Analytical Valuation of Barrier Interest Rate Options Under Market Models

Abstract

Classic interest rate models, such as those by Vasicek-Hull-White and Heath-Jarrow-Morton, assume one or more underlying stochastic processes of a particular form, with Gaussian disturbances. Interest-dependent securities are then priced under the assumption of market equilibrium, given the rates processes. These models are theoretically elegant but they can be hard to implement in practice because the probability distributions for rates at future discrete dates are no longer Gaussian. Practitioners have largely turned to “market models,” e.g., the LIBOR Market Model (LMM) of Brace, Gatarek, and Musiela, and the Swap Market Model (SMM) of Jamshidian, that do not model the instantaneous evolution of spot rates, but rather the behavior of the forward rates for the specific future dates on which a security’s cash flows will occur. This allows the use of the basic Black model for individual caplets and easy calibration of the model to the market. In this article, Wu and Chen develop closed-form valuation equations for caps, floors, and swaps with barriers within the LMM and SMM frameworks. The key is to model the joint distribution of the rate on each future payment date and the maximum or minimum level the forward may reach over the time period up to that date. Monte Carlo simulation confirms that the closed-form equations provide very accurate pricing.