Calibration in Finance: All You Need to Know

What is Calibration?

Calibration is knows as selecting model parameters to ensure that the theoretical values for exchange-traded contracts that your model generates match the market prices at any given moment in time exactly or as closely as possible. It is comparable to the opposite of fitting historical time series parameters. It is common because you are removing opportunities for arbitrage if you match pricing exactly.

A Short Example

You know which interest rate model is your favourite, but you’re not sure how to set the model’s parameters. You see that the markets for bonds, swaps, and combinations of swaps are extremely liquid and probably highly efficient. Thus, you set up the model’s parameters such that, for these basic instruments, the theoretical output of your model equals their market values.

A Deep Dive into Calibration

Almost all financial models have one or more parameters that are impossible to measure with accuracy. Volatility is the parameter in the Black-Scholes model, which is the most basic non-trivial instance. How can we determine that parameter’s value if we are unable to measure it? For the model is meaningless if we don’t know what it is worth.

There are two obvious approaches. Using historical data is one option; using price data from today is the other.

Using Historical Data

Let’s observe the first approach in operation. Maybe look at equities data to get a sense of how volatile something is. That has the drawback of being inevitably retrograde, utilising historical data. This may not apply in the future. This could result in prices that are out of line with the market, which is another issue. For instance, you’d like to purchase a specific option. When you apply your estimated 27% volatility to price the option, you arrive at $15. Nevertheless, that option’s market value is $19.
Do you still want to purchase it? Either you determine that your estimate of volatility is off, or the option is mispriced.

Using Current Price

The alternative approach is to essentially presume that the market prices of traded instruments contain information. In the example above, we query what level of volatility we should add to a calculation in order to obtain the $19 “correct” price. After that, we price other instruments using that figure. Instead of using any historical data, we have in this instance calibrated our model using an instantaneous snapshot of the market at a certain point in time.

All markets need calibration, but it’s typically more intricate than in the straightforward example above. Dozens of parameters or even complete functions in interest rate models may be selected by comparing them to the market.

As a result, calibration is frequently time-consuming. An example of an inverse problem is calibration, where the problem is to determine the parameters, but we know the answer (the prices of simple contracts). For example, inverse issues are notoriously hard because of their extreme sensitivity to initial conditions.

Calibration may deceive you by implying that your pricing are accurate. How can you determine which model is superior, for instance, if you calibrate one model to a set of vanilla contracts and then another model to the same set of vanillas? Both pricing vanillas today accurately. How, though, will they fare tomorrow?
Must you adjust your calibration? How do you determine which price to use when pricing an exotic contract using the two distinct models? What is the best way to determine which hedging ratios are better? How will you even be able to determine if you have gained or lost money?

Related Readings

• Modern Portfolio Theory in Finance
• Arbitrage in Quantitative Finance: All You Need To Know
• Modelling Approaches in Quantitative Finance: All You Need To Know
• Put-Call Parity: All You Need To Know