Journal of the Korean Society for Industrial and Applied Mathematics

  • 제27권3호
  • /
  • Pages.180-193
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  • 2023
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  • 1226-9433(pISSN)
  • /
  • 1229-0645(eISSN)
한국산업응용수학회 (Korean Society for Industrial and Applied Mathematics)
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APPROXIMATION FORMULAS FOR SHORT-MATURITY NEAR-THE-MONEY IMPLIED VOLATILITIES IN THE HESTON AND SABR MODELS

  • HYUNMOOK CHOI (DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL UNIVERSITY) ;
  • HYUNGBIN PARK (DEPARTMENT OF MATHEMATICAL SCIENCES AND RESEARCH INSTITUTE OF MATHEMATICS, SEOUL NATIONAL UNIVERSITY) ;
  • HOSUNG RYU (DEPARTMENT OF MATHEMATICAL SCIENCES, SEOUL NATIONAL UNIVERSITY)
  • 투고 : 2023.03.17
  • 심사 : 2023.08.19
  • 발행 : 2023.09.25
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초록

Approximating the implied volatilities and estimating the model parameters are important topics in quantitative finance. This study proposes an approximation formula for short-maturity near-the-money implied volatilities in stochastic volatility models. A general second-order nonlinear PDE for implied volatility is derived in terms of time-to-maturity and log-moneyness from the Feyman-Kac formula. Using regularity conditions and the Taylor expansion, an approximation formula for implied volatility is obtained for short-maturity nearthe-money call options in two stochastic volatility models: Heston model and SABR model. In addition, we proposed a novel numerical method to estimate model parameters. This method reduces the number of model parameters that should be estimated. Generating sample data on log-moneyness, time-to-maturity, and implied volatility, we estimate the model parameters fitting the sample data in the above two models. Our method provides parameter estimates that are close to true values.

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과제정보

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1011675 and No. 2022R1A5A6000840). Financial support from the Institute for Research in Finance and Economics of Seoul National University is gratefully acknowledged.

참고문헌

  1. M. Forde, A. Jacquier and R. Lee, The small-time smile and term structure of implied volatility under the Heston model, SIAM Journal on Financial Mathematics, 3(1) (2012), 690-708. https://doi.org/10.1137/110830241
  2. F. Le Floc'h and G. J. Kennedy, Explicit SABR calibration through simple expansions, available at SSRN 2467231, (2014).
  3. A. Medvedev and O. Scaillet, Approximation and calibration of short-term implied volatilities under jump-diffusion stochastic volatility, The Review of Financial Studies, 20(2) (2007), 427-459. https://doi.org/10.1093/rfs/hhl013
  4. A. Medvedev, Implied volatility at expiration, Swiss Finance Institute Research Paper No. 08-04 (2008).
  5. J. Feng, M. Forde and J-P. Fouque, Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model, SIAM Journal on Financial Mathematics, 1(1) (2010), 126-141. https://doi.org/10.1137/090745465
  6. J. Gatheral, E. P. Hsu, P. Laurence, C. Ouyang and T-H. Wang, Asymptotics of implied volatility in local volatility models, Mathematical Finance, 22(4) (2012), 591-620. https://doi.org/10.1111/j.1467-9965.2010.00472.x
  7. E. Alos, R. De Santiago and J. Vives, Calibration of stochastic volatility models via second-order approximation: The Heston case, International Journal of Theoretical and Applied Finance, 18(6) (2015), 1-31. https://doi.org/10.1142/S0219024915500363
  8. Y. Cui, S. del Bano Rollin and G. Germano, Full and fast calibration of the Heston stochastic volatility model, European Journal of Operational Research, 263(2) (2017), 625-638. https://doi.org/10.1016/j.ejor.2017.05.018
  9. S.L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, The Review of Financial Studies, 6(2) (1993), 327-343. https://doi.org/10.1093/rfs/6.2.327
  10. P. S. Hagan, D. Kumar, A. Lesniewski and D. Woodward, Managing smile risk, Wilmott, 1 (2002), 84-108.
  11. L. B. G. Andersen and M. Lake, Robust high-precision option pricing by Fourier transforms: Contour deformations and double-exponential quadrature, Available at SSRN 3231626, (2018)
  12. A. Friedman, Partial differential equations of parabolic type, Courier Dover Publications, 2008.