D. Y. Tangman

University of Mauritius, Moka Village, Moka, Mauritius

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Publications (14)8.61 Total impact

  • [show abstract] [hide abstract]
    ABSTRACT: Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.
    European Journal of Operational Research 01/2013; 224(1):219–226. · 2.04 Impact Factor
  • Nawdha Thakoor, Desire Yannick Tangman, Muddun Bhuruth
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    ABSTRACT: Binomial and trinomial lattices are popular techniques for pricing financial options. These methods work well for European and American options, but for barrier options, the need to place a tree node very close to a barrier brings diffi�culties in their implementations and a large number of time steps are usually required when the barrier is close to the current asset price. A �finite diff�erence implementation is simpler and we propose a fourth-order numerical scheme for continuously and discretely monitored barriers. We demonstrate the superior performance of our technique over existing procedures for the Black-Scholes model and we then price barriers under the constant elasticity of variance (CEV) di�ffusion. Continuously monitored barriers under CEV admit an analytical solution but evaluation via this formula is not straightforward. Furthermore, discretely monitored barriers have to be priced numerically. Our main contribution is therefore a highly accurate and effi�cient numerical scheme for barrier options under CEV and we provide several numerical examples to illustrate the merit of the new technique.
    Journal of Computational and Applied Mathematics 01/2013; · 0.99 Impact Factor
  • N. Thakoor, D. Y. Tangman, M. Bhuruth
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    ABSTRACT: The empirically observed negative relationship between a stock price and its return volatility can be captured by the constant elasticity of variance option pricing model. For European options, closed form expressions involve the non-central chi-square distribution whose computation can be slow when the elasticity factor is close to one, volatility is low or time to maturity is small. We present a fast numerical scheme based on a high-order compact discretisation which accurately computes the option price. Various numerical examples indicate that for comparable computational times, the option price computed with the scheme has higher accuracy than the Crank-Nicolson numerical solution. The scheme accurately computes the hedging parameters and is stable for strongly negative values of the elasticity factor.
    Applied Mathematics Letters 08/2012; · 1.50 Impact Factor
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    Nawdha Thakoor, D. Yannick Tangman, Muddun Bhuruth
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    ABSTRACT: The constant elasticity of variance (CEV) model has the ability to capture the empirically observed negative relationship between a stock price and its return volatility. New numerical methods are developed for the pricing of European and American options under the CEV model. For European options, we develop a high-order compact (HOC) scheme which yields highly accurate prices. The numerical results exhibit fourth-order convergence and we provide evidence that for comparable running times, the HOC numerical solutions are more accurate than the Crank-Nicolson solutions. For American options, few numerical pricing techniques have been proposed under CEV. An operator splitting algorithm for pricing options is introduced and we give numerical illustrations of the merit of our approach for pricing American options.
    International Congress on Computational and Applied Mathematics; 07/2012
  • A. A. E. F. Saib, D. Y. Tangman, M. Bhuruth
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    ABSTRACT: A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.
    International Journal of Computer Mathematics - IJCM. 01/2012;
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    ABSTRACT: (Top 25 Download in Science-Direct) http://top25.sciencedirect.com/subject/economics-econometrics-and-finance/10/journal/finance-research-letters/15446123/archive/33 A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.
    Finance Research Letters 01/2011; · 0.33 Impact Factor
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    ABSTRACT: A new computational method for approximating prices of zerocoupon bonds and bond option prices under general Chan–Karolyi– Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.
    Finance Research Letters 01/2011; 8:206 - 212. · 0.33 Impact Factor
  • Nisha Rambeerich, Desire Yannick Tangman, Muddun Bhuruth
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    ABSTRACT: Under infinite activity Lévy models, American option prices can be obtained by solving a partial integro-differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time-stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank–Nicolson (CN) method and an implicit–explicit method in conjunction with an extrapolation (IMEX-Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE-based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX-Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point-wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX-Extrap achieves a precision of the order of 10−4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10−5 in 237.891 s and 22.772 s, respectively. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:809–829, 2011
