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17
Option pricing under a mixedexponential jump diffusion model
 Management Science
, 2011
"... This paper aims to extend the analytical tractability of the Black–Scholes model to alternative models witharbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixedexponential distribution, which is a weighted average of expo ..."
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This paper aims to extend the analytical tractability of the Black–Scholes model to alternative models witharbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixedexponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and doubleexponential jump diffusion models, because the mixedexponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavytailed distributions. The mixedexponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for pathdependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a highorder integrodifferential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day. Key words: jump diffusion; mixedexponential distributions; lookback options; barrier options; Merton’s normal jump diffusion model; first passage times
Componentwise splitting methods for pricing American options under stochastic volatility
 Int. J. Theor. Appl. Finance
, 2007
"... Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple qu ..."
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Abstract A linear complementarity problem (LCP) is formulated for the price of American options under the Bates model which combines the Heston stochastic volatility model and the Merton jumpdiffusion model. A finite difference discretization is described for the partial derivatives and a simple quadrature is used for the integral term due to jumps. A componentwise splitting method is generalized for the Bates model. It is leads to solution of sequence of onedimensional LCPs which can be solved very efficiently using the Brennan and Schwartz algorithm. The numerical experiments demonstrate the componentwise splitting method to be essentially as accurate as the PSOR method, but order of magnitude faster. Furthermore, pricing under the Bates model is less than twice more expensive computationally than under the Heston model in the experiments. 1
An iterative method for pricing American options under jumpdiffusion models,
 Appl. Numer. Math.
, 2011
"... Abstract We consider the numerical pricing of American options under the Bates model which adds lognormally distributed jumps for the asset value to the Heston stochastic volatility model. A linear complementarity problem (LCP) is formulated where partial derivatives are discretized using finite d ..."
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Abstract We consider the numerical pricing of American options under the Bates model which adds lognormally distributed jumps for the asset value to the Heston stochastic volatility model. A linear complementarity problem (LCP) is formulated where partial derivatives are discretized using finite differences and the integral resulting from the jumps is evaluated using simple quadrature. A rapidly converging fixed point iteration is described for the LCP, where each iterate requires the solution of an LCP. These are easily solved using a projected algebraic multigrid (PAMG) method. The numerical experiments demonstrate the efficiency of the proposed approach. Furthermore, they show that the PAMG method leads to better scalability than the projected SOR (PSOR) method when the discretization is refined.
A New Spectral Element Method for Pricing European Options Under the Black–Scholes and Merton Jump Diffusion Models
, 2011
"... Abstract We present a new spectral element method for solving partial integrodifferential equations for pricing European options under the Black–Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear syste ..."
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Abstract We present a new spectral element method for solving partial integrodifferential equations for pricing European options under the Black–Scholes and Merton jump diffusion models. Our main contributions are: (i) using an optimal set of orthogonal polynomial bases to yield banded linear systems and to achieve spectral accuracy; (ii) using Laguerre functions for the approximations on the semiinfinite domain, to avoid the domain truncation; and (iii) deriving a rigorous proof of stability for the time discretizations of European put options under both the Black–Scholes model and the Merton jump diffusion model. The new method is flexible for handling different boundary conditions and nonsmooth initial conditions for various contingent claims. Numerical examples are presented to demonstrate the efficiency and accuracy of the new method.
Variational Methods in Derivatives Pricing
, 2008
"... When underlying financial variables follow a Markov jumpdiffusion process, the value function of a derivative security satisfies a partial integrodifferential equation (PIDE) for Europeanstyle exercise or a partial integrodifferential variational inequality (PIDVI) for Americanstyle exercise. U ..."
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When underlying financial variables follow a Markov jumpdiffusion process, the value function of a derivative security satisfies a partial integrodifferential equation (PIDE) for Europeanstyle exercise or a partial integrodifferential variational inequality (PIDVI) for Americanstyle exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PIDE or the PIDVI numerically. In this chapter we briefly survey a computational method for the valuation of options in jumpdiffusion models based on: (1) converting the PIDE or PIDVI to a variational (weak) form; (2) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (3) integrating the resulting system of ODEs in time. To introduce the method, we start with the basic examples of European, barrier, and American
A Lattice Algorithm for Pricing Moving Average Barrier Options
 JOURNAL OF ECONOMIC DYNAMICS AND CONTROL
, 2009
"... This paper presents a lattice algorithm for pricing both European and Americanstyle moving average barrier options (MABOs). We develop a finitedimensional partial differential equation (PDE) model for discretely monitored MABOs and solve it numerically by using a forward shooting grid method. The ..."
