The authors present a new simple algorithm to approximate weakly stochastic differential equation... more The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [10][16]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast.
[1] Shigeo Kusuoka, "Approximation of Expectation of Diffusion Process and Mathematical Finance,"... more [1] Shigeo Kusuoka, "Approximation of Expectation of Diffusion Process and Mathematical Finance," Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, "Cubature on Wiener Space," Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, "Weak approximation of stochastic differential equations and application to derivative pricing," Applied Mathematical Finance, Volume 15, Issue 2 April 2008, pages 107--121
Abstract. Anew class of $s$-dimensional uniformly distributed sequences called the generalized va... more Abstract. Anew class of $s$-dimensional uniformly distributed sequences called the generalized van der Corput sequence is defined. The sequence is constructed by using the generalized number system based on an integer matrix whose all eigenvalues reside out of the unit circle. In this talk, we show that by using the generalized van der Corput sequence we can calculate numerical integrations with the convergence speed $O(1/N) $ when integrands satisfy some regularity conditions. We also apply the sequence to anumerical integration problem and test effectiveness of the sequence. 1Introduction We can consider the van der Corput seqeunce to be an orbit of the origin under the adding machine transformation that is accompanied by an expand-ing one dimensional linear transformation [2,6-8]. Following this principle of regarding the van der Corput sequence as an orbit of the adding machine transformation, we can generalize the van der Corput sequence in various directions [1,4,6,7]. In this...
The authors report on the construction of a new algorithm for the weak approximation of stochasti... more The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge-Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.
This study proposes new higher-order discretization methods of forward-backward stochastic differ... more This study proposes new higher-order discretization methods of forward-backward stochastic differential equations. In the proposed methods, the forward component is discretized using the Kusuoka-Lyons-Ninomiya-Victoir scheme with discrete random variables and the backward component using a higher-order numerical integration method consistent with the discretization method of the forward component, by use of the tree based branching algorithm. The proposed methods are applied to the XVA pricing, in particular to the credit valuation adjustment. The numerical results show that the expected theoretical order and computational efficiency could be achieved.
It is well known that low-discrepancy sequences and their discrepancy play essential roles in qua... more It is well known that low-discrepancy sequences and their discrepancy play essential roles in quasi Monte Carlo methods [5]. In this paper, a new class of low-discrepancy sequences N β is constructed by using the ergodic theoretical transformation which is called β-adic ...
Journal of the Operations Research Society of Japan, 1998
In this paper, we trace Paskov's experiment, investigate the characteristics and the generation a... more In this paper, we trace Paskov's experiment, investigate the characteristics and the generation algorithm of the pseudo-random number generator in question, and explain why the wrong convergences occur. We then present a method for avQiding such wrong convergences. A variance reduction procedure is applied, tegether with a method for obtaining more precise values, and its correctness is examined. We also investigate whether seatistical tests for pseuderamdom numbers can detect the cause of wrong convergences.
Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications, 2016
The main theme of this research is numerical verification of applicability of a higher-order appr... more The main theme of this research is numerical verification of applicability of a higher-order approximation to pricing barrier options, which is a both mathematically and practically important path-dependent type problem in mathematical finance. The authors successfully extend two types of algorithms called NV and NN. Both algorithms are based on a higher-order approximation scheme called Kusuoka approximation and have been shown to attain second-order approximation of stochastic differential equations as long as applied to European-type problems. In extending the algorithms, the authors apply a function representing the probability of hitting a boundary. In the numerical experiments, these two algorithms are compared with the Euler-Maruyama scheme which is one of the most popular first-order approximation schemes. As a result, it is demonstrated that the speed of calculation of these two algorithms could be much higher than that of the Euler-Maruyama scheme extended by the same way. It is concluded that one of the keys to improvement of the results is the construction of the function for calculation of the probability of hitting a boundary.
Stochastic Processes and Applications to Mathematical Finance, 2004
The authors apply a new simulation method of diffusion processes to finance problems and show thi... more The authors apply a new simulation method of diffusion processes to finance problems and show this new method realizes extremely fast calculation in some cases.
The Kusuoka approximation is a new simulation scheme for di¬usion processes which are solutions o... more The Kusuoka approximation is a new simulation scheme for di¬usion processes which are solutions of SDE with smooth coe¯cients. The author had reported that the Kusuoka approximation realizes several thousands times faster calculation of some nancial derivative pricing problems than the Euler-Maruyama approximation does. In this paper, the author applied TBBA to the Kusuoka approximation and succeeded in several hundreds times faster calculation than naive Monte Carlo sampling.
We apply a new simulation scheme proposed by Kusuoka to finance problems. By using this method, w... more We apply a new simulation scheme proposed by Kusuoka to finance problems. By using this method, we achieve 6500 times faster simulation than traditional Euler-Maruyama scheme.
The authors report on the construction of a new algorithm for the weak approximation of stochasti... more The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge-Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex... more In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is possible by using conventional Monte Carlo ...
Working Papers are a series of manuscripts in their draft form. They are not intended for circula... more Working Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Working Papers may not be reproduced or distributed without the written consent of the author.
The authors present a new simple algorithm to approximate weakly stochastic differential equation... more The authors present a new simple algorithm to approximate weakly stochastic differential equations in the spirit of [10][16]. They apply it to the problem of pricing Asian options under the Heston stochastic volatility model, and compare it with other known methods. It is shown that the combination of the suggested algorithm and quasi-Monte Carlo methods makes computations extremely fast.
