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123
Efficient simulation of the Heston stochastic volatility model. Working paper, Banc of America Securities
, 2007
"... Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for timediscreti ..."
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Cited by 72 (1 self)
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Stochastic volatility models are increasingly important in practical derivatives pricing applications, yet relatively little work has been undertaken in the development of practical Monte Carlo simulation methods for this class of models. This paper considers several new algorithms for timediscretization and Monte Carlo simulation of Hestontype stochastic volatility models. The algorithms are based on a careful analysis of the properties of affine stochastic volatility diffusions, and are straightforward and quick to implement and execute. Tests on realistic model parameterizations reveal that the computational efficiency and robustness of the simulation schemes proposed in the paper compare very favorably to existing methods. 1
Convergence of numerical methods for stochastic differential equations in mathematical finance
, 1204
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Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients
"... We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Mo ..."
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Cited by 46 (15 self)
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We consider the numerical solution of elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification for groundwater flow. We describe a novel variance reduction technique for the standard Monte Carlo method, called the multilevel Monte Carlo method. The main result is that in certain circumstances the asymptotic cost of solving the stochastic problem is a constant (but moderately large) multiple of the cost of solving the deterministic problem. Numerical calculations demonstrating the effectiveness of the method for one and twodimensional model problems arising in groundwater flow are presented. 1
HIGH ORDER DISCRETIZATION SCHEMES FOR THE CIR PROCESS: APPLICATION TO AFFINE TERM STRUCTURE AND HESTON MODELS
"... Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya a ..."
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Cited by 32 (3 self)
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Abstract. This paper presents weak second and third order schemes for the CoxIngersollRoss (CIR) process, without any restriction on its parameters. At the same time, it gives a general recursive construction method for getting weak second order schemes that extend the one introduced by Ninomiya and Victoir. Combine both these results, this allows us to propose a second order scheme for more general affine diffusions. Simulation examples are given to illustrate the convergence of these schemes on CIR and Heston models.
Pricing Options in JumpDiffusion Models: An Extrapolation Approach
, 2008
"... We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization sc ..."
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Cited by 17 (3 self)
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We propose a new computational method for the valuation of options in jumpdiffusion models. The option value function for European and barrier options satisfies a partial integrodifferential equation (PIDE). This PIDE is commonly integrated in time by implicitexplicit (IMEX) time discretization schemes, where the differential (diffusion) term is treated implicitly, while the integral (jump) term is treated explicitly. In particular, the popular IMEX Euler scheme is firstorder accurate in time. Secondorder accuracy in time can be achieved by using the IMEX midpoint scheme. In contrast to the above approaches, we propose a new highorder time discretization scheme for the PIDE based on the extrapolation approach to the solution of ODEs that also treats the diffusion term implicitly and the jump term explicitly. The scheme is simple to implement, can be added to any PIDE solver based on the IMEX Euler scheme, and is remarkably fast and accurate. We demonstrate our approach on the examples of Merton’s and Kou’s jumpdiffusion models, the diffusionextended variance gamma model, as well as the twodimensional DuffiePanSingleton model with correlated and contemporaneous jumps in the stock price and its volatility. By way of example, pricing a oneyear doublebarrier option in Kou’s jumpdiffusion model, our scheme attains accuracy of 10−5 in 72 time steps (in 0.05 seconds). In contrast, it takes the firstorder IMEX Euler scheme more than 1.3 million time steps (in 873 seconds) and the secondorder IMEX midpoint scheme 768 time steps (in 0.49 seconds) to attain the same accuracy. Our scheme is also well suited for Bermudan options. Combining simplicity of implementation and remarkable gains in computational efficiency, we expect this method to be very attractive
Fast strong approximation MonteCarlo schemes for stochastic volatility models
 Working Paper, ABN AMRO
, 2006
"... Abstract Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a meanreverting CEV process or as a transformed OrnsteinUhlenbeck process. For ..."
