Is the gaussian the real representation of extrinsic value derived from Black and Scholes? Should it be lognormal?
From an idea of quantpie I took the following graph: It takes the lognormal of an underlying with an expected value of 10 (40% volatility and r=0) and divides the x-axis into intervals, every interval with its probability. The total summation of every single product gives 10, i.d. the expected value. Then it chooses a strike K, at the money. The summation truncated at strike K is the N(d1)*S of BSM, the strike K multiplied by the same probability is the N(d2)*K of BSM, and the result is the option value, which can be defined as an "expected intrinsic value". With the strike moving OTM the histograms decrease following a lognormal shaped tail. What makes me think is that, from ATM to deep OTM, the extrinsic value of an option follows a lognormal, but from ATM to deep ITM what kind of distribution do we have?
P.s. I hope I made it clear, I'm from Italy and I write very seldom in English.