(To be divided into sub-FAQs later!)

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.dWtaking 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).dWThe 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).dWBlack 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).dWSandmann 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

I think literature on interest rate models is very rich here, just remember only significative one - factor models in brief: 1. dr = mu dt + sigma dZ with mu and sigma constant Merton 1973 2. dr = alfa(iota - r) dt + sigma dZ with alfa, iota, sigma constants Vasicek 1977 3. dr = sigma r dZ with sigma constant Dothan 1978 4. dr = alfa (iota -r) dt + sigma r^1/2 dZ alfa, iota, sigma constants Cox Ingersoll Ross 1985 5. dr = iota(t) dt + sigma dZ, iota(t) function, sigma constant Ho-Lee 1986 6. dr = [b(t) + (sigma'(t)/sigma(t)) log (r)]rd(t) + sigma (t)r dZ continous BDT 1990 with b(t), sigma(t) functions 7. dr = [b(t) - a(t)r] dt + sigma (t) dZ a(t), b(t), sigma(t) functions Jamisdian,Hull/White 1990 8. = [ b(t) - a (t) logr] r dt + sigma (t)r dZ a(t),b(t), sigma(t) functions Hull White-Black-Karasinski

Those are the one-factor short rate models.You can make two factor models by adding two of these processes together, like two Vasicek processes...Of course, there are other models which try not just to model the short rate, but rather the term structure using a no-arbitrage argument on a market of zeros. The most popular of these is the Libor Market Model, but you may also see the string model and string forward model at UCLA...

Onother well known model is Ritchken & Sankarasubramanian or Cheyette model (or skew short rate model)which under some regularity conditions of the volatility structure one can make it Markovian (avoid path-dependency) in a bi-dimensional diffusion system.

A sub-FAQ: what is the simplest way to describe the differences between HJM, BGM/LMM and Longstaff & Schwartz's string market models?

what is the key strength of Cheyette model ?is that really used by all houss ?

Singleton and Dai have a good review paper on the term structure research

Hi, could anyone please provide a paper, which explains the cheyette model.a weblink would be very helpful. Thanks. Peter

- horacioaliaga
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Stochastic volatility Libor Market Model

- cosmologist
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My two cents on the topic for the beginners.I would strongly suggest to buy two books1) Brigo & Mercurio - Read the relevant part. You will notice that for exotic interest rate products which are complex and difficult to price, you would require a model which would adequately model the forwards. The libor Market Model would beat the most. There are two more models which are easier to implement and makes use of sensible maths which you can master in a couple of readings SABR and LGM(linear gaussian Markov ) , both by Pat Hagan. LGM and HW are two sides of the same coin with LGM being far more sophisticated. You don't need to write codes for it. Bloomberg has implemented LGM, i think. You need to understand the theory and the calibration part.2) If you would like to graduate to LIBOR with all its complexity, Buy another book Libor Market Model - Gatarek is one of the authors.I can bet that you won't need any more book if you can finish these two books and the two more models which I mentioned.I can bet that it will take a lot of time for a beginner to finish these two books. If you(the beginner) mange to finish the above stuffs, please call me. I will treat you to a nice lunch/dinner. Choice is yours.P.S. - I assume that the beginner started her financial mathematics journey with the BLACK BOOK( a book with the black cover). It had nice codes. I started with the first edition. there have been reprints and broken into two parts. If no, then i strongly suggest to borrow and read that book.

- cosmologist
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duplicate post deleted

Last edited by cosmologist on November 4th, 2008, 11:00 pm, edited 1 time in total.

It seems like lately some of the G7 ex-Japan might be racing to test the "BOJ model" of interest rates: bring the overnight rate to 0% and see what happens to the rest of the curve.

- Cuchulainn
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QuoteOriginally posted by: exotiqIt seems like lately some of the G7 ex-Japan might be racing to test the "BOJ model" of interest rates: bring the overnight rate to 0% and see what happens to the rest of the curve.What about the positive definite matrix whose diagonal term is sqrt(1.0 + r*delta_T) and when r = 0 it starts to oscillate? And no more convection terms! Can r become < 0?

Last edited by Cuchulainn on December 3rd, 2008, 11:00 pm, edited 1 time in total.

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- ViveLesMaths
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what about infinite dimensional noise models, like using 2nd order parabolic SPDE or even more generally stochastic differential equations in an (infinite dimensional) Hilbert space (like Rama Cont - Modeling Term Structure Dynamics: An Infinite Dimensional Approach) ?