Suppose you are interested in the value of an American option (in the case where the value is strictly larger than the european one, hence with dividends or so)Suppose I run a Monte Carlo simulation naively and generate a path of the asset and then pick the best day to exercise my American option and the value of this run (forget about interest rates for a moment). If I then take the average value (in the risk-neutral simulations) over those simulations. Can you tell in advance how this value will relate to the actual correct one?I have currently no implementation ready for the least-squares approach for MC, but since this seems like a silly question, I know people have an understanding of this. Will the answer be general, or does it depend on the actual payout (put, call or even exotic with american features)? Can you actually prove this than mathematically?I would be very happy to hear morethanks

QuoteOriginally posted by: pleoniSuppose you are interested in the value of an American option (in the case where the value is strictly larger than the european one, hence with dividends or so)Suppose I run a Monte Carlo simulation naively and generate a path of the asset and then pick the best day to exercise my American option and the value of this run (forget about interest rates for a moment). If I then take the average value (in the risk-neutral simulations) over those simulations. Can you tell in advance how this value will relate to the actual correct one?If I understand this correctly, you're going to take the maximum payoff on each realization and then average them.It sounds like you're pricing a lookback option

american option should be something like max_{tau} E[ f(S_{tau}}] where tau runs over the stopping times. You can't move the max inside the expectation to take E[ max_{tau} f(S_{tau})].