Abstract
Motivated by statistical tests on historical data that confirm the normal distribution assumption on the spreads between major constant maturity swap (CMS) indexes, we propose an easy-to-implement two-factor model for valuing CMS spread link instruments, in which each forward CMS spread rate is modeled as a Gaussian process under its relevant measure, and is related to the lognormal martingale process of a corresponding maturity forward LIBOR rate through a Brownian motion. An illustrating example is provided. Closed-form solutions for CMS spread options are derived.
Highlights
A Constant Maturity Swap ( constant maturity swap (CMS)) spread derivative is a financial instrument whose payoff is a function of the spread between two swap rates of different maturities
Motivated by statistical tests on historical data that confirm the normal distribution assumption on the spreads between major constant maturity swap (CMS) indexes, we propose an easy-to-implement two-factor model for valuing CMS spread link instruments, in which each forward CMS spread rate is modeled as a Gaussian process under its relevant measure, and is related to the lognormal martingale process of a corresponding maturity forward LIBOR rate through a Brownian motion
Cient between Si t the t, correlation and coeffii t 1 i t 2 1 2 is the orthogonal complement of Equation (1) defines the stochastic differential equation of the forward CMS spread rate under its relevant measure, the Ti 1 -forward measure. This equation can lead to closed-form solutions to value financial derivatives that depend on a CMS spread rate at a single maturity date such as CMS spread caplets, floorlets, and digital options
Summary
A Constant Maturity Swap ( CMS) spread derivative is a financial instrument whose payoff is a function of the spread between two swap rates of different maturities (e.g., the 10-year swap rate minus the 2year swap rate). The difficulty arises from the fact that unlike a single interest rate, a CMS spread rate can allow both positive and negative values, as the yield curve moves in a way that any part can be either flat, upward or downward sloping This feature adds an extra complication in the pricing of derivative instruments for which a CMS spread rate is the underlying. TCHUINDJO who approximate the value of CMS spread options in the standard lognormal LIBOR market model with deterministic and stochastic volatilities respectively This current approach has the advantage to help understand the influence of various model parameters—in particular the correlation between the two rates used to calculate the spread.
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