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Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

Author

Listed:
  • Julien Hok

    (Crédit Agricole CIB)

  • Shih-Hau Tan

    (Cuemacro)

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 challenging and time-consuming because of the multi-dimensional nature of the problem. A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an accurate calibration and provides an efficient implementation algorithm. The essential idea is based on solving the derived forward equation satisfied by $$P(t, S, r) \mathcal {Z}(t, S, r)$$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 $$\mathcal {Z}(t, S, r)$$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. It reduces by one order the computations costs and then allows to speed up significantly the calibration procedure. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.

Suggested Citation

  • Julien Hok & Shih-Hau Tan, 2019. "Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 609-637, December.
    • Handle: RePEc:spr:decfin:v:42:y:2019:i:2:d:10.1007_s10203-019-00232-3
    DOI: 10.1007/s10203-019-00232-3
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    References listed on IDEAS

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    1. E. Benhamou & E. Gobet & M. Miri, 2012. "Analytical formulas for a local volatility model with stochastic rates," Quantitative Finance, Taylor & Francis Journals, vol. 12(2), pages 185-198, September.
    2. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    3. E. Benhamou & E. Gobet & M. Miri, 2009. "Smart expansion and fast calibration for jump diffusions," Finance and Stochastics, Springer, vol. 13(4), pages 563-589, September.
    4. Emmanuel Gobet & Julien Hok, 2014. "Expansion Formulas For Bivariate Payoffs With Application To Best-Of Options On Equity And Inflation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 17(02), pages 1-32.
    5. 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.
    6. Andrey Itkin, 2017. "Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps," Applied Mathematical Finance, Taylor & Francis Journals, vol. 24(6), pages 485-519, November.
    7. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion Formulas For European Quanto Options In A Local Volatility Fx-Libor Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(02), pages 1-43, March.
    8. Mark Rubinstein., 1994. "Implied Binomial Trees," Research Program in Finance Working Papers RPF-232, University of California at Berkeley.
    9. Griselda Deelstra & Gr�gory Ray�e, 2013. "Local Volatility Pricing Models for Long-Dated FX Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
    10. Emanuel Derman & Iraj Kani, 1998. "Stochastic Implied Trees: Arbitrage Pricing with Stochastic Term and Strike Structure of Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 61-110.
    11. Rubinstein, Mark, 1994. "Implied Binomial Trees," Journal of Finance, American Finance Association, vol. 49(3), pages 771-818, July.
    12. Julien Hok & Philip Ngare & Antonis Papapantoleon, 2018. "Expansion formulas for European quanto options in a local volatility FX-LIBOR model," Papers 1801.01205, arXiv.org, revised Apr 2018.
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    Cited by:

    1. Elisa Alòs & Maria Elvira Mancino & Tai-Ho Wang, 2019. "Volatility and volatility-linked derivatives: estimation, modeling, and pricing," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 321-349, December.
    2. Julien Hok & Sergei Kucherenko, 2021. "Pricing and Risk Analysis in Hyperbolic Local Volatility Model with Quasi Monte Carlo," Papers 2106.08421, arXiv.org.

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    More about this item

    Keywords

    Local volatility model; Stochastic interest rates; Hybrid; Calibration; Forward Fokker–Planck-type equation; Alternating direction implicit (ADI) method;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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