We present a method for conditional time series forecasting based on an adaptation of the recent deep convolutional WaveNet architecture. The proposed network contains stacks of dilated convolutions that allow it to access a broad range of historical data when forecasting. It also uses a rectified linear unit (ReLU) activation function, and conditioning is performed by applying multiple convolutional filters in parallel to separate time series, which allows for the fast processing of data and the exploitation of the correlation structure between the multivariate time series. We test and analyze the performance of the convolutional network both unconditionally and conditionally for financial time series forecasting using the Standard & Poor’s 500 index, the volatility index, the Chicago Board Options Exchange interest rate and several exchange rates, and we extensively compare its performance with those of the well-known autoregressive model and a long short-term memory network. We show that a convolutional network is well suited to regression-type problems and is able to effectively learn dependencies in and between the series without the need for long historical time series, that it is a time-efficient and easy-to-implement alternative to recurrent-type networks, and that it tends to outperform linear and recurrent models.
{"title":"Dilated Convolutional Neural Networks for Time Series Forecasting","authors":"A. Borovykh, S. Bohté, C. Oosterlee","doi":"10.21314/JCF.2019.358","DOIUrl":"https://doi.org/10.21314/JCF.2019.358","url":null,"abstract":"We present a method for conditional time series forecasting based on an adaptation of the recent deep convolutional WaveNet architecture. The proposed network contains stacks of dilated convolutions that allow it to access a broad range of historical data when forecasting. It also uses a rectified linear unit (ReLU) activation function, and conditioning is performed by applying multiple convolutional filters in parallel to separate time series, which allows for the fast processing of data and the exploitation of the correlation structure between the multivariate time series. We test and analyze the performance of the convolutional network both unconditionally and conditionally for financial time series forecasting using the Standard & Poor’s 500 index, the volatility index, the Chicago Board Options Exchange interest rate and several exchange rates, and we extensively compare its performance with those of the well-known autoregressive model and a long short-term memory network. We show that a convolutional network is well suited to regression-type problems and is able to effectively learn dependencies in and between the series without the need for long historical time series, that it is a time-efficient and easy-to-implement alternative to recurrent-type networks, and that it tends to outperform linear and recurrent models.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49022722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new approach to bounding financial derivative prices in regime-switching market models from both above and below. We derive sufficient conditions under which a particular class of functions act as bounds for the prices of financial derivatives in regime-switching market models. Using these sufficient conditions, we then formulate, in a general setting, optimization problems whose solutions can be identified with tight upper and lower bounds. The problems are made numerically tractable by imposing polynomial structures and employing results from the theory of sum-of-squares polynomials to arrive at a semidefinite programming problem that is implementable by existing software. The bounds obtained take the form of smooth polynomial functions and are valid for a continuous range of initial times and states. Moreover, they are obtained without recourse to sample path simulation or discretization of the temporal or spatial variables. We demonstrate the effectiveness of the proposed method on European-, barrier- and American-style options in several regime-switching settings with and without jumps.
{"title":"Polynomial Upper and Lower Bounds for Financial Derivative Price Functions under Regime-Switching","authors":"Louis Bhim, Ray Kawai","doi":"10.21314/JCF.2018.352","DOIUrl":"https://doi.org/10.21314/JCF.2018.352","url":null,"abstract":"We present a new approach to bounding financial derivative prices in regime-switching market models from both above and below. We derive sufficient conditions under which a particular class of functions act as bounds for the prices of financial derivatives in regime-switching market models. Using these sufficient conditions, we then formulate, in a general setting, optimization problems whose solutions can be identified with tight upper and lower bounds. The problems are made numerically tractable by imposing polynomial structures and employing results from the theory of sum-of-squares polynomials to arrive at a semidefinite programming problem that is implementable by existing software. The bounds obtained take the form of smooth polynomial functions and are valid for a continuous range of initial times and states. Moreover, they are obtained without recourse to sample path simulation or discretization of the temporal or spatial variables. We demonstrate the effectiveness of the proposed method on European-, barrier- and American-style options in several regime-switching settings with and without jumps.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46134969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops efficient importance sampling schemes for a class of jump–diffusion processes that are commonly used for modeling stock prices. For such financial models, related option pricing problems are often difficult, especially when the option under study is out-of-the-money and there are multiple underlying assets. Even though analytical pricing formulas do exist in a few very simple cases, often analysts must resort to numerical methods or Monte Carlo simulation. We demonstrate that efficient and easy-to-implement importance sampling schemes can be constructed via the method of cross-entropy combined with the expectation–maximization algorithm, when the alternative sampling distributions are chosen from the family of exponentially tilted distributions or their mixtures. Theoretical justification is given by characterizing the limiting behavior of the cross-entropy algorithm under appropriate scaling. Numerical experiments on vanilla options, path-dependent options and rainbow options are also performed to illustrate the use of this technology.
