Empirical Economics

, Volume 57, Issue 6, pp 1935–1958 | Cite as

Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure

  • Jan PospíšilEmail author
  • Tomáš Sobotka
  • Philipp Ziegler


In this paper, we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also an impact of the long memory parameter is measured for the approximative fractional SV model (FSV). For the first time, the robustness of calibrated models is measured using bootstrapping methods on market data and Monte Carlo filtering techniques. In contrast to several other sensitivity analysis approaches for SV models, the newly proposed methodology does not require independence of calibrated parameters—an assumption that is typically not satisfied in practice. Empirical study is performed on a data set of Apple Inc. equity options traded in four different days in April and May 2015. In particular, the results for Heston, Bates and approximative FSV models are provided.


Robustness analysis Sensitivity analysis Stochastic volatility models Bootstrapping Monte Carlo filtering 

Mathematics Subject Classification

62F35 62F40 91G20 91G70 

JEL Classification

C52 C58 C12 G12 



This work was supported by the GACR Grant 14-11559S Analysis of Fractional Stochastic Volatility Models and their Grid Implementation. Computational resources were provided by the CESNET LM2015042 and the CERIT Scientific Cloud LM2015085, provided under the programme “Projects of Large Research, Development, and Innovations Infrastructures”.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Barndorff-Nielsen OE, Shephard N (2006) Econometrics of testing for jumps in financial economics using bipower variation. J Financ Econom 4(1):1–30. CrossRefGoogle Scholar
  2. Bates DS (1996) Jumps and stochastic volatility: exchange rate processes implicit in Deutsche mark options. Rev Financ Stud 9(1):69–107. CrossRefGoogle Scholar
  3. Baustian F, Mrázek M, Pospíšil J, Sobotka T (2017) Unifying pricing formula for several stochastic volatility models with jumps. Appl Stoch Models Bus Ind 33(4):422–442. CrossRefGoogle Scholar
  4. Bayer C, Friz P, Gatheral J (2016) Pricing under rough volatility. Quant Finance 16(6):887–904. CrossRefGoogle Scholar
  5. Benhamou E, Gobet E, Miri M (2010) Time dependent Heston model. SIAM J Financ Math 1(1):289–325. CrossRefGoogle Scholar
  6. Bianchetti M, Kucherenko S, Scoleri S (2015) Pricing and risk management with high-dimensional quasi Monte Carlo and global sensitivity analysis. Wilmott 2015(78):46–70. CrossRefGoogle Scholar
  7. Black FS, Scholes MS (1973) The pricing of options and corporate liabilities. J Polit Econ 81(3):637–654. CrossRefGoogle Scholar
  8. Campolongo F, Cariboni J, Schoutens W (2006) The importance of jumps in pricing European options. Reliab Eng Syst Saf 91(10):1148–1154. CrossRefGoogle Scholar
  9. Carr P, Wu L (2003) What type of process underlies options? A simple robust test. J Finance 58(6):2581–2610. CrossRefGoogle Scholar
  10. Chernick MR (2008) Bootstrap methods: a guide for practitioners and researchers, 2nd edn. Wiley series in probability and statistics. Wiley, HobokenGoogle Scholar
  11. Christoffersen P, Heston S, Jacobs K (2009) The shape and term structure of the index option smirk: why multifactor stochastic volatility models work so well. Manag Sci 55(12):1914–1932. CrossRefGoogle Scholar
  12. Creel M, Kristensen D (2015) ABC of SV: Limited information likelihood inference in stochastic volatility jump-diffusion models. J Empir Finance 31:85–108. CrossRefGoogle Scholar
  13. De Marco S, Martini C (2012) The term structure of implied volatility in symmetric models with applications to Heston. Int J Theor Appl Finance 15(04):1250,026. CrossRefGoogle Scholar
  14. Detlefsen K, Härdle WK (2007) Calibration risk for exotic options. J Deriv 14(4):47–63. CrossRefGoogle Scholar
  15. Duffie D, Pan J, Singleton K (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68(6):1343–1376. CrossRefGoogle Scholar
  16. Efron B (1979) Bootstrap methods: another look at the jackknife. Ann Stat 7(1):1–26CrossRefGoogle Scholar
  17. Efron B (1982) The jackknife, the bootstrap and other resampling plans, CBMS-NSF Regional Conference Series in Applied Mathematics, vol 38. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PAGoogle Scholar
  18. Elices A (2008) Models with time-dependent parameters using transform methods: application to Heston’s model. arXiv:0708.2020
  19. Fukasawa M (2011) Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch 15(4):635–654. CrossRefGoogle Scholar
  20. Hájek J, Šidák Z, Sen PK (1999) Theory of rank tests, 2nd edn. Probability and mathematical statistics. Academic Press Inc., San DiegoGoogle Scholar
  21. Heston SL (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 6(2):327–343. CrossRefGoogle Scholar
  22. Hwang E, Shin DW (2014) A bootstrap test for jumps in financial economics. Econom Lett 125(1):74–78. CrossRefGoogle Scholar
  23. Kim PJ (1976) The smirnov distribution. Ann Inst Stat Math 28(1):267–275. CrossRefGoogle Scholar
  24. Lewis AL (2000) Option valuation under stochastic volatility: with mathematica code. Finance Press, Newport BeachGoogle Scholar
  25. Mikhailov S, Nögel U (2003) Heston’s stochastic volatility model—implementation, calibration and some extensions. Wilmott Mag 2003(July):74–79Google Scholar
  26. Mrázek M, Pospíšil J, Sobotka T (2016) On calibration of stochastic and fractional stochastic volatility models. Eur J Oper Res 254(3):1036–1046. CrossRefGoogle Scholar
  27. Osajima Y (2007) The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model.
  28. Pospíšil J, Sobotka T (2016) Market calibration under a long memory stochastic volatility model. Appl Math Finance 23(5):323–343. CrossRefGoogle Scholar
  29. Saltelli A, Tarantola S, Campolongo F, Ratto M (2004) Sensitivity analysis in practice: a guide to assessing scientific models. Wiley, ChichesterGoogle Scholar
  30. Saltelli A, Ratto M, Andres T, Campolongo F, Cariboni J, Gatelli D, Saisana M, Tarantola S (2008) Global sensitivity analysis: the primer. Wiley, ChichesterGoogle Scholar
  31. Senger O, Celik AK (2013) A Monte Carlo simulation study for Kolmogorov–Smirnov two-sample test under the precondition of heterogeneity: upon the changes on the probabilities of statistical power and type I error rates with respect to skewness measure. J Stat Econom Methods 2(4):1–16Google Scholar
  32. Shreve SE (2004) Stochastic calculus for finance. Springer Finance. Springer, New YorkCrossRefGoogle Scholar
  33. Yekutieli I (2004) Implementation of the Heston model for the pricing of FX options. Tech. rep, Bloomberg LPGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.NTIS - New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPlzeňCzech Republic
  2. 2.Department of MathematicsUniversity of RostockRostockGermany

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