Prague Economic Papers 2009, 18(4):309-326 | DOI: 10.18267/j.pep.356
Valuation of Convexity Related Interest Rate Derivatives
- University of Economics, Prague.
We investigate valuation of derivatives with payoff deined 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 inite 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.
Klíčová slova: interest rate derivatives, Libor in arrears, constant maturity swap, valuation models, convexity adjustment
JEL classification: C13, E43, E47, G13
Zveřejněno: 1. leden 2009 Zobrazit citaci
Reference
- Aitchinson J., Brown J. A. C. (1966), The Lognormal Distribution. Cambridge University Press.
- Benhamou, E. (2000a), "Pricing Convexity Adjustment with Wiener Chaos." London School of Economics, SSRN Working Paper.
Přejít k původnímu zdroji...
- Benhamou, E. (2000b), "A Martingale Result for Convexity Adjustment in the Blal Pricing Model." London School of Economics, SSRN Working Paper.
Přejít k původnímu zdroji...
- Brotherton-Ratcliffe, R., Iben, B. (1993), "Yield Curve Applications of Swap Products." In Schwartz, R. J., Smith, C. W., Advanced Strategies in Financial Risk Management, New York Institute of Finance, New York, pp. 400-450.
- Black, F., Scholes, M. (1973), "The Pricing of Options and Corporate Liabilities." Journal of Political Economy 81, pp.637-659.
Přejít k původnímu zdroji...
- Gatarek, D. (2003), "Constant Maturity Swaps, Forward Measure, and LIBOR Market Model." Deloitte & Touche, SSRN Working Paper.
- Hagan, P. S. (2003), "Convexity Conundrums: Pricing CMS Swaps, Caps, and Floors." Wilmott Magazine, March, pp. 38-44.
Přejít k původnímu zdroji...
- Henrard, M. (2007). "CMS Swaps in Separable One-Factor Gaussian LLM and HJM Model." BIS, SSRN Working Paper.
- Hunt, P. J., Kennedy, J. E. (2007), Financial Derivatives in Theory and Practice. Wiley, 2000.
- Harrison, J. M., Pliska, S. R. (1981), "Martingales and Stochastic Integrals in the Theory of Continuous Trading." Stochastic Processes and Their Applications, pp. 55-61.
Přejít k původnímu zdroji...
- Hull, J. C. (2006), Options, Futures, and Other Derivatives. 6th Edition, Pearson Prentice Hall.
- Jílek, J. (2006), Finanční a komoditní deriváty. Praha: Grada.
- Li, A., Raghavan, V. R. (1996), "Libor-in-Arrears Swaps." Journal of Derivatives, Vol. 3 (3), pp. 44-48.
Přejít k původnímu zdroji...
- Málek, J. (2005), Dynamika úrokových měr a úrokové deriváty. Praha: Ekopress.
- Mičulka, J. (2007), "Analysis of the Derivative Market in the Czech Republic." University of Economics, Prague, Diploma Thesis.
- Mercurio, F., Pallavicini, A. (2006), "Swaption Skews and Convexity Adjustments." Bana IMI, SSRN Working Paper.
- Musiela, M., Rutkowski M. (1997), Martingale Methods in Financial Modeling. Springer-Verlag.
Přejít k původnímu zdroji...
- Pelsser, A. (2003), "Mathematical Foundation of Convexity Correction." Quantitative Finance 3, pp.59-65.
Přejít k původnímu zdroji...
- Vojtek, M. (2004), "Calibration of Interest Rate Models - Transition Market Case." CERGE-EI Working Paper, No. 237.
Přejít k původnímu zdroji...
Tento článek je publikován v režimu tzv. otevřeného přístupu k vědeckým informacím (Open Access), který je distribuován pod licencí Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY NC ND 4.0), která umožňuje nekomerční distribuci, reprodukci a změny, pokud je původní dílo řádně ocitováno. Není povolena distribuce, reprodukce nebo změna, která není v souladu s podmínkami této licence.