12.7 Calibration to market volatilities


12.7 Calibration to market volatilities

One of the main advantages of the BGM model is that calibration no longer involves the translation of unobservable state variables (such as instantaneous spot and forward rates) into quantities observed in the market. The volatility parameter 2 ( t, T j ) in (12.31) may be directly calibrated to the observed Black volatilities. However 2 ( t, T j ) is the volatility for a single caplet, while quoted Black cap volatilities are in fact ˜average' volatilities over a series of caplets. In order to derive individual caplet volatilities from quoted Black cap volatilities, assumptions may need to be made about the relationships between individual caplet volatilities. These caplet volatilities, which represent volatilities of individual forward rates, are referred to as forward-forward volatilities.

In the derivation of the above pricing formulae, the volatility function ³ is n -dimensional, allowing for n volatility factors (sources of uncertainty). These factors can be identified from market data along similar lines detailed in Chapter 15 which details calibration of the HJM approach.

The inconsistency within market caplet and swaption prices remains. All market prices of caplets and swaptions are derived from a model which simultaneously assumes a lognormal distribution for the underlying remains. While the lognormality assumption is valid for any individual caplet or swaption, joint lognormality of all caplets and swaptions is impossible . The pricing impact of this internal inconsistency is likely to be small [ 45 ]. While one cannot exactly calibrate a single model to all caplet and swaption volatilities several optimised hybrid approaches have been suggested (e.g. [ 46 ]).




Interest Rate Modelling
Interest Rate Modelling (Finance and Capital Markets Series)
ISBN: 1403934703
EAN: 2147483647
Year: 2004
Pages: 132

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