A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical ...A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.展开更多
Background:This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary(EEB).This model is based on a closed-form solution J-formula for pricing Euro...Background:This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary(EEB).This model is based on a closed-form solution J-formula for pricing European options,defined in the study by Jerbi(Quantitative Finance,15:2041-2052,2015).The J-am pricing formula is a solution of the Black&Scholes(BS)PDE with an additional function called f as a second member and with limit conditions adapted to the American option context.The aforesaid function f represents the cash flows resulting from an early exercise of the option.Methods:This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J-am,BS-am,and Heston-am models.These three models are based on the J-formula by Jerbi(Quantitative Finance,15:2041-2052,2015),BS model,and Heston(Rev Financ Stud,6:327-343,1993)model,respectively.This study performs a general algorithm leading to the EEB and to the American option price for the three models.Results:After implementing the algorithms,we compare the three aforesaid models in terms of pricing and the EEB curve.In particular,we examine the equivalence between J-am and Heston-am as an extension of the equivalence studied by Jerbi(Quantitative Finance,15:2041-2052,2015).This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model.Conclusions:We deduce that our model J-am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies.The required CPU time to compute the solution is significantly less in the case of the J-am model compared with to the Heston-am model.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
We study the approximation of variational inequality related to American options problem. A simple proof to asymptotic behavior is also given using the theta time scheme combined with a finite element spatial approxim...We study the approximation of variational inequality related to American options problem. A simple proof to asymptotic behavior is also given using the theta time scheme combined with a finite element spatial approximation in uniform norm, which enables us to locate free boundary in practice.展开更多
In this paper, we use a modified path simulation method for valuation of Asian American Options. This method is a modification of the path simulation model proposed by Tiley. We assume that the behavior of the log ret...In this paper, we use a modified path simulation method for valuation of Asian American Options. This method is a modification of the path simulation model proposed by Tiley. We assume that the behavior of the log return of the underlying assets follows the Variance Gamma (VG) process, since its distribution is heavy tail and leptokurtic. We provide sensitivity analysis of this method and compare the obtained prices to Asian European option prices.展开更多
Geometric Brownian Motion (GBM) is widely used to model the asset price dynamics. Option price models such as the Black-Sholes and the binomial tree models rely on the assumption that the underlying asset price dynami...Geometric Brownian Motion (GBM) is widely used to model the asset price dynamics. Option price models such as the Black-Sholes and the binomial tree models rely on the assumption that the underlying asset price dynamics follow the GBM. Modeling the asset price dynamics by using the GBM implies that the log return of assets at particular time is normally distributed. Many studies on real data in the markets showed that the GBM fails to capture the characteristic features of asset price dynamics that exhibit heavy tails and excess kurtosis. In our study, a class of Levy process, which is called a variance gamma (VG) process, performs much better than GBM model for modeling the dynamics of those stock indices. However, valuation of financial instruments, e.g. options, under the VG process has not been well developed. Here, we propose a new approach to the valuation of European option. It is based on the conditional distribution of the VG process. We also apply the path simulation model to value American options by assuming the underlying asset log return follow the VG process. Such a model is similar with that proposed by Tiley [1]. Simulation study shows that the proposed method performs well in term of the option price.展开更多
We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes ...We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.展开更多
In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation (PDE) was invesigated in a fixed domain for valuing two assets of American (call-max/put-min) options...In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation (PDE) was invesigated in a fixed domain for valuing two assets of American (call-max/put-min) options. From the viscosity solution of a PDE, a unique viscosity solution was obtained for the semilinear Black-Scholes PDE.展开更多
We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce...We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.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
Mathematically, the Black-Scholes model of American option pricing is a free boundary problem of partial differential equation. It is well known that this model is a nonlinear problem, and it has no closed form soluti...Mathematically, the Black-Scholes model of American option pricing is a free boundary problem of partial differential equation. It is well known that this model is a nonlinear problem, and it has no closed form solution. We can only obtain an approximate solution by numerical method, but the precision and stability are hard to control, because the singularity at the exercise boundary near expiration date has a great effect on precision and stability for numerical method. We propose a new numerical method, FDA method, to solve the American option pricing problem, which combines advantages the Semi-Analytical Method and the Front-Fixed Difference Method. Using the FDA method overcomes the difficulty resulting from the singularity at the terminal of optimal exercise boundary. A large amount of calculation shows that the FDA method is more accurate and stable than other numerical methods.展开更多
