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Recovering Local Volatility Functions Of Forward Libor Rates

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  • Grace Kuan

    (University of Warwick)

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    Abstract

    It is commonly observed in the market that implied volatilities of standard European options vary with strike levels and expiration dates. The former is usually referred to as volatility skew and the later is volatility term structure. The idea of implied pricing is to recover the dynamics of the underlying asset from market prices of liquid options prices and use the information to price and hedge less liquid products. In this paper, we apply implied pricing in the interest rate market and use market cap prices to back out the local volatility functions of the forward LIBOR rate processes. The recovered dynamics of forward LIBOR rates reveal the market's expectation toward interest rates and they can be used to price other exotic interest rate options.The implied pricing methods developed so far mainly focus on the application in the equity market and foreign exchange market. The complexity of implementing implied methods to interest rate options lies in the fact that, usually in interest rate models, both the infinitesimal drift and volatility of the interest rate process are unknown. To save the computation of the drift, we work with the framework of forward LIBOR rate model in [3] and [4], where only the local volatility functions need to be approximated. We use spline functional approach suggested by Coleman, Li and Verma [2] to recover the local volatility. It is assumed to be a function of the time and forward LIBOR rate and represented by the tensor product splines. Given this representation, we use finite difference methods to solve the partial differential equation satisfied by caplet prices. The parameters of the splines are found by fitting the market caplet prices. The advantage of using forward LIBOR rate model is, given the local volatility functions, the drifts of forward LIBOR rates under the spot LIBOR measure or terminal forward measure can be easily obtained for the one-factor model.The paper is organised as follows. Section 2 gives an overview to implied pricing methods developed in both equity market and interest rate market. In section 3, we will have a brief review to the forward LIBOR rate model and describe the numerical procedure to recover local volatilities. Section 4 includes two computation examples. In the first example, the market caplet prices are simulated with extended forward LIBOR model developed by Andersen and Andreasen [1]. It shows that the method is able to recover the constant elasticity variance volatility structure accurately. In the second example, the method is applied to the market data of three months GBP LIBOR cap prices. The recovered local volatility functions appear non-linear in both variables of time and forward LIBOR rates.

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    Bibliographic Info

    Paper provided by Society for Computational Economics in its series Computing in Economics and Finance 2000 with number 255.

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    Date of creation: 05 Jul 2000
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    Handle: RePEc:sce:scecf0:255

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    Postal: CEF 2000, Departament d'Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas, 25,27, 08005, Barcelona, Spain
    Fax: +34 93 542 17 46
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    Web page: http://enginy.upf.es/SCE/
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    1. Frank Milne & Dilip Madan, 1994. "Contingent Claims Valued And Hedged By Pricing And Investing In A Basis," Working Papers 1158, Queen's University, Department of Economics.
    2. Merton, Robert C., 1975. "Option pricing when underlying stock returns are discontinuous," Working papers 787-75., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    3. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. " Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-30, March.
    4. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    5. Dilip B. Madan & Frank Milne, 1992. "Contingent Claims Valued and Hedged by Pricing and Investment in a Basis," Working Papers 868, Queen's University, Department of Economics.
    6. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, 04.
    7. Ait-Sahalia, Yacine, 1996. "Nonparametric Pricing of Interest Rate Derivative Securities," Econometrica, Econometric Society, vol. 64(3), pages 527-60, May.
    8. Chan, K C, et al, 1992. " An Empirical Comparison of Alternative Models of the Short-Term Interest Rate," Journal of Finance, American Finance Association, vol. 47(3), pages 1209-27, July.
    9. Constantinides, George M, 1992. "A Theory of the Nominal Term Structure of Interest Rates," Review of Financial Studies, Society for Financial Studies, vol. 5(4), pages 531-52.
    10. Marek Rutkowski & Marek Musiela, 1997. "Continuous-time term structure models: Forward measure approach (*)," Finance and Stochastics, Springer, vol. 1(4), pages 261-291.
    11. Melick, William R. & Thomas, Charles P., 1997. "Recovering an Asset's Implied PDF from Option Prices: An Application to Crude Oil during the Gulf Crisis," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 32(01), pages 91-115, March.
    12. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    13. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
    14. Joshua Rosenberg & Robert F. Engle, 2000. "Empirical Pricing Kernels," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-014, New York University, Leonard N. Stern School of Business-.
    15. Joshua Rosenberg, 1999. "Empirical Tests of Interest Rate Model Pricing Kernels," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-015, New York University, Leonard N. Stern School of Business-.
    16. Amin, Kaushik I. & Morton, Andrew J., 1994. "Implied volatility functions in arbitrage-free term structure models," Journal of Financial Economics, Elsevier, vol. 35(2), pages 141-180, April.
    17. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
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