CIR Interest-Rate Dynamics and Their Closed-Form Results

Summary

The document discusses the Cox–Ingersoll–Ross short-rate process, which combines mean-reverting drift with volatility that scales with the square root of the rate. It responds to a question about converting that process to standard Brownian motion and notes that the model does not have a closed-form solution for the rate path. An integrating-factor representation expresses the rate using a stochastic integral whose integrand still depends on the rate itself.

The response points to other tractable results: the conditional rate distribution is noncentral chi-squared, conditional moments can be computed, and closed-form prices are available for zero-coupon bonds and European options on them. These results distinguish pathwise solution from distributional and pricing formulas. The document offers no derivation, parameter conditions, or worked example, and its displayed expression should be checked carefully before reuse because its parameter notation does not match the process as stated.

Key ideas

• The CIR process has mean-reverting drift and rate-dependent volatility proportional to the square root of the rate.
• An integrating-factor expression still contains a stochastic integral driven by the rate process.
• The conditional rate distribution is noncentral chi-squared.
• Conditional moments and prices for zero-coupon bonds and European bond options have closed-form results.

Tags

# Cox Ingersoll Ross (1985) Model


# Cox Ingersoll Ross (1985) Model












How can I convert the following process to a standard Brownian Motion?

$$\mathrm{d}r_t=(a-br_t)\mathrm{d}t+\sigma\sqrt{r_t}\mathrm{d}W_t$$

## Answer by Kevin (score 2)

https://quant.stackexchange.com/a/51113

I am not quite sure I get your question. You cannot solve the model in closed-form. What you get is that \begin{align*} r_t=r_0e^{-a t}+\frac{b}{a}(1- e^{-a t})+\sigma e^{-at}\int_0^te^{a u} \sqrt{r_u}\mathrm{d}W_u. \end{align*} Furthermore, you can get that $r_t$ follows a (non-central) chi-squared distribution and can compute the (conditional) moments of $r_t$ and obtain closed-form solutions for the prices of zero-coupon bonds and European-style zero-coupon bond options.

You may want to have a look at "Interest rate models" from Brigo and Mercurio which is an excellent reference.

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