# Is Vasicek risk neutral?

I am a bit new to this, and am trying to understand the concepts of the risk neutrality in interest-rate models.

What I can't seem to understand is why the Vasicek model is risk-neutral? Following some logic in Wilmott's books about risk drift and risk-neutral drift, the drift term ($dr = \mbox{drift} \times dt + \mbox{vol} \times dX$) has to be compensated with a $\mbox{lambda} \times \mbox{vol}$ factor, i.e. market price at risk times volatility. This is not the case for the Vasicek model.

I see that there is some big concept I am missing, and as I said I am new to this so anyone who can explain this to me, please, I would be very grateful.

Risk-neutrality isn't really a property of a model. Instead, it describes a certain calibration of a model (almost always represented by an SDE).

We say a model has been calibrated to risk-neutral probabilities if

• model parameters can be inferred from traded security prices, and
• there's some anti-arbitrage assumption and hedging scheme available for those traded security prices

The machinery is frequently abused to say that a risk-compensated model (with a nontrivial term for market price of risk) has been calibrated risk-neutrally. That's mainly because the math is all the same. Note also that even in the absence of hedging arguments, one can still often make large-$N$ portfolio arguments for security prices to all be priced on a consistent calibration.

Now, in the specific case of the Vasicek model, we can imagine calibrating its parameters to, say, swap and swaption prices. We won't match them all, but we'll do our best. And to the extent the model represents reality, any new securities we see in the market can be fairly priced using the model.

If we were instead calibrating the Vasicek model to a time series of overnight rates, our calibration would not be risk-neutral, and would be more appropriate for risk computations.

Finally, note that the short rate in the Vasicek (or similar) model is not an investable security. Market price of risk would not be applied directly to it.

• It's still not clear to me why calibrating means you get the risk neutral parameters. Is there anywhere else this is covered? Commented May 15, 2021 at 20:44

Vasicek model has parameters, which allow it to be calibrated to market prices (this means it becomes risk-neutral) or, if you'd like to, to history (and it becomes real-world model).

Example of calibration to history see here: http://www.sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model/

Me thoughts on calibration to market see here: http://guseynovrv.wordpress.com/2013/02/11/calibration-of-vasicek-model-part-iii-fitting-to-market/

UPD. Few words more: change of drift with market price of risk so that to obtain risk neutral process is possible with IR models, but is not practical. See example here: http://www.ressources-actuarielles.net/EXT/ISFA/1226.nsf/0/0daceb518d4ed890c12576fe00412e59/$FILE/MPR%20Ahmad-IS27v2.pdf That's why calibration is the main tool with IR models. • Your first sentence is in contradiction with the first link you provide, i.e., the Vasicek model is calibrated to historical data using the risk neutral dynamics. Commented Jul 27, 2015 at 20:21 • How can you be so certain that calibration to market prices will lead to being risk neutral? Commented May 2, 2021 at 20:03 Risk Aversion is a General Economics concept which is widely used in Finance. One example is that given a specific uncertain return$X_t$, a risk neutral investor would only care about the expected (average) return and hence would not require risk premia based on the volatility of that return$Var(X_t)\$.

A risk averse individual on the other hand would also care about the variation of the return and not only its expected value. As a result, he/she would require a premium that depends on that volatility.

I am not particularly familiar with the Vasicek paper that you refer to but most likely it has to do with the preferences of the agents when they evaluate their returns and that a risk premia for volatility of those returns is not required.