Abstract
The relation between CKLS model and CIR model will be investigatedin this paper. It will be shown that under a suitabletransformation, any CKLS model of order $\frac{1}{2} 1$ corresponds to a CIR model under a new probabilityspace. Moreover, the explicit solution and the precise distributionof the CKLS model at any time $t$ are obtained under the newprobability measure. The moment estimation of CKLS model will be given finally.
Highlights
Suppose that (Ω, F, P; (Ft)t≥0) is a right continuous filtered probability space satisfying the usual conditions
The relation between CKLS model and CIR model will be investigated in this paper
We can not solve equation (1) explicitly except for some special cases (e.g., γ = 1), but by a suitable measure transform, we can get the explicit solution of our CKLS model under the new probability space
Summary
Suppose that (Ω, F , P; (Ft)t≥0) is a right continuous filtered probability space satisfying the usual conditions. In 1992, Chan, Karolyi, Longstaff and Sanders (Chan, Karolyi, Longstaff & Sanders, 1992) suggested modelling the behavior of the instantaneous interest rate by the following stochastic differential equation drt = (a − brt)dt + σrtγdBt, This so called CKLS model is a very important model in both theory and application, it contains many important models in finance. We can not solve equation (1) explicitly except for some special cases (e.g., γ = 1), but by a suitable measure transform, we can get the explicit solution of our CKLS model under the new probability space. Under the probability measure Q, the precise density function of the solution rt of equation (1) is gt(x) = gδ,ζ( f (x)/L)| f ′(x)|/L, ∀x > 0, where f is the same as in Theorem 1, gδ,ζ is the density of the non-central chi-square law with δ degrees of freedom and parameter ζ, δ. We give the moment estimation of CKLS model, which is our Theorem 4
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