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Valuation of Convexity Related Interest Rate Derivatives

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  • Jiří Witzany

Abstract

We investigate valuation of derivatives with payoff defined as a nonlinear though close to linear function of tradable underlying assets. Interest rate derivatives involving Libor or swap rates in arrears, i.e. rates paid at wrong time, are a typical example. It is generally tempting to replace the future unknown interest rates with the forward rates. We show rigorously that indeed this is not possible in the case of Libor or swap rates in arrears. We introduce formally the notion of linear plain vanilla derivatives as those that can be replicated by a finite set of elementary operations and show that derivatives involving the rates in arrears are not (linear) plain vanilla. We also study the issue of valuation of such derivatives. Beside the popular convexity adjustment formula, we develop an improved two or more variable adjustment formula applicable in particular on swap rates in arrears. Finally, we get a precise fully analytical formula based on the usual assumption of log-normality of the relevant tradable underlying assets applicable to a wide class of convexity related derivatives. We illustrate the techniques and different results on a case study of a real life controversial exotic swap.

Suggested Citation

  • Jiří Witzany, 2009. "Valuation of Convexity Related Interest Rate Derivatives," Prague Economic Papers, University of Economics, Prague, vol. 2009(4), pages 309-326.
  • Handle: RePEc:prg:jnlpep:v:2009:y:2009:i:4:id:356:p:309-326
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    References listed on IDEAS

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    1. Eric Benhamou, 2000. "Pricing Convexity Adjustment with Wiener Chaos," FMG Discussion Papers dp351, Financial Markets Group.
    2. Eric Benhamou, 2002. "A Martingale Result for Convexity Adjustment in the Black Pricing Model," Finance 0212005, EconWPA.
    3. Martin Vojtek, 2004. "Calibration of Interest Rate Models - Transition Market Case," Finance 0410015, EconWPA.
    4. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
    5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    6. Henrard, Marc, 2007. "CMS swaps in separable one-factor Gaussian LLM and HJM model," MPRA Paper 3228, University Library of Munich, Germany.
    7. A. Pelsser, 2003. "Mathematical foundation of convexity correction," Quantitative Finance, Taylor & Francis Journals, vol. 3(1), pages 59-65.
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    Cited by:

    1. Martin Pohl, 2012. "Czech Swap Market in the Crisis Period," Prague Economic Papers, University of Economics, Prague, vol. 2012(1), pages 101-122.

    More about this item

    Keywords

    interest rate derivatives; Libor in arrears; constant maturity swap; valuation models; convexity adjustment;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • E43 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Interest Rates: Determination, Term Structure, and Effects
    • E47 - Macroeconomics and Monetary Economics - - Money and Interest Rates - - - Forecasting and Simulation: Models and Applications
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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