Fwd
Subtype of Delta DefThe forward delta Δᶠ of an fx option on the spot fx rate s of the currency pair FOR/DOM, with FOR the foreign currency and DOM the domestic currency, is defined as the first derivative ∂V(f)/∂f, where:
V(f) is the option's price expressed as a function of the forward fx rate f according to the Black Scholes formula Black Scholes FX formula
It can be shown that it equals (expressed as a function of K,σ,φ):
Δᶠ(K,σ,φ) = φN(φd₊)
Solving for K yields:
K = fe-φN⁻¹(φΔᶠ)στ¹ᐟ² + ½σ²τ
The put-call parity relation Call Price - Put Price = s - KDᵈ implies that the implied vol σ is the same for a call and put on the same strike K and then the above formula for Δᶠ leads to the put-call delta parity relation:
Δᶠ(K,σ,+1) - Δᶠ(K,σ,-1) = 1
Since Δᶠ(K,σ,-1) is always negative, the above can be written:
Δᶠ(K,σ,+1) + |Δᶠ(K,σ,-1)| = 1
This relation implies the following:
Consider a certain strike K for which the correspond call has a delta of 0.75. This call is referred to as a 75-delta call.
Then the put at the same strike K must have a delta of -0.25. This put is referred to as a 25-delta put.
As mentioned above, since this put has the same strike as the call, it must also have the same implied vol σ
The conclusion is that the 75-delta call must have the same implied vol as the 25-delta put.
Note, this relation holds only when "delta" is defined as "forward delta".
The meaning of symbols and more details in Black Scholes FX formula
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