# Constant Maturity Swaps in the LIBOR Market Model

A **constant maturity swap (CMS)** is a derivative with a payoff that is based on a swap rate of a specific maturity. For example, while a regular floating rate note might pay semi-annual coupons based on semi-annual fixings of 6-month USD LIBOR, a CMS note might pay semi-annual coupons based on semi-annual fixings of the 10-year semi-annual swap rate. Note, however, that the coupon frequency need not match that of the underlying swap rate: the note might pay semi-annual coupons based on fixings of the 10-year annual swap rate, for example.

Whereas a regular floating rate (e.g., 6-month LIBOR) contains information about short-term interest rates, a CMS rate (e.g., a 10-year or 20-year semi-annual swap rate) contains information about the overall level of the yield curve. Derivatives based on a CMS rate are therefore traded by parties who wish to take a view on future changes in the level of the yield curve. On the other hand, the spread between two CMS rates (e.g., the 20-year CMS rate minus the 2-year CMS rate) contains information on the slope of the yield curve; for that reason certain CMS spread instruments are sometimes called steepeners. Derivatives based on a CMS spread are therefore traded by parties who wish to take a view on future relative changes in different parts of the yield curve.

Standard approaches to **pricing CMS derivatives** must take into account convexity and timing corrections. The convexity adjustment arises since the expected payoff is calculated in a world which is forward risk-neutral with respect to a zero-coupon bond (the expected bond price equals the forward bond price). In that world, the expected underlying swap rate (upon which the payoff is based) does not equal the forward swap rate. The convexity is just the difference between the expected swap rate and the forward swap rate. The timing adjustment arises since the CMS rate is usually fixed at the beginning of each coupon period, but paid at the end.

In addition to this, various CMS derivatives (options, range accruals and inverse floaters) are sensitive to the volatility smile i.e., the dependence of the volatility of the CMS rate on the strike. Assuming a simple lognormal evolution of forward and/or swap rates will not necessarily be good enough.

For these reasons, the FINCAD approach to pricing CMS derivatives is to use the **LIBOR Market Model (LMM)** with extensions to allow for non-lognormal evolution of the underlying LIBOR forward rates. Using Monte Carlo simulation, we can obtain the forward LIBOR rates under a single measure, and calculate the required CMS rates from these forward rates. This method does not require any convexity adjustment. In addition, it is easy to take into account non-lognormal evolution of the forward rates.

The LIBOR Market model must of course be properly calibrated and it is relatively easy to calibrate to the smile of caplets (forward rates) and/or swaptions (swap rates). In order to price smile-dependent CMS derivatives, the LMM should ideally be calibrated to the smile of CMS options, but to first order the calibration can performed using swaptions that expire on CMS rate fixing dates at a range of strikes, where the swap underlying the swaption is the same swap used to set the CMS rate. Obtaining this swaption data from the market might not always be possible, so the LMM may need to be calibrated to the caplet smile as well or instead.

Monte Carlo simulations are then used to evolve forward rates, from which CMS rates can be calculated. Convexity and timing corrections to forward CMS rates are not necessary because the Monte Carlo simulation evolves rates forward in time using drifts corresponding to the terminal measure, and calculates expectations in this measure directly. The Monte Carlo simulation can handle arbitrary payoffs and can therefore be used to price a wide variety of CMS derivatives, including Bermudan CMS spread options. Bermudan exercise in Monte Carlo can be modeled using the Least Squares Monte Carlo (LSMC) algorithm proposed by Longstaff and Schwartz.

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