American options can be exercised prior to the date of expiration, the valuation of American options then constitutes a free boundary value problem. How to determine the free boundary, i.e. the optimal exercise price,...American options can be exercised prior to the date of expiration, the valuation of American options then constitutes a free boundary value problem. How to determine the free boundary, i.e. the optimal exercise price, is a key problem. In this paper, a nonlinear equation is given. The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.展开更多
One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments mus...One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments must be further developed to adapt to the new complexity,with short runtimes and efficient use of memory space.Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.In this paper we combine an ADI(alternating direction implicit)scheme for the temporal discretization with a sparse grid approach and the combination technique.The later approach considerably reduces the number of“spatial”grid points.The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach,since it can easily be adapted to other more complex models.展开更多
This paper studies the nonlinear variational inequality with integro-differential term arising from valuation of American style double barrier option. First, the authors use the penalty method to transform the variati...This paper studies the nonlinear variational inequality with integro-differential term arising from valuation of American style double barrier option. First, the authors use the penalty method to transform the variational inequality into a nonlinear parabolic initial boundary problem(i.e., penalty problem). Second, the existence and uniqueness of solution to the penalty problem are proved by using the Scheafer fixed point theory. Third, the authors prove the existence of variational inequality' solution by showing the fact that the penalized PDE converges to the variational inequality. The uniqueness of solution to the variational inequality is also proved by contradiction.展开更多
基金This work has been supported by the Spanish Ministerio de Economía,Industria y Competitividad(MINECO),the Agencia Estatal de Investigación(AEI)and Fondo Europeo de Desarrollo Regional(FEDER UE)grant MTM2017-89664-P.
文摘A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.
文摘Background:This study develops a new model called J-am for pricing American options and for determining the related early exercise boundary(EEB).This model is based on a closed-form solution J-formula for pricing European options,defined in the study by Jerbi(Quantitative Finance,15:2041-2052,2015).The J-am pricing formula is a solution of the Black&Scholes(BS)PDE with an additional function called f as a second member and with limit conditions adapted to the American option context.The aforesaid function f represents the cash flows resulting from an early exercise of the option.Methods:This study develops the theoretical formulas of the early exercise premium value related to three American option pricing models called J-am,BS-am,and Heston-am models.These three models are based on the J-formula by Jerbi(Quantitative Finance,15:2041-2052,2015),BS model,and Heston(Rev Financ Stud,6:327-343,1993)model,respectively.This study performs a general algorithm leading to the EEB and to the American option price for the three models.Results:After implementing the algorithms,we compare the three aforesaid models in terms of pricing and the EEB curve.In particular,we examine the equivalence between J-am and Heston-am as an extension of the equivalence studied by Jerbi(Quantitative Finance,15:2041-2052,2015).This equivalence is interesting since it can reduce a bi-dimensional model to an equivalent uni-dimensional model.Conclusions:We deduce that our model J-am exactly fits the Heston-am one for certain parameters values to be optimized and that all the theoretical results conform with the empirical studies.The required CPU time to compute the solution is significantly less in the case of the J-am model compared with to the Heston-am model.
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%.
文摘We study the approximation of variational inequality related to American options problem. A simple proof to asymptotic behavior is also given using the theta time scheme combined with a finite element spatial approximation in uniform norm, which enables us to locate free boundary in practice.
文摘In this paper, we use a modified path simulation method for valuation of Asian American Options. This method is a modification of the path simulation model proposed by Tiley. We assume that the behavior of the log return of the underlying assets follows the Variance Gamma (VG) process, since its distribution is heavy tail and leptokurtic. We provide sensitivity analysis of this method and compare the obtained prices to Asian European option prices.
文摘Geometric Brownian Motion (GBM) is widely used to model the asset price dynamics. Option price models such as the Black-Sholes and the binomial tree models rely on the assumption that the underlying asset price dynamics follow the GBM. Modeling the asset price dynamics by using the GBM implies that the log return of assets at particular time is normally distributed. Many studies on real data in the markets showed that the GBM fails to capture the characteristic features of asset price dynamics that exhibit heavy tails and excess kurtosis. In our study, a class of Levy process, which is called a variance gamma (VG) process, performs much better than GBM model for modeling the dynamics of those stock indices. However, valuation of financial instruments, e.g. options, under the VG process has not been well developed. Here, we propose a new approach to the valuation of European option. It is based on the conditional distribution of the VG process. We also apply the path simulation model to value American options by assuming the underlying asset log return follow the VG process. Such a model is similar with that proposed by Tiley [1]. Simulation study shows that the proposed method performs well in term of the option price.
