Stock prices will rarely follow the assumed model but, when stock's transaction times are dense in the interval t_0,T), they determine risk neutral probability (-ies) {\cal P}^* for the stock  price at time T . The remaining available risk neutral probabilities at T correspond to stock prices with different jumps-variability . The findings indicate that {\cal P}^* may be a mixture. The necessary and suficient condition used to obtain {\cal P}^* is related with the flow of information and concepts in Market Manipulation; it contributes in understanding the  relation between market  informational efficiency and the arbitrage-free option pricing methodology.  {\cal P}^*-price C for the stock's European call option expiring at T is also obtained. For "calm" stock prices, C coincides with the Black-Scholes-Merton price and confirms its universal validity without stock price modeling assumptions. Additional results for calm stock : a) show that volatility's role is fundamental in the call's transaction, b) clarify the behaviors of the trader and the call's buyerand c) confirm quantitatively that the buyer 's price carries an exponentially increasing volatility  premium.
The price of a stock will rarely follow the assumed model and a curious investor or a Regulatory Authority may wish to obtain a probability model the prices support. A risk neutral probability {\cal P}^* for the stock 's price at time T is determined in closed form from the prices before T without assuming a price model . The findings indicate that {\cal P}^* may be a mixture. Under mild conditions on the prices the necessary and sufficient condition to obtain {\cal P}^* is the coincidence at T of the stock price ranges assumed by the stock's trader and buyer. This result clarifies the relation between market 's informational efficiency and the arbitrage-free option pricing methodology. It also shows that in an incomplete market there are risk neutral probabilities not supported by each stock and their use can be limited. {\cal P}^*-price C for the stock's European call option expiring at T is obtained. Among other results it is shown for "calm" prices, like the log-normal, that i) C is the Black-Scholes-Merton price thus confirming its validity for various stock prices, ii) the buyer 's price carries an exponentially increasing volatility premium and its difference with C provides a measure of the market risk premium.
R
Stock prices
The price of a stock
D
but, when stock's transaction times are dense in the interval
R
t_0,T), they determine
and a curious investor or a Regulatory Authority may wish to obtain a probability model the prices support. A
D
(-ies)
A
's
R
. The remaining available risk neutral probabilities at T correspond to stock prices with different jumps-variability
is determined in closed form from the prices before T without assuming a price model
R
The necessary and suficient condition used
Under mild conditions on the prices the necessary and sufficient condition
D
related with the flow of information and concepts in Market Manipulation; it contributes in understanding
A
coincidence at T of the stock price ranges assumed by the stock's trader and buyer. This result clarifies the
A
's
A
It also shows that in an incomplete market there are risk neutral probabilities not supported by each stock and their use can be limited.
R
also obtained. For
obtained. Among other results it is shown for
R
stock prices, C coincides with
prices, like the log-normal, that i) C is
R
and confirms its universal validity without stock price modeling assumptions. Additional results for calm stock : a) show that volatility's role is fundamental in the call's transaction, b) clarify the behaviors of the trader and the call's buyerand c) confirm quantitatively that the buyer
thus confirming its validity for various stock prices, ii) the buyer
A
premium and its difference with C provides a measure of the market risk