Before revision:  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.
After revision:  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 }. It coincides for "calm" share prices with the Black-Scholes-Merton (B-S-M)price after translation.