Calibration Methods

Prices of liquid financial instruments are given by the market and are determined by supply-and-demand. Calibrating a model means finding numerical values of its parameters such that the prices of market instruments computed within the model, at a given time, coincide with their market prices. Liquid market prices are thus actually used by models in the “reverse-engineering” mode that consists in calibrating a model to market prices. Once calibrated to the market, a model can be used for Greeking and/or for pricing more exotic claims (Greeking means computing risk sensitivities in order to set-up a related hedge). Calibration thus corresponds to estimation of a model. However, in finance the term “estimation” specifically refers to statistical estimation, i.e. estimation based on historical data by maximum likelihood or any other statistical procedure. Statistical estimation is thus backward looking, whereas calibration is forward looking, since derivative prices at the current time are based on the views of the market regarding the future dynamics of the underlyings. It is generally acknowledged that, whenever option data are available, it is better to use them to calibrate the model than to estimate a model statistically on past data. The simplest example of a calibration problem is encountered in Chap. 5 , where we discuss the notions of the implied volatility of an option and the implied correlation of a CDO tranche. In these cases the calibration problem is easy since there is only one parameter to “calibrate” to only one market quote. But can this really be called calibration? Well, it depends on what one wants to do. The exercise for the bank is, given a product it’s interested in, to identify a number of risk factors, select a number of hedging instruments also responsive to these, and to devise a model consistent for everybody (derivative and its hedging assets, which in some cases can be derivatives themselves) such that the risk of the position can be monitored in this model. But wait! Where has the real world gone? The model only exists in our heads until it is calibrated to the market. The least requirement is that the current price of the derivative and its hedging assets in the model are consistent with the ones observed today in the market. For instance, if it is only about hedging a vanilla option with the underlying stock, calibration in the sense of fitting a Black–Scholes implied volatility to that option’s price may be enough. If there are other derivatives among the hedging assets, then it’s not only a matter of calibration to one price, but of co-calibration of the whole set of instruments in use. Moreover (co-)calibration at a given time is only a first step, which can always be achieved in simple models such as local volatility models. These tell us that the volatility of a stock S is random (which it is), but only as a function of S. This would mean that options can be perfectly hedged by their underlying. Do you really believe in this? You shouldn’t, since it would simply contradict the existence of derivative markets (which would be useless if derivatives could be synthetized in terms of their underlyings). What we are missing here is the dynamics, namely we need a (co-)calibratable model with, in addition, the right dynamics, or the right Greeks. Now there is a “meta-theorem” in financial modeling stating that right Greeks are stable Greeks, namely Greeks which are stable when the model is recalibrated to the market every day. Which is a matter of calibration again, but across time (stability of the recalibrated parameters). So: calibration, co-calibration and re-calibration, the master equation of financial modeling!