5.2 Choice of generating process


5.2 Choice of generating process

There exists empirical evidence to support both the random walk and the elastic random walk as generating processes for stochastic factors within a macro-economic system. Therefore we may conclude that the generating process for the short- term interest rate is adequately described by:

where a ( t ), b ( t ) and ƒ ( t ) are either constants or functions of time. a ( t )+ b ( t ) r ( t ) is the stochastic [1] instantaneous drift and ƒ ( t ) the deterministic instantaneous variance of r ( t ). The behaviour of r ( t ) is determined by the value of b ( t ) since, for:

  • b < 0, r tends to ˆ’

  • b = 0, the generating process for r simplifies to a random walk,

  • b > 0, r explodes in finite time since it is repelled by the level ˆ’ .

Under the random walk generating process, short-term interest rates drift to positive and negative infinity with probability one. The elastic random walk with b < 0 eliminates this problem. It does, however, allow transient occurrences of negative interest rates and hence is not an appropriate model when short-term interest rates are close to zero. Negative interest rates are completely eliminated by setting the variance coefficient proportional to r ± , ± > 0. In the case of the Cox, Ingersoll and Ross model [ 18 ], ± = ½ (see Chapter 2). This creates a natural reflecting barrier at r = 0, but introduces mathematical complexity which is difficult to implement in the multivariate case where the underlying factors are stochastic. Langetieg makes use of an elastic random walk process, with the assumption that the short-term interest rate is sufficiently above zero to make the probability of negative interest rates, in finite time, negligible.

[1] It is stochastic due to the functional dependence on r ( t ).




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

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