A COMMON MARKET MEASURE FOR LIBOR AND PRICING CAPS, FLOORS AND SWAPS IN A FIELD THEORY OF FORWARD INTEREST RATES

Abstract

The main result of this paper is that a martingale evolution can be chosen for LIBOR such that, by appropriately fixing the drift, all LIBOR interest rates have a common market measure. LIBOR is described using a quantum field theory model, and a common measure is seen to emerge naturally for such models. To elaborate how the martingale for the LIBOR belongs to the general class of numeraires for the forward interest rates, two other numeraires are considered, namely the money market measure that makes the evolution of the zero coupon bonds a martingale, and the forward measure for which the forward bond price is a martingale. The price of an interest rate cap is computed for all three numeraires, and is shown to be numeraire invariant. Put-call parity is discussed in some detail and shown to emerge due to some nontrivial properties of the numeraires. Some properties of swaps, and their relation to caps and floors, are briefly discussed.