## Black Scholes FX formula

The

**defines the price (i.e. present value)**

*Black Scholes FX formula***of a European fx option on a currency pair**

*V(s)***, with**

*FOR/DOM***the foreign currency and**

*FOR***the domestic currency, as a function of the spot fx rate**

*DOM***according to:**

*s*V(s) = φ[DᶠsN(φd₊) - DᵈKN(φd₋)]

where

**represents the number of**

*s***units required to pay on trade date in order to receive 1**

*DOM***unit on the same date.**

*FOR***= 1 for call and -1 for put**

*φ***(**

*Dᶠ***) is the foreign (domestic) discount factor for a maturity equal to the option's expiry**

*Dᵈ*

*T***is the strike**

*K***is the cumulative normal distribution function**

*N(x)*d± = [ln(f/K) ± ½σ²τ]/(στ¹ᐟ²)

where

**is the fx forward for maturity**

*f***that may be calculated by the parity rates relation:**

*T*f = sDᶠ/Dᵈ

**is the time in annual units until maturity**

*τ*

*T***is the averaged volatility until**

*σ***so that στ¹ᐟ² is the standard deviation of ln(s) at T**

*T*If we use the rates parity relation to express

**as fDᵈ/Dᶠ, the option price becomes a function of the forward fx rate**

*s***, as follows:**

*f*V(f) = φDᵈ[fN(φd₊) - KN(φd₋)]

The option's spot delta

**is defined as the first derivative**

*Δˢ***and can be shown that it equals:**

*∂V(s)/∂s*Δˢ = φDᶠN(φd₊)

Financially, it represents the number of

**units that need to be held as a hedge against a short position on a**

*FOR***call option on one**

*FOR/DOM***unit.**

*FOR*The above equation can be solved for

**to give:**

*K*K = fe

^{-φN⁻¹(φΔˢ/Dᶠ)στ¹ᐟ² + ½σ²τ}

Viewing

**as a function**

*Δˢ***of**

*Δˢ(φ)***, the following put-call parity relationship holds:**

*φ*Δˢ(+1) - Δˢ(-1) = Dᶠ

The option's forward delta

**is defined as the first derivative**

*Δᶠ***and can be shown using the chain rule that it equals:**

*∂V(f)/∂f*Δᶠ = φN(φd₊)

Financially, it represents the number of forward

**units that need to be held as a hedge against a short position on a**

*FOR***call option on one**

*FOR/DOM***unit.**

*FOR*The above equation can be solved for

**to give:**

*K*K = fe

^{-φN⁻¹(φΔᶠ)στ¹ᐟ² + ½σ²τ}

Viewing

**as a function**

*Δˢ***of**

*Δˢ(φ)***, the following put-call parity relationship holds:**

*φ*Δᶠ(+1) - Δᶠ(-1) = 1

The option's premium-adjusted spot delta

**is defined so that it represents the number of**

*Δˢₚₐ***units that need to be held as a hedge against a short position on a**

*FOR***option on one**

*FOR/DOM***unit in the case where the option's premium is paid in**

*FOR***currency.**

*FOR*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

**analytically and a numerical procedure must be employed.**

*K*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

**as a function**

*Δˢₚₐ***of**

*Δˢₚₐ(φ)***, the following put-call parity relationship holds:**

*φ*Δˢₚₐ(+1) - Δˢₚₐ(-1) = DᶠK/f

The option's premium-adjusted forward delta

**is defined so that it represents the number of forward**

*Δᶠₚₐ***units that need to be held as a hedge against a short position on a**

*FOR***option on one**

*FOR/DOM***unit in the case where the option's premium is paid in**

*FOR***currency.**

*FOR*It can be shown that it equals:

Δᶠₚₐ = φ(K/f)N(φd₋)

Regarding solving the above equation wrt

**, the same comments apply as in the premium-adjusted spot delta case above.**

*K*Viewing

**as a function**

*Δᶠₚₐ***of**

*Δᶠₚₐ(φ)***, the following put-call parity relationship holds:**

*φ*Δˢₚₐ(+1) - Δˢₚₐ(-1) = K/f