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The typical representation of the extrinsic value of an option is the following: enter image description here

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: enter image description here 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.

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    $\begingroup$ Assuming that by extrinsic value derived from Black and Scholes you mean the difference between the BS-price and the intrinsic value $(S-e^{-rt}K)^+$. This is neither normal nor lognormal. It may look similar to your graph though. $\endgroup$
    – Kurt G.
    Commented May 26, 2022 at 16:08
  • $\begingroup$ Thank you very much. So the normal is a good approximation $\endgroup$
    – Giovanni Berardi
    Commented May 26, 2022 at 17:56
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    $\begingroup$ Have you tried how good that approximation is ? Looking roughly similar is all I get. $\endgroup$
    – Kurt G.
    Commented May 26, 2022 at 18:14
  • $\begingroup$ There is a square in the normal and none in @KurtG. formula, I say they are quite different. $\endgroup$
    – Bob Jansen ♦
    Commented May 26, 2022 at 18:37
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    $\begingroup$ @GiovanniBerardi . I voted to close the question. Because we have an explicit formula for extrinsic value in the BSM case I fail to see why we need to approximate it by some PDF. $\endgroup$
    – Kurt G.
    Commented May 27, 2022 at 19:00

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The formula for the BSM call price is well-known, hence, the exact formula for the "extrinsic value" (more commonly called time-value) $\text{EV}$ is well known: $$ \text{EV}=SN(d_1)-e^{-rt}KN(d_2)-(S-e^{-rt}K)^+\quad\text{ where }\quad d_{1,2}=\frac{\log(S/K)+rt\pm\sigma^2t/2}{\sigma\sqrt{t}}\,. $$ With $K=10,r=0,t=1,\sigma=40\%$ I fitted a normal and a lognormal density to the extrinsic value $\text{EV}$ to the best of my ability (the $x$-axis is the spot price $S$):

enter image description here

enter image description here

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    $\begingroup$ +1 It is clear from these graphs that the curve is quite different from the normal and log-normal distributions. $\endgroup$
    – nbbo2
    Commented May 27, 2022 at 20:27
  • $\begingroup$ @Kurt G. Thank you very much $\endgroup$
    – Giovanni Berardi
    Commented May 27, 2022 at 21:14

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