Grzelak, Lech and Oosterlee, Kees (2010): On crosscurrency models with stochastic volatility and correlated interest rates.

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Abstract
We construct multicurrency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We frst deal with a foreign exchange (FX) model of Hestontype, in which the domestic and foreign interest rates are generated by the shortrate process of HullWhite [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaceddiffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semiclosed form approximations which lead to effcient calibration of the multicurrency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity FXinterest rate hybrid payoffs.
Item Type:  MPRA Paper 

Original Title:  On crosscurrency models with stochastic volatility and correlated interest rates 
Language:  English 
Keywords:  Foreignexchange (FX); stochastic volatility; Heston model; stochastic interest rates; interest rate smile; forward characteristic function; hybrids; affne diffusion; effcient 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 
Item ID:  23020 
Depositing User:  Lech A. Grzelak 
Date Deposited:  04. Jun 2010 10:22 
Last Modified:  12. Feb 2013 09:16 
References:  L.B.G. Andersen, J. Andreasen, Volatility Skews and Extensions of the Libor Market Model. Appl. Math. Finance, 1(7): 1–32, 2000. L.B.G. Andersen, J. Andreasen, Volatile Volatilities. Risk, 15(12): 163–168, 2002. J. Andreasen, Closed Form Pricing of FX Options Under Stochastic Rates and Volatility, Presentation at Global Derivatives Conference 2006, Paris, 911 May 2006. A. Antonov, T. Misirpashaev, Efficient Calibration to FX Options by Markovian Projection in CrossCurrency LIBOR Market Models. SSRN working paper, 2006. A. Antonov, M. Arneguy, N. Audet, Markovian Projection to a Displaced Volatility Heston Model. SSRN working paper, 2008. C. Beveridge, M. Joshi, Interpolation Schemes in the DisplacedDiffusion Libor Market Model and the Efficient Pricing and Greeks for Callable Range Accruals. SSRN working paper, 2009. D. Brigo, F. Mercurio, Interest Rate Models Theory and Practice: With Smile, Inflation and Credit. Springer Finance, second edition, 2007. O. Caps, On the Valuation of PowerReverse Duals and EquityRates Hybrids. Presentation at Frankfurt MathFinance Conference, Derivatives and Risk Management in Theory and Practice, Frankfurt, 2627 May 2007. J.C. Cox, J.E. Ingersoll, S.A. Ross, A Theory of the Term Structure of Interest Rates. Econometrica, 53(2): 385–407, 1985. M.H.A. Davis, V. MataixPastor, ArbitrageFree Interpolation of the Swap Curve. IJTAF, 12(7): 969–1005, 2009. D. Duffie, J. Pan, K. Singleton, Transform Analysis and Asset Pricing for Affine JumpDiffusions. Econometrica, 68: 1343–1376, 2000. F. Fang, C.W. Oosterlee, A Novel Pricing Method for European Options Based on FourierCosine Series Expansions. SIAM J. Sci. Comput., 31: 826848, 2008. H. Geman, N. El Karoui, J.C. Rochet, Changes of Num´eraire, Changes of Probability Measures and Pricing of Options. J. App. Prob., 32: 443–548, 1996. A. Giese, On the Pricing of AutoCallable Equity Securities in the Presence of Stochastic Volatility and Stochastic Interest Rates. Presentation at MathFinanceWorkshop: Derivatives and Risk Management in Theory and Practice, Frankfurt, 2004. P. Glasserman, X. Zhao, Fast Greeks by Simulation in Forward LIBOR Models. J. Comp. Fin., 3(1): 5–39, 1999. L.A. Grzelak, C.W. Oosterlee, On the Heston Model with Stochastic Interest Rates. Techn. Report 0905, Delft Univ. Techn., the Netherlands, SSRN working paper, 2009. L.A. Grzelak, C.W. Oosterlee, An EquityInterest Rate Hybrid Model With Stochastic Volatility and Interest Rate Smile. Techn. Report 1001, Delft Univ. Techn., the Netherlands, SSRN working paper, 2010. A.v. Haastrecht, A. Pelsser, Generic Pricing of FX, Inflation and Stock Options Under Stochastic Interest Rates and Stochastic Volatility. SSRN working paper, 2009. D. Heath, R.A. Jarrow, A. Morton, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation. Econometrica, 1(60): 77–105, 1992. S. Heston, A Closedform Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. Rev. Fin. Studies, 6(2): 327–343, 1993. J. Hull, A. White, Using HullWhite Interest Rate Trees. J. Derivatives, 3(3): 26– 36, 1996. 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. C. Hunter, Hybrid Derivatives. The Euromoney Derivatives and Risk Management Handbook, 2005. P. J¨ackel, R. Rebonato, Linking Caplet and Swaption Volatilities in a BGM Framework: Approximate Solutions. J. Comp. Fin., 6(4): 41–60, 2000. A. Kawai, P. J¨ackel, An Asymptotic FX Option Formula in the Cross Currency Libor Market Model. Wilmott, 7484, March 2007. P. Mikkelsen, CrossCurrency LIBOR Market Models. Center for Analytica Finance Aarhus School of Business, working paper no.85. 2001. M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling. Springer Finance, first edition, 1997. V.V. Piterbarg, Time to Smile, Risk, 18(5): 7175, 2005. V.V. Piterbarg, Computing Deltas of Callable Libor Exotics in Forward Libor Models. J. Comp. Fin., 7(2): 107–144, 2004. V.V. Piterbarg, Smiling Hybrids, Risk, 19: 6671, 2006. E. Schlögl, ArbitrageFree 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. E. Schlögl, A Multicurrency Extension of the Lognormal Interest Rate Market Models. Finance and Stochastics, Springer, 6(2): 173196, 2002. J. Sippel, S. Ohkoshi, All Power to PRDC notes. Risk, 15(11): 531533, 2002. S. Shreve, Stochastic Calculus for Finance II: ContinuosTime Models. New York: Springer, 2004. A. Takahashi, K. Takehara, Fourier Transform Method with an Asymptotic Expansion Approach: an Application to Currency Options. IJTAF, 11(4): 381–401, 2008. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/23020 