{"title":"Importance Sampling for Jump–Diffusions Via Cross-Entropy","authors":"R. Rieke, Weiming Sun, Hui Wang","doi":"10.21314/JCF.2018.349","DOIUrl":"https://doi.org/10.21314/JCF.2018.349","url":null,"abstract":"This paper develops efficient importance sampling schemes for a class of jump–diffusion processes that are commonly used for modeling stock prices. For such financial models, related option pricing problems are often difficult, especially when the option under study is out-of-the-money and there are multiple underlying assets. Even though analytical pricing formulas do exist in a few very simple cases, often analysts must resort to numerical methods or Monte Carlo simulation. We demonstrate that efficient and easy-to-implement importance sampling schemes can be constructed via the method of cross-entropy combined with the expectation–maximization algorithm, when the alternative sampling distributions are chosen from the family of exponentially tilted distributions or their mixtures. Theoretical justification is given by characterizing the limiting behavior of the cross-entropy algorithm under appropriate scaling. Numerical experiments on vanilla options, path-dependent options and rainbow options are also performed to illustrate the use of this technology.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46016847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In general, no analytical formulas exist for pricing discretely monitored exotic options, even when a geometric Brownian motion governs the risk-neutral underlying. While specialized numerical algorithms exist for pricing particular contracts, few can be applied universally with consistent success and with general Lévy dynamics. This paper develops a general methodology for pricing early exercise and exotic financial options by extending the recently developed PROJ method. We are able to efficiently obtain accurate values for complex products including Bermudan/American options, Bermudan barrier options, survival probabilities and credit default swaps by value recursion, European barrier and lookback/hindsight options by density recursion, and arithmetic Asian options by characteristic function recursion. This paper presents a unified approach to tackling these and related problems. Algorithms are provided for each option type, along with a demonstration of convergence. We also provide a large set of reference prices for exotic, American and European options under Black-Scholes-Merton, Normal Inverse Gaussian, Kou’s double exponential jump diffusion, Variance Gamma, KoBoL/CGMY and Merton’s jump diffusion models.
