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16
A novel pricing method for European options based on Fouriercosine series expansions
 SIAM J. SCI. COMPUT
, 2008
"... Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, ..."
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Cited by 57 (14 self)
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Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The key insight is in the close relation of the characteristic function with the series coefficients of the Fouriercosine expansion of the density function. In most cases, the convergence rate of the COS method is exponential and the computational complexity is linear. Its range of application covers different underlying dynamics, including Lévy processes and the Heston stochastic volatility model, and various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal with European options in particular. In a followup paper we will present its application to options with earlyexercise features.
A Fast and Accurate FFTBased Method for Pricing EarlyExercise Options under Lévy Processes
 SIAM JOURNAL OF SCIENTIFIC COMPUTING
, 2008
"... A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula by reco ..."
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Cited by 36 (9 self)
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A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the wellknown riskneutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within many regular affine models, among which the class of exponential Lévy models. For an Mtimes exercisable Bermudan option, the overall complexity is O(MN log 2 (N)) with N grid points used to discretise the price of the underlying asset. American options are priced efficiently by applying Richardson extrapolation to the prices of Bermudan options.
Pricing earlyexercise and discrete barrier options by Fouriercosine series expansions
 Numerische Mathematik
"... We present a pricing method based on Fouriercosine expansions for earlyexercise and discretelymonitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional pr ..."
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Cited by 28 (8 self)
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We present a pricing method based on Fouriercosine expansions for earlyexercise and discretelymonitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth (C ∞ [a, b] ∈ R) transitional probability density functions. The computational complexity is O((M − 1)N log N) with N a (small) number of terms from the series expansion, and M, the number of earlyexercise/monitoring dates. This paper is the followup of [22] in which we presented the impressive performance of the Fouriercosine series method for European options. 1
A novel option pricing method based on Fouriercosine series expansions, submitted
, 2008
"... Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The convergence rate of the COS method is exponential and the computational complexity is linear. It has a wide range of applicability for different underlying dynamics, incl ..."
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Cited by 9 (5 self)
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Here we develop an option pricing method for European options based on the Fouriercosine series, and call it the COS method. The convergence rate of the COS method is exponential and the computational complexity is linear. It has a wide range of applicability for different underlying dynamics, including Lévy processes and Heston’s stochastic volatility model, and for various types of option contracts. We will present the method and its applications in two separate parts. The first one is this paper, where we deal in particular with European options. In a followup paper, part II, we will present its application to options with earlyexercise features. 1
A new look at shortterm implied volatility in asset price models with jumps
 IN MATHEMATICAL FINANCE
, 2013
"... We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, w ..."
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Cited by 6 (1 self)
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We analyse the behaviour of the implied volatility smile for options close to expiry in the exponential Lévy class of asset price models with jumps. We introduce a new renormalisation of the strike variable with the property that the implied volatility converges to a nonconstant limiting shape, which is a function of both the diffusion component of the process and the jump activity (BlumenthalGetoor) index of the jump component. Our limiting implied volatility formula relates the jump activity of the underlying asset price process to the short end of the implied volatility surface and sheds new light on the difference between finite and infinite variation jumps from the viewpoint of option prices: in the latter, the wings of the limiting smile are determined by the jump activity indices of the positive and negative jumps, whereas in the former, the wings have a constant modelindependent slope. This result gives a theoretical justification for the preference of the infinite variation Lévy models over the finite variation ones in the calibration based on shortmaturity option prices.
Iterative Methods for the Solution of the Singular Control Formulation of a GMWB Pricing Problem
"... Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a Guaranteed Minimum Withdrawal Benefit (GMWB), where the underlying asset is assumed to follow ..."
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Cited by 3 (2 self)
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Discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep. We consider a singular stochastic control problem arising in pricing a Guaranteed Minimum Withdrawal Benefit (GMWB), where the underlying asset is assumed to follow a jump diffusion process. We use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges. Our methods take advantage of the special structure of the GMWB problem in order to obtain a rapidly convergent iteration. The direct control method has a scaling parameter which must be set by the user. We give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of roundoff error. Example computations verify that these estimates are of the correct order. Finally, we compare the scaled direct control formulation to a formulation based on a block version of the penalty method [15]. We show that the scaled direct control method has some advantages over the penalty method.
