However, the behavior of financial assets in the real-world might be substantially different to the evolution used in a risk-neutral context.
For instance, if I want to estimate the real-world probability of an equity asset reaching certain thresholds, which models and calibration techniques could be used?
In particular, some questions that may arise in the estimation of real-world probabilities are:
- Calibration: Should real-world probabilities be calibrated to current market prices or, alternative, historical data should be used for this type of estimation?
- No-arbitrage conditions: Could they be relaxed or they still play a role in the assessment of real-world probabilities?
- Expected returns: Assuming that I have already estimated the expected return of an asset $\mu$, how accurate would be a real-world estimation that combines a widely used evolution model (e.g. Geometric Brownian motion), with the use of $\mu$ instead of the risk free rate $r$?
Per comments, I understand that in order to estimate real-world probabilities:
- I should use expected returns instead of the risk-free rate.
- The asset evolution should still respect the no-arbitrage conditions (i.e: the real-world dynamics should still reproduce the current prices of vanilla options).
However, if I just use $\mu$ instead of $r$, but leave unchanged all other inputs, the estimated asset behavior might not be consistent with the market prices of contingent claims. In particular, if I just change $r$ by $\mu$ (with $\mu>r$) the underlying asset dynamics will lead to call prices above its current market price, and put prices below its market price.
Therefore, in addition to use expected returns, which other adjustment might be needed in order to properly estimate real-world probabilities?
Any papers or references regarding real-world estimation will be greatly appreciated.