Analytical pricing of American Put options on a Zero Coupon Bond in the Heath-Jarrow-Morton model

We study the optimal stopping problem of pricing an American Put option on a Zero Coupon Bond (ZCB) in Musiela's parametrization of the Heath-Jarrow-Morton (HJM) model for forward interest rates. First we show regularity properties of the price function by probabilistic methods. Then we find an infinite dimensional variational formulation of the pricing problem by approximating the original optimal stopping problem by finite dimensional ones, after a suitable smoothing of the payoff. As expected, the first time the price of the American bond option equals the payoff is shown to be optimal.