Assume I'm an investor that wants to sell exotic put options. No one else is selling my kind of put option, so I need to determine my own "Market Price" through Monte Carlo simulation. I know that by the law of one price, this should hold:
$$P_t = E^Q[P_t|\mathcal{F}_t] = E^P[P_t|\mathcal{F}_t]$$
In my risk neutral Monte Carlo valuation, I model my stock price as:
$$dS = rS_tdt + \sigma S_tdW_t$$
In my real world Monte Carlo valuation, I model my stock price as:
$$dS = \mu S_tdt + \sigma S_tdW_t$$
Just thinking about this intuitively though, the put option valued under my real world Monte Carlo simulation will be way cheaper than the put option under my risk neutral simulations, because the growth rate is so much higher. So what am I missing here? Am I wrong in my first statement, that expectation under the P and Q measures are equal, or am I formulating my second statements incorrectly?