# Which interest rate model for which product

Given the multitude of existing interest rate models (ranging from simple to very complex) it would be interesting to know when the additional complexity actually makes sense.

The models I have in mind:

• Simple one factor (e.g. HW)
• Two factor models
• Two or one factor models with stochastic volatility
• LIBOR-Market, HJM, SABR and SABR-LMM model

Are there any rules of thumb to decide which model to use for which product? (Perhaps there is some book dealing with that topic that I am not aware of)

Edit 22.02.2014:

While trying to answer the question myself I found the follwing very interesting paper on the emperical comparison of interest rate models. Here the authors mainly compare how well the different models can fit market data and hit the relevant market pries after being calibrated.

Thus a follow up Question: (that is also related to the question on model validation)

Does it suffice to hit the market pries spot on after calibration for a model to
qualify for being used in pricing for an instrument ? Or are there other aspects to be
considered? (computational speed, statistical fittness, robustness of the hedges)


Let's assume my model fits the market data really well - backtesting however shows that the hedges it provides don't work that well. Also the model might not be able to statistially fit the path of the underlying. E.g. mean reversion can be observed in some markets but not in others. One could argue that risk neutrality does not necessarily entail meaningful real-world scenarios etc.

I have good theoretical grasp of the models but have mainly used them for risk management (thus generating paths and analysing what happens to a portfolio or the balance sheet of an enterprise)

• I look forward to hearing from someone with more front office experience than me, but the rule of thumb as I understand it is that one must use a model that can capture the dynamics and risks to which your product is exposed. So, a one-factor model is fine for a cap, since the cap is basically exposed to the risk of the level of the yield curve, and that's it. However, it's not sufficient for a spread option, since the dynamics of the spread can't be adequately modeled with one factor. I, too, would appreciate a more rigorous reference for this though! – experquisite Feb 20 '14 at 15:47
• Still your explanation is a good start. I work alot with interest rate models in Risk-Mangement but here one is mostly interested in the "real-world"-measure - whatever that might be. Most books I know only introduce you to the instruments and don't give much advice when to use them. – Probilitator Feb 20 '14 at 17:19
• I have been googling for a while now - can't seem to find anything substential. The text book give me the tools but not the manual to use them ... – Probilitator Feb 22 '14 at 18:26
• One criteria could be that the model produce realistic values. Hull-White for instance can lead to negative interest rates. – klon Mar 5 '14 at 12:52
• The model to use depends on the purpose of the pricing. If you're long and short caps at different maturities, you need a model that captures the term structure of volatilities. If you are trading out of money vols, then you need a model that captures the volatility vs strike dynamics. In reality the risk management applies to the whole portfolio, thus you can say models should be most complicated to capture the dynamics between the different factors. Though there is a trade-off of the model risk. – adam Mar 8 '14 at 13:12

The model of choice depends on the purpose of the exercise. In general there are two types of models:

1. Equilibrium models: These are general used use for "fitting" the spot curve to the discount function available in the market. So different models will give you different yield curves. One can use this information to see the relative value of implementing a strategy. Examples are Vasicek two factor model. Since the yield curve produced by these models will not be same as the actual yield curve hence they are usually not used for derivatives pricing (you won't get exact market prices of bonds using a calibrated model here).

2. Risk - Neutral models: These are the second class of models. They fit the current yield curve exactly and are used to price derivative securities like caps/floors/swaptions etc. Libor Market Models would fall in this category. Example, Black-Derman-Toy, String model (Longstaff-Schwartz et al) etc. You can use BDT to construct binomial trees and get the price of the derivative security. String model can be used to simulate yield curves using Monte Carlo simulation and price securities in the process.

On factors:

As per literature some researchers have used Principal Component Analysis on yield curve data and found that for yield curve, almost all of variation can be captured using 4 principal components: 1. Level (parallel shift) 2. Slope (tilt) 3. Curvature 4. Money Market Factor (short end)

Some complex models use some/all of these factors to model the yield curves.

• Well, it still doesn't really answer the question. – SmallChess Aug 21 '14 at 4:25
• @Taran could you elaborate further on examples between 1) and 2) models and how to spot them – Permian Sep 8 '18 at 22:43

Here is a list of model attributes that are necessary for a derivatives model to qualify for use in pricing and hedging: 1) exact fit to liquid yield curve inputs and good interpolation scheme in between 2) good fit to relevant volatility inputs. (if a product has exposure to volatility points all through the grid, then the model needs to fit well everywhere) 3) good fit to the market skew, if relevant for the product 4) computation time is important if the portfolio is large 5) generates reasonable hedges - note that the choice for the skew dynamic heavily inputs what hedges are generated 6) the implementation is user friendly That's just a few