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$$
v_0 = s N(d_1) - ke^{-rt} N(d_2),
$$
where $N$ is the standard normal cumluative distribution function,
$d_1 = (\log(s/k) + (r + σ^2/2)t)/σ\sqrt{t}$, and $d_2 = d_1 - σ\sqrt{t}$.
Fischer Black ($k$, $f$, $σ$, $t$)
$$
v_t = f N(d_1) - k N(d_2),
$$
where $f = se^{rt}$ is forward price and $v_t = v_0 e^{rt}$.
Positive underlying ($k$, $f$, $s$)
$$
v = f P_s(F \ge k) - k P(F\ge k)
$$
where $dP_s/dP = ε_s(X) = e^{s X - κ(s)}$, $F = fε_s(X)$, and
$κ(s) = \log E[e^{s X - κ(s)}]$ is the cumulant.
For any differentiable function $ν$ let $v = E[ν(F)]$, then
$$
∂_f v = E[ν'(F) ∂_f F] E[ν'(F) e^{sX - s^2/2}] = E_s[ν'(F)].
$$
This establishes the formula for option delta without any turmoil.
Share Measure
The forward value of an option paying $ν(F)$ in some currency at
expiration is $E[ν(F)]$.
In terms of shares of $F$, $ν_s(F) = E[ν(F)F/E[F]$.
Share measure for positive underlyings $E_s$ is defined by $E_s[ν(F)] = E[ν(F) F/E[F]]$.
Note $F > 0$ and $E_s[1] = 1$ so share measure is a probability measure.
It shows up in the formula for valuing a call
$$
\begin{aligned}
E[(F - k)^+] &= E[(F - k)1(F\ge k)] \
&= EF1(F\ge k) - k P(F\ge k) \
&= fP_s(F\ge k) - k P(F\ge k).
\end{aligned}
$$
Every positive random variables $F$ can be written $F = f e^{s X - κ(s)}$
where $X$ is a random variable with mean 0 and variance 1 and
$κ(s) = \log E[e^{sX}]$ is the cumulant of $X$.
Note $f = E[F]$ and $s^2 = \Var(\log F)$.
Exercise. Clearly $\log(F/E[F]) = m + sX$ for some random variable $X$ with mean 0 and variance 1.
Show $E[F] = f$ implies $m = -κ(s)$.
If we let $ε_s(x) = e^{s x - κ(s)}$, so $F = fε_s(X)$, this can be written
$E_s[ν(F)] = E[ν(F) ε_s(X)]$ and we see share measure is the Esscher transform.
The cumulative distribution of $F$ under this measure is
$$
P_s(F\le y) = P_s(X\le x) = E[1(X\le x) e^{sX - κ(s)}]
$$
where $x = x(y) = ε_s^{-1}(y/f) = (\log y/f + κ(s))/s$ is the moneyness
of $y$. Note $ε_s(x(y)) = y/f$.
Greeks
Let $ν(F)$ be the option payoff at expiration. The forward value of
the option is $v = E[ν(F)]$.
Delta is the derivative of value with respect to the forward
$$
∂_f v = E[ν'(F) ∂_f F] = E[ν'(F) ε_s(X)]
$$
since $∂_f F = ε_s(X)$.
Gamma is the second derivative with respect to the forward
$$
∂_f^2 v = E[ν''(F)ε_s^2(X)].
$$
Vega is the derivative with respect to vol
$$
∂_s v = E[ν'(F) ∂_s F] = E[ν'(F)F(X - κ'(s))]
$$
since $∂_s F = F(X - κ'(s))$.
The inverse of option value as a function of vol is the implied vol.
Put and Call
A put option pays $ν(F) = (k - F)^+ = \max{k - F,0}$ at expiration
and has value $p = E[(k - F)^+]$.
A call option pays $ν(F) = (F - k)^+$ at expiration
and has value $c = E[(F - k)^+]$.
Note $(F - k)^+ - (k - F)^+ = F - k$ is a forward with strike$k$ so
all models satisfy put-call parity: $c - p = f - k$.
Call delta is $∂_f c = ∂_f p + 1$ and call gamma equals put gamma $∂_f^2 c = ∂_f^2 p$.
We also have $∂_s c - ∂_s p = 0$ so call vega equals put vega.
The value of a put is
$$
p = E[(k - F)^+] = kP(F\le k) - f P_s(F\le k).
$$
Put delta is
$$
∂_f p = E[-1(F\le k)ε_s(X)] = -P_s(F\le k).
$$
since $∂_f F = ε_s(X)$.
Gamma for either a put or call is
$$
∂_f^2 p = E[δ_k(F)ε_s(X)^2] = E_s[δ_k(F)ε_s(X)]
$$
where $δ_k$ is a point mass at $k$.