Efficient solution of backward jumpdiffusion PIDEs with splitting and matrix exponentials
 Journal of Computational
"... We propose a new, unified approach to solving jumpdiffusion partial integrodifferential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a secondorder operator splitting on financial processes (diffusion and jumps) is applied to thes ..."
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Cited by 3 (3 self)
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We propose a new, unified approach to solving jumpdiffusion partial integrodifferential equations (PIDEs) that often appear in mathematical finance. Our method consists of the following steps. First, a secondorder operator splitting on financial processes (diffusion and jumps) is applied to these PIDEs. To solve the diffusion equation, we use standard finitedifference methods, which for multidimensional problems could also include splitting on various dimensions. For the jump part, we transform the jump integral into a pseudodifferential operator. Then for various jump models we show how to construct an appropriate first and second order approximation on a grid which supersets the grid that we used for the diffusion part. These approximations make the scheme to be unconditionally stable in time and preserve positivity of the solution which is computed either via a matrix exponential, or via Páde approximation of the matrix exponent. Various numerical experiments are provided to justify these results. 1
Finite Volume Difference Scheme for a Degenerate Parabolic Equation in the ZeroCoupon Bond PricingI
"... In this paper we solve numerically a degenerate parabolic equation with dynamical boundary conditions of zerocoupon bond pricing. First, we discuss some properties of the differential equation. Then, starting from the divergent form of the equation we implement the finitevolume method of S. Wang [ ..."
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Cited by 2 (2 self)
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In this paper we solve numerically a degenerate parabolic equation with dynamical boundary conditions of zerocoupon bond pricing. First, we discuss some properties of the differential equation. Then, starting from the divergent form of the equation we implement the finitevolume method of S. Wang [16] to discretize the differential problem. We show that the system matrix of the discretization scheme is a Mmatrix, so that the discretization is monotone. This provides the nonnegativity of the price with respect to time if the initial distribution is nonnegative. Numerical experiments demonstrate the efficiency of our difference scheme near the ends of the interval where the degeneration occurs.
A Numerical Study of Radial Basis Function Based Methods for Options Pricing under the One Dimen sion Jumpdiffusion Model
, 2011
"... Abstract The aim of this paper is to show how option prices in the Jumpdiffusion models, mainly on the Merton and Kou models, can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integrodifferential equa ..."
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Cited by 1 (0 self)
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Abstract The aim of this paper is to show how option prices in the Jumpdiffusion models, mainly on the Merton and Kou models, can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integrodifferential equation (PIDE) in onedimension for the American vanilla put and the European vanilla call/put options on dividendpaying stocks. The radial basis function we select is the Cubic Spline. We also propose a simple numerical algorithm for finding a finite computational range of an improper integral term in the PIDE so that the accuracy of approximation of the integral can be improved. Moreover, we use a numerical technique called factorization of the Cubic Spline to avoid inverting the illconditioned Cubic Spline interpolant. Finally, we will show numerically that in the European case the solution is second order accurate for the spatial and time variables, while in the American case it is second order accurate for spatial variables and first order accurate for time variables.
A Fast Method for Pricing EarlyExercise Options with the FFT
"... Abstract. A fast and accurate method for pricing early exercise options in computational finance is presented in this paper. The main idea is to reformulate the wellknown riskneutral valuation formula by recognizing that it is a convolution. This novel pricing method, which we name the ‘CONV ’ met ..."
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Abstract. A fast and accurate method for pricing early exercise options in computational finance is presented in this paper. The main idea is to reformulate the wellknown riskneutral valuation formula by recognizing that it is a convolution. This novel pricing method, which we name the ‘CONV ’ method for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Lévy models, including the exponentially affine jumpdiffusion models. For an Mtimes exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretize the price of the underlying asset. It is also shown that American options can be very efficiently computed by combining Richardson extrapolation to the CONV method.