Black–Scholes Time Value and Its Shape Across Moneyness
Summary
The document examines whether an option’s extrinsic value, also called time value, follows a normal or lognormal shape as the underlying price changes. It describes an interpretation of the Black–Scholes call formula using a lognormal distribution of future prices: the payoff-weighted terms correspond to truncated expected underlying value and discounted strike value. Their difference gives the call value before intrinsic value is removed.
The response states the exact expression for time value as the Black–Scholes call price less intrinsic value and reports fitting normal and lognormal densities to that value as a function of spot price. The excerpt provides the formula and fitting setup, but not the resulting comparison or fit quality. It therefore does not establish that time value itself follows either distribution, and the distinction between a distribution of future prices and the shape of option value across spot levels remains central.
Key ideas
- Black–Scholes call time value equals the call price minus intrinsic value.
- The underlying’s risk-neutral terminal price is modeled as lognormal in the standard framework.
- The call formula combines a truncated expected asset value with a discounted strike term.
- A lognormal distribution for terminal prices does not by itself prove that time value across spot prices is lognormal.
- The excerpt describes density fitting but omits its results.
Tags
Full text
# Option Extrinsic value representation
# Option Extrinsic value representation
The typical representation of the extrinsic value of an option is the following:
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.
## Answer by Kurt G. (score 3, accepted)
https://quant.stackexchange.com/a/71053
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 $S$):Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.