9.1 The lognormality assumption


9.1 The lognormality assumption

Ideally one wants a process for the short- term interest rate such that negative interest rates are prevented, but the zero level may be reached and maintained for extended periods of time. None of the processes examined thus far, i.e. normal, lognormal and square root processes, satisfy both these requirements. A lognormal process does not admit a zero interest rate, while the square root process makes the zero level a reflecting barrier .

BK use a lognormal process. A lognormal distribution is fully described by its mean and variance, which are functions of time, so we have a different lognormal distribution of the short-term interest rate at each future time. When mean reversion is combined with a lognormal model, we have three time-dependent factors, an example being the BDT model:

However, here ( t ) is a function of ƒ ( t ). Dropping this functional dependence, and letting ¼ ( t ) be the target interest rate, i.e. the reversion level, the BK model may be written as:

where ( t ) is the speed of the mean reversion and ƒ ( t ) the local volatility, i.e. the volatility of the short-term interest rate. BK calibrate their model to the initial observed interest rate and volatility term structures as well as the observed cap curve. The cap curve gives the prices of at-the-money caps, which pay the difference between the forward rate (strike) and the realised short-term interest rate at maturity. BK do not attempt to specify a process which accurately depicts the evolution of the short-term interest rate, but rather a short-term interest rate process which can be fitted to observed market prices and hence used to price securities in a consistent manner. The future risk-neutral distribution of the short-term interest rate generated by the model is not the true distribution, but rather a distribution which leads to correct option prices.




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

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