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28
Twodimensional Fourier cosine series expansion method for pricing financial options
 SIAM Journal on Scientific Computing
"... the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing twocolor rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponen ..."
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the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing twocolor rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponentially in the number of terms in the Fourier cosine series summations; otherwise we achieve algebraic convergence. The use of an FFT algorithm, for asset prices modeled by Lévy processes, makes the algorithm highly efficient. We perform extensive numerical experiments.
Fast valuation and calibration of credit default swaps under Lévy dynamics
 Journal of Computational Finance
"... In this paper we address the issue of finding an efficient and flexible numerical approach for calculating survival/default probabilities and pricing Credit Default Swaps under advanced jump dynamics. We have chosen to use the firm’s value approach, modeling the firm’s value by an exponential Lévy m ..."
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In this paper we address the issue of finding an efficient and flexible numerical approach for calculating survival/default probabilities and pricing Credit Default Swaps under advanced jump dynamics. We have chosen to use the firm’s value approach, modeling the firm’s value by an exponential Lévy model. For this approach the default event is defined as a first passage of a barrier and it is therefore possible to exploit a numerical technique developed to price barrier options under Lévy models to calculate the default probabilities. The method presented is based on the Fouriercosine series expansion of the underlying model’s density function.
Oosterlee, An efficient pricing algorithm for swing options based on Fourier cosine expansions, Internal report TU
, 2010
"... Swing options give contract holders the right to modify amounts of future delivery of certain commodities, such as electricity or gas. In this paper, we assume that these options can be exercised at any time before the end of the contract, and more than once. However, a recovery time between any two ..."
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Swing options give contract holders the right to modify amounts of future delivery of certain commodities, such as electricity or gas. In this paper, we assume that these options can be exercised at any time before the end of the contract, and more than once. However, a recovery time between any two consecutive exercise dates is incorporated as a constraint to avoid continuous exercise. We introduce an efficient way of pricing these swing options, based on the Fourier cosine expansion method, which is especially suitable when the underlying is modeled by a Lévy process. 1
Extension of Stochastic Volatility Equity Models with HullWhite Interest Rate Process
, 2008
"... We present an extension of the stochastic volatility equity models by a stochastic HullWhite interest rate component. We place this system of stochastic differential equations in the class of affine jump diffusion linear quadratic jumpdiffusion processes (Duffie, Pan and Singleton [11], Cheng and ..."
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Cited by 4 (2 self)
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We present an extension of the stochastic volatility equity models by a stochastic HullWhite interest rate component. We place this system of stochastic differential equations in the class of affine jump diffusion linear quadratic jumpdiffusion processes (Duffie, Pan and Singleton [11], Cheng and Scaillet [8]) so that the pricing of European products can be efficiently done within the Fourier cosine expansion pricing framework [12]. We also apply the model to price some hybrid structured derivatives, which combine the different asset classes: equity and interest rate.
Oosterlee. Acceleration of Option Pricing Technique on Graphics Processing Units
, 2010
"... The acceleration of an option pricing technique based on Fourier cosine expansions on the Graphics Processing Unit (GPU) is reported. European options, in particular with multiple strikes, and Bermudan options will be discussed. The influence of the number of terms in the Fourier cosine series expan ..."
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The acceleration of an option pricing technique based on Fourier cosine expansions on the Graphics Processing Unit (GPU) is reported. European options, in particular with multiple strikes, and Bermudan options will be discussed. The influence of the number of terms in the Fourier cosine series expansion, the number of strikes, as well as the number of exercise dates for Bermudan options, are explored. We also give details about the different ways of implementing on a GPU. Numerical examples include asset price processes based on a Lévy process of infinite activity and the stochastic volatility Heston model. Furthermore, we discuss the issue of precision on the present GPU systems.
The Stochastic Grid Bundling Method: Efficient Pricing of Bermudan Options and their Greeks
, 2013
"... This paper describes a practical simulationbased algorithm, which we call the Stochastic Grid Bundling Method (SGBM) for pricing multidimensional Bermudan (i.e. discretely exercisable) options. The method generates a direct estimator of the option price, an optimal earlyexercise policy as well as ..."
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This paper describes a practical simulationbased algorithm, which we call the Stochastic Grid Bundling Method (SGBM) for pricing multidimensional Bermudan (i.e. discretely exercisable) options. The method generates a direct estimator of the option price, an optimal earlyexercise policy as well as a lower bound value for the option price. An advantage of SGBM is that the method can be used for fast approximation of the Greeks (i.e., derivatives with respect to the underlyingspot prices, such as delta, gamma, etc) for Bermudanstyle options. Computational results for various multidimensional Bermudan options demonstrate the simplicity and efficiency of the algorithm proposed. 1
On the application of spectral filters in a Fourier option pricing technique
, 2013
"... When Fourier techniques are employed to specific option pricing cases from computational finance with nonsmooth functions, the socalled Gibbs phenomenon may become apparent. This seriously impacts the efficiency and accuracy of the pricing. For example, the Variance Gamma asset price process gives ..."
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When Fourier techniques are employed to specific option pricing cases from computational finance with nonsmooth functions, the socalled Gibbs phenomenon may become apparent. This seriously impacts the efficiency and accuracy of the pricing. For example, the Variance Gamma asset price process gives rise to algebraically decaying Fourier coefficients, resulting in a slowly converging Fourier series. We apply spectral filters to achieve faster convergence. FilteringiscarriedoutinFourierspace; theseries coefficients arepremultipliedbyadecreasing filter, which does not add significant computational cost. Tests with different filters show how the algebraic index of convergence is improved.
On the Fourier cosine series expansion (COS) method for stochastic control problems. Working paper
, 2011
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A Fouriercosine method for an efficient computation of solutions to BSDEs
, 2013
"... We develop a Fourier method to solve backward stochastic differential equations (BSDEs). General thetadiscretization of the timeintegrands leads to an induction scheme with conditional expectations. These are approximated by using Fouriercosine series expansions, relying on the availability of a ..."
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We develop a Fourier method to solve backward stochastic differential equations (BSDEs). General thetadiscretization of the timeintegrands leads to an induction scheme with conditional expectations. These are approximated by using Fouriercosine series expansions, relying on the availability of a characteristic function. The method is applied to BSDEs with jumps. Numerical experiments demonstrate the applicability of BSDEs in financial and economic problems and show fast convergence of our efficient probabilistic numerical method.
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"... Abstract. We develop an efficient Fourierbased numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The twodimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C. ..."
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Abstract. We develop an efficient Fourierbased numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The twodimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826–848, F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27–62], and highorder quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence.