基于Jain提出的高阶紧致有限差分格式(high order compact of Jain,HOCJ),结合卷积积分(convolution integral)与快速傅里叶变换(FFT),构建了一种新颖的数值方法,简称HOCJ-CF,并用于Bates模型下美式看跌期权定价.针对期权定价偏积分微...基于Jain提出的高阶紧致有限差分格式(high order compact of Jain,HOCJ),结合卷积积分(convolution integral)与快速傅里叶变换(FFT),构建了一种新颖的数值方法,简称HOCJ-CF,并用于Bates模型下美式看跌期权定价.针对期权定价偏积分微分方程(PIDE)的微分项,首先将其拆分成三个子偏微分方程(sub-PDE),然后分别应用Numerov离散方法,衍生出具有空间四阶精度和时间二阶精度的HOCJ格式;积分项则将其转化成卷积积分,并运用FFT.在相同模型参数设置下,数值结果验证了新方法在精度、收敛率及效率相比IMEX格式的优越性.展开更多
主要研究了一类状态转换下美式跳扩散期权定价模型的修正Crank-Nicolson拟合有限体积法并且给出收敛性分析.文章所构造的新方法是对[Gan X T,Yin J F,Li R,Fitted finite volume method for pricing American options under regime-swit...主要研究了一类状态转换下美式跳扩散期权定价模型的修正Crank-Nicolson拟合有限体积法并且给出收敛性分析.文章所构造的新方法是对[Gan X T,Yin J F,Li R,Fitted finite volume method for pricing American options under regime-switching.jump-diffusion models based on penalty method.Adv.Appl.Math.Mech.,2020,12(3):748-773]中时间方向上Crank-Nicolson格式的改进.同时,还对求解非线性系统迭代方法的收敛性证明进行了补充.最后,数值实验验证了新方法的有效性.展开更多
期权是以金融产品作为行权品种的交易合约。随着期权交易规模和交易量的迅速增长,期权定价的计算量越来越大,在传统CPU平台上对期权进行定价变得越来越困难。图形处理器(GPU)平台的出现和发展为解决期权定价计算提供了解决方案。在GPU...期权是以金融产品作为行权品种的交易合约。随着期权交易规模和交易量的迅速增长,期权定价的计算量越来越大,在传统CPU平台上对期权进行定价变得越来越困难。图形处理器(GPU)平台的出现和发展为解决期权定价计算提供了解决方案。在GPU上使用最小二乘蒙特卡罗算法(Least Squares Monte Carlo,LSM)实现了对一维和四维美式期权定价计算:首先利用CURAND库产生大量随机数,然后并行化期权标的价格变化路径,最后对最小二乘法和贴现定价进行并行化。为提高GPU平台上LSM方法的计算效率,对整个过程进行了优化。实际测试结果表明,在CPU+GPU上实现一维和四维美式期权定价相对CPU平台的加速比最高分别达到20.275和47.538,且比其他文献的方法整体性能有较大的提升。展开更多
A new method using nonlinear regression to approximate the option price based on approximate dynamic programming is proposed. As a result a representation of the American option price is obtained as a solution to the ...A new method using nonlinear regression to approximate the option price based on approximate dynamic programming is proposed. As a result a representation of the American option price is obtained as a solution to the dual minimization problem. In addition, an available Q-value iteration algorithm in practice is given.展开更多
This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American optio...This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.展开更多
We present a parallel algorithm that computes the ask and bid prices of an American option when proportional transaction costs apply to trading in the underlying asset. The algorithm computes the prices on recombining...We present a parallel algorithm that computes the ask and bid prices of an American option when proportional transaction costs apply to trading in the underlying asset. The algorithm computes the prices on recombining binomial trees, and is designed for modern multi-core processors. Although parallel option pricing has been well studied, none of the existing approaches takes transaction costs into consideration. The algorithm that we propose partitions a binomial tree into blocks. In any round of computation a block is further partitioned into regions which are assigned to distinct processors. To minimise load imbalance the assignment of nodes to processors is dynamically adjusted before each new round starts. Synchronisation is required both within a round and between two successive rounds. The parallel speedup of the algorithm is proportional to the number of processors used. The parallel algorithm was implemented in C/C++ via POSIX Threads, and was tested on a machine with 8 processors. In the pricing of an American put option, the parallel speedup against an efficient sequential implementation was 5.26 using 8 processors and 1500 time steps, achieving a parallel efficiency of 65.75%.展开更多
