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**Black, Fischer, Piotr Karasinski, "***Bond and Option Pricing When Short Rates are Lognormal*", Financial Analysts Journal, Vol. 47, No. 4, (July-August 1991), pp. 52-59.
**Abstract:** This article describes a one-factor model for bond and option pricing that is based on the short-term interest rate and that allows the target rate, mean reversion and local volatility to vary deterministically through time. For any horizon, the distribution of possible short rates is lognormal, so the rate neither falls below zero nor reflects off a barrier at zero. A model like this allows one to match the yield curve, the volatility curve and the cap curve. Surprisingly, adding to future local volatility lowers the volatility curve. A conventional binary tree with probabilities of 0.5 but variable time spacing is used to value bonds and options. When the inputs are constant, the slope of the yield curve starts out positive and ends up negative, while its curvature shifts from negative to positive. Even when mean reversion is zero, the volatility curve has a negative slope. The differential cap curve rises steeply (through the effects of volatility) and then falls steeply (through the effects of discounting and a falling forward rate used as the strike price).
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