# What is the purpose of short rate models?

Just venturing into quantitative finance and studying short rate models (Vasicek, CIR, Hull-White etc.). Wanted to ask a very simple intuitive question. How would a practitioner use these models? I understand that they are used to simulate the future price of the short rate because the time series generated by these models have properties similar to those observed in historical rate time series - by running Monte Carlos with these models you can construct a distribution for the path of the short rate. Fine. But you could also write an ARMA (or some variant including GARCH) model which has the same properties (mean reversion, known variance) and produced forecasts which are the same as the expectation of the short rate model. My questions are:

• Am I correct in thinking of the short rate models purely as an approximation/description tool and NOT as forecasting models (like VAR, for instance)? This seems intuitive as there is no information in short rate models except for the history of the time series.

• How would a practitioner use these models to do something useful?

Thanks!

• They are not forecasting models, they are used for the valuation of options, which require an estimate of how much interest rates can change in the future (i.e. the volatilty, not the direction, of future interest rates). These models are used to crank out alternative paths for interest rates in the future, for the valuation of option-like interest rate derivatives. The expected path is not of any great interest for this purpose, it is 'what range of things could plausibly happen'. May 7 '18 at 13:10
• As @AlexC has explained, short-rate models are used to price interest rate derivatives, not to predict the future path of interest rates. They are most heavily used for pricing vanilla derivatives that depend on a single rate, instead of the whole curve; they are also used for pricing hybrid derivatives (e.g. a derivative that depends on a stock price and an interest rate). May 7 '18 at 13:29
• @AlexC how do they model crank out alternate paths exactly??? Apr 21 at 22:05

Short rate models were first used in the 1970s and 1980s to try to fit and explain the term structure of interest rates - they went beyond simple parametric shapes (polynomials and exponential forms). They were not used for pricing as the fact that these short-rate models (Vasicek, CIR and Ho-Lee) had only two or three free parameters meant that they could not exactly fit the term structure of interest rates.

This lack of fit was not seen as a problem by their early users because what a short-rate model gave was a way to capture the relationship between the shape of the term structure of interest rates and the term structure of interest rate volatility. This was important if you wanted to understand the value of bond convexity.

A multifactor version of such a model might also be argued to have some "economic" properties that might make one think that it captures the relationship between various points on the curve and deviation from this curve may be considered to be a "mispricing". In the early 1990s, hedge funds such as LTCM used multifactor extensions of these models in this way and used them to put on massive "convergence trades". Some people still use them like this.

Concerning their use in derivative pricing, in the early 1990s, caps and floors and European style swaptions could all be priced using Black's model. However, for more exotic products, and also for these products, a more complete model was required. However these short-rate models did not refit the initial term structure of interest rates and so could not be used.

This problem was solved by Heath, Jarrow and Morton in the late eighties (published 1990) who showed how to construct a drift that would ensure a fit to the initial term structure of interest rates making it arbitrage-free. Although HJM is based on a forward curve, it is also possible to apply this to a short-rate model. So Hull and White showed how to do this to Vasicek's short-rate model, and others showed how to do this to other rate processes such as Black-Derman-Toy and Black Karasinski.

Although forward rate models such as BGM are now quite dominant, arbitrage-free short rate models still play a large role in derivative pricing. HW is still popular due to it having a fast analytical solution to the bond price. HW and others are also used for multi-callable products as they can be more easily implemented on binomial and trinomial trees than forward rate models which rely more on Monte Carlo techniques.

I might get down-voted for this, but in my opinion, short-rate models are not very useful for any practical pricing problems in today's finance. Even for simple vanilla rate derivatives (i.e. Caplet or Floorlet), the Libor Market Model framework (just focusing on one particular forward Libor rate) would be more useful and the preferred way to price.

Short rate models might only be used in Risk, to provide a range of possible future paths of interest rates: within the risk framework, one would simulate a distribution of potential future values of interest rates, then value a portfolio of derivatives on these paths (note that the pricing model would again not be a short-rate model, even if the paths are generated by a short-rate model).

Typically in risk, once you re-value your portfolio of derivatives on all simulated future interest rate paths, you'd be interested in a percentile of this distribution, say the 97.5th percentile. This could be used in counterparty credit risk, to monitor credit limits: the management might have set credit limits against every single counterparty that the bank trades with, and they want to be satisfied that with "97.5% probability, across all future interest rate paths, these credit limits won't be breached".

(Of course, the calibration of the short rate model used to simulate the future paths is subjective. The model choice is subjective also. Therefore the 97.5% probability of "comfort" is always "Bayesian": which you can be sure most management in banks won't fully comprehend).

• Aren't for example Bermudan swaptions priced still using HW 2-factor? Oct 7 '20 at 16:49
• Wouldn't these be priced using LMM, under American-type MC? Oct 7 '20 at 16:51
• Could be LMM used as well, but I've heard short rate models are used for these as well as more tractable. I suppose it depends on the shop. Probably you're better versed in IRDs than I am. Oct 7 '20 at 16:53
• @ilovevolatility: I doubt that :) But thank you for thinking that. Oct 7 '20 at 16:56

Long story short, the main reason of a short rate model is to provide an analytical solution for the zero coupon bond $$P(t, T)$$, given by the following expectation:

$$P(t, T) = E_t^Q \left[ \exp \left( - \int_t^T r(s) ds \right) \right].$$

Otherwise, when pricing interest rate derivatives using Monte Carlo simulations, you would have to perform Monte Carlo simulations on top Monte Carlo simulations, which is computationally prohibitive.

As said above, it is not a forecasting model.