Numerical Inversion of Laplace Transforms of Probability Distributions

Abate, Joseph; Whitt, Ward

doi:10.1287/ijoc.7.1.36

Cited by 586 publications

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“…Instead of using ArrowDebreu securities to span the space of European-style contingent claims written on a scalar diffusion process, we introduce a concept of eigensecurities or eigenvectors of the pricing operator, as fundamental building blocks in our approach. 1 Eigensecurities diagonalize the pricing operator. All other European-style contingent claims with square-integrable payoffs are represented as portfolios of eigensecurities.…”

Section: Introductionmentioning

confidence: 99%

“…In the scalar diffusion context, this static pricing equation can be interpreted as a second-order ordinary differential equation (ODE) of the Sturm-Liouville type. 3 Secondly, in cases where the state space is a Þnite interval with unmixed boundary conditions (e.g., absorbing or reßecting), the spectrum of the associated Sturm-Liouville problem is guaranteed to 1 The foundations of semigroup pricing theory in a general Markov context are developed by Duffie (1985), Duffie and Garman (1985), and Garman (1985). See Ethier and Kurtz (1986) for the semigroup approach to Markov processes.…”

Section: Introductionmentioning

confidence: 99%

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Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Давыдов¹,

Linetsky

2003

Operations Research

146

122

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This paper develops an eigenfunction expansion approach to pricing options on scalar diffusion processes. All derivative securities are unbundled into portfolios of primitive securities termed eigensecurities. Eigensecurities are eigenvectors of the pricing operator (present value operator). Pricing is then immediate by the linearity property of the pricing operator and the eigenvector property of eigensecurities. To illustrate the computational power of the method, we develop two applications: pricing vanilla, single-and double-barrier options under the constant elasticity of variance (CEV) process and interest rate knock-out options in the Cox-IngersollRoss (CIR) term-structure model.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Давыдов¹,

Linetsky

2003

Operations Research

146

122

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“…Our tool supports two Laplace transform inversion algorithms, which are briefly outlined below: the Euler technique [12] and the Laguerre method [13] with modifications summarised in [2].…”

Section: Distribution Representation and Laplace Inversionmentioning

confidence: 99%

“…Euler summation is employed to accelerate the convergence of the alternating series infinite sum, so we calculate the sum of the first n terms explicitly and use Euler summation to calculate the next m. To give an accuracy of 10 −8 we set A = 19.1, n = 20 and m = 12 (compared with A = 19.1, n = 15 and m = 11 in [12]). …”

Section: Summary Of Euler Inversionmentioning

confidence: 99%

Iterative convergence of passage-time densities in semi-Markov performance models

Bradley

Wilson

2005

Performance Evaluation

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“…Avellandea and Wu [3] used a lattice method. Labart and Lelong [9] used an inversion formula based on the Abate and Whitt [1] method, while Bernard, Courtois, and Quittard-Pinon [4] obtained numerical prices by approximating the Laplace transforms using a linear combination of fractional functions. In this paper, we used a different method to obtain the option price without numerically inverting its Laplace transform.…”

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confidence: 99%

Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time

Dassios¹,

Lim²

2013

SIAM J. Finan. Math.

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Abstract. In this paper, we obtain the density function of the single barrier one-sided Parisian stopping time.The problem reduces to that of solving a Volterra integral equation of the first kind, where a recursive solution is consequently obtained. 1. Introduction. Parisian options were first introduced by Chesney, Jeanblanc-Picque, and Yor [5]. They are path-dependent options whose payoff depends not only on the final value of the underlying asset, but also on the path trajectory of the underlying asset above or below a predetermined barrier L. The owner of a Parisian down-and-out call loses the option when the underlying asset price S reaches the level L and remains constantly below this level for a time interval longer than D, while for a Parisian down-and-in call, the same event gives the owner the right to exercise the option. Parisian options are a kind of barrier option. However, it has the advantage of not being as easily manipulated by an influential agent as a simple barrier option, and thus is a guarantee against easy arbitrage.No explicit pricing formula is known for this type of option. Previous literature has largely focused on using Laplace transforms to price Parisian options. In [5,7,10], the problem is reduced to finding the Laplace transform of the Parisian stopping time, which is the first time the length of the excursion reaches level D. In [5], the Laplace transform of the stopping time was obtained using the Brownian meander and Azema martingale, while Dassios and Wu [7] introduced a perturbed Brownian motion and a semi-Markov model to obtain the Laplace transform. In both of these, an explicit form of the Laplace transform of the distribution of the Parisian stopping time and consequently that of the option price is found. Other methods of pricing Parisian options include the PDE method, studied by Haber, Schonbucher, and Wilmott [8]. There exist also other types of Parisian options. Cumulative Parisian options,

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Numerical Inversion of Laplace Transforms of Probability Distributions

Cited by 586 publications

References 0 publications

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach

Iterative convergence of passage-time densities in semi-Markov performance models

Parisian Option Pricing: A Recursive Solution for the Density of the Parisian Stopping Time

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