The last two decades were marked by a rapidly growing interest in the modelling of bond market. One of the major achievements in this area was a new approach to the term structure modelling proposed by D. Heath, R. Jarrow and A. Morton in their seminal paper published in 1992. One of the main features of the HJM methodology is that it covers a large variety of previously proposed models, and it provides a unified approach to the modelling of instantaneous interest rates and to the valuation of interest-rate sensitive derivatives. It is also notable that, by construction, the HJM model fits the initial yield curve. The HJM methodology appeared to be very successful both from the theoretical and practical viewpoints. However, since the HJM approach to the term structure modelling is based on an arbitrage-free dynamics of the instantaneous continuously compounded forward rates, it requires a certain degree of smoothness with respect to the tenor of the bond prices and their volatilities. For this reason, working with such models is not always convenient. An alternative construction of an arbitrage-free family of bond prices, making no reference to instantaneous rates, is in some circumstances more suitable. The first step in this direction was done by K. Sandmann and D. Sondermann (1993), who focused on the effective annual interest rate. This approach was further developed in ground-breaking papers by K. Miltersen, K. Sandmann and D. Sondermann (1997) and A. Brace, D. Gatarek and M. Musiela (1997), who proposed to focus on a direct modelling of forward LIBORs. Their works were later extended by F. Jamshidian (1997) to the case of co-terminal forward swap rates. The main goal in these papers was to produce arbitrage-free term structure models supporting the market practice of pricing typical interest-rate derivatives (caps and swaptions) through a suitable version of the Black futures formula. This practical requirement enforces the lognormality of the forward LIBOR (or swap) rate under the corresponding LIBOR (or swap) martingale measure. The goal of the talk is to present the most important features of market models of LIBORs and swap rates developed in the above-mentioned papers.

Market Models of LIBORS and Swap Rates:

• Notes (PDF file)
• Slides (PDF file)

References:

• A. Brace, D. Gatarek and M. Musiela, The market model of interest rate dynamics, Mathematical Finance 7 (1997), 127-154.
• F. Jamshidian, LIBOR and swap market models and measures, Finance and Stochastics 1 (1997), 293-330.
• K. Miltersen, K. Sandmann and D. Sondermann, Closed form solutions for term structure derivatives with log-normal interest rates, The Journal of Finance 52 (1997), 409-430.
• M. Musiela and M. Rutkowski, Continuous-time term structure models: Forward measure approach, Finance and Stochastics 1 (1997), 261-291.
• M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling, Springer, Berlin Heidelberg New York, 1997 [2nd edition, 2005].

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