Calibrating Market Price of Risk in a CIR Term-Structure Model

Summary

The document raises a fixed-income modeling question about how to infer the market price of risk in a Cox–Ingersoll–Ross short-rate framework. It describes a workflow in which historical short-rate data are used to estimate model parameters, after which the zero-coupon bond pricing formula and the current yield curve are used to calibrate the risk premium. The bond price is expressed in affine form, with time-dependent coefficients multiplying an exponential function of the current short rate.

The author asks whether calibration amounts to minimizing the difference between model-implied and market zero-coupon bond prices, and how yield-curve observations enter that process. No answer or calibration procedure is included, so the document does not establish an objective function, parameter-identification approach, or empirical result. It is useful as a statement of the distinction between estimating short-rate dynamics and matching risk-adjusted bond prices, but readers need additional material to implement the calibration.

Key ideas

• The CIR model expresses zero-coupon bond prices using time-dependent affine coefficients and the current short rate.
• Historical short-rate data can be used to estimate the model’s dynamic parameters.
• The yield curve contains market bond prices or yields that may inform risk-premium calibration.
• The document asks how to match model prices to market prices but does not provide a solution.

Tags

# short rate, yield curve and zero-coupon bond price formula under CIR mode: How to calibrate the market price of risk


# short rate, yield curve and zero-coupon bond price formula under CIR mode: How to calibrate the market price of risk












I recently read a document posted by a user in QF, who said that "In the past, I have calibrated simple short rate models to the term structure by using maximum likelihood to get the parameters of the Vasicek/CIR sde and then use the ZCB formula and the current yield curve to calibrate the market price of risk."

I can not fully understand how having a knowledge of ZCB formula and yield curve leads to finding the market price of risk. For example, if we are working under a CIR model for the short rate, we need to know the ZCB price formula, which has the form of $A(t, T) e^{-B(t, T)r_t}$. Then, where can I use the data on the yield curve to calibrate the market price of risk? Should I say that the market price of risk is obtained by minimizing the difference between the ZCB model price and ZCB market price? I still do not know where I should use yield curve.

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