Explain the treatment of the dynamics of the volatility term for the Vasicek interest rate model
What will be an ideal response?
Let us begin by noting that there have been several formulations of the dynamics of the volatility term. If volatility is not assumed to depend on time, then ?(r,t) = ?(r). In general, the dynamics of the volatility term can be specified as follows:
?r?dz
where ? is equal to the constant elasticity of variance and z is a random term with dz denoting a random process. The above equation is called the constant elasticity of variance model (CEV model). The CEV model allows us to distinguish between the different specifications of the dynamics of the volatility term for the various interest-rate models suggested by researchers.
For the Vasicek interest rate model, we look at the case for ? = 0 . Substituting this value for ? into the above equation, we get the following model identified by Vasicek who first proposed it:
? = 0: ?(r,t) = ? (Vasicek specification of CEV model).
In the Vasicek specification, volatility is independent of the level of the short rate as in the equation of dr = bdt + ?dz (where b is the drift term) and is referred to as the normal model. In the normal model, it is possible for negative interest rates to be generated.
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