TY - JOUR

T1 - Diffusion approximation of Lévy processes with a view towards finance

AU - Kiessling, Jonas

AU - Tempone, Raul

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2011/1

Y1 - 2011/1

N2 - Let the (log-)prices of a collection of securities be given by a d-dimensional Lévy process X t having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(X T)]. Let X̄ T be a finite activity approximation to X T, where diffusion is introduced to approximate jumps smaller than a given truncation level ∈ > 0. The main result of this work is a derivation of an error expansion for the resulting model error, E[g(X T) - g(X̄ T)], with computable leading order term. Our estimate depends both on the choice of truncation level ∈ and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error. Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure. © de Gruyter 2011.

AB - Let the (log-)prices of a collection of securities be given by a d-dimensional Lévy process X t having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(X T)]. Let X̄ T be a finite activity approximation to X T, where diffusion is introduced to approximate jumps smaller than a given truncation level ∈ > 0. The main result of this work is a derivation of an error expansion for the resulting model error, E[g(X T) - g(X̄ T)], with computable leading order term. Our estimate depends both on the choice of truncation level ∈ and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error. Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure. © de Gruyter 2011.

UR - http://hdl.handle.net/10754/561696

UR - https://www.degruyter.com/doi/10.1515/mcma.2011.003

UR - http://www.scopus.com/inward/record.url?scp=84858679675&partnerID=8YFLogxK

U2 - 10.1515/MCMA.2011.003

DO - 10.1515/MCMA.2011.003

M3 - Article

VL - 17

SP - 11

EP - 45

JO - Monte Carlo Methods and Applications

JF - Monte Carlo Methods and Applications

SN - 0929-9629

IS - 1

ER -