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This paper presents a lattice algorithm for pricing both European and Americanstyle moving average barrier options (MABOs). We develop a finitedimensional partial differential equation (PDE) model for discretely monitored MABOs and solve it numerically by using a forward shooting grid method. The modeling PDE for continuously monitored MABOs has infinite dimensions and cannot be solved directly by any existing numerical method. We find their approximate values indirectly by using an extrapolation technique with the prices of discretely monitored MABOs. Numerical experiments show that our algorithm is very efficient.
Chapter 2 JumpDiffusion Models for Asset Pricing in Financial Engineering
 In: Handbooks in Operations Research and Management
, 2007
"... In this survey we shall focus on the following issues related to jumpdiffusion models for asset pricing in financial engineering. (1) The controversy over tailweight of distributions. (2) Identifying a riskneutral pricing measure by using the rational expectations equilibrium. (3) Using Laplace ..."
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In this survey we shall focus on the following issues related to jumpdiffusion models for asset pricing in financial engineering. (1) The controversy over tailweight of distributions. (2) Identifying a riskneutral pricing measure by using the rational expectations equilibrium. (3) Using Laplace transforms to pricing options, including European call/put options, pathdependent options, such as barrier and lookback options. (4) Difficulties associated with the partial integrodifferential equations related to barriercrossing problems. (5) Analytical approximations for finitehorizon American options with jump risk. (6) Multivariate jumpdiffusion models. 1
Inverse transform method for simulating Lévy processes and discrete Asian options pricing
 in: “Proceedings of the 2011 Winter Simulation Conference
, 2011
"... The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inv ..."
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The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inverse transform method, when combined with a Hilbert transform approach for computing the cdf of the Lévy increment, is reliable and efficient. The Hilbert transform representation for the cdf is easy to implement and highly accurate, with approximation errors decaying exponentially. The inverse transform method can be combined with quasiMonte Carlo methods and variance reduction techniques to greatly increase the efficiency of the scheme. As an illustration, discrete Asian options pricing in the CGMY model is considered, where the combination of the Hilbert transform inversion of characteristic functions, quasiMonte Carlo methods and the control variate technique proves to be very efficient. 1
INVERSE TRANSFORM METHOD FOR SIMULATING LEVY PROCESSES AND DISCRETE ASIAN OPTIONS PRICING
"... The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inverse ..."
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The simulation of a Lévy process on a discrete time grid reduces to simulating from the distribution of a Lévy increment. For a general Lévy process with no explicit transition density, it is often desirable to simulate from the characteristic function of the Lévy increment. We show that the inverse transform method, when combined with a Hilbert transform approach for computing the cdf of the Lévy increment, is reliable and efficient. The Hilbert transform representation for the cdf is easy to implement and highly accurate, with approximation errors decaying exponentially. The inverse transform method can be combined with quasiMonte Carlo methods and variance reduction techniques to greatly increase the efficiency of the scheme. As an illustration, discrete Asian options pricing in the CGMY model is considered, where the combination of the Hilbert transform inversion of characteristic functions, quasiMonte Carlo methods and the control variate technique proves to be very efficient. 1
Pricing HighDimensional Bermudan Options using VarianceReduced Monte Carlo Methods
"... We present a numerical method for pricing Bermudan options depending on a large number of underlyings. American option prices can be approximated with the same method by choosing a sufficiently large number of exercise dates. The asset prices are modeled with exponential timeinhomogeneous jumpdiffu ..."
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We present a numerical method for pricing Bermudan options depending on a large number of underlyings. American option prices can be approximated with the same method by choosing a sufficiently large number of exercise dates. The asset prices are modeled with exponential timeinhomogeneous jumpdiffusion processes. We improve the leastsquares Monte Carlo method proposed by Longstaff and Schwartz introducing an efficient variance reduction scheme. A control variable is obtained from a lowdimensional approximation of the multivariate Bermudan option. To this end, we adapt a model reduction method called proper orthogonal decomposition (POD), which is closely related to principal component analysis, to the case of Bermudan options. Our goal is to make use of the correlation structure of the assets in an optimal way. We compute the expectation of the control variable by either solving a lowdimensional partial integrodifferential equation or by applying Fourier methods. The POD approximation can also be used as a candidate for the minimizing martingale in the dual pricing approach suggested by Rogers. We evaluate both approaches in numerical experiments. Key Words American options, dimension reduction, proper orthogonal decomposition, regressionbased Monte Carlo, Fourier methods 1