Black Scholes FX formula


The Black Scholes FX formula defines the price (i.e. present value) V(s) of a European fx option on a currency pair FOR/DOM, with FOR the foreign currency and DOM the domestic currency, as a function of the spot fx rate s according to:
V(s) = φ[DᶠsN(φd₊) - DᵈKN(φd₋)]
where
s represents the number of DOM units required to pay on
trade date in order to receive 1 FOR unit on the same date.
φ = 1 for call and -1 for put
Dᶠ (Dᵈ) is the foreign (domestic) discount factor for a maturity equal to the option's expiry T
K is the strike
N(x) is the cumulative normal distribution function
d± = [ln(f/K) ± ½σ²τ]/(στ¹ᐟ²)
where
f is the fx forward for maturity T that may be calculated by the parity rates relation:
f = sDᶠ/Dᵈ
τ is the time in annual units until maturity T
σ is the averaged volatility until T so that στ¹ᐟ² is the standard deviation of ln(s) at T

If we use the rates parity relation to express s as fDᵈ/Dᶠ, the option price becomes a function of the forward fx rate f, as follows:
V(f) = φDᵈ[fN(φd₊) - KN(φd₋)]

The option's spot delta Δˢ is defined as the first derivative ∂V(s)/∂s and can be shown that it equals:
Δˢ = φDᶠN(φd₊)
Financially, it represents the number of FOR units that need to be held as a hedge against a short position on a FOR/DOM call option on one FOR unit.
The above equation can be solved for K to give:
K = fe-φN⁻¹(φΔˢ/Dᶠ)στ¹ᐟ² + ½σ²τ
Viewing Δˢ as a function Δˢ(φ) of φ, the following put-call parity relationship holds:
Δˢ(+1) - Δˢ(-1) = Dᶠ

The option's forward delta Δᶠ is defined as the first derivative ∂V(f)/∂f and can be shown using the chain rule that it equals:
Δᶠ = φN(φd₊)
Financially, it represents the number of forward FOR units that need to be held as a hedge against a short position on a FOR/DOM call option on one FOR unit.
The above equation can be solved for K to give:
K = fe-φN⁻¹(φΔᶠ)στ¹ᐟ² + ½σ²τ
Viewing Δˢ as a function Δˢ(φ) of φ, the following put-call parity relationship holds:
Δᶠ(+1) - Δᶠ(-1) = 1

The option's premium-adjusted spot delta Δˢₚₐ is defined so that it represents the number of FOR units that need to be held as a hedge against a short position on a FOR/DOM option on one FOR unit in the case where the option's premium is paid in FOR currency.
It can be shown that it equals:
Δˢₚₐ = Δˢ - V/s
which results to the final expression:
Δˢₚₐ = φDᶠ(K/f)N(φd₋)
The above equation cannnot be solved for K analytically and a numerical procedure must be employed.
The call (φ = +1) case is more difficult because then the function is not monotone, which means that two strikes will generally correspond to a given delta.
The standard practice is to limit the function's domain on strikes located to the right of the delta maximum.
Viewing Δˢₚₐ as a function Δˢₚₐ(φ) of φ, the following put-call parity relationship holds:
Δˢₚₐ(+1) - Δˢₚₐ(-1) = DᶠK/f

The option's premium-adjusted forward delta Δᶠₚₐ is defined so that it represents the number of forward FOR units that need to be held as a hedge against a short position on a FOR/DOM option on one FOR unit in the case where the option's premium is paid in FOR currency.
It can be shown that it equals:
Δᶠₚₐ = φ(K/f)N(φd₋)
Regarding solving the above equation wrt K, the same comments apply as in the premium-adjusted spot delta case above.
Viewing Δᶠₚₐ as a function Δᶠₚₐ(φ) of φ, the following put-call parity relationship holds:
Δˢₚₐ(+1) - Δˢₚₐ(-1) = K/f