An exact analytical solution for discrete barrier options

Fusai, Gianluca; Abrahams, I. David; Sgarra, Carlo

doi:10.1007/s00780-005-0170-y

Cited by 80 publications

(64 citation statements)

References 37 publications

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“…The Fourier transform and the z-transform, and also their inverses, can be interchanged because the z-transform is a power series in q which converges uniformly in a closed and bounded set given by the radius of convergence ρ [22]. In Equation (36) the inverse z-transform is performed before the inverse Fourier transform to minimize the computational cost.…”

Section: Lookback Optionsmentioning

confidence: 99%

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

2016

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The Wiener-Hopf factorization of a complex function arises in a variety of fields in applied mathematics such as probability, finance, insurance, queuing theory, radio engineering and fluid mechanics. The factorization fully characterizes the distribution of functionals of a random walk or a Lévy process, such as the maximum, the minimum and hitting times. Here we propose a constructive procedure for the computation of the Wiener-Hopf factors, valid for both single and double barriers, based on the combined use of the Hilbert and the z-transform. The numerical implementation can be simply performed via the fast Fourier transform and the Euler summation. Given that the information in the Wiener-Hopf factors is strictly related to the distributions of the first passage times, as a concrete application in mathematical finance we consider the pricing of discretely monitored exotic options, such as lookback and barrier options, when the underlying price evolves according to an exponential Lévy process. We show that the computational cost of our procedure is independent of the number of monitoring dates and * Corresponding author Email addresses: gianluca.fusai@unipmn.it, gianluca.fusai.1@city.ac.uk (Gianluca Fusai), g.germano@ucl.ac.uk, g.germano@lse.ac.uk (Guido Germano), daniele.marazzina@polimi.it (Daniele Marazzina) Preprint submitted to ElsevierNovember 27, 2015the error decays exponentially with the number of grid points.

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Section: Lookback Optionsmentioning

confidence: 99%

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

2016

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“…To overcome the mis-pricing, a wide variety of numerical techniques have been proposed in the literature, including recent noteworthy additions [5][6][7][8][9][10][11][12][13][14][15]. This work is a revision of an article by two of the authors [16] in which an exact analytic expression for the down-out option price was obtained as the solution to the Black-Scholes PDE.…”

Section: Discrete Monitoringmentioning

confidence: 99%

Pricing financial claims contingent upon an underlying asset monitored at discrete times

2007

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Exotic option contracts typically specify a contingency upon an underlying asset price monitored at a discrete set of times. Yet, techniques used to price such options routinely assume continuous monitoring leading to often substantial price discrepancies. A brief review of relevant option-pricing methods is presented. The pricing problem is transformed into one of Wiener-Hopf type using a z-transform in time and a Fourier transform in the logarithm of asset prices. The Wiener-Hopf technique is used to obtain probabilistic identities for the related random walks killed by an absorbing boundary. An accurate and efficient approximation is obtained using Padé approximants and an approximate inverse z-transform based on the trapezoidal rule. For simplicity, European barrier options in a Gaussian Black-Scholes framework are used to exemplify the technique (for which exact analytic expressions are obtained). Extensions to different option contracts and options driven by other Lévy processes are discussed.

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“…• (Fusai et al, 2006) due to highly complex, non-linear temporal and spatial characteristics of these types of problems. Thus, this paper also relates to solving these problems.…”

Section: Introductionmentioning

confidence: 99%

Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options

2009

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A large number of financial engineering problems involve non-linear equations with non-linear or timedependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS) equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF) method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.Keywords: financial engineering; radial basis functions; diffusional method; barrier options. ResumoMuitos problemas de engenharia financeira envolvem equações não-lineares com condições de contorno não-lineares ou dependentes do tempo. Apesar de soluções analíticas disponíveis, várias formas clássicas e modificadas da conhecida equação de Black-Scholes (BS) requerem soluções numéricas rápidas e acuradas. Este trabalho introduz o método de função de base radial (RBF) aplicado à solução da equação BS com condições de contorno não-lineares relacionadas a opções de barreira dependentes da trajetória. Além disso, explora-se o método difusional para solucionar equações advectivo-difusivas quanto à sua efetividade para solucionar equações BS. Utilizam-se funções de base radial Cúbica e Thin-Plate Spline (TPS), aplicadas à solução de problemas de opções de barreiras. Os resultados numéricos, quando comparados com as soluções analíticas, permitem afirmar que o método RBF é muito acurado e fácil de ser implementado. O método difusional associado ao método RBF leva aos mesmos resultados obtidos pela formulação clássica da equação de Black-Scholes.

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An exact analytical solution for discrete barrier options

Cited by 80 publications

References 37 publications

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

Spitzer Identity, Wiener-Hopf Factorization and Pricing of Discretely Monitored Exotic Options

Pricing financial claims contingent upon an underlying asset monitored at discrete times

Use of radial basis functions for meshless numerical solutions applied to financial engineering barrier options

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