{"title":"American and exotic option pricing with jump diffusions and other Levy processes","authors":"Justin Lars Kirkby","doi":"10.21314/jcf.2018.355","DOIUrl":"https://doi.org/10.21314/jcf.2018.355","url":null,"abstract":"In general, no analytical formulas exist for pricing discretely monitored exotic options, even when a geometric Brownian motion governs the risk-neutral underlying. While specialized numerical algorithms exist for pricing particular contracts, few can be applied universally with consistent success and with general Lévy dynamics. This paper develops a general methodology for pricing early exercise and exotic financial options by extending the recently developed PROJ method. We are able to efficiently obtain accurate values for complex products including Bermudan/American options, Bermudan barrier options, survival probabilities and credit default swaps by value recursion, European barrier and lookback/hindsight options by density recursion, and arithmetic Asian options by characteristic function recursion. This paper presents a unified approach to tackling these and related problems. Algorithms are provided for each option type, along with a demonstration of convergence. We also provide a large set of reference prices for exotic, American and European options under Black-Scholes-Merton, Normal Inverse Gaussian, Kou’s double exponential jump diffusion, Variance Gamma, KoBoL/CGMY and Merton’s jump diffusion models.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"24 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Monte Carlo payoff smoothing for pricing autocallable instruments","authors":"Frank Koster,Achim Rehmet","doi":"10.21314/jcf.2018.340","DOIUrl":"https://doi.org/10.21314/jcf.2018.340","url":null,"abstract":"","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"23 1","pages":"59-77"},"PeriodicalIF":0.9,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138537329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting. The first ingredient in our approach is the dimensional-adaptive sparse grid combination technique, which, in the case of underlying models with stochastic volatilities, allows for inhomogeneous discretization levels of the dimensional axes. Thus, by applying the dimensional-adaptive combination technique to such problems, one may achieve higher numerical efficiency. We combine this approach with a stretched grid discretization that is derived from the underlying’s stochastic differential equation (SDE) in a general manner. This stretching enables us to employ efficient geometrical multigrid solvers, even for the strong anisotropic convection and diffusion coefficients that frequently occur in application. Our combination of the dimensional-adaptive sparse grid combination technique with SDE-based grid stretching and an efficient multigrid solver represents a new approach designed to enable derivative pricing by directly solving PDEs in higher dimensions than were possible before. The numerical results outlined in the paper demonstrate the efficacy of this new approach and of our implementation method, which entails pricing various derivatives with up to twelve dimensions in a general and simple manner.
{"title":"Pricing Multidimensional Financial Derivatives with Stochastic Volatilities Using the Dimensional-Adaptive Combination Technique","authors":"J. Benk, D. Pflüger","doi":"10.21314/JCF.2017.335","DOIUrl":"https://doi.org/10.21314/JCF.2017.335","url":null,"abstract":"In this paper, we present a new and general approach to price derivatives based on the Black–Scholes partial differential equation (BS-PDE) in a multidimensional setting. The first ingredient in our approach is the dimensional-adaptive sparse grid combination technique, which, in the case of underlying models with stochastic volatilities, allows for inhomogeneous discretization levels of the dimensional axes. Thus, by applying the dimensional-adaptive combination technique to such problems, one may achieve higher numerical efficiency. We combine this approach with a stretched grid discretization that is derived from the underlying’s stochastic differential equation (SDE) in a general manner. This stretching enables us to employ efficient geometrical multigrid solvers, even for the strong anisotropic convection and diffusion coefficients that frequently occur in application. Our combination of the dimensional-adaptive sparse grid combination technique with SDE-based grid stretching and an efficient multigrid solver represents a new approach designed to enable derivative pricing by directly solving PDEs in higher dimensions than were possible before. The numerical results outlined in the paper demonstrate the efficacy of this new approach and of our implementation method, which entails pricing various derivatives with up to twelve dimensions in a general and simple manner.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45786596","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maria do Rosario Grossinho, D. Ševčovič, Yaser Faghan Kord
In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over relaxation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs.
