Financial Modelling with Jump Processes

Cont, Rama; Tankov, Peter

doi:10.1201/9780203485217

Cited by 1,241 publications

(627 citation statements)

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“…However, if the order of the process and/or the interval between two sampling dates are small, parabolic iFT is significant faster than flat (refined) iFFT, which is due to the fact that the integrand of the Fourier inversion in flat iFT decays very slowly at infinity, and too many terms may be needed to satisfy the For an exposition of the general theory of Lévy processes and their applications to pricing derivative securities, we refer the reader to [10,58] and [17,30,61], respectively.…”

Section: Resultsmentioning

confidence: 99%

“…For an introduction to applications of these models applied to finance, we refer the reader to S. Boyarchenko and Levendorskiȋ [17] and Cont and Tankov [30]. Pricing the continuously sampled geometric average options in exponential Lévy models is easy and quite straightforward (see S.Boyarchenko and Levendorskiȋ [17]) but pricing of arithmetic Asians presents serious mathematical and computational difficulties.…”

Section: Pricing In Exponential Lévy Modelsmentioning

confidence: 99%

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Pricing of Discretely Sampled Asian Options Under Levy Processes

Xie

2012

SSRN Journal

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Abstract. We develop a new method for pricing options on discretely sampled arithmetic average in exponential Lévy models. The main idea is the reduction to a backward induction procedure for the difference W n between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, W n is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method.

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Section: Resultsmentioning

confidence: 99%

Section: Pricing In Exponential Lévy Modelsmentioning

confidence: 99%

Pricing of Discretely Sampled Asian Options Under Levy Processes

Xie

2012

SSRN Journal

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“…Normal inverse Gaussian (NIG) processes is one the most popular Lévy process for modeling financial prices (see, e.g., [10] for more information). This process is defined as…”

Section: Time-changed Normal Inverse Gaussian Lévy Modelsmentioning

confidence: 99%

“…This model is an infinite-jump activity process with jumps activity β = 1. The increments of Z can be simulated by a rejection-type method (see [10], pp. 183).…”

Section: Time-changed Normal Inverse Gaussian Lévy Modelsmentioning

confidence: 99%

Statistical estimation of Lévy-type stochastic volatility models

Figueroa-López

2010

Ann Finance

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Volatility clustering and leverage are two of the most prominent features of the dynamics of asset prices. In order to incorporate these features as well as the typical fat-tails of the log return distributions, several types of exponential Lévy models driven by random clocks have been proposed in the literature, providing with a viable alternative to the classical stochastic volatility approach based on SDEs driven by Wiener processes. This paper has two main contributions. First, using a threshold type estimators based on high-frequency discrete observations of the process, we consider the recovery problem of the underlying random clock of the process. We show consistency of our estimator in the mean-square sense, extending former results in the literature for more general Lévy processes and for irregular sampling schemes. Secondly, we illustrate empirically the estimation of the random clock, the Blumenthal-Geetor index of jump activity, and the spectral Lévy measure of the process using intraday high-frequency data.

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“…For Lévy processes, this amounts to solving a certain linear partial integral differential equation (PIDE) with nonlocal Dirichlet conditions (see [9,Proposition 12.6, Section 12.2] and [10]). In the case of Lévy processes the PIDE approach has the advantage that we can utilize the fast Fourier transform (FFT) to perform efficient computation of the convolutions involved; however, the method also involves the truncation of the state space (the real line in our case) and discretization in the x and t variables, and the resulting errors are not easy to control.…”

Section: Introductionmentioning

confidence: 99%

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Hurd

Kuznetsov

2009

J. Appl. Probab.

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In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

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Financial Modelling with Jump Processes

Cited by 1,241 publications

References 0 publications

Pricing of Discretely Sampled Asian Options Under Levy Processes

Pricing of Discretely Sampled Asian Options Under Levy Processes

Statistical estimation of Lévy-type stochastic volatility models

On the First Passage time for Brownian Motion Subordinated by a Lévy Process

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