Black–Scholes Decomposition of Option Value into Forward and Time Components
Summary
The document examines a rearrangement of the Black–Scholes call price that separates a forward-related component from a term proportional to the difference between the normal cumulative probabilities at d1 and d2. It connects that difference to the option’s time value, while a response clarifies that this is a nontraditional decomposition: the forward component can be negative for an out-of-the-money call, so it is not ordinary intrinsic value.
For small volatility scaled by the square root of time, the document approximates the probability difference using the normal density and relates the resulting time component to vega. It further observes that at the money, the forward component vanishes under the stated setup and the call value is represented by the time component. These are approximations and interpretations within Black–Scholes assumptions; the discussion does not establish them as general option-pricing identities beyond that framework.
Key ideas
- The Black–Scholes call price can be rearranged into a forward-related term and a term involving the difference between the d1 and d2 probabilities.
- The forward-related term in this decomposition is not conventional intrinsic value and can be negative for an out-of-the-money call.
- For small volatility scaled by the square root of time, the probability difference is approximated using the normal density.
- The associated time-value component is linked to vega under the stated approximation.
- The at-the-money interpretation depends on the setup and Black–Scholes assumptions.
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Full text
# Option time value is Nd1-Nd2
# Option time value is Nd1-Nd2
I can't find the below statement anywhere (rearrangement of Black-Scholes formula) :
$C(0, S) = e^{-rT}N_2[F-K] + [N_1-N_2]S$
$F$ being the forward, it reads as a straightforward decomposition to intrinsic value (1st term) and extrinsic/time value (2nd term). This may answer the famous question what is the difference between $Nd_1$ and $Nd_2$ (mathematical difference and the difference in meaning too): The difference is the time value of the option.
Edit:
Just wanna add that for small log-normal volatility $\sigma\sqrt{T} < 1 $ : $$N_1 - N_2 = N(d_1) -N(d_1 - \sigma\sqrt{T}) \approx \sigma\sqrt{T}n_1$$ Hence, as $\mathcal{Vega} = S\sqrt{T}n_1$ the "speculative" time value is $$ [N_1 - N_2]S = \sigma\mathcal{Vega} = \sigma\sqrt{T}n_1S $$
And: $$N_2 \approx N_1 - \frac{\sigma\mathcal{Vega}}{S} = \Delta - \frac{\sigma\mathcal{Vega}}{S} $$
Thus for small $\sigma\sqrt{T}$ : $$C = \left[\Delta - \frac{\sigma\mathcal{Vega}}{S} \right] [F - K] + \sigma\mathcal{Vega}$$ The "intrinsic value" of the 1st term is not negative for OTM as mentioned in the comment (bc delta $\approx$ 0 and vega > 0).
ATM the 1st term (intrinsic value) is zero so the price is linear in volatility and is purely speculative (think of vega as a proxy for the bid-ask spread). Also, the ATM vega is maximal $\mathcal{Vega}_{max} = 0.4S\sqrt{T}$ which makes the ATM price equals to the maximal time value of the option, both equal to $0.4S\sigma\sqrt{T}$.
The difference $N_1 - N_2$, the time value and the vega (vega cash) normalized by S are three sides of the same coin.
## Answer by nbbo2 (score 9)
https://quant.stackexchange.com/a/75485
That's nice. Starting from
$$C = e^{-r T}N_2 (F-K) + (N_1 - N_2) S$$
we can substitute $F= e^{r T}S$ (no dividend case) so we get
$$C = e^{-r T}N_2(e^{r T} S-K) + (N_1 - N_2) S =$$
$$= N_1 S -e^{-r T}N_2 K $$
which is just the Black Scholes 1971 formula.
The first term $e^{-r T}N_2 (F-K)$ is a new definition of "intrinsic value", different from the traditional one, you could call it the "forward intrinsic value" or something like that. It is the present value of the forward minus the strike, times the probability $N_2$ (roughy speaking the probability of exercise). It could be negative for an OTM call (weird). Then the second one $ (N_1 - N_2) S$ is the corresponding form of time value which again deserves a new name (the "speculative value"?).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.