Calibration of local volatility model with stochastic interest rates by efficient numerical PDE methods

  • Julien HokEmail author
  • Shih-Hau Tan


Long-maturity options or a wide class of hybrid products are evaluated using a local volatility-type modelling for the asset price S(t) with a stochastic interest rate r(t). The calibration of the local volatility function is challenging and time-consuming because of the multi-dimensional nature of the problem. A key requirement of any equity hybrid derivatives pricing model is the ability to rapidly and accurately calibrate to vanilla option prices. In this paper, we develop a calibration technique based on a partial differential equation (PDE) approach which allows an accurate calibration and provides an efficient implementation algorithm. The essential idea is based on solving the derived forward equation satisfied by \(P(t, S, r) \mathcal {Z}(t, S, r)\), where P(tSr) represents the risk-neutral probability density of (S(t), r(t)) and \(\mathcal {Z}(t, S, r)\) the projection of the stochastic discounting factor in the state variables (S(t), r(t)). The solution provides effective and sufficient information for the calibration and pricing. The PDE solver is constructed by using ADI (alternative direction implicit) method based on an extension of the Peaceman–Rachford scheme. Furthermore, an efficient algorithm to compute all the corrective terms in the local volatility function due to the stochastic interest rates is proposed by using the PDE solutions and grid points. It reduces by one order the computations costs and then allows to speed up significantly the calibration procedure. Different numerical experiments are examined and compared to demonstrate the results of our theoretical analysis.


Local volatility model Stochastic interest rates Hybrid Calibration Forward Fokker–Planck-type equation Alternating direction implicit (ADI)  method 

JEL Classification




The authors thank Prof. Kevin Parrott from the University of Greenwich and the participants at QuantMinds International conference 2018 for their fruitful comments. The authors also thank the referees and the associate editor for their constructive feedbacks and remarks to improve the quality of this paper.


