14.2 Calibrating to interest rate term structure only


14.2 Calibrating to interest rate term structure only

As discussed in Chapter 8, the rate of mean reversion is a function of the short-term interest rate volatility. This implies that, by specifying the functional form of the time-dependent short-term interest rate volatility, we are simultaneously specifying the shape of the term structure of volatilities and vice versa. This may not be a desirable property, since future volatility term structures may be distorted , taking on unreasonable characteristics. For this reason many practitioners allow for a constant short-term interest rate volatility parameter when using the BDT model. This implies that the mean reversion speed is zero and the model is calibrated to the initial interest rate term structure only. A constant volatility parameter may also prove optimal in markets where a reliable term structure of implied interest rate volatilities is not available and historical volatilities are seen to be poor proxies for implied volatilities.

Letting ƒ ( t ) = ƒ where ƒ is constant, the continuous time short rate dynamics reduce to:

and from (8.17) in Chapter 8, the discrete time representation of the shortterm interest rate becomes:

Hence calibration of the binomial tree to market data reduces to finding u ( i ) at each time step. Given a value of ƒ we may calibrate the binomial tree such that market- observed discount bond prices are retrieved. Since the bond price is independent of ƒ , this gives no clue as to the correctness or otherwise of the chosen ƒ value. To ascertain the correctness of the ƒ parameter we need to make use of another set of security prices which depend on interest rate volatility (i.e. ƒ ).




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

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