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Calibration of Local Volatility Model with Stochastic Interest Rates by Efficient Numerical PDE Method

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  • Julien Hok
  • Shih-Hau Tan

Abstract

Long maturity options or a wide class of hybrid products are evaluated using a local volatility type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is usually time-consuming because of the multi-dimensional nature of the problem. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an efficient implementation. The essential idea is based on solving the derived forward equation satisfied by P(t; S; r)Z(t; S; r), where P(t; S; r) represents the risk neutral probability density of (S(t); r(t)) and Z(t; S; r) the projection of the stochastic discounting factor in the state variables (S(t); r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (Alternative Direction Implicit) method based on an extension of the Peaceman-Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Suggested Citation

  • Julien Hok & Shih-Hau Tan, 2018. "Calibration of Local Volatility Model with Stochastic Interest Rates by Efficient Numerical PDE Method," Papers 1803.03941, arXiv.org.
    • Handle: RePEc:arx:papers:1803.03941
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    References listed on IDEAS

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    1. Hull, John & White, Alan, 1993. "One-Factor Interest-Rate Models and the Valuation of Interest-Rate Derivative Securities," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(2), pages 235-254, June.
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    Cited by:

    1. Orcan Ogetbil & Narayan Ganesan & Bernhard Hientzsch, 2020. "Calibrating Local Volatility Models with Stochastic Drift and Diffusion," Papers 2009.14764, arXiv.org, revised May 2023.

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