The Garman-Kohlhagen Formula for Pricing Currency Options: Practice Problems and Solutions

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 36

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 36 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10

here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here.

The Garman-Kohlhagen Formula is a variant on the Black-Scholes option pricing formula, applied to finding the prices of currency options.

We note that the prepaid forward price for a given currency (our underlying asset in this case) can be expressed as F^P_0,T(x) = x₀e^-fT, where x_t is the exchange rate (in "domestic currency" per unit of "foreign" currency at time t), T is the forward's time to expiration, and f (r_f in McDonald's book) is the "foreign" currency risk-free interest rate. Here the "foreign" currency risk-free interest rate f is analogous to the continuously compounded dividend yield ∂ in the Black-Scholes equation.

The Garman-Kohlhagen Formula for the price of a call option is

C(x, K, σ, r, T, f) = xe^-fTN(d₁) - Ke^-rTN(d₂)

where d₁ = [ln(x/K) + (r - f + 0.5σ²)T]/[σ√(T)] and d₂ = d₁ - σ√(T)

The Garman-Kohlhagen formula for the put price is

P(x, K, σ, r, T, f) = Ke^-rTN(-d₂) - xe^-fTN(-d₁)

We can also get the put formula via put-call parity:
P(x, K, σ, r, T, f) = C(x, K, σ, r, T, f) + Ke^-rT - xe^-fT

Meaning of variables:

x = currency exchange rate (in "domestic currency" per unit of "foreign" currency at time t).

K = strike price (strike exchange rate) of the option.

C = call option price.