  1. Andreasen, J., Andersen, L.: Volatility skews and extensions of the libor market model. Appl. Math. Finance 1, 1 (2000)Google Scholar
  2. Atlan, M.: Localiszing volatilies. Working paper, Laboratoire de Probabilites Universite Pierre et Marie Currie (2006)Google Scholar
  3. Avellaneda, M., Laurence, P.: Quantitative Modeling of Derivative Securities, from Theory to Practice. CRC, Boca Raton (2000)Google Scholar
  4. Benhamou, E., Gruz, A., Rivoira, A.: Stochastic Interest Rates for Local Volatility Hybrids Models. Wilmott, Letchworth (2008)Google Scholar
  5. Benhamou, E., Gobet, E., Miri, M.: Smart expansion and fast calibration for jump diffusion. Finance Stoch. 13(4), 563–589 (2009)Google Scholar
  6. Benhamou, E., Gobet, E., Miri, M.: Analytical formulas for local volatility model with stochastic rates. Quant. Finance 12, 185–198 (2012)Google Scholar
  7. Bompis, R., Hok, J.: Forward implied volatility expansion in time-dependent local volatility models. ESAIM Proc. 45, 88–97 (2014)Google Scholar
  8. Brigo, D., Mercurio, F.: Interest Rate Models-Theory and Practice With Smile, Inflation and Credit. Springer, Berlin (2006)Google Scholar
  9. Clark, I.J.: Foreign Exchange Option Pricing: A Practitioner’s Guide. Wiley, Hoboken (2011)Google Scholar
  10. Dang, D.M., Christara, C., Jackson, K.R., Lakhany, A.: A PDE pricing framework for cross-currency interest rate derivatives. AIP Conf. Proc. 1281(1), 2371–2380 (2010)Google Scholar
  11. Deelstra, G., Rayee, G.: Local volatility pricing models for long-dated FX derivatives. Appl. Math. Finance 20(4), 380–402 (2012)Google Scholar
  12. Derman, E., Kani, I.: Stochastic implied trees: arbitrage pricing with stochastic term and strike structure of volatility. Int. J Theor. Appl. Finance 01, 61–110 (1998)Google Scholar
  13. Dupire, B.: Pricing with a smile. Risk 7(1), 18–20 (1994)Google Scholar
  14. Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, Berlin (2003)Google Scholar
  15. Gobet, E., Hok, J.: Expansion formulas for best-of option on equity and inflation. Int. J. Theor. Appl. Finance 17, 1450010 (2014)Google Scholar
  16. Guyon, J., Labordere, P.H.: The smile calibration problem solved. Working paper (2011)Google Scholar
  17. Haddou, Z. A.: Stochastic Local Volatility and High Performance Computing. Master thesis (2012)Google Scholar
  18. Haowen, F.: European option pricing formula under stochastic interest rate. Prog. Appl. Math. 4(1), 14–21 (2012)Google Scholar
  19. Hok, J., Papapantoleon, A., Ngare, P.: Expansion formulas for European quanto options in a local volatility FX-LIBOR model. Int. J. Theor. Appl. Finance 1(2), 1850017 (2018)Google Scholar
  20. Hull, J., White, A.: One factor interest rate models and the valuation of interest rate derivative securities. J. Financ. Quant. Anal. 28, 235–254 (1993)Google Scholar
  21. in’t Hout, K.J., Maarten, W.: Convergence of the modified Craig-Sneyd scheme for two-dimensional convection-diffusion equations with mixed derivative term. J. Comput. Appl. Math. 296, 170–180 (2016)Google Scholar
  22. Itkin, A.: Modeling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps. Appl. Math. Finance 2, 485–519 (2017)Google Scholar
  23. Jackel, P.: Hyperbolic local volatility. Working paper (2010)Google Scholar
  24. Joshi, M., Ranasinghe, N.: Local volatility under stochastic interest rates using mixture models. Working paper, SSRN 2780072 (2016)Google Scholar
  25. Kim, Y.J.: Option pricing under stochastic interest rates: an empirical investigation. Asia Pac. Financ. Mark. 9, 23–44 (2002)Google Scholar
  26. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)Google Scholar
  27. Lamberton, D., Lapeyre, B.: Introduction au calcul stochastique applique a la finance, 3rd edn. Ellipses, Paris (2012)Google Scholar
  28. Maarten, W., in’t Hout, K.J.: An adjoint method for the exact calibration of stochastic local volatility models. J. Comput. Sci. 24, 182–194 (2017)Google Scholar
  29. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling, 2nd edn. Springer, Berlin (2005)Google Scholar
  30. Oksendal, B.K.: Stochastic Differential Equations: An Introduction with Applications. Springer, Berlin (2003)Google Scholar
  31. Overhaus, M., Bermudez, A., Buehler, H., Ferraris, A., Jordinson, C., Lamnouar, A.: Equity Hybrid Derivatives. Wiley, Hoboken (2007)Google Scholar
  32. Peaceman Jr., D.W., Rachford, H.H.: The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)Google Scholar
  33. Piterbarg, V.: Stochastic volatility model with time dependent skew. Appl. Math. Finance 12, 147–185 (2005a)Google Scholar
  34. Piterbarg, V.: Time to smile. Risk. 71–75 (2005b)Google Scholar
  35. Piterbarg, V.: Smiling hybrids. Risk Mag. 19, 66–71 (2006)Google Scholar
  36. Ren, Y., Madan, D., Qian, M.Q.: Calibrating and pricing with embedded local volatility models. Risk 20, 138–143 (2007)Google Scholar
  37. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (2001)Google Scholar
  38. Rubinstein, M.: Implied binomial trees. J. Finance 49, 771–818 (1994)Google Scholar
  39. Saichev, A., Woyczynski, W.: Distributions in the Physical and Engineering Sciences, Linear and Nonlinear Dynamics in Continuous Media, vol. 2. Birkhäuser, Basel (2013)Google Scholar
  40. Sepp, A.: Stochastic Local Volatility Models: Theory and Implementation. Working paper (2010)Google Scholar
  41. Shreve, S.: Stochastic Calculus for Finance II. Springer, Berlin (2004)Google Scholar
  42. Tian, Y., Zhu, Z., Lee, G., Klebaner, F., Hamza, K.: Calibrating and pricing with a stochastic-local volatility model. J Deriv. 22(3), 21–39 (2015)Google Scholar
  43. Windcliff, H., Forsyth, P.A., Vetzal, K.R.: Analysis of the stability of the linear boundary condition for the Black–Scholes equation. J. Comput. Finance 8, 65–92 (2004)Google Scholar

Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2019

Authors and Affiliations

  1. 1.Crédit Agricole CIBLondonUK
  2. 2.CuemacroLondonUK

Personalised recommendations