Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates





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    • Miltersen is from Odense Universitet, Denmark. Sandmann is from Johannes-Gutenberg Universität, Mainz, and Sondermann is from Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany. An earlier version of this article, Sandmann, Sondermann, and Miltersen (1994), was presented at the Seventh Annual European Futures Research Symposium in Bonn and at Norwegian School of Economics and Business Administration, Bergen, Norway. This version of the article was presented at the Isaac Newton Institute at Cambridge University during the Financial Mathematics Program. We are grateful to an anonymous referee and to Simon Babbs, Darrell Duffie, Claus Munk, Sven Rady, Erik Schlögl, and the two CBOT discussants, Chris Veld and Frans de Roon, for comments and assistance. Financial support from the Danish Natural and Social Science Research Councils for the first author and from the Deutsche Forschungsgemeinschaft at Rheinische Friedrich-Wilhelms-Universität Bonn for all three authors is gratefully acknowledged.


We derive a unified model that gives closed form solutions for caps and floors written on interest rates as well as puts and calls written on zero-coupon bonds. The crucial assumption is that simple interest rates over a fixed finite period that matches the contract, which we want to price, are log-normally distributed. Moreover, this assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility.