Interest rate models
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Spot rate models
Black (1976) and Rendleman and Bartter (1980)
Lognormally distributed short rate
.
Such lognormality is not able to capture the mean-reverting property of interest rates.
Vasicek (1977)
The short rate follows a mean-reverting process with constant parameters
.
However, in this model short rates can become negative.
Cox-Ingersoll-Ross (CIR) (1985)
They added a square-root diffusion term to the Vasicek model
.
is then distributed as chi-square.
No-arbitrage models
These take the initial term (and perhaps volatility) structure as inputs.
Ho and Lee (1986)
They pioneered an arbitrage-free lattice approach for interest rate models. They studied a binomial version of
.
In this 
is chosen to match the initial term structure.
Heath, Jarrow and Morton (HJM) (1990, 1992)
They extended the Ho and Lee model in three directions:
1) they chose forward rates as basic building blocks,
2) they incorporated capability of continuous trading and
3) they allowed for multiple factors.
The HJM model is also consistent with any volatility structure, taking it as input.
Dibvig (1988)
He studied the Ho and Lee model in the HJM framework for the case of two factors.
Hull and White (1990)
They extended the Vasicek model to fit both the current term structure and volatilities of interest rates. In their model the short rate follows a normal mean-reverting process with time-dependent parameters:
.
The model is popular in practice for it produces closed-form solutions for bond prices.
Black, Derman and Toy (1990)
They combined the mean-reverting behaviour of the short rate with the lognormal distribution. The major appeal of the model is the transparent calibration procedure to the yield and volatility curves. However, the cost for that is mutually dependent mean-reversion and volatility terms:
.
Black and Karasinski (1991)
They rectified this shortcoming of the BDT model by allowing for independent parameters.
.
Sandmann and Sondermann (1993)
They studied a general arbitrage-free model dynamically incorporating properties of both normal and lognormal models:
.
(Thus, the interest rate process can't become negative and doesn't explode.)
Brace, Gatarek and Musiela (1997) and Jamshidian (1997) (BGM/J)
They developed a unifying framework, the Market Model, based on HJM, for forward LIBOR rates. Due to the assumption of simple compounding of LIBOR rates (versus continuous compounding of forward rates) the variance term can take the form:
.
(In the case of continuous compunding, this would result in an exploding process.)
The approach is arbitrage-free and has closed-form solutions for European swaptions.
