11.5 The problem with forward rates


11.5 The problem with forward rates

By (11.32) we see that the forward rate process is completely specified by the volatility functions ƒ i ( ·, ·), i =1, , n . Consider a framework with one source of uncertainty and so one volatility parameter. For practical implementation it is desirable to apply a lognormal volatility structure for all forward rates [ 45 ]. This is because market prices of caps and swaptions assume a lognormal structure of forward rates. Hence set ƒ 1 ( t, T )= ƒ f ( t, T ), where ƒ > 0 is a constant. However, under this volatility structure (11.32) becomes:

Here, the drift of the forward rate grows as the square of the forward rate [ 49 ] and causes the forward rate to explode in finite time. Therefore for calibration purposes an upper bound needs to be imposed:

This problem is not particular to the HJM model, but rather a characteristic of all lognormal models of instantaneous forward rates.




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