[1] Shigeo Kusuoka, "Approximation of Expectation of Diffusion Process and Mathematical Finance,"... more [1] Shigeo Kusuoka, "Approximation of Expectation of Diffusion Process and Mathematical Finance," Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998 (T. Sunada, ed.), vol. 31 2001, pp. 147--165. [2] Terry Lyons and Nicolas Victoir, "Cubature on Wiener Space," Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460 (2004), pp. 169--198. [3] Syoiti Ninomiya, Nicolas Victoir, "Weak approximation of stochastic differential equations and application to derivative pricing," Applied Mathematical Finance, Volume 15, Issue 2 April 2008, pages 107--121
Abstract. Anew class of $s$-dimensional uniformly distributed sequences called the generalized va... more Abstract. Anew class of $s$-dimensional uniformly distributed sequences called the generalized van der Corput sequence is defined. The sequence is constructed by using the generalized number system based on an integer matrix whose all eigenvalues reside out of the unit circle. In this talk, we show that by using the generalized van der Corput sequence we can calculate numerical integrations with the convergence speed $O(1/N) $ when integrands satisfy some regularity conditions. We also apply the sequence to anumerical integration problem and test effectiveness of the sequence. 1Introduction We can consider the van der Corput seqeunce to be an orbit of the origin under the adding machine transformation that is accompanied by an expand-ing one dimensional linear transformation [2,6-8]. Following this principle of regarding the van der Corput sequence as an orbit of the adding machine transformation, we can generalize the van der Corput sequence in various directions [1,4,6,7]. In this...
The authors report on the construction of a new algorithm for the weak approximation of stochasti... more The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge-Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.
This study proposes new higher-order discretization methods of forward-backward stochastic differ... more This study proposes new higher-order discretization methods of forward-backward stochastic differential equations. In the proposed methods, the forward component is discretized using the Kusuoka-Lyons-Ninomiya-Victoir scheme with discrete random variables and the backward component using a higher-order numerical integration method consistent with the discretization method of the forward component, by use of the tree based branching algorithm. The proposed methods are applied to the XVA pricing, in particular to the credit valuation adjustment. The numerical results show that the expected theoretical order and computational efficiency could be achieved.
It is well known that low-discrepancy sequences and their discrepancy play essential roles in qua... more It is well known that low-discrepancy sequences and their discrepancy play essential roles in quasi Monte Carlo methods [5]. In this paper, a new class of low-discrepancy sequences N β is constructed by using the ergodic theoretical transformation which is called β-adic ...
Journal of the Operations Research Society of Japan, 1998
In this paper, we trace Paskov's experiment, investigate the characteristics and the generation a... more In this paper, we trace Paskov's experiment, investigate the characteristics and the generation algorithm of the pseudo-random number generator in question, and explain why the wrong convergences occur. We then present a method for avQiding such wrong convergences. A variance reduction procedure is applied, tegether with a method for obtaining more precise values, and its correctness is examined. We also investigate whether seatistical tests for pseuderamdom numbers can detect the cause of wrong convergences.
Proceedings of the ISCIE International Symposium on Stochastic Systems Theory and its Applications, 2016
The main theme of this research is numerical verification of applicability of a higher-order appr... more The main theme of this research is numerical verification of applicability of a higher-order approximation to pricing barrier options, which is a both mathematically and practically important path-dependent type problem in mathematical finance. The authors successfully extend two types of algorithms called NV and NN. Both algorithms are based on a higher-order approximation scheme called Kusuoka approximation and have been shown to attain second-order approximation of stochastic differential equations as long as applied to European-type problems. In extending the algorithms, the authors apply a function representing the probability of hitting a boundary. In the numerical experiments, these two algorithms are compared with the Euler-Maruyama scheme which is one of the most popular first-order approximation schemes. As a result, it is demonstrated that the speed of calculation of these two algorithms could be much higher than that of the Euler-Maruyama scheme extended by the same way. It is concluded that one of the keys to improvement of the results is the construction of the function for calculation of the probability of hitting a boundary.
Stochastic Processes and Applications to Mathematical Finance, 2004
The authors apply a new simulation method of diffusion processes to finance problems and show thi... more The authors apply a new simulation method of diffusion processes to finance problems and show this new method realizes extremely fast calculation in some cases.
The Kusuoka approximation is a new simulation scheme for di¬usion processes which are solutions o... more The Kusuoka approximation is a new simulation scheme for di¬usion processes which are solutions of SDE with smooth coe¯cients. The author had reported that the Kusuoka approximation realizes several thousands times faster calculation of some nancial derivative pricing problems than the Euler-Maruyama approximation does. In this paper, the author applied TBBA to the Kusuoka approximation and succeeded in several hundreds times faster calculation than naive Monte Carlo sampling.
We apply a new simulation scheme proposed by Kusuoka to finance problems. By using this method, w... more We apply a new simulation scheme proposed by Kusuoka to finance problems. By using this method, we achieve 6500 times faster simulation than traditional Euler-Maruyama scheme.
The authors report on the construction of a new algorithm for the weak approximation of stochasti... more The authors report on the construction of a new algorithm for the weak approximation of stochastic differential equations. In this algorithm, an ODE-valued random variable whose average approximates the solution of the given stochastic differential equation is constructed by using the notion of free Lie algebras. It is proved that the classical Runge-Kutta method for ODEs is directly applicable to the ODE drawn from the random variable. In a numerical experiment, this is applied to the problem of pricing Asian options under the Heston stochastic volatility model. Compared with some other methods, this algorithm is significantly faster.
In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex... more In this paper, we investigate the feasibility of using low-discrepancy sequences to allow complex derivatives, such as mortgage-backed securities (MBSs) and exotic options, to be calculated considerably faster than is possible by using conventional Monte Carlo ...
Working Papers are a series of manuscripts in their draft form. They are not intended for circula... more Working Papers are a series of manuscripts in their draft form. They are not intended for circulation or distribution except as indicated by the author. For that reason Working Papers may not be reproduced or distributed without the written consent of the author.
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