# Estimating market price of interest rate risk under CIR model

My goal is to find the market price of risk associated with the interest rate under the CIR model whose stochastic differential equation under the physical measure is given: $$\begin{eqnarray}\label{ref3} dr^{\mathbb{P}}_t = \theta (\kappa - r_t)dt + \sigma \sqrt{r^{\mathbb{P}}_t}dW^{\mathbb{P}}_t, \end{eqnarray}$$ Applying the Girsanov´s theorem allows finding the dynamics of the CIR model under a risk-neutral measure, which is given by: $$$$\label{ref5} dr_t^{\mathbb{Q}} = \theta^*(\kappa^* - r_t^{\mathbb{Q}})dt + \sigma \sqrt{r_t^{\mathbb{Q}}}dW_t^{\mathbb{Q}}$$$$
with new transformed parameters $$\theta^*$$ and $$m^*$$, which are defined by $$$$\label{ref6} \theta^* = \theta + \lambda, \; \kappa^* = \frac{\theta \kappa}{\theta + \lambda}$$$$ where $$\lambda$$ is interpreted as the market price of risk. In order to find a value for $$\lambda$$, we first need to find the estimations of physical parameters by using historical data. Afterwards, we need to use zero-coupon bond prices to calibrate the model for $$\lambda$$.

My problem is how to use zero-coupon bond price to calibrate the model for $$\lambda$$ through estimated parameters from the previous stage. I can find the estimations for physical parameters, but I can not fully understand how to perform the last process in practice.

• Does this help ? May 9, 2022 at 0:14
• Thank you. I have already seen this paper, but I guessed it is not in simple language. However, I need to read it carefully. What I expect is something in <d-nb.info/1027388515/34>, pages 13-14. May 9, 2022 at 6:53
• In that paper you linked: S. Zeytun, A. Gupta, A Comparative Study of the Vasicek and the CIR Model of the Short Rate, on p. 13-14, the authors seem to do exactly what you wanted. Why the QSE question ? And why another paper ? May 9, 2022 at 11:22
• Yes, but the thing I am stuck with is how to find the zero-coupon bond price, I mean the real data. May 10, 2022 at 12:34
• For example, if I consider 3-month maturity US monthly or daily Treasury bill data, then I have to consider zero-coupon bonds with a maturity 3-month (I.e., US treasury securities). After estimating the model parameters by using, for example, MLE method, applied to the 3-month maturity US monthly or daily Treasury bill data, I need to calibrate the zero-coupon bond price, which under CIR model is of the form $$P(t, T) = A(t, T) e^{-B(t, T )r_t}$$ for $\lambda$ by using the nonlinear least square method. May 10, 2022 at 12:34