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Pricing Step Options under the CEV and other Solvable Diffusion Models

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  • Giuseppe Campolieti
  • Roman N. Makarov
  • Karl Wouterloot

Abstract

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.

Suggested Citation

  • Giuseppe Campolieti & Roman N. Makarov & Karl Wouterloot, 2013. "Pricing Step Options under the CEV and other Solvable Diffusion Models," Papers 1302.3771, arXiv.org.
  • Handle: RePEc:arx:papers:1302.3771
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    File URL: http://arxiv.org/pdf/1302.3771
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    References listed on IDEAS

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    1. Vadim Linetsky, 1999. "Step Options," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 55-96.
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    Cited by:

    1. Roman N. Makarov, 2016. "Modeling liquidation risk with occupation times," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(04), pages 1-11, December.
    2. Zhou, Jiang & Wu, Lan & Bai, Yang, 2017. "Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications," Statistics & Probability Letters, Elsevier, vol. 125(C), pages 80-90.

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