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Abstract Numerical integration methods for stochastic volatility models in financial markets are discussed. We concentrate on two classes of stochastic volatility models where the volatility is either directly given by a meanreverting CEV process or as a transformed OrnsteinUhlenbeck process. For the latter, we introduce a new model based on a simple hyperbolic transformation. Various numerical methods for integrating meanreverting CEV processes are analysed and compared with respect to positivity preservation and efficiency. Moreover, we develop a simple and robust integration scheme for the twodimensional system using the strong convergence behaviour as an indicator for the approximation quality. This method, which we refer to as the IJK (4.47) scheme, is applicable to all types of stochastic volatility models and can be employed as a dropin replacement for the standard logEuler procedure. Acknowledgment: The authors thank Vladimir Piterbarg and an anonymous referee for helpful comments and suggestions.
Gamma Expansion of the Heston Stochastic Volatility Model
"... We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We ..."
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Cited by 14 (1 self)
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We derive an explicit representation of the transitions of the Heston stochastic volatility model and use it for fast and accurate simulation of the model. Of particular interest is the integral of the variance process over an interval, conditional on the level of the variance at the endpoints. We give an explicit representation of this quantity in terms of infinite sums and mixtures of gamma random variables. The increments of the variance process are themselves mixtures of gamma random variables. The representation of the integrated conditional variance applies the PitmanYor decomposition of Bessel bridges. We combine this representation with the BroadieKaya exact simulation method and use it to circumvent the most timeconsuming step in that method.
Efficient, almost exact simulation of the Heston stochastic volatility model
 International Journal of Theoretical and Applied Finance
, 2010
"... Efficient, almost exact simulation of the Heston stochastic volatility model A. van Haastrecht1 2 and A.A.J. Pelsser3. ..."
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Cited by 12 (1 self)
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Efficient, almost exact simulation of the Heston stochastic volatility model A. van Haastrecht1 2 and A.A.J. Pelsser3.
Sensitivity Estimates From Characteristic Functions
, 2007
"... The likelihood ration method (LRM) is a technique for estimating derivatives of expectations through simulation. LRM estimators are constructed from the derivatives of probability densities of inputs to a simulation. We investigate the application of the likelihood ratio method for sensitivity estim ..."
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Cited by 12 (3 self)
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The likelihood ration method (LRM) is a technique for estimating derivatives of expectations through simulation. LRM estimators are constructed from the derivatives of probability densities of inputs to a simulation. We investigate the application of the likelihood ratio method for sensitivity estimation when the relevant densities for the underlying model are known only through their characteristic functions or Laplace transforms. This problem arises in financial applications, where sensitivities are used for managing risk and where a substantial class of models have transition densities known only through their transforms. We quantify various sources of errors arising when numerical transform inversion is used to sample through the characteristic function and to evaluate the density and its derivative, as required in LRM. This analysis provides guidance for setting parameters in the method to accelerate convergence. 1
An eulertype method for the strong approximation of the cox–ingersoll–ross process
 Proceedings of the Royal Society A Engineering Science
"... Abstract. We analyze the strong approximation of the CoxIngersollRoss (CIR) process in the regime where the process does not hit zero by a positivity preserving driftimplicit Eulertype method. As an error criterion we use the pth mean of the maximum distance between the CIR process and its appr ..."
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Cited by 12 (2 self)
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Abstract. We analyze the strong approximation of the CoxIngersollRoss (CIR) process in the regime where the process does not hit zero by a positivity preserving driftimplicit Eulertype method. As an error criterion we use the pth mean of the maximum distance between the CIR process and its approximation on a finite time interval. We show that under mild assumptions on the parameters of the CIR process the proposed method attains, up to a logarithmic term, the convergence of order 1/2. This agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations (SDEs) with Lipschitz coefficients – despite the fact that the CIR process has a nonLipschitz diffusion coefficient.