi-asset options, when log-returns jointly follow a multivariate tempered stable distribution. These lead to processes that are more realistic than the better known Brownian motion and stable processes. Further, we introduce the diagonal tempered stable model, which is parsimonious but allows for rich dependence between assets. Here, the number of parameters only grows linearly as the dimension increases, which makes it tractable in higher dimensions and avoids the so-called “curse of dimensionality.” As an illustration, we apply the model to price multi-asset options in two, three, and four dimensions. Detailed goodness-of-fit methods show that our model fits the data very well.

Solving multi-dimensional European option pricing problems by integrals of the inverse quadratic radial basis function on non-uniform meshes

This paper explores multi-asset options as a means to diversify portfolios, mitigating risk across various assets. We present a numerical method using radial basis function-generated finite difference solvers via integrals of the inverse quadratic kernel. Our method introduces new weights for the task we are dealing with. We derive and compute analytical solutions to approximate function derivatives on three-node stencils with non-uniform and uniform distances. Our findings highlight the convergence order of the proposed analytical weights. Numerical examples illustrate the theory.

Numerical investigation of high-dimensional option pricing PDEs by utilizing a hybrid radial basis function - finite difference procedure

The target of this research is to resolve high-dimensional partial differential equations (PDEs) for multi-asset options, modeled as parabolic time-dependent PDEs. We present a hybrid radial basis function - finite difference (RBF-FD) solver, which combines the advantages of Gaussian and multiquadric functions. Additionally, we employ the Krylov subspace method on the resulting system of ordinary differential equations, reducing the computation load for finding the numerical solution. Computational tests, up to seven dimensions, and comparisons support the superiority of our hybrid solver.

Using Time‐Space Double Radial Basis Function Method to Solve High‐Dimensional PDEs Arising from Multiasset Option Pricing

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.

Deep Learning Approach for Multi-Asset Option Pricing

Deep Learning Approach for Multi-Asset Option Pricing

Total Value Adjustment of Multi-Asset Derivatives under Multivariate CGMY Processes

Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today’s context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives.

Covariance dependent kernels, a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.

Meshless methods for American option pricing through Physics-Informed Neural Networks

Nowadays, Deep Learning is drastically revolutionizing financial research as well as industry. Many methods have been discussed in the last few years, mainly related to option pricing. In fact, traditional approaches such as Monte Carlo simulation or finite difference methods are seriously harmed by multi-dimensional underlying and path dependency. Thus, dealing with particular contracts such as American multi-asset options is still rough. This paper addresses such a problem by pricing said put options with a novel meshless methodology, named Physics-Informed Neural Networks (PINNs), based on Artificial Intelligence. PINN paradigm has been recently introduced in Deep Learning literature. It exploits the theoretical background of the universal approximation theorem for neural networks to solve Partial Differential Equations numerically. This Deep Learning meshless method incorporates the equation and its initial and boundary conditions thanks to a specially designed loss function. We develop a suitable PINN for the proposed problem by introducing an algorithmic trick for improving the convergence of the free boundary problem. Furthermore, the worthiness of the proposal is assessed by several experiments concerned with single and multi-asset options. Finally, a parametric model is built to benefit further studies of option value behaviour related to particular market conditions.

An operator splitting method for multi-asset options with the Feynman-Kac formula

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively.

Optimal damping with a hierarchical adaptive quadrature for efficient Fourier pricing of multi-asset options in Lévy models

Optimal damping with a hierarchical adaptive quadrature for efficient Fourier pricing of multi-asset options in Lévy models

A Penalty Method for American Multi-Asset Option Problems

A Penalty Method for American Multi-Asset Option Problems

Multi-Asset Option Pricing Using Normal Tempered Stable Processes with Stochastic Correlation

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.