{"title":"Pricing American Call Options Using the Black–Scholes Equation with a Nonlinear Volatility Function","authors":"Maria do Rosario Grossinho, D. Ševčovič, Yaser Faghan Kord","doi":"10.21314/jcf.2020.379","DOIUrl":"https://doi.org/10.21314/jcf.2020.379","url":null,"abstract":"In this paper we investigate a nonlinear generalization of the Black-Scholes equation for pricing American style call options in which the volatility term may depend on the underlying asset price and the Gamma of the option. We propose a numerical method for pricing American style call options by means of transformation of the free boundary problem for a nonlinear Black-Scholes equation into the so-called Gamma variational inequality with the new variable depending on the Gamma of the option. We apply a modified projective successive over relaxation method in order to construct an effective numerical scheme for discretization of the Gamma variational inequality. Finally, we present several computational examples for the nonlinear Black-Scholes equation for pricing American style call option under presence of variable transaction costs.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44805875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we develop a new method for implied volatility surface construction for FX options. The methodology is based on the local variance gamma model developed by Carr (2008). Our approach is to solve a simplified "one-step" version of the Dupire equation analytically under the assumption of a continuous five parameter diffusion function. The unique solution to this equation can be interpreted as a continuous representation of option prices, defined for strikes in an arbitrarily large range. The derived price functions are C^2 -positive, arbitrage-free by construction, and they do not depend on the strike discretization. By using a least-square approach, we calibrate price functions to Reuters quoted FX volatility smiles. Our results suggest that the model allows for very rapid calibration; using a Levenberg-Marquardt algorithm we measure the average calibration time to less than 1 ms for one expiry on a standard personal computer.We also extend our model to allow for interpolation between maturities and present sufficient conditions for absence of calendar spread arbitrage. In order to generate the whole implied volatility surface, we suggest a simple, fast and yet market-consistent technique allowing for arbitrage-free interpolation of calibrated price functions in the maturity dimension.The methodology is tested against EURUSD and EURSEK options, where we show that the model has the capability to produce volatility surfaces which fit market quotes with an error of few volatility basis points. We then apply the methodology to pricing variance swaps.
{"title":"Local Variance Gamma Revisited","authors":"Markus Falck, M. Deryabin","doi":"10.2139/ssrn.2659728","DOIUrl":"https://doi.org/10.2139/ssrn.2659728","url":null,"abstract":"In this paper we develop a new method for implied volatility surface construction for FX options. The methodology is based on the local variance gamma model developed by Carr (2008). Our approach is to solve a simplified \"one-step\" version of the Dupire equation analytically under the assumption of a continuous five parameter diffusion function. The unique solution to this equation can be interpreted as a continuous representation of option prices, defined for strikes in an arbitrarily large range. The derived price functions are C^2 -positive, arbitrage-free by construction, and they do not depend on the strike discretization. By using a least-square approach, we calibrate price functions to Reuters quoted FX volatility smiles. Our results suggest that the model allows for very rapid calibration; using a Levenberg-Marquardt algorithm we measure the average calibration time to less than 1 ms for one expiry on a standard personal computer.We also extend our model to allow for interpolation between maturities and present sufficient conditions for absence of calendar spread arbitrage. In order to generate the whole implied volatility surface, we suggest a simple, fast and yet market-consistent technique allowing for arbitrage-free interpolation of calibrated price functions in the maturity dimension.The methodology is tested against EURUSD and EURSEK options, where we show that the model has the capability to produce volatility surfaces which fit market quotes with an error of few volatility basis points. We then apply the methodology to pricing variance swaps.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2017-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42433294","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the Delta. When options on a basket - be it a stock index, a cross-foreign exchange rate or an interest rate spread - are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. They may also fail to be admissible, i.e., positive semi-definite. We explain how, by combining the particle method presented in "The smile calibration problem solved" by Guyon and Henry-Labordere (2011) with a simple affine transform, we can build all the calibrated local correlation models. The two existing models appear as special cases (if admissible). For the first time, one can now choose a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging and risk-management of multi-asset derivatives. This technique is generalized to calibrate models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility and local correlation. Numerical results show the wide variety of calibrated local correlations and give insight into a difficult (still unsolved) problem: finding lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.
