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Alternative results for option pricing and implied volatility in jump-diffusion models using Mellin transforms

Published online by Cambridge University Press:  06 December 2016

T. RAY LI  and
MARIANITO R. RODRIGO
T. RAY LI
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, New South Wales, Australia email: tl115@uowmail.edu.au, marianito_rodrigo@uow.edu.au
MARIANITO R. RODRIGO
Affiliation:
School of Mathematics and Applied Statistics, University of Wollongong, New South Wales, Australia email: tl115@uowmail.edu.au, marianito_rodrigo@uow.edu.au
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Abstract

In this article, we use Mellin transforms to derive alternative results for option pricing and implied volatility estimation when the underlying asset price is governed by jump-diffusion dynamics. The current well known results are restrictive since the jump is assumed to follow a predetermined distribution (e.g., lognormal or double exponential). However, the results we present are general since we do not specify a particular jump-diffusion model within the derivations. In particular, we construct and derive an exact solution to the option pricing problem in a general jump-diffusion framework via Mellin transforms. This approach of Mellin transforms is further extended to derive a Dupire-like partial integro-differential equation, which ultimately yields an implied volatility estimator for assets subjected to instantaneous jumps in the price. Numerical simulations are provided to show the accuracy of the estimator.

Keywords

Mellin transformBlack–Scholes partial differential equationjump-diffusion modelimplied volatility estimationDupire equation
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 5 , October 2017 , pp. 789 - 826
Copyright
Copyright © Cambridge University Press 2016 

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