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Home arrow Business & Finance arrow Frequently Asked Questions in Quantitative Finance

What is Calibration?

Short Answer

Calibration means choosing parameters in your model so that the theoretical prices for exchange-traded contracts output from your model match exactly, or as closely as possible, the market prices at an instant in time. In a sense it is the opposite of fitting parameters to historical time series. If you match prices exactly then you are eliminating arbitrage opportunities, and this is why it is popular.

Example

You have your favourite interest rate model, but you don't know how to decide what the parameters in the model should be. You realize that the bonds, swaps and swaptions markets are very liquid, and presumably very efficient. So you choose your parameters in the model so that your model's theoretical output for these simple instruments is the same as their market prices.

Long answer

Almost all financial models have some parameter(s) that can't be measured accurately. In the simplest non-trivial case, the Black-Scholes model, that parameter is volatility. If we can't measure that parameter how can we decide on its value? For if we don't have an idea of its value then the model is useless.

Two ways spring to mind. One is to use historical data, the other is to use today's price data.

Let's see the first method in action. Examine, perhaps, equity data to try to estimate what volatility is. The problem with that is that it is necessarily backward looking, using data from the past. This might not be relevant to the future. Another problem with this is that it might give prices that are inconsistent with the market. For example, you are interested in buying a certain option. You think volatility is 27%, so you use that number to price the option, and the price you get is $15. However, the market price of that option is $19. Are you still interested in buying it? You can either decide that the option is incorrectly priced or that your volatility estimate is wrong.

The other method is to assume, effectively, that there is information in the market prices of traded instruments. In the above example we ask what volatility must we put into a formula to get the 'correct' price of $19. We then use that number to price other instruments. In this case we have calibrated our model to an instantaneous snapshot of the market at one moment in time, rather than to any information from the past.

Calibration is common in all markets, but is usually more complicated than in the simple example above. Interest rate models may have dozens of parameters or even entire functions to be chosen by matching with the market.

Calibration can therefore often be time consuming. Calibration is an example of an inverse problem, in which we know the answer (the prices of simple contracts) and want to find the problem (the parameters). Inverse problems are notoriously difficult, for example being very sensitive to initial conditions.

Calibration can be misleading, since it suggests that your prices are correct. For example, if you calibrate a model to a set of vanilla contracts, and then calibrate a different model to the same set of vanillas, how do you know which model is better? Both correctly price vanillas today. But how will they perform tomorrow? Will you have to recalibrate? If you use the two different models to price an exotic contract how do you know which price to use? How do you know which gives the better hedge ratios? How will you even know whether you have made money or lost it?

References and Further Reading

Schoutens, W, Simons, E & Tistaert, J 2004 A perfect calibration! Now what? Wilmott magazine, March 66-78

What is Option Adjusted Spread?

Short answer

The Option Adjusted Spread (OAS) is the constant spread added to a forward or yield curve to match the market price of some complex instrument and the present value of all its cash flows.

Example

Analyses using Option Adjusted Spreads are common in

Mortgage-Backed Securities (MBS).

Long answer

We know from Jensen's Inequality that if there is any convexity (or optionality) together with randomness in a product and model then we have to be careful about not treating random quantities as deterministic, we may miss inherent value. In the case of the Mortgage-Backed Security we have two main sources of randomness, interest rates and prepayment. If we treat these two quantities as deterministic, saying that forward rates and prepayment rates are both fixed, then we will incorrectly value the contract, there will be additional value in the combination of randomness in these two quantities and convexity within the instrument. Treating them as deterministic will give the wrong value. Assuming that MBSs are valued better in the market than in this naive fashion, then there is bound to be a difference between the naive (deterministic) value and the market value. To allow for this one makes a parallel shift in rates and revalues the contract until the theoretical deterministic value and the market price match. The shift in the curve that ensured this is then the OAS. So Option Adjusted Spread just means the spread by which you have to adjust rates to allow for optionality (convexity).

There are problems with this analysis, however. It can be problematic when the instrument is not monotonic in the quantity that has been assumed deterministic. See bastard greeks.

References and Further Reading

Hull, JC 2006 Options, Futures and Other Derivatives, Pearson

 
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