A parallel and pipelined implementation of a pascal-simplex based multi-asset option pricer on FPGA using OpenCL

Multi-asset option pricing is a financial tool that offers a competitive advantage in the global financial markets. To achieve the commercial advantage, performance and accuracy are two key desiderata of such tools. Techniques for financial option pricing have been actively researched for some time. Combined with recent advances in heterogeneous computing, new computing methods of lower latency option pricing have become available. Field-Programmable Gate Array technology offers a powerful framework for building architectures for highly parallel option pricing. This paper presents a versatile, high performance implementation of an accurate multi-asset option pricer on an Intel Stratix 10 GX device. The implementation is highly flexible for the number of assets supported, the maximum tree depth, and the pipelining parameters. The theory that this architecture is based on is that of a recombining multinomial tree approach, which is a generalization of the binomial tree model. This novel architecture was implemented on Field-Programmable Gate Array technology. The implementation presented in this paper enables the computation of an option price on up to 7 assets and can compute up to 43 times faster than a software implementation in a general-purpose processor.

DISCRETIZATION PROCESSING OF FINANCIAL RISK MANAGEMENT USING STOCHASTIC DIFFERENTIAL EQUATION SIMULATION METHOD

The purposes of this paper are to improve the scientific processing level of risk management in the financial field, enrich the application range of mathematical models in financial calculations, and comprehensively discuss the theories and concepts of mathematical finance and stochastic differential equations. More importantly, the common option pricing issues in financial risk management have been researched using the forward–backward stochastic differential equation. The fully discrete and uncoupled forward–backward stochastic differential equation is employed to analyze the spread option and the better-of option, the complicated multi-asset options. Results demonstrate that the fully discrete and uncoupled forward–backward stochastic differential equations can effectively price the spread option and the better-of option. Simulation by the MATLAB software suggests that the value of spread option pricing is 0.0264, and the value of the better-of option pricing is 0.0251. The above results can provide scientific and useful references for the subsequent application research on forward–backward stochastic differential equations in the financial field; simultaneously, they also have important practical significance for researching on and developing the financial risk management.

A robust numerical method for single and multi-asset option pricing

<abstract><p>In this work, we numerically solve some different single and multi-asset European options with the finite difference method (FDM) and take the advantages of the antithetic variate method in Monte Carlo simulation (AMC) as a variance reduction technique in comparison to the standard Monte Carlo simulation (MC) in the end point of the domain, and the linear boundary condition has been implemented in other boundaries. We also apply the grid stretching transformation to make a non-equidistance discretization with more nodal points around the strike price (K) which is the non-smooth point in the payoff function to reduce the numerical errors around this point and have more accurate results. Superiority of our method (GS&AMC) will be demonstrated by comparison with the finite difference scheme with the equidistance discretization and the linear boundary conditions (Equi&L), the grid stretching discretization around K with linear boundary conditions (GS&L) and also the equidistance discretization with combination of the standard Monte Carlo simulation at the end point of the domain (Equi&MC). Furthermore, the root mean square errors (RMSE) of these four schemes in the whole region and the most interesting region which is around the strike price, have been compared.</p></abstract>

Static Replication of European Multi-Asset Options with Homogeneous Payoff

ABSTRACT The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by [Carr, Peter, and Dilip Madan. 1998. “Towards a Theory of Volatility Trading.” In Volatility: New Estimation Techniques for Pricing Derivatives, Vol. 29, edited by Robert A. Jarrow, 417–427. Risk books.] extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.

A novel local meshless scheme based on the radial basis function for pricing multi-asset options

‎A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model‎. ‎The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction‎. ‎Therefore‎, ‎it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation‎. ‎We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach‎. ‎Moreover‎, ‎based on a review performed in the current study‎, ‎the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced‎. ‎As seen in numerical experiments‎, ‎it has a low computational cost and generates higher accuracy‎. ‎Finally‎, ‎the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎. ‎Finally‎, ‎we conclude that the proposed local RBF scheme is very versatile so that it can be used easily for Solving numerous models and obstacles not just in the finance sector‎, ‎as well as in other fields of engineering and science‎.

Static Replication of European Multi-Asset Options with Homogeneous Payoff

The replication of any European contingent claim by a static continuous portfolio of calls and puts, formally proven by Carr and Madan (1998), extends to multi-asset claims with absolutely homogeneous payoff. Using sophisticated tools from integral geometry, we show how such claims may be replicated with a continuum of vanilla basket calls and derive closed-form solutions to replicate two-asset best-of and worst-of options. We also derive a novel mathematical formula to invert the Radon transform which we apply to obtain a tractable expression of the joint implied distribution. Consequently, a large class of multi-asset options admit a model-free price enforced by arbitrage, just as single-asset European claims do.