Calibrating the Market Price of Risk in a CIR Interest Rate Model

Summary

The document sets out a calibration question for the Cox-Ingersoll-Ross short-rate model. It begins with mean-reverting interest-rate dynamics under the physical probability measure and describes a change to risk-neutral dynamics in which the mean-reversion parameters are transformed while volatility retains its square-root form. The market price of risk is represented by a parameter that links the physical and risk-neutral specifications.

The proposed workflow is to estimate physical-measure parameters from historical rate data, then use zero-coupon bond prices to calibrate the market price of risk through the transformed risk-neutral parameters. The author asks how to carry out that second step in practice. No calibration procedure, bond-pricing equations, data example, or estimated parameter values are supplied, so the text frames the problem rather than solving it. Applying the idea would require specifying the model's bond-price relation and fitting it to observed maturities, with care about assumptions and whether the chosen parameterization is adequate for the data.

Key ideas

• The CIR model describes short rates with mean reversion and volatility proportional to the square root of the rate.
• A change from the physical to risk-neutral measure transforms the drift parameters.
• The proposed workflow estimates physical parameters from historical rate data and calibrates risk premia using bond prices.
• The document poses the calibration problem but does not provide the practical fitting method or an empirical example.

Tags

# Estimating market price of interest rate risk under CIR model


# 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: \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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.