SFB 303 Discussion Paper No. B - 094

Author: Sandmann, Klaus
Title: An Intertemporal Interest Rate Market Model: Complete Markets
Abstract: In 1973 F. Black and M. Scholes deduced their famous formula for the pricing of European-Call-Options. Since then there have been many important new developments in option pricing. Options are special examples of contingent claims. J. Michael Harrison and David M. Kreps (1983) analyzed the valuation of contingent claims and gave a characterization in terms of martingale measures. In the following by extending this J. M. Harrison and S. R. Pliska (1981,1983) opened up the possibility of using the martingale theory in option pricing.

This paper is based on the work of J. M. Harrison and D. M. Kreps and makes use of the whole technique developed by them. In contrast to their work it is intended to model a market which depends only on the interest rate structure. So interest rate is the only source of uncertainty and dynamic force in the market. In this framework the valuation of coupon paying debt securities is characterized in terms of martingale measures like in Harrison and Kreps. The first model assumes that there exists a known finite number of trading dates and coupons are paid at these dates. As coupon rates are not negative, but may be zero, it is assumed without loss of generality, that coupon payments take place at every trading date. The second model is a continuous time generalization, that is trading and coupon payments occur continuously. Again a martingale characterization is derived. This second part of the paper makes use of the results obtained by J. M. Harrison and S. R. Pliska (1981,1983) and transfers them to the interest rate market model described here. As a conclusion the market model is complete if and only if one equivalent martingale measure exists. With the result from Yor (1978) it follows that in this model the class of interest rate processes is limited to the Wiener and Poisson martingales.
Creation-Date: April 1988
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