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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.

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1 Answer 1

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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"?).

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  • $\begingroup$ I think it gives a safer (more diversified ) dynamic hedging strategy : trade dynamically $N_2$ in a forward contract and only $(N_1-N_2)$ in the stock instead of $N_1$. Less money is required for hedging and the delta gap risk may be lower. But not sure if it makes sense in practice ?! $\endgroup$
    – bigInner
    Commented May 9, 2023 at 13:50
  • $\begingroup$ It is a novel (to me) decomposition of the options value into two parts. But since it is just a reshuffling of terms the value and the delta are the same, I would think. And it is a little confusing that it depends on both F and S. $\endgroup$
    – nbbo2
    Commented May 9, 2023 at 15:45
  • $\begingroup$ It is actually the same delta but redistributed between Forward contract and the spot. Around the money you rely on stock and far from the money on the forward contract $\endgroup$
    – bigInner
    Commented May 9, 2023 at 16:37

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