Please correct me if any of my following statements are false.
My understanding as to why we use Risk Neutral Analysis is that it makes life easy, and ultimately, allows use to come to a closed form solution for pricing options. Using risk neutral pricing also requires us to construct a risk free portfolio by hedging away the risks of any stochastic processes we have.
With that said, a derivative's price is the present value of future expected payoffs. Let's say I want to play around with the assumptions behind the underlying process. The typical underlying process is:
dS = μSdt + σSdX where dX is equal to sqrt(dt) * norm(0,1)
In the Black Scholes model, μ is assumed to equal r, since we are in the risk neutral world. What if instead of the above, I want to create my own right hand side of the equation, so that:
dS = "custom function"
Obviously this means I need to estimate additional parameters, which is frowned upon by most people who believe in the efficient market hypothesis. But if I were to do that, and then use Monte Carlo simulation to construct the ending distribution of stock prices, and then calculate option prices based on that distribution, would my methodology still be sound? For example, for a vanilla call option, I just find the E[max(PV(S-K), 0] using my custom ending distribution.
My goal is to start with the BSM underlying process and have it produce option prices. Then, I want to slightly tweak the underlying process (like adding different factors, changing the stochastic term, etc...) to see how it affects option prices. I don't want to have to use risk neutral parameters since my math is not strong enough to form riskless portfolios. Just want to make sure this is a sound process and I'm not breaking any mathematics rules or major economic theories.