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A Modern View on Merton's Jump-Diffusion Model

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Abstract

Merton has provided a formula for the price of a European call option on a single stock where the stock price process contains a continuous Poisson jump component, in addition to a continuous log-normally distributed component. In Merton's analysis, the jump-risk is not priced. Thus the distribution of the jump-arrivals and the jump-sizes do not change under the change of measure. We go onto introduce a Radon-Nikodym derivative process that induces the change of measure from the market measure to an equivalent martingale measure. The choice of parameters in the Radon-Nikodym derivative allows us to price the option under different financial-economic scenarios. We introduce a hedging argument that eliminates the jump-risk in some sort of averaged sense, and derive an integro-partial differential equation of the option price that is related to the one obtained by Merton.

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File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp287.pdf
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Paper provided by Quantitative Finance Research Centre, University of Technology, Sydney in its series Research Paper Series with number 287.

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Length: 14
Date of creation: 01 Jan 2011
Date of revision:
Handle: RePEc:uts:rpaper:287

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Keywords: financial derivatives; compound Poisson processes; equivalent martingale measure; hedging portfolio;

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  1. Aase, Knut K., 1988. "Contingent claims valuation when the security price is a combination of an Ito process and a random point process," Stochastic Processes and their Applications, Elsevier, vol. 28(2), pages 185-220, June.
  2. Robert A. Jarrow, 2009. "The Term Structure of Interest Rates," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 69-96, November.
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Cited by:
  1. Scalas, Enrico & Politi, Mauro, 2012. "A parsimonious model for intraday European option pricing," Economics Discussion Papers 2012-14, Kiel Institute for the World Economy.

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