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Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models

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  • Kristensen, Dennis
  • Mele, Antonio

Abstract

We develop a new approach to approximating asset prices in the context of continuous-time models. For any pricing model that lacks a closed-form solution, we provide a closed-form approximate solution, which relies on the expansion of the intractable model around an “auxiliary” one. We derive an expression for the difference between the true (but unknown) price and the auxiliary one, which we approximate in closed-form, and use to create increasingly improved refinements to the initial mispricing induced by the auxiliary model. The approach is intuitive, simple to implement, and leads to fast and extremely accurate approximations. We illustrate this method in a variety of contexts including option pricing with stochastic volatility, computation of Greeks, and the term structure of interest rates.

Suggested Citation

  • Kristensen, Dennis & Mele, Antonio, 2011. "Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models," Journal of Financial Economics, Elsevier, vol. 102(2), pages 390-415.
    • Handle: RePEc:eee:jfinec:v:102:y:2011:i:2:p:390-415
    DOI: 10.1016/j.jfineco.2011.05.007
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    3. Yang, Nian & Chen, Nan & Wan, Xiangwei, 2019. "A new delta expansion for multivariate diffusions via the Itô-Taylor expansion," Journal of Econometrics, Elsevier, vol. 209(2), pages 256-288.
    4. Kailin Ding & Zhenyu Cui & Xiaoguang Yang, 2023. "Pricing arithmetic Asian and Amerasian options: A diffusion operator integral expansion approach," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 43(2), pages 217-241, February.
    5. Cosma, Antonio & Galluccio, Stefano & Scaillet, Olivier, 2012. "Valuing American options using fast recursive projections," Working Papers unige:41856, University of Geneva, Geneva School of Economics and Management.
    6. Michael Kurz, 2018. "Closed-form approximations in derivatives pricing: The Kristensen-Mele approach," Papers 1804.08904, arXiv.org.
    7. Aït-Sahalia, Yacine & Li, Chenxu & Li, Chen Xu, 2021. "Closed-form implied volatility surfaces for stochastic volatility models with jumps," Journal of Econometrics, Elsevier, vol. 222(1), pages 364-392.
    8. Dennis Kristensen & Young Jun Lee & Antonio Mele, 2023. "Closed-form approximations of moments and densities of continuous-time Markov models," Papers 2308.09009, arXiv.org.
    9. Wan, Xiangwei & Yang, Nian, 2021. "Hermite expansion of transition densities and European option prices for multivariate diffusions with jumps," Journal of Economic Dynamics and Control, Elsevier, vol. 125(C).
    10. Xiu, Dacheng, 2014. "Hermite polynomial based expansion of European option prices," Journal of Econometrics, Elsevier, vol. 179(2), pages 158-177.
    11. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    12. Juan Arismendi, 2014. "A Multi-Asset Option Approximation for General Stochastic Processes," ICMA Centre Discussion Papers in Finance icma-dp2014-03, Henley Business School, University of Reading.
    13. Choi, Seungmoon, 2015. "Explicit form of approximate transition probability density functions of diffusion processes," Journal of Econometrics, Elsevier, vol. 187(1), pages 57-73.
    14. Filipović, Damir & Mayerhofer, Eberhard & Schneider, Paul, 2013. "Density approximations for multivariate affine jump-diffusion processes," Journal of Econometrics, Elsevier, vol. 176(2), pages 93-111.
    15. Jarno Talponen, 2018. "Matching distributions: Recovery of implied physical densities from option prices," Papers 1803.03996, arXiv.org.
    16. Recchioni, Maria Cristina & Iori, Giulia & Tedeschi, Gabriele & Ouellette, Michelle S., 2021. "The complete Gaussian kernel in the multi-factor Heston model: Option pricing and implied volatility applications," European Journal of Operational Research, Elsevier, vol. 293(1), pages 336-360.
    17. João Pedro Vidal Nunes & Pedro Miguel Silva Prazeres, 2014. "Pricing Swaptions Under Multifactor Gaussian Hjm Models," Mathematical Finance, Wiley Blackwell, vol. 24(4), pages 762-789, October.
    18. Azusa Takeyama & Nick Constantinou & Dmitri Vinogradov, 2012. "A Framework for Extracting the Probability of Default from Stock Option Prices," IMES Discussion Paper Series 12-E-14, Institute for Monetary and Economic Studies, Bank of Japan.
    19. Dong Hwan Oh & Andrew J. Patton, 2021. "Better the Devil You Know: Improved Forecasts from Imperfect Models," Finance and Economics Discussion Series 2021-071, Board of Governors of the Federal Reserve System (U.S.).
    20. Cosma, Antonio & Galluccio, Stefano & Pederzoli, Paola & Scaillet, Olivier, 2020. "Early Exercise Decision in American Options with Dividends, Stochastic Volatility, and Jumps," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 55(1), pages 331-356, February.

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    More about this item

    Keywords

    Continuous-time models; Option pricing theory; Stochastic volatility; Closed-form approximations;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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