A new method to obtain risk neutral probability, without stochastic calculus and price modeling, confirms the universal validity of Black-Scholes-Merton formula and volatility's role

Abstract: A new method is proposed to obtain the risk neutral probability of share prices without stochastic calculus and price modeling, via an embedding of the price return modeling problem in Le Cam's statistical experiments framework. Strategies-probabilities $P_{t_0,n}$ and $P_{T,n}$ are thus determined and used, respectively,for the trader selling the share's European call option at time $t_0$ and for the buyer who may exercise it in the future, at $T; \ n$ increases with the number of share's transactions in $[t_0,T].$ When the transaction times are dense in $[t_0,T]$ it is shown, with mild conditions, that under each of these probabilities $\log \frac{S_T}{S_{t_0}}$ has infinitely divisible distribution and in particular normal distribution for "calm" share; $S_t$ is the share's price at time $t.$ The price of the share's call is the limit of the expected values of the call's payoff under the translated $P_{t_0,n}.$ It coincides for "calm" share prices with the Black-Scholes-Merton formula with variance not necessarily proportional to $(T-t_0),$ thus confirming formula's universal validity without model assumptions. Additional results clarify volatility's role in the transaction and the behaviors of the trader and the buyer. Traders may use the pricing formulae after estimation of the unknown parameters.