Vega for a put or call is
$$
∂_s p = -E[1(F\le k) F (X - κ'(s))] = -f E_s[1(F\le k) (X - κ'(s))].
$$
Distribution
Let $Φ(x) = P(X\le x)$ be the cumulative distribution functions of $X$
and $Φ_s(x) = P_s(X\le x) = E[1(X\le x)ε_s(X)]$ be the share cdf where
$ε_s(x) = e^{sx - κ(s)}$. Of course $Φ(x) = Φ_0(s)$.
Let $Ψ_s(y) = P_s(F\le y) = Φ_s(x)$ be the share cumulative distribution function of $F$
where $y = fε_s(x)$. The share density function is
$$
ψ_s(y) = φ_s(x) ∂x/∂y = φ_s(x)/ys
$$
since $∂y/∂x = ys$.
We also have
$$
∂_s Φ_s(x) = E[1(X\le x)ε_s(X)(X - κ'(s))] = E_s[1(X\le x) (X - κ'(s))].
$$
In terms of the distribution function for $X$, the value is
$$
p = k Φ(x(k)) - f Φ_s(x(k),
$$
put delta is
$$
∂_f p = -Φ_s(x(k)),
$$
put and call gamma is
$$
∂_f^2 p = E_s[δ_k(F)ε_s(X)] = ψ_s(k) ε_s(x(k)) = (φ_s(x(k))/ks) (k/f) = φ_s(x(k))/fs.
$$
Since $φ_s(x) = φ(x) ε_s(x)$ and ε_s(x(y)) = y/f$, $∂_f^2 p
and put and call vega is
$$
∂_s p = -f E_s[1(F\le k) (X - κ'(s))] = -f ∂_s Φ_s(x(k)).
$$
Black Model
In the Black modes $F = fe^{sX - s^2/2}$ where $X$ is standard normal.
Recall if $X$ is standard normal then $E[g(X) e^{sX}] = e^{s^2/2}E[g(X + s)]$.
Using $g(x) = 1$ we see $κ(s) = s^2/2$. Using $g(X) = 1(X\le x)$
we get $Φ_s(x) = P(X + s \le x) = Φ(x - s)$
and $∂Φ_s(x)/∂s = -φ(x - s) = -φ_s(x)$.
Put value is
$$
p = k Φ(x(k)) - f Φ(x(k) - s)
$$
where $x(k) = \log(k/f)/s + s/2$.
Exercise. Show $x(k) = \log(k/f)/s + s/2 = -d_2$ and $x(k) - s = \log(k/f)/s - s/2 = -d_1$.
Put delta is
$$
∂_f p = -Φ_s(x(k)) = -Φ(x(k) - s).
$$
Gamma is
$$
∂_f^2 p = φ_s(x(k))/fs.
$$
Vega is
$$
∂_s v = -f ∂_s Φ_s(x(k)) = fφ_s(x(k)).
$$
Digital
A digital put has payoff $ν(F) = 1(F \le k)$ and
a digital call has payoff $ν(F) = 1(F > k)$ with values.
Since $1(F \le k) + 1(F > k) = 1$ we have digital put-call parity $p + c =
1$ where $p$ is the digital put value and $c$ is the digital call value.
$$
p = P(F \le k), c = P(F > k) = 1 - p.
$$
Digital put delta is
$$
∂_f p = -E[δ_k(F)ε_s(X)] = -E_s[δ_k(F)]
$$
Digital gamma is
$$
∂_f^2 p = E[δ'_k(F)ε_s(X)^2] = E_s[δ'_k(F) ε_s(X)].
$$
Digital put vega is
$$
∂_s p = -E[\delta_k(F)F(X - s)] = -f E_s[δ_k(F) (X - κ(s))].
$$
Parameters
The Black-Scholes/Merton values and greeks can be calculated in terms of
the parameters $f$ and $s$ using the
chain rule. For example, the Black model takes $X$ to be standard normal
and vol $s = σ \sqrt{t}$ where $σ$ is the standard Black volatilty
and $t$ is time in years to expiration. In this case standard vega
is $∂_σ E[ν(F)] = ∂_s E[ν(F)] ∂_σ s = ∂_s E[ν(F)]\sqrt{t}$.
The Black-Merton/Scholes model uses spot prices instead of forward.
If a risk-free bond has realized return $R$ over the period the value of the underlying
at expiration is $U = Rue^{sX - κ(s)}$. Since $F = U$ we have $f = Ru$.
The spot value of the option is $v_0 = E[ν(U)]/R$. We have
$$
∂_u v_0 = E[ν'(U) ∂_u U]/R
= E[ν'(F) ∂_f F ∂_u f]/R
= E[ν'(F) ∂_f F R]/R = ∂_f v.
$$
Spot and forward delta are equal but the spot gamma is
$$
∂_u^2 v_0 = ∂_u ∂_f v = ∂_f^2 v ∂_u f = ∂_f^2 v R.
$$
Spot vega is
$$
∂_s v_0 = E[ν'(U) ∂_s U]/R = E[ν'(F) ∂_s F/R = ∂_s v/R.
$$