文摘We give a new way to price American options by using Samuelson’s formula. We first obtain the option price corresponding to a European option at time t, weighing it by the probability that the underlying asset takes the value S at time t. We then use Samuelson’s formula with this factor which is given by the solution of the Fokker-Planck (Kolmogorov) equation for the transition probability density. The main advantage of this approach is that we can systematically introduce the effect of macroeconomic factors. If a macroeconomic framework is given by a dynamical system in the form of a set of ordinary differential equations we only have to solve a partial differential equation for the transition probability density. In this context, we verify, for the sake of consistency, that this formula coincides with the Black-Scholes model and compare several numerical implementations.
基金Project supported by the National Natural Science Foundation of China (Grant No.10271072)
文摘In this paper, by using the optimal stopping theory, the semilinear Black-Scholes partial differential equation (PDE) was invesigated in a fixed domain for valuing two assets of American (call-max/put-min) options. From the viscosity solution of a PDE, a unique viscosity solution was obtained for the semilinear Black-Scholes PDE.
基金Acknowledgements The authors would like to thank the anonymous referees for their careflll reading of the manuscript and their valuable comments. The authors also wish to thank the High Performance Computing Center of Jilin University and C, omputing Center of ,lilin Province for essential support. This work was supported by the National Natural Science Foundation of China Grant Nos. 11271157, 11371171), the Open Project Program of the State Key Lab of CAD&CG (A1302) of Zhejiang University, the Scientific Research Foundation for bleturned Scholars, Ministry of Education of China. and UIBE (11QD17).
文摘We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one- dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.
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.
基金This work was supported by a joint grant from the National Natural Science Foundation of China (Grant No. 60131160743) and Hong Kong (Grant No. N CityU102/01) .
文摘Mathematically, the Black-Scholes model of American option pricing is a free boundary problem of partial differential equation. It is well known that this model is a nonlinear problem, and it has no closed form solution. We can only obtain an approximate solution by numerical method, but the precision and stability are hard to control, because the singularity at the exercise boundary near expiration date has a great effect on precision and stability for numerical method. We propose a new numerical method, FDA method, to solve the American option pricing problem, which combines advantages the Semi-Analytical Method and the Front-Fixed Difference Method. Using the FDA method overcomes the difficulty resulting from the singularity at the terminal of optimal exercise boundary. A large amount of calculation shows that the FDA method is more accurate and stable than other numerical methods.
基金Supported by National Science Foundation of China
文摘American options can be exercised prior to the date of expiration, the valuation of American options then constitutes a free boundary value problem. How to determine the free boundary, i.e. the optimal exercise price, is a key problem. In this paper, a nonlinear equation is given. The free boundary can be obtained by solving the nonlinear equation and the numerical results are better.
基金supported by the bilateral German-Slovakian Project MATTHIAS–Modelling and Approximation Tools and Techniques for Hamilton-Jacobi-Bellman equations in finance and Innovative Approach to their Solution,financed by DAAD and the Slovakian Ministry of Education.Further the authors acknowledge partial support from the bilateral German-Portuguese Project FRACTAL–FRActional models and CompuTationAL Finance financed by DAAD and the CRUP–Conselho de Reitores das Universidades Portuguesas.
文摘One goal of financial research is to determine fair prices on the financial market.As financial models and the data sets on which they are based are becoming ever larger and thus more complex,financial instruments must be further developed to adapt to the new complexity,with short runtimes and efficient use of memory space.Here we show the effects of combining known strategies and incorporating new ideas to further improve numerical techniques in computational finance.In this paper we combine an ADI(alternating direction implicit)scheme for the temporal discretization with a sparse grid approach and the combination technique.The later approach considerably reduces the number of“spatial”grid points.The presented standard financial problem for the valuation of American options using the Heston model is chosen to illustrate the advantages of our approach,since it can easily be adapted to other more complex models.
基金supported by the National Natural Science Foundations of China under Grant No.71201013the National Science Fund for Distinguished Young Scholars of China under Grant No.70825006+1 种基金the Program for Changjiang Scholars and Innovative Research Team in University under Grant No.IRT0916the National Natural Science Innovation Research Group of China under Grant No.71221001
基金supported by the National Science Foundation of China under Grant Nos.71171164 and 70471057the Doctorate Foundation of Northwestern Polytechnical University under Grant No.CX201235
文摘This paper studies the nonlinear variational inequality with integro-differential term arising from valuation of American style double barrier option. First, the authors use the penalty method to transform the variational inequality into a nonlinear parabolic initial boundary problem(i.e., penalty problem). Second, the existence and uniqueness of solution to the penalty problem are proved by using the Scheafer fixed point theory. Third, the authors prove the existence of variational inequality' solution by showing the fact that the penalized PDE converges to the variational inequality. The uniqueness of solution to the variational inequality is also proved by contradiction.