The embedding of the stock price return modeling problem in Le Cam's statistical experiments framework suggests strategies-probabilities, obtained from the traded stock prices in the time interval t0,T%DIFDELCMD < ]%%% , for the agent selling the stock 's European call option at  t_0 and for the buyer who may exercise it at  T . The nature of these probabilities is justified by the slight dependence of stock returns and the weak Efficient Market Hypothesis (Fama,1965) . When the transaction times are dense in [ t0 ,T] it is shown, with mild conditions, that under each of these probabilities log S_T/S_{t_0 divisible distribution and in particular normal distribution for "calm" stock ; S_t is the stock 's price at time t. The price of the stock's European option is the expected cost of the agent at t_0 obtained using the distribution of log S_T/S _{t_0  }. For calm stock, the price coincides with the Black-Scholes-Merton (B-S-M)price after translation. The strike price determined by the agent does not give arbitrage opportunity to the buyer. Additional results clarify volatility's role in the buyer's behavior and establish a connection between European option pricing and Bayes risk. The results justify the extensive use of the B-S-M price and provide traders with a new tool .
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 %DIFDELCMD < ]%%% . Strategies-probabilities P_{t_0,n '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 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 .
R
The
A new method is proposed to obtain the risk neutral probability of share prices without stochastic calculus and price modeling, via an
D
stock
D
suggests strategies-probabilities, obtained from the traded stock prices in the time interval
D
t0,T
R
, for the agent selling the stock
. Strategies-probabilities P_{t_0,n
A
time
R
at
in the future, at T; \ n increases with the number of share's transactions in
A
t_0,
D
. The nature of these probabilities is justified by the slight dependence of stock returns and the weak Efficient Market Hypothesis (Fama,1965)
R
t0
t_0
R
log S_T/S_{t_0
\log \frac{S_T
R
stock
share
R
stock
share
R
stock's European option is the expected cost of the agent at t_0 obtained using the distribution of log S_T/S
share's call is the limit of the expected values of the call's payoff under the translated P
A
,n
R
For calm stock, the price coincides
It coincides for "calm" share prices
R
(B-S-M)price after translation. The strike price determined by the agent does not give arbitrage opportunity to the buyer.
formula with variance not necessarily proportional to (T-t_0), thus confirming formula's universal validity without model assumptions.
R
buyer's behavior and establish a connection between European option pricing and Bayes risk. The results justify the extensive use of
transaction and the behaviors of the trader and
R
B-S-M price and provide traders with a new tool
buyer. Traders may use the pricing formulae after estimation of the unknown parameters