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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: \begin{equation}\label{ref5} dr_t^{\mathbb{Q}} = \theta^*(\kappa^* - r_t^{\mathbb{Q}})dt + \sigma \sqrt{r_t^{\mathbb{Q}}}dW_t^{\mathbb{Q}} \end{equation}
with new transformed parameters $\theta^*$ and $m^*$, which are defined by \begin{equation}\label{ref6} \theta^* = \theta + \lambda, \; \kappa^* = \frac{\theta \kappa}{\theta + \lambda} \end{equation} 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.

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  • $\begingroup$ Does this help ? $\endgroup$
    – Kurt G.
    May 9, 2022 at 0:14
  • $\begingroup$ 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. $\endgroup$
    – user53249
    May 9, 2022 at 6:53
  • $\begingroup$ 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 ? $\endgroup$
    – Kurt G.
    May 9, 2022 at 11:22
  • $\begingroup$ Yes, but the thing I am stuck with is how to find the zero-coupon bond price, I mean the real data. $\endgroup$
    – user53249
    May 10, 2022 at 12:34
  • $\begingroup$ 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 \begin{equation} P(t, T) = A(t, T) e^{-B(t, T )r_t} \end{equation} for $\lambda$ by using the nonlinear least square method. $\endgroup$
    – user53249
    May 10, 2022 at 12:34

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