For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over...For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.展开更多
文摘基于Jain提出的高阶紧致有限差分格式(high order compact of Jain,HOCJ),结合卷积积分(convolution integral)与快速傅里叶变换(FFT),构建了一种新颖的数值方法,简称HOCJ-CF,并用于Bates模型下美式看跌期权定价.针对期权定价偏积分微分方程(PIDE)的微分项,首先将其拆分成三个子偏微分方程(sub-PDE),然后分别应用Numerov离散方法,衍生出具有空间四阶精度和时间二阶精度的HOCJ格式;积分项则将其转化成卷积积分,并运用FFT.在相同模型参数设置下,数值结果验证了新方法在精度、收敛率及效率相比IMEX格式的优越性.
文摘主要研究了一类状态转换下美式跳扩散期权定价模型的修正Crank-Nicolson拟合有限体积法并且给出收敛性分析.文章所构造的新方法是对[Gan X T,Yin J F,Li R,Fitted finite volume method for pricing American options under regime-switching.jump-diffusion models based on penalty method.Adv.Appl.Math.Mech.,2020,12(3):748-773]中时间方向上Crank-Nicolson格式的改进.同时,还对求解非线性系统迭代方法的收敛性证明进行了补充.最后,数值实验验证了新方法的有效性.
文摘期权是以金融产品作为行权品种的交易合约。随着期权交易规模和交易量的迅速增长,期权定价的计算量越来越大,在传统CPU平台上对期权进行定价变得越来越困难。图形处理器(GPU)平台的出现和发展为解决期权定价计算提供了解决方案。在GPU上使用最小二乘蒙特卡罗算法(Least Squares Monte Carlo,LSM)实现了对一维和四维美式期权定价计算:首先利用CURAND库产生大量随机数,然后并行化期权标的价格变化路径,最后对最小二乘法和贴现定价进行并行化。为提高GPU平台上LSM方法的计算效率,对整个过程进行了优化。实际测试结果表明,在CPU+GPU上实现一维和四维美式期权定价相对CPU平台的加速比最高分别达到20.275和47.538,且比其他文献的方法整体性能有较大的提升。
文摘A new method using nonlinear regression to approximate the option price based on approximate dynamic programming is proposed. As a result a representation of the American option price is obtained as a solution to the dual minimization problem. In addition, an available Q-value iteration algorithm in practice is given.
文摘This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
文摘We present a parallel algorithm that computes the ask and bid prices of an American option when proportional transaction costs apply to trading in the underlying asset. The algorithm computes the prices on recombining binomial trees, and is designed for modern multi-core processors. Although parallel option pricing has been well studied, none of the existing approaches takes transaction costs into consideration. The algorithm that we propose partitions a binomial tree into blocks. In any round of computation a block is further partitioned into regions which are assigned to distinct processors. To minimise load imbalance the assignment of nodes to processors is dynamically adjusted before each new round starts. Synchronisation is required both within a round and between two successive rounds. The parallel speedup of the algorithm is proportional to the number of processors used. The parallel algorithm was implemented in C/C++ via POSIX Threads, and was tested on a machine with 8 processors. In the pricing of an American put option, the parallel speedup against an efficient sequential implementation was 5.26 using 8 processors and 1500 time steps, achieving a parallel efficiency of 65.75%.
基金Supported by the National Natural Science Foundation of China(Grant No.11431002)
文摘For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.
