Higher Order Forward Rate Agreements and the Smoothness of the Term Structure

Abstract

This paper proposes linear higher order conditions on the term structure that allow to compute valuation bounds for any deterministic cash stream. Starting from bounds on the forward rate curve and its derivatives, which are nonlinear in the discount factors, we derive linear conditions that are only slightly less restrictive than the nonlinear conditions. The linearization of the term structure constraints has two advantages. First, the valuation bounds can be computed by highly developed LP solvers. Second, the constraints have an economic meaning as auxiliary cash streams. Thus, price discrepancies can be easily translated into profitable trading strategies. Depending on the choice of the constraints on the forward rate curve the valuation bounds for cash streams can be very wide or very close. Arbitrage bounds are a special case of our general valuation bounds. On the other end of the extreme, the valuation bounds on the term structure itself behave like quadratic splines in the forward rate curve if the third order parameters are chosen in a restrictive way. The higher order conditions on the term structure are related to extremal event statistics of short-term interest rates. This puts the resulting valuation bounds conceptually (albeit not mathematically) close to methods like value at risk. Methods that calibrate a single price system to observed prices abound. Needed are valuation bounds that are based solely on economic assumptions. The valuation bounds under the linear higher order conditions on the term structure generalize arbitrage bounds and provide a sharper method when the arbitrage principle is too weak.