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.

  • $\begingroup$ These models are typically solved under the risk neutral measure, it is not necessary to specify the real measure that affects the market price of risk. But in principle this is the (constant) Sharpe ratio of the bonds under the real measure. This will be clearer if you do the pricing instead under the real measure by specifying a stochastic discount factor. $\endgroup$
    – fes
    May 19, 2022 at 18:53
  • $\begingroup$ Thank you for your reply. But I am still confused by the sentences written by that user. His idea for obtaining the market price of risk is very close to what I am thinking but not very clear. $\endgroup$
    – user53249
    May 20, 2022 at 11:33


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