Speedup of Calibration and Pricing with SABR Models: From Equities to Interest Rates Derivatives

  • Ana María Ferreiro
  • José A. García-Rodríguez
  • José G. López-Salas
  • Carlos VázquezEmail author
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 135)


In the more classical models for equities and interest rates evolution, constant volatility is usually assumed. However, in practice the volatilities are not constant in financial markets and different models allowing a varying local or stochastic volatility also appear in the literature. Particularly, here we consider the SABR model that has been first introduced in a paper by Hagan and coworkers, where an asymptotic closed-form formula for the implied volatility of European plain-vanilla options with short maturities is proposed. More recently, different works (Mercurio and Morini, Modeling Interest Rates: Advances in Derivatives Pricing, Risk Books 2009; Hagan and Lesniewski, LIBOR market model with SABR style stochastic volatility. Working Paper., 2008; Rebonato, A time-homogeneous SABR-consistent extension of the LMM. Risk, 2008) have extended the use of SABR model in the context of LIBOR market models for the evolution of forward rates (SABR-LMM). One drawback of these models in practice comes from the increase of computational cost, mainly due to the growth of model parameters to be calibrated. Additionally, sometimes either it is not always possible to compute an analytical approximation for the implied volatility or its expression results to be very complex, so that numerical methods (for example, Monte Carlo in the calibration process) have to be used. In this work we mainly review some recently proposed global optimization techniques based on Simulated Annealing (SA) algorithms and its implementation on Graphics Processing Units (GPUs) in order to highly speed up the calibration and pricing of different kinds of options and interest rate derivatives. Finally, we present some examples corresponding to real market data.


SABR volatility models SABR/LIBOR market models Parallel simulated annealing GPUs 


  1. 1.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)CrossRefGoogle Scholar
  2. 2.
    Brace, A., Gatarek, D., Musiela, M.: The Market model of interest rate dynamics. Math. Financ. 7(2), 127–155 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fernández, J.L., Ferreiro, A.M., García, J.A., López-Salas, J.G., Vázquez, C.: Static and dynamic SABR stochastic volatility models: calibration and option pricing using GPUs. Math. Comput. Simul. 94, 55–75 (2013)CrossRefGoogle Scholar
  4. 4.
    Ferreiro, A.M., García, J.A., López-Salas, J.G., Vázquez, C.: An efficient implementation of parallel simulated annealing algorithm in GPUs. J. Glob. Optim. 57(3), 863–890 (2013)CrossRefzbMATHGoogle Scholar
  5. 5.
    Ferreiro, A.M., García, J.A., López-Salas, J.G., Vázquez, C.: SABR/LIBOR market models: pricing and calibration for some interest rate derivatives. Appl. Math. Comput. 242, 65–89 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hagan, P., Lesniewski, A.: LIBOR market model with SABR style stochastic volatility. Working Paper. (2008)
  7. 7.
    Hagan, P.S., Kumar, D., Lesniewski, A.S., Woodward, D.E.: Managing smile risk. Wilmott Magazine (2002)Google Scholar
  8. 8.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220, 671–680 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mercurio, F., Morini, M.: No-Arbitrage dynamics for a tractable SABR term structure Libor model. In: Modeling Interest Rates: Advances in Derivatives Pricing. Risk Books, London, UK (2009)Google Scholar
  10. 10.
    MPI: A message-passing interface standard. Message Passing Interface Forum.
  11. 11.
    Oblój, J.: Fine-tune your smile: correction to Hagan et al. Wilmott Magazine (2008)Google Scholar
  12. 12.
  13. 13.
    Osajima, Y.: The asymptotic expansion formula of implied volatility for dynamic SABR model and FX hybrid model. Report UTMS 2006-29. pp. 24. Graduate School of Mathematical Sciences. University of Tokyo (2006).Google Scholar
  14. 14.
    Rebonato, R.: A time-homogeneous SABR-consistent extension of the LMM. Risk 20, 102–106 (2007)Google Scholar
  15. 15.
    Rebonato, R., Mckay, K., White, R.: The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives, 1st edn. Wiley, Chichester (2009)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ana María Ferreiro
    • 1
  • José A. García-Rodríguez
    • 1
  • José G. López-Salas
    • 1
  • Carlos Vázquez
    • 1
    Email author
  1. 1.Faculty of Informatics, Department of MathematicsCoruñaSpain

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