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A Simple Stochastic Rate Model for Rate Equity Hybrid Products

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  • Eberlein
  • Madan
  • Pistorius
  • Yor

Abstract

A positive spot rate model driven by a gamma process and correlated with equity is introduced and calibrated via closed forms for the joint characteristic function for the rate r , its integral y and the logarithm of the stock price s under the T -forward measure. The law of the triple is expressed as a nonlinear transform of three independent processes, a gamma process, a variance gamma process and a Wiener integral with respect to the Dirichlet process. The generalized Stieltjes transform of the Wiener integral with respect to the Dirichlet process is derived in closed form. Inversion of this transform using Schwarz (2005, The generalized Stieltjes transform and its inverse, Journal of Mathematical Physics , 46(1), doi: 10.1063/1.1825077) makes large step simulations possible. Valuing functions are built and hedged using quantization and high dimensional interpolation methods. The hedging objective is taken to be capital minimization as described by Carr, Madan and Vicente Alvarez (2011, Markets, profits, capital, leverage and returns, Journal of Risk , 14(1), pp. 95--122).

Suggested Citation

  • Eberlein & Madan & Pistorius & Yor, 2013. "A Simple Stochastic Rate Model for Rate Equity Hybrid Products," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(5), pages 461-488, November.
    • Handle: RePEc:taf:apmtfi:v:20:y:2013:i:5:p:461-488
    DOI: 10.1080/1350486X.2013.770240
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    Cited by:

    1. Yoshihiro Shirai, 2023. "A Levy-driven Ornstein-Uhlenbeck process for the valuation of credit index swaptions," Papers 2301.05332, arXiv.org, revised Oct 2023.
    2. Yoshihiro Shirai, 2023. "Acceptable Bilateral Gamma Parameters," Papers 2301.05333, arXiv.org.
    3. Imdade Chitou & Gilles Dufrénot & Julien Esposito, 2021. "Linking Covid-19 epidemic and emerging market OAS: Evidence using dynamic copulas and Pareto distributions," Working Papers halshs-03297198, HAL.
    4. Fan Jiang & Xin Zang & Jingping Yang, 2020. "Asymptotic expansion for the transition densities of stochastic differential equations driven by the gamma processes," Papers 2003.06218, arXiv.org.
    5. Yoshihiro Shirai, 2022. "Extreme Measures in Continuous Time Conic Finace," Papers 2210.13671, arXiv.org, revised Oct 2023.

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