THE EVALUATION OF AMERICAN OPTION PRICES UNDER STOCHASTIC VOLATILITY AND JUMP-DIFFUSION DYNAMICS USING THE METHOD OF LINES

This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop...

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Bibliographic Details
Published in:International journal of theoretical and applied finance 2009-05, Vol.12 (3), p.393-425
Main Authors: CHIARELLA, CARL, KANG, BODA, MEYER, GUNTER H., ZIOGAS, ANDREW
Format: Article
Language:English
Subjects:
Algorithms
Asset pricing
Derivatives
Financial engineering
Financial models
Mathematical finance
Numerical analysis
Option pricing
Put & call options
Securities prices
Stochastic models
Studies
Volatility
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Summary:This paper considers the problem of numerically evaluating American option prices when the dynamics of the underlying are driven by both stochastic volatility following the square root process of Heston [18], and by a Poisson jump process of the type originally introduced by Merton [25]. We develop a method of lines algorithm to evaluate the price as well as the delta and gamma of the option, thereby extending the method developed by Meyer [26] for the case of jump-diffusion dynamics. The accuracy of the method is tested against two numerical methods that directly solve the integro-partial differential pricing equation. The first is an extension to the jump-diffusion situation of the componentwise splitting method of Ikonen and Toivanen [21]. The second method is a Crank-Nicolson scheme that is solved using projected successive over relaxation and which is taken as the benchmark for the price. The relative efficiency of these methods for computing the American call option price, delta, gamma and free boundary is analysed. If one seeks an algorithm that gives not only the price but also the delta and gamma to the same level of accuracy for a given computational effort then the method of lines seems to perform best amongst the methods considered.
ISSN:0219-0249
1793-6322
DOI:10.1142/S0219024909005270