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Black’s model in a negative interest rate environment, with application to OTC derivatives

Author

Listed:
  • Riccardo Bramante

    (Università Cattolica del Sacro Cuore)

  • Gimmi Dallago

    (Allitude S.p.A.)

  • Silvia Facchinetti

    (Università Cattolica del Sacro Cuore)

Abstract

The most common application of Black’s formula is interest rate derivatives pricing. Black’s model, a variant of Black-Scholes option pricing model, was first introduced by Fischer Black in 1976. In recent market conditions, where global interest rates are at very low levels and in some markets are currently zero or negative, Black model—in its canonical form—fails to price interest rate options since positive interest rates are assumed in its formula. In this paper we propose a heuristic method that, without explicit assumptions about the forward rate generating process, extends the cumulative standard normal distribution domain to negative interest rates and allows Black’s model to work in the conventional way. Furthermore, we provide the derivations of the so called five Greek letters that enable finance professionals to evaluate the sensitivity of an option to various parameters. Along with the description of the methodology, we present an extensive simulation study and a comparison with the Normal model which is widely used in the negative environment option pricing problems.

Suggested Citation

  • Riccardo Bramante & Gimmi Dallago & Silvia Facchinetti, 2022. "Black’s model in a negative interest rate environment, with application to OTC derivatives," Computational Management Science, Springer, vol. 19(1), pages 25-39, January.
    • Handle: RePEc:spr:comgts:v:19:y:2022:i:1:d:10.1007_s10287-021-00408-6
    DOI: 10.1007/s10287-021-00408-6
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    References listed on IDEAS

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    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Pier Giuseppe Giribone & Simone Ligato & Martina Mulas, 2017. "The effects of negative interest rates on the estimation of option sensitivities: The impact of switching from a log-normal to a normal model," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-42, March.
    3. Maria Cristina Recchioni & Yu Sun & Gabriele Tedeschi, 2017. "Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 17(8), pages 1257-1275, August.
    4. Fries, Christian P. & Nigbur, Tobias & Seeger, Norman, 2017. "Displaced relative changes in historical simulation: Application to risk measures of interest rates with phases of negative rates," Journal of Empirical Finance, Elsevier, vol. 42(C), pages 175-198.
    5. Harriet Jackson, 2015. "The International Experience with Negative Policy Rates," Discussion Papers 15-13, Bank of Canada.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
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