scholarly journals Retrieving Lévy Processes from Option Prices: Regularization of an Ill-posed Inverse Problem

2006 ◽  
Vol 45 (1) ◽  
pp. 1-25 ◽  
Author(s):  
Rama Cont ◽  
Peter Tankov
Keyword(s):  
Inverse Problem ◽  
Lévy Processes ◽  
Levy Processes ◽  
Option Prices ◽  
Ill Posed

APPROXIMATING LÉVY PROCESSES WITH A VIEW TO OPTION PRICING

2010 ◽  
Vol 13 (01) ◽  
pp. 63-91 ◽  
Author(s):  
JOHN CROSBY ◽  
NOLWENN LE SAUX ◽  
ALEKSANDAR MIJATOVIĆ
Keyword(s):  
Option Pricing ◽  
Lévy Processes ◽  
Jump Diffusion ◽  
Levy Processes ◽  
Poisson Processes ◽  
Exotic Options ◽  
Option Prices ◽  
Barrier Option ◽  
Systematic Methodology ◽  
The Mean

We examine how to approximate a Lévy process by a hyperexponential jump-diffusion (HEJD) process, composed of Brownian motion and of an arbitrary number of sums of compound Poisson processes with double exponentially distributed jumps. This approximation will facilitate the pricing of exotic options since HEJD processes have a degree of tractability that other Lévy processes do not have. The idea behind this approximation has been applied to option pricing by Asmussen et al. (2007) and Jeannin and Pistorius (2008). In this paper we introduce a more systematic methodology for constructing this approximation which allow us to compute the intensity rates, the mean jump sizes and the volatility of the approximating HEJD process (almost) analytically. Our methodology is very easy to implement. We compute vanilla option prices and barrier option prices using the approximating HEJD process and we compare our results to those obtained from other methodologies in the literature. We demonstrate that our methodology gives very accurate option prices and that these prices are more accurate than those obtained from existing methodologies for approximating Lévy processes by HEJD processes.

On correlating Lévy processes

2010 ◽  
Vol 13 (1) ◽  
pp. 3-16 ◽  
Author(s):  
Ernst Eberlein ◽  
Dilip Madan
Keyword(s):  
Lévy Processes ◽  
Levy Processes

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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