在多资产模型中，允许相关性是局部的，即依赖于状态的，可以通过合并Delta中的相关性移动来实现更好的对冲。当一篮子期权——无论是股指、跨外汇汇率还是利差——进行流动性交易时，人们可能希望校准与这些期权价格的本地相关性。到目前为止，文献中只提出了两种特殊的解决方案。两者都对资产价值施加了相关性矩阵的特定依赖，这是人们没有理由经历的。它们也可能不能被接受，即肯定半定。我们解释了如何将Guyon和Henry-Labordere(2011)的“the smile calibration problem solved”中提出的粒子方法与简单的仿射变换相结合，构建所有校准的局部相关模型。现有的两种模型是特例(如果允许的话)。现在，人们第一次可以选择校准的局部相关性，以适应相关倾斜的观点，或重现历史相关性，或匹配一些外来期权价格，从而改善多资产衍生品的定价、对冲和风险管理。将该方法推广到组合随机利率、随机股息收益率、局部随机波动率和局部相关的模型中。数值结果显示了各种校准的局部相关性，并深入了解了一个困难(仍未解决)的问题:给定一篮子及其组成部分的隐含波动率的整个表面，找到一般多资产期权价格的下界/上界。
{"title":"Calibration of Local Correlation Models to Basket Smiles","authors":"Julien Guyon","doi":"10.21314/JCF.2016.326","DOIUrl":"https://doi.org/10.21314/JCF.2016.326","url":null,"abstract":"Allowing correlation to be local, i.e., state-dependent, in multi-asset models allows better hedging by incorporating correlation moves in the Delta. When options on a basket - be it a stock index, a cross-foreign exchange rate or an interest rate spread - are liquidly traded, one may want to calibrate a local correlation to these option prices. Only two particular solutions have been suggested so far in the literature. Both impose a particular dependency of the correlation matrix on the asset values that one has no reason to undergo. They may also fail to be admissible, i.e., positive semi-definite. We explain how, by combining the particle method presented in \"The smile calibration problem solved\" by Guyon and Henry-Labordere (2011) with a simple affine transform, we can build all the calibrated local correlation models. The two existing models appear as special cases (if admissible). For the first time, one can now choose a calibrated local correlation in order to fit a view on the correlation skew, or reproduce historical correlation, or match some exotic option prices, thus improving the pricing, hedging and risk-management of multi-asset derivatives. This technique is generalized to calibrate models that combine stochastic interest rates, stochastic dividend yield, local stochastic volatility and local correlation. Numerical results show the wide variety of calibrated local correlations and give insight into a difficult (still unsolved) problem: finding lower bounds/upper bounds on general multi-asset option prices given the whole surfaces of implied volatilities of a basket and its constituents.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67702475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a hybrid spatial finite-difference/pseudospectral discretization for European option-pricing problems under the Heston and Heston–Hull–White models. In the direction of the underlying asset, where the payoff profile is nonsmooth, we use a standard central second-order finite-difference scheme, whereas we use a Chebyshev collocation method in the other spatial dimensions. In the time domain, we employ alternating direction implicit schemes to efficiently decompose the system matrix into simpler one-dimensional problems. This approach allows us to compute numerical solutions, which are second-order accurate in time and exhibit spectral accuracy in the spatial domains except for the asset direction. The numerical experiments reveal that the proposed scheme outperforms the standard second-order finite-difference scheme in terms of accuracy versus runtime and shows an unconditionally stable behavior.
{"title":"Hybrid Finite–Difference/Pseudospectral Methods for the Heston and Heston–Hull–White Partial Differential Equations","authors":"Christian Hendricks, M. Ehrhardt, M. Günther","doi":"10.21314/JCF.2018.342","DOIUrl":"https://doi.org/10.21314/JCF.2018.342","url":null,"abstract":"We propose a hybrid spatial finite-difference/pseudospectral discretization for European option-pricing problems under the Heston and Heston–Hull–White models. In the direction of the underlying asset, where the payoff profile is nonsmooth, we use a standard central second-order finite-difference scheme, whereas we use a Chebyshev collocation method in the other spatial dimensions. In the time domain, we employ alternating direction implicit schemes to efficiently decompose the system matrix into simpler one-dimensional problems. This approach allows us to compute numerical solutions, which are second-order accurate in time and exhibit spectral accuracy in the spatial domains except for the asset direction. The numerical experiments reveal that the proposed scheme outperforms the standard second-order finite-difference scheme in terms of accuracy versus runtime and shows an unconditionally stable behavior.","PeriodicalId":51731,"journal":{"name":"Journal of Computational Finance","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2016-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67703162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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