Integro-differential equations for option prices in exponential Lévy models

Cont, Rama; Voltchkova, Ekaterina

doi:10.1007/s00780-005-0153-z

Cited by 143 publications

(109 citation statements)

References 27 publications

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“…Then, the Kolmogorov backward equations for these processes are considered as one of the useful methods for option pricing. See Cont & Voltchkova (2005); Garreau & Kopriva (2013); Lee et al (2012) and the references therein.…”

Section: R\{0}mentioning

confidence: 99%

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

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Section: R\{0}mentioning

confidence: 99%

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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“…Repeating essentially the derivations from [6] or [12], we obtain the following Partial Integro-Differential Equation (PIDE) for the call prices (see, for example, equation (13) in [6])…”

Section: Option Prices In Exponential Additive Modelsmentioning

confidence: 99%

Tangent Lévy market models

Carmona

Nadtochiy

2011

Finance Stoch

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In this paper, we introduce a new class of models for the time evolution of the prices of call options of all strikes and maturities. We capture the information contained in the option prices in the density of some time-inhomogeneous Lévy measure (an alternative to the implied volatility surface), and we set this static code-book in motion by means of stochastic dynamics of Itôs type in a function space, creating what we call a tangent Lévy model. We then provide the consistency conditions, namely, we show that the call prices produced by a given dynamic code-book (dynamic Lévy density) coincide with the conditional expectations of the respective payoffs if and only if certain restrictions on the dynamics of the code-book are satisfied (including a drift conditionà la HJM). We then provide an existence result, which allows us to construct a large class of tangent Lévy models, and describe a specific example for the sake of illustration.

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“…However, in the case of barrier options the proof of regularity is much more involved [12] than for European ones. The following result, based on Bensoussan and Lions [5], allows to obtain a martingale representation for barrier options under further assumptions:…”

Section: Barrier Optionsmentioning

confidence: 99%

“…The European option price (12) can then be written as the expectation of a Lévy process Z t = log(X t /X 0 ):…”

Section: Hedging With the Underlying In An Exponential Lévy Modelmentioning

confidence: 99%

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Hedging with Options in Models with Jumps

Cont

Tankov

Voltchkova

2007

Stochastic Analysis and Applications

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Abel Symposium 2005 on Stochastic analysis and applications in honor of Kiyosi Ito's 90th birthday. AbstractWe consider the problem of hedging a contingent claim, in a market where prices of traded assets can undergo jumps, by trading in the underlying asset and a set of traded options. We give a general expression for the hedging strategy which minimizes the variance of the hedging error, in terms of integral representations of the options involved. This formula is then applied to compute hedge ratios for common options in various models with jumps, leading to easily computable expressions. The performance of these hedging strategies is assessed through numerical experiments.

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Integro-differential equations for option prices in exponential Lévy models

Cited by 143 publications

References 27 publications

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tangent Lévy market models

Hedging with Options in Models with Jumps

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