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The Black 76 model is one of the standard models for interest rate derivatives like pricing caps, floors, swaptions, etc.

The Black 76 model is given as $$dF_t = \sigma F_t dW_t$$ so it models the dynamics of the forward rate $F_t$ which implies a certain term structure. Why is the Black 76 model not considered an interest rate model (like Vasicek) in the literature even though it is used for pricing interest rate derivatives?

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  • $\begingroup$ I'd say that it is considered an interest rate model. What make you think that it is not? $\endgroup$ – MarinD Aug 22 '16 at 21:16
  • $\begingroup$ E.g. in Hull's Options, Futures, and other Derivatives, 9e it's not mentioned in chapter 31 (Interest Rate Derivatives: Models of the Short Rate) at all, nor is Black-Scholes. Ok, it doesn't model the rate directly, but the forward. But I'm wondering why it's not mentioned in literature as model for the term structure as you can describe the term structure by rates, discount factors or forwards. $\endgroup$ – dnl Aug 22 '16 at 21:38
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Black's model for interest rate derivatives is a perfectly acceptable starting model for pricing interest rate derivatives in a forward measure using a lognormal distribution assumption with just one time horizon.

The only caveat is that a model that evolves the forward rate curve through time (in order to price more complex derivatives than a cap/floor) needs a drift adjustment to ensure that the model is arbitrage free. How this should be done was not made clear until the later work of Heath, Jarrow and Morton in about 1990. Hence such models, and especially the lognormal forward rate version, are now classified as HJM models.

In the real-world, Black's forward rate model is now only used as a translator to convert cap/floor prices to implied volatility and back. Most dealers price and risk-manage using models which take the interest rate volatility skew into account such as SABR and stochastic/local volatility models.

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