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Chapter 1 Introduction
The strategy of banks and insurance companies using short term borrowing to financing long term debt is the usually policy. The exposure to long term interest rate is the principal focus of financial institutions for investment. Due to the volatile interest rate environment over the past decade, the risk management of the assets and liabilities has raised the great attention of risk managers, especially for those who work in the financial institutions with long-term liability. Therefore, the swap rate derivatives have become more and more popular among the interest rate derivatives.
For example, the payoff of the CMS spread option is based on the difference of the swap rates with different maturities. The payoff of the CMS ratchet option is based on the difference between the current swap rate and the previous swap rates with one period lagging. In this paper, we will discuss the pricing and the hedging both for these two derivatives.
LMM is a popular forward-rate model for pricing the interest rate derivatives, CMS derivatives included. The advantage of this model is that it will generates a set of forward-LIBOR rates (FLRs), which are directly observable in the market and whose volatilities are linked to the tradable contracts. Each FLR is modeled by a lognormal process under its forward measure. Under the setting, Musiela and Rutkowski (1997) provide a Black-like Cap pricing formula.
In opposition to the LMM, the HJM model (Heath et al., 1992) is another forward-rate model generating the evolution of the entire yield curves, where the forward-rates dynamics are fully specified through their instantaneous volatility structure. Based on HJM model, Mercurio and Pallavicini (2005) suggest pricing the
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derivatives of CMS by the three factors mixing Gaussian model and calibrating the correlation structure of the instantaneous forward rates via the CMS spread options.
Brace et al. (1997) overcome some existing technical problems associated with the lognormal version of the HJM model and further prove that the HJM model is consistent to the LMM. In the HJM model, the state variables of the forward-rate dynamics are unobservable from the market data and the instantaneous volatility structure is difficult to calibrate from the OTC derivatives.
In addition to LMM, the swap market model (SMM) is another popular market model. Analogous to the Cap pricing formula, Jamshidian (1997) proves that the processes of FSR are martingales when the “present value of a basis point” taken as the numeraire, and upon these assumptions derive a Black-like pricing formula for the swpation. Although the volatility of each FSR can be utilized, the correlation matrix of the FSR is not characterized under the SMM and there is no suitable method to fit it. There are two essences for a good financial model. One is that the state variable is observable from the market data. The other is that the parameters in the model can be easily calibrated to the financial instruments and be hedged through the financial instruments.
Although we can capture the volatility structure of the FLRs, we are unable to capture the covariance structure from the market data. Rebonato (2004) provides a swaption volatility approximated formula under the LMM and therefore establish the link between the LMM and SMM. It guarantees the covariance matrix fitted from the market quotation of the swaption. Therefore, LMM is the unifying model of the interest rate which is capable of encompassing the global properties of the swap market model. Due to these features, in this paper we choose the FLR to illustrate the yield curve rather than the FSR. The LMM can be applied not only to price the CMS
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spread option but also to the CMS ratchet options. There are wider applications for the LMM than the co-initial swap market model in Galluccio and Hunter (2004).
The traditional methodology for pricing the CMS products by the LMM is to simulate the swap rates which will later be converted into the forward LIBOR rates.
However, the simulation is very time consuming. In this paper, we investigate the approximated dynamic process of the FSR under the LMM with the characteristic which the FSR is approximated to the linear combination of the FLRs.
Based on the approximated dynamics of the FSR, we derivate the analytic formulas for the valuation of CMS spread options and CMS ratchet options. In empirical studies, the covariance matrix of FLR is often rank one, and this permits us to adopt the one factor LMM to price CMS products involving more than two reference rates. Under the framework, the correlations of each FSR are all one in one factor LMM. It avoids the complicated numerical computation resembling the spread option. From the result of the numerical analysis, the errors between the Monte Carlo simulation and the approximated formula would be small.
The rest of this paper is organized as follows. Chapter 2 reviews some essential result related to the LMM and SMM and further integrate the two interest market model. In chapter 3, we propose the approximated dynamic process of the forward swap rate under the LMM, and briefly introduce the contracts of the CMS spread option and CMS ratchet option. Following the no-arbitrage theorem, we derive the approximated closed formulas for the CMS derivatives in the chapter 4. In chapter 5, we compute the errors between the Monte Carlo simulation and the approximation and provide an efficient manner for hedge. The conclusion remarks are in chapter 6.