Cox-Ingersoll-Ross Stochastic Clocks

In this post, we consider stochastic clocks whose instantaneous rate of activity follows a square root process. We present all intermediate steps in the derivation of the characteristic function of the corresponding time change. The corresponding steps closely resemble those in the derivation of the zero-coupon bond price in the Cox et al. (1985) model for the short-term interest rate.

In a previous post we introduced the general concept of stochastic clocks. Here, we consider the case where the instantaneous rate of activity process has the dynamics

where is a standard one-dimensional Brownian motion and are constants. The corresponding time-change is given by

Laplace Transform of the Time-Change

We fix a and start by computing the Laplace transform

When , we recognize this as the time price of a zero-coupon bond with maturity in in the Cox et al. (1985) model; see e.g. Musiela and Rutkowski (2005). Since solutions to stochastic differential equations have the Markov property, it follows that

where is the -algebra generated by . Now, since is adapted to the filtration , it follows that there exists a function such that .

Note that for , we have

It follows that the process defined as

is a -martingale. Using the Itō product rule, we find that its differential is given by

Here, we used that the first factor of is a process of bounded variation. Furthermore, we assume that so that the Itō formula can be applied. It follows from the martingale property that the drift term in the above differential has to vanish. We obtain the PDE

with terminal condition for all . We postulate a solution of the form

where

We also require that such that for all and the terminal condition is satisfied. Substituting into the PDE yields

Here, the second step follows from being strictly positive. Since this PDE has to hold for all values of , the term that multiplies it has to be equal to zero and we obtain the first ODE

with terminal condition . Setting this term equal to zero in the original PDE yields the second ODE

with terminal condition .

Solving the ODEs

The first ODE is of the Riccati type and can thus be simplified using the standard transformations for this class. We start by defining

such that

and get

We now set

such that

and get

or

This is a homogeneous second order linear ODE with constant coefficients and can be solved using standard methods. We note that has been fixed and make another substitution by defining such that with and . We get

The characteristic equation is

with roots

We thus have the general solution

with

and for some constants and to be determined. We obtian the solution to the Riccati ODE by back substituting

By the terminal condition , it follows that

Thus,

and

We get

To solve for , we first note that

which can be solved by integration

By the terminal condition , it follows that and we get

Putting Everything Together

Putting everything together, we find that the Laplace transform of the integrated time-change is given by

The characteristic function of is

where

This expression is equivalent to the ones given in Carr et al. (2003) and Schoutens (2003), who both define slightly differently.

References

Carr, Peter, Hélyette Geman, Dilip B. Madan, and Marc Yor (2003) “Stochastic Volatility for Lévy Processes”, Mathematical Finance, Vol. 13, No. 3, pp. 345-382

Cox, John C., Jonathan Ingersoll Jr., and Stephen A. Ross (1985) “A Theory of the Term Structure of Interest Rates”, Econometrica, VOl. 53, No. 2, pp. 385-407

Musiela, Marek and Marek Rutkowski (2005) Martingale Methods in Financial Modelling: Springer

Schoutens, Wim (2003) Lévy Processes in Finance: John Wiley & Sons