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?
EDIT 1: 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?
- Historical fit: Which models could be used to calibrate the real-world evolution of an asset to its past behavior?
- 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 $r$?
EDIT 2: Per comments, I understand that in order to estimate real-world probabilities I need to use the expected return $\mu$ instead of the risk free rate $r$. In addition, the asset real-world evolution should be still calibrated to reproduce current market prices.
However, if I just use an estimated $\mu$ instead of $r$, but leave unchanged all other inputs, the new asset behavior might not be consistent with the market prices of contingent claims. In particular, if I take a traditional pricing model and the only modification that I make is to change $r$ by $\mu$, the asset evolution described by this model will (generally) lead to call prices above its current market price, and put prices below its market price.
Therefore, in addition of substituting $r$ by $\mu$, which other adjustment might be needed in order to properly estimate real-world probabilities.