Interest Rate Model

Earliest one-factor short-rate models:

-- Black (1976) and Rendleman and Bartter (1980) - with lognormally distributed short rate
dr = mu.rdt + sig.rdW

However, assumption of lognormality was immediately critisized as not able to capture mean-reverting
property of interest rates.

-- Vasicek (1977) - short rate following a normal mean-reverting process with constant parameters
dr = theta.(a - r)dt + sig.dW

Drawback: short rates can run negative.

-- Cox-Ingersoll-Ross (CIR) (1985) added square-root diffusion term to the Vasicek model
dr = theta.(a - r)dt + sig.sqrt(r)dW

r is then distributed as chi-square.


No-arbitrage models:
take initial term (and volatility) structure as inputs.

Ho and Lee (1986) pioneered an arbitrage-free lattice approach for IR models. They studied binomial version of:
dr = mu(t)dt + sig.dW
taking initial term structure of IRs as input.

Heath, Jarrow and Morton (HJM) (1990, 1992) extended the Ho and Lee model in three directions:
1) they chose forward rates as basic building blocks,
2) incorporated capability of continuous trading and
3) allowed for multiple factors.
The HJM model is also consistent with any volatility structure, taking it as input.

Dibvig (1988) studied the Ho and Lee model in the HJM framework for the case of two factors.

Hull and White (1990) 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:
dr = theta(t).(a(t) - r)dt + sig(t).dW
The model is popular in practice for it produces closed-form solutions for bond prices.

Black, Derman and Toy (1990) the mean-reverting behaviour of the short rate was for the first time combined with 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:
d(ln_r) = (a(t) + {sig'(t)/sig(t)}.ln_r)dt + sig(t).dW

Black and Karasinski (1991) rectified this shortcoming of the BDT model by allowing for independent parameters.
d(ln_r) = (a(t) - theta(t).ln_r)dt + sig(t).dW

Sandmann and Sondermann (1993) studied a general arbitrage-free model dynamically incorporating properties of both normal and lognormal models:
R = ln(1+r), dR = mu(t).Rdt + sig(t).RdW
(Thus, IR process can't run negative and doesn't explode.)

Brace, Gatarek and Musiela (1997) and Jamshidian (1997) (BGM/J) developed a unifying framework, the Market Model, based on HJM, for forward LIBOR rates. Due to the assumption of simple compounding of LIBOR rates (vs continuous compounding of fwd rates) the variance term can take the form:
sig(t,T) = g(t,T).L(t,T)
(in case of continuous compunding, this would result in an exploding process)
The approach is arbitrage-free and has closed-form solutions for European swaptions.
LIBOR rate proces might also be seen as a discrete approximation of fwd rate, due to:
L(t,T) --> f(t,T), as tenor --> 0

출처  : Wilmott