An Equity-Interest Rate Hybrid Model With Stochastic Volatility and the Interest Rate Smile

Grzelak, Lech and Oosterlee, Kees (2010): An Equity-Interest Rate Hybrid Model With Stochastic Volatility and the Interest Rate Smile.

Abstract

We define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero correlation between the main processes. By an appropriate change of measure the dimension of the corresponding pricing PDE can be greatly reduced. We place by a number of approximations the model in the class of affine processes [DPS00], for which we then provide the corresponding forward characteristic function. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds.

• Item Type: MPRA Paper
• Language: English
• Keywords: hybrid models; Heston equity model; Libor Market Model with stochastic volatility; displaced diffusion; affine diffusion; fast calibration.
• Subjects: G - Financial Economics > G1 - General Financial Markets; F - International Economics > F3 - International Finance; G - Financial Economics > G1 - General Financial Markets > G13 - Contingent Pricing ; Futures Pricing

References

[AA00] L.B.G. Andersen, J. Andreasen, Volatility Skews and Extensions of the Libor Market Model. Appl. Math. Finance, 1(7): 1–32, 2000.

[AA02] L.B.G. Andersen, J. Andreasen, Volatile Volatilities. Risk, 15(12): 163–168, 2002.

[ABR05] L.B.G. Andersen, R. Brotherton-Ratcliffe, Extended Libor Market Models with Stochastic Volatility. J. Comp. Fin., 9(1): 1–40, 2005.

[BJ09] C. Beveridge, M. Joshi, Interpolation Schemes in the Displaced-Diffusion Libor Market Model and The Efficient Pricing and Greeks for Callable Range Accruals. SSRN working paper, 2009.

[BK91] F. Black, P. Karasinski, Bond and Option Pricing when Short Rates are Lognormal. Financial Analysys J., 4(47): 52–59, 1991.

[BS73] F. Black, M. Scholes, The Pricing of Options and Corporate Liabilities. J. Political Economy, 81(3): 637–654, 1973.

[BM07] D. Brigo, F. Mercurio, Interest Rate Models- Theory and Practice: With Smile, Inflation and Credit. 2nd ed. Springer, New York, 2007.

[BGM97] A. Brace, D. Gatarek, M. Musiela, The Market Model of Interest Rate Dynamics. Math. Finance, 7(2): 127–155, 1997.

[CM99] P.P. Carr, D.B. Madan, Option Valuation Using the Fast Fourier Transform. J. Comp. Fin., 2(4): 61–73, 1999.

[CIR85] J.C. Cox, J.E. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates. Econometrica, 53(2): 385–407, 1985.

[DMP09] M.H.A. Davis, V. Mataix-Pastor, Arbitrage-Free Interpolation of the Swap Curve. IJTAF, 12(7): 969–1005, 2009.

[DPS00] D. Duffie, J. Pan, K. Singleton, Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6): 1343–1376, 2000.

[Duf01] D. Dufresne, The Integrated Square-Root Process. Working paper, University of Montreal, 2001.

[FO08] F. Fang, C.W. Oosterlee, A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions. SIAM J. Sci. Comput., 31(2): 826–848, 2008.

[GZ99] P. Glasserman, X. Zhao, Fast Greeks by Simulation in Forward LIBOR Models. J. Comp. Fin., 3(1): 5–39, 1999.

[GO09] L.A. Grzelak, C.W. Oosterlee, On the Heston Model with Stochastic Interest Rates. Techn. Report 09-05, Delft Univ. Techn., the Netherlands, SSRN working paper, 2009.

[GOW09] L.A. Grzelak, C.W. Oosterlee, S. v. Weeren, Efficient Option Pricing with Multi-Factor Equity Interest Rate Hybrid Models. Techn. Report 09-07, Delft Univ. Techn., the Netherlands, SSRN working paper, 2009.

[Hes93] S. Heston, A Closed-form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Rev. Fin. Studies, 6(2): 327–343, 1993.

[HW96] J. Hull, A. White, Using Hull-White Interest Rate Trees. J. Derivatives, 3(3): 26–36, 1996.

[HW00] J. Hull, A. White, Forward Rate Volatilities, Swap Rate Volatilities and the Implementation of the Libor Market Model. J. Fixed Income, 10(2): 46–62, 2000.

[Hun05] C. Hunter, Hybrid Derivatives. The Euromoney Derivatives and Risk Management Handbook, 2005.

[JR00] P. J¨ackel, R. Rebonato, Linking Caplet and Swaption Volatilities in a BGM Framework: Approximate Solutions. J. Comp. Fin., 6(4): 41–60, 2000. 19

[Jam91] F. Jamshidian, Bond and Option Evaluation in the Gausian Interest Rate Model. Research in Finance, 9: 131–170, 1991.

[Jam97] F. Jamshidian, Libor and Swap Market Models and Measures. Financ. Stoch., 1(4): 293–330, 1997.

[Kum36] E.E. Kummer, ¨Uber die Hypergeometrische Reihe F(a; b; x). J. reine angew. Math., 15: 39–83, 1936.

[Lee04] R. Lee, Option Pricing by Transform Methods: Extensions, Unification, and Error Control. J. Comp. Fin., 7(3): 51–86, 2004.

[Lew00] A.L. Lewis, Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach, California, USA, 2000.

[MM09] F. Mercurio, M. Morini, No-Arbitrage Dynamics for a Tractable SABR Term Structure Libor Model Modelling Interest Rates: Advances in Derivatives Pricing, edited by F. Mercurio, Risk Books, 2009.

[MSS97] K.R. Miltersen, K. Sandmann, D. Sondermann, Closed Form Solutions for Term Structure Derivatives with Lognormal Interest Rates. J. of Finance, 52(1): 409– 430, 1997.

[MR97] M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling. Springer Finance, 1st edition, New York, 1997.

[Pit03] V. Piterbarg, A Stochastic Volatility Forward Libor Model with a Term Structure of Volatility Smiles. SSRN Working Paper, 2003.

[Pit04] V. Piterbarg, Computing Deltas of Callable Libor Exotics in Forward Libor Models. J. Comp. Fin., 7(2): 107–144, 2004.

[Reb02] R. Rebonato, Modern Pricing of Interest Rate Derivatives: the Libor Market Model and Beyond. Princeton University Press, UK 2002.

[Reb04] R. Rebonato, Volatility and Correlation. The Perfect Hedger and the Fox. John Wiley & Sons, Ltd, second edition, 2004.

[Sch02] E. Schl¨ogl, Arbitrage-Free Interpolation in Models of A Market Observable Interest Rates. In K. Sandmann and P. Sch¨onbucher, editors, Advances in Finance and Stochastic: Essays in Honour of Dieter Sondermann. 197–218. Springer Verlag, Heidelberg, 2002.

[WZ08] L. Wu, F. Zhang, Fast Swaption Pricing Under the Market Model with a Square- Root Volatility Process. Quant. Fin., 8(2): 163–180, 2008.