    Journal of Futures Markets 11/2010; 31(9):809 - 829. · 0.46 Impact Factor
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    ABSTRACT: We analyze a Simpler GMRES variant of augmented GMRES with implicit restarting for solving nonsymmetric linear systems with small eigenvalues. The use of a shifted Arnoldi process in the Simpler GMRES variant for computing Arnoldi basis vectors has the advantage of not requiring an upper Hessenberg factorization and this often leads to cheaper implementations. However the use of a non-orthogonal basis has been identified as a potential weakness of the Simpler GMRES algorithm. Augmented variants of GMRES also employ non-orthogonal basis vectors since approximate eigenvectors are added to the Arnoldi basis vectors at the end of a cycle and in case the approximate eigenvectors are ill-conditioned, this may have an adverse effect on the accuracy of the computed solution. This problem is the focus of our paper where we analyze the shifted Arnoldi implementation of augmented GMRES with implicit restarting and compare its performance and accuracy with that based on the Arnoldi process. We show that augmented Simpler GMRES with implicit restarting involves a transformation matrix which leads to an efficient implementation and we theoretically show that our implementation generates the same subspace as the corresponding GMRES variant. We describe various numerical tests that indicate that in cases where both variants are successful, our method based on Simpler GMRES keeps comparable accuracy as the augmented GMRES variant. Also, the Simpler GMRES variants perform better in terms of computational time required.
    Computer Science and its Applications - ICCSA 2010. 01/2010; 6017:570-585.
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    D. Y. Tangman, A. Peer, N. Rambeerich, M. Bhuruth
    Journal of Computational Finance. 07/2009;
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    ABSTRACT: We consider exponential time integration schemes for fast numerical pricing of European, American, barrier and butterfly options when the stock price follows a dynamics described by a jump-diffusion process. The resulting pricing equation which is in the form of a partial integro-differential equation is approximated in space using finite elements. Our methods require the computation of a single matrix exponential and we demonstrate using a wide range of numerical tests that the combination of exponential integrators and finite element discretisations with quadratic basis functions leads to highly accurate algorithms for cases when the jump magnitude is Gaussian. Comparison with other time-stepping methods are also carried out to illustrate the effectiveness of our methods
    Journal of Computational and Applied Mathematics 03/2009; 224:668-678. · 0.99 Impact Factor
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    D.Y. Tangman, A. Gopaul, M. Bhuruth
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    ABSTRACT: We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black–Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.
    Journal of Computational and Applied Mathematics 01/2008; · 0.99 Impact Factor
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    D.Y. Tangman, A. Gopaul, M. Bhuruth
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    ABSTRACT: We describe an improvement of Han and Wu’s algorithm [H. Han, X.Wu, A fast numerical method for the Black–Scholes equation of American options, SIAM J. Numer. Anal. 41 (6) (2003) 2081–2095] for American options. A high-order optimal compact scheme is used to discretise the transformed Black–Scholes PDE under a singularity separating framework. A more accurate free boundary location based on the smooth pasting condition and the use of a non-uniform grid with a modified tridiagonal solver lead to an efficient implementation of the free boundary value problem. Extensive numerical experiments show that the new finite difference algorithm converges rapidly and numerical solutions with good accuracy are obtained. Comparisons with some recently proposed methods for the American options problem are carried out to show the advantage of our numerical method.
    Journal of Computational and Applied Mathematics 01/2008; · 0.99 Impact Factor
  • D.Y. Tangman, A. Gopaul, M. Bhuruth
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    ABSTRACT: We consider exponential time differencing (ETD) schemes for the numerical pricing of options. Special treatments for the implementation of the boundary conditions that arise in finance are described. We show that only one explicit time step computation gives unconditional second order accuracy for European, Barrier and Butterfly spread options under both Black–Scholes geometric Brownian motion model and Merton's jump diffusion model with constant coefficients. In comparison, the commonly used Crank–Nicolson scheme is shown to be only conditionally stable due to lack of L0-stability. Finally, we describe how the use of spectral spatial discretisation based on a Chebychev grid point concentration strategy gives fourth order accurate option prices for both the Black–Scholes and Merton's jump–diffusion model.
    Applied Numerical Mathematics. 01/2008;