Analytical Pricing of American Bond Options in the Heath-Jarrow-Morton Model
AbstractIn this paper we study the optimal stopping problem of pricing an American Put option on a Zero Coupon Bond (ZCB) in the Heath-Jarrow-Morton (HJM) framework for the forward interest rate. In particular we consider its Musiela's parametrization to guarantee a Markovian setting. Hence we are in an infinite dimensional setting, in which the forward rate curve is described by a SDE in a suitable Hilbert space. In order to find an infinite dimensional variational formulation of the pricing problem, we extend some results on infinite dimensional optimal stopping and variational inequalities recently obtained in . The proof goes through three main steps. First we regularize the American bond option's payoff by adopting usual smoothing arguments. Next we approximate the infinite dimensional dynamics by finite dimensional ones to which we associate suitable optimal stopping problems in R^n. Then, by taking the limit as n goes to infinity and by removing the smoothing on the payoff, we obtain an infinite dimensional variational inequality for the price of the American bond option. Moreover, the first time at which the price of the American bond option equals the payoff turns out to be an optimal exercise time.
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Bibliographic InfoPaper provided by arXiv.org in its series Papers with number 1212.0781.
Date of creation: Dec 2012
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Web page: http://arxiv.org/
This paper has been announced in the following NEP Reports:
- NEP-ALL-2012-12-10 (All new papers)
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- Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
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