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The Black Model and the Pricing of Options on Assets, Futures and Interest Rates

We establish a general necessary and sufficient for the Black model to correctly price, to underprice, or to overprice European-style options. The condition for the Black model to hold is that the product of the asset price probability distribution and the pricing kernel is lognormal. This condition is applied to value stock options in particular cases where the pricing kernel exhibits declining or increasing elasticity, options on bonds where bond prices are lognormal, and options on interest rates where interest rates follow a Miltersen, Sandmann and Sondermann (1997)-Brace, Gatarek and Musiela (1997) process. We show that the Libor Market Model is just a special case of this class of Black models.

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