Modelling and Calibration:

Models:

This paragraph explains the types of models used in the finance industry for people who have no knowledge of quantitative finance.

The market conditions will make the modeller state assumptions that will lead to a partial differential equation.

In order to solve this partial differential equation (PDE), the modeller will select the appropriate methodology.

Roughly we can distinguish 3 main types of models:

· Closed formula models: for example Black and Scholes, Black 76, etc.

They have strong assumptions about volatilities and market behaviour. They are simple to implement and they imply a one to one relationship between volatility and prices (hence why traders usually quote options in vol terms).

· Monte Carlo Simulations:

Mainly used for equity complex derivative valuations. In particular when the payoff is path dependant (which means that it is a function of where the market was in different points in time). For example average options. The modeller can make some simple or complex assumptions about correlations and volatilities.

· Interest rate modelling:

When the modeller needs to value instruments that require a term structure, he uses this type of model, which is computer intensive and requires more advanced maths. There are a wide range of these models, short rate models, Libor models, etc. They are based on assuming the behaviour for the variable we are simulating; for example (Hull-White). The models are categorised by their family (i.e. underlying assumptions) and the number of factors they have. The number of factors is the number of stochastic variables.

This last category usually requires calibration.

This is a rough way of presenting financial models. Quants enjoy mixing Monte Carlo with interest rate modelling and other combinations of models.

Calibration:

This paragraph explains the types of models used in the finance industry for people who have no knowledge of calibration. Please note that these examples have been oversimplified.

If the modeller wants to value a Bermudan swaption and the he has decided to use a 1 factor Hull and White model. The short rate behaviour is therefore following a mean reverting process: .

In order to value his Bermudan swaption the modeller needs to estimate a, b and. One way of estimating those parameters is to “calibrate to the market data”. Basically, admitting that the chosen model is the correct model. One way of handling this is to admit that it should be able to value correctly European options as well as Bermudan ones. Therefore if we input the European swaptions volatilities, an optimisation process should give us the “best fits” for estimating a, b and that will output good proxies for the European swaptions volatilities we entered. The iterative method to do this is called calibration. You can measure how good the calibration is by calculating the error on European swaption prices. Every time the European swaption volatilities change, the model should be recalibrated.

Calibration

This example explains what calibration is.

Calibration is one of the more subtle and difficult processes in the day to day life of a modeller / trader. A model can “refuse” to calibrate certain days.

1. In practice, it is very unlikely that a modeller can recalibrate after each market move. However it is good practice to recalibrate every day.
2. The more factors, the harder it is to calibrate.
3. Some models are known to be difficult to calibrate; for example BGM is very unstable because of calibration.
4. Calibration can move valuations significantly.

5. There are two schools of thought on calibration:

· The modeller can calibrate each instrument with only the relevant European swaptions to the instrument. For example, if the Bermudan swaption is a 10 year one, the modeller could decide to only use European swaption volatilities up to 10 years. So they “tailor” the calibration to the instrument they are interested to value. This will give a better fit if they have more than one instrument to value, it may not give “consistent” valuations.

· The modeller can decide to calibrate all instruments on the “full volatility matrix”. This will probably give bigger errors than the previous methods but give consistent price and is recommended for finance purposes.