I want to extract CIR parameters from monthly LIBOR data in the EULER-MARYAMA method in MATLAB language. I found the data but I can't extract parameters from it. What is the process? What is the formula?


As You Know, CIR model is the square root process given by the following stochastic differential equation $$d{{r}_{t}}=\kappa (\theta -{{r}_{t}})dt+\sigma \sqrt{{{r}_{t}}}d{{W}_{t}}$$ Let $\Theta=(\kappa,\theta,\sigma)$. It is well-known that conditional on a realized value of $r_t$, the random variable $2c_t\,r_{t+\Delta t}$ follows a non-central chi-square distribution with $d = 4\kappa\theta/\sigma^2$ degrees of freedom and non-centrality parameter $2c_t\,r_te^{−κ\Delta t}$, where $$c_t=\frac{2\kappa }{{{\sigma }^{2}}\,[1-{{e}^{-\kappa \Delta t}}]}$$ Indeed the density of $r_{t+\Delta t}$ is $$P(r_{t+\Delta t}|r_t;\Theta)=c\,e^{-u-v}(\frac{u}{v})^{\frac{q}{2}}I_q(2\sqrt{uv})$$ where

$$\,\,\,u_t=c_t\,r_te^{−κ\Delta t}$$ $$v_t=c_t\,r_{t+\Delta t}$$ $$\,\,\,\,\,\,\,q=\frac{2\kappa\theta}{\sigma^2}-1$$ and $I_q(2\sqrt{uv})$ is modified Bessel function of the first kind and of order $q$. The transition density has been originally derived in this.

Parameter estimation is carried out on interest rate time series with N observations We consider equally spaced observations with $\Delta t$ time. The likelihood function for interest rate time series with $N$ observations is step $$L(\Theta )=\prod\limits_{i=1}^{N-1}{P({{t}_{t+\Delta t}}}|\,{{r}_{t}}\,;\,\Theta )$$ It is computationally convenient to work with the log-likelihood function $$\ln L(\Theta )=\sum\limits_{i=1}^{N-1}{\ln P({{t}_{t+\Delta t}}}\,|{{r}_{t}}\,;\,\Theta )$$ from which we easily derive the log-likelihood function of the CIR process $$\ln L(\Theta )=(N-1)\ln c+\sum\limits_{i=1}^{N-1}{\left( -{{u}_{{{t}_{i}}}}-{{v}_{{{t}_{i}}}}+\frac{1}{2}q\,\ln \left( \frac{{{v}_{{{t}_{i+1}}}}}{{{u}_{{{t}_{i}}}}} \right)+\ln {{I}_{q}}(\sqrt{2{{u}_{{{t}_{i}}}}{{v}_{{{t}_{i+1}}}}} \right)}$$ You can find maximum likelihood estimates $\widehat{\Theta }$ of parameter vector $\Theta$ by maximizing the log-likelihood function last equation over its parameter space: $$\widehat{\Theta }=arg\,\underset{\Theta }{\mathop{max}}\,\,\ln \,L(\Theta )$$

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.