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Efficient European and American option pricing under a jump-diffusion process

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  • Marcellino Gaudenzi
  • Alice Spangaro
  • Patrizia Stucchi

Abstract

When the underlying asset displays oscillations, spikes or heavy-tailed distributions, the lognormal diffusion process (for which Black and Scholes developed their momentous option pricing formula) is inadequate: in order to overcome these real world difficulties many models have been developed. Merton proposed a jump-diffusion model, where the dynamics of the price of the underlying are subject to variations due to a Brownian process and also to possible jumps, driven by a compound Poisson process. Merton's model admits a series solution for the European option price, and there have been a lot of attempts to obtain a discretisation of the Merton model with tree methods in order to price American or more complex options, e. g. Amin, the $O(n^3)$ procedure by Hilliard and Schwartz and the $O(n^{2.5})$ procedure by Dai et al. Here, starting from the implementation of the seven-nodes procedure by Hilliard and Schwartz, we prove theoretically that it is possible to reduce the complexity to $O(n \ln n)$ in the European case and $O(n^2 \ln n)$ in the American put case. These theoretical results can be obtained through suitable truncation of the lattice structure and the proofs provide closed formulas for the truncation limitations.

Suggested Citation

  • Marcellino Gaudenzi & Alice Spangaro & Patrizia Stucchi, 2017. "Efficient European and American option pricing under a jump-diffusion process," Papers 1712.08137, arXiv.org.
  • Handle: RePEc:arx:papers:1712.08137
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    References listed on IDEAS

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    3. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    4. Amin, Kaushik I, 1993. "Jump Diffusion Option Valuation in Discrete Time," Journal of Finance, American Finance Association, vol. 48(5), pages 1833-1863, December.
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