Closed Form Term Structure Derivatives in a Heath-Jarrow- Morton Model with Log-Normal Annually Compounded Interest Rates
Starting with observable annually compounded forward rates we derive a term structure model of interest rates. The model relies upon the assumption that a specific set of annually compounded forward rates is log-normally distributed. We derive solutions for interest rate caps and floors as well as puts and calls written on zero-coupon bonds. In particular, for caplets with payment periods of same length as the compounding period (in our paper we have chosen one year, but it could be as well three or six months with quarterly or biannual compounding) we obtain the same Black formula as often used by market practioners, however, without making the unrealistic assumption that forward rates are independent of the accummulation process. Moreover, the log-normal assumption is shown to be consistent with the Heath-Jarrow-Morton model for a specific choice of volatility.
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