The Dynamics of the Forward Interest Rate Curve with Stochastic String Shocks

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Abstract
This article offers a new class of models for the term structure of forward interest rates. We allow each instantaneous forward rate to be driven by a different stochastic shock, but constrain the shocks so that the forward rate curve is kept continuous. We term the shocks to the forward curve “stochastic string shocks,” and construct then as solutions to stochastic partial differential equations. This article offers a variety of parameterizations that can produce, with parsimony, any correlation pattern among forward rates of different maturities. We derive the no-arbitrage condition on the drift of forward rates shocked by stochastic strings and show how to price interest rate derivatives. Although derivatives can be easily priced, they can be perfectly hedged only by trading in an infinite number of bonds of all maturities. We show that the strings model is consistent with any panel dataset of bond prices, and does not require the addition of error terms in econometric models. Finally, we empirically calibrate some versions of the model and price the delivery option embedded in long bond futures. We show that the delivery option is much more valuable in string models than in a similar one-factor model. This is due to the greater variety of shapes of the term structure that string models can produce, which induces more changes in the cheapest-to-deliver bond.