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

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.

σ = annual exchange rate volatility.

r = annual continuously compounded "domestic" currency risk-free interest rate.

T = time to expiration.

f = annual continuously compounded "foreign" currency risk-free interest rate.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, p. 381.

Problem GKFPCO1. One piece of stone from Yap (YPS) currently trades for 800 cigarettes (CG). You own a cigarette-denominated YPS call option with time to expiration of 5 years and strike price of 850 CG. The annual continuously compounded cigarette-denominated risk-free interest rate is 0.23. The annual continuously compounded YPS-denominated risk-free interest rate is 0.05. The exchange rate volatility relevant for the Garman-Kohlhagen Formula is 0.1. Find the value of d₁ in the Garman-Kohlhagen Formula for such a call option.

Solution GKFPCO1. We use the formula d₁ = [ln(x/K) + (r - f + 0.5σ²)T]/[σ√(T)], where x = 800, K = 850, r = 0.23, f = 0.05, σ = 0.1, T = 5. Thus,

d₁ = [ln(x/K) + (r - f + 0.5σ²)T]/[σ√(T)] = [ln(800/850) + (0.23 - 0.05 + 0.5*0.1²)5]/[0.1√(5)] =

d₁ = 3.865604207

Problem GKFPCO2. One piece of stone from Yap (YPS) currently trades for 800 cigarettes (CG). You own a cigarette-denominated YPS call option with time to expiration of 5 years and strike price of 850 CG. The annual continuously compounded cigarette-denominated risk-free interest rate is 0.23. The annual continuously compounded YPS-denominated risk-free interest rate is 0.05. The exchange rate volatility relevant for the Garman-Kohlhagen Formula is 0.1. Find the value of d₂ in the Garman-Kohlhagen Formula for such a call option.

Solution GKFPCO2. We use the formula d₂ = d₁ - σ√(T), where σ = 0.1, T = 5, and d₁ = 3.865604207 from Solution GKFPCO1. Thus, d₂ = 3.865604207 - 0.1√(5) = d₂ = 3.641997409

Problem GKFPCO3. One piece of stone from Yap (YPS) currently trades for 800 cigarettes (CG). You own a cigarette-denominated YPS call option with time to expiration of 5 years and strike price of 850 CG. The annual continuously compounded cigarette-denominated risk-free interest rate is 0.23. The annual continuously compounded YPS-denominated risk-free interest rate is 0.05. The exchange rate volatility relevant for the Garman-Kohlhagen Formula is 0.1. Use the Garman-Kohlhagen Formula to find the price of such a call option.

Solution GKFPCO3. We use the formula C(x, K, σ, r, T, f) = xe^-fTN(d₁) - Ke^-rTN(d₂)

We are given that x = 800, K = 850, r = 0.23, f = 0.05, σ = 0.1, T = 5.

Furthermore, from Solutions GKFPCO1-2, d₁ = 3.865604207 and d₂ = 3.641997409.

In MS Excel, using the input "=NormSDist(3.865604207)", we find that N(d₁) = 0.999944572

In MS Excel, using the input "=NormSDist(3.641997409)", we find that N(d₂) = 0.9998647

Thus, C(x, K, σ, r, T, f) = 800e^-0.05*50.999944572 - 850e^-0.23*50.9998647 =

C(x, K, σ, r, T, f) = $353.9012534

Problem GKFPCO4. One piece of stone from Yap (YPS) currently trades for 800 cigarettes (CG). Amon-Ra owns a cigarette-denominated YPS put option with time to expiration of 5 years and strike price of 850 CG. The annual continuously compounded cigarette-denominated risk-free interest rate is 0.23. The annual continuously compounded YPS-denominated risk-free interest rate is 0.05. The exchange rate volatility relevant for the Garman-Kohlhagen Formula is 0.1. Use the Garman-Kohlhagen Formula to find the price of such a put option.

Solution GKFPCO4. Since the corresponding call option price is known (from Solution BSFOSDD3), we can use put-call parity to find the put option price.

P(x, K, σ, r, T, f) = C(x, K, σ, r, T, f) + Ke^-rT - xe^-fT = 353.9012534 + 850e^-0.23*5 - 800e^-0.05*5 =

P(x, K, σ, r, T, f) = $0.0018809151

Problem GKFPCO5. For a currency option priced using the Garman-Kohlhagen Formula, volatility σ suddenly increased by a factor of 2, leaving all other factors the same. It is the case that d₁ = (0.2 + 8σ²)/4σ. By how much has d₂ changed as a result of the volatility increase?

Solution GKFPCO5. We note that the denominator of the equation for d₁ is 4σ = σ√(T).

Thus, d₂ = d₁ - σ√(T) = d₁ - 4σ = (0.2 + 8σ²)/4σ - 4σ = 0.2/4σ + 2σ - 4σ = d₂ = 1/20σ - 2σ

If σ increases to 2σ, the increase in d₂ is [1/(20*2σ) - 2(2σ)] - [1/20σ - 2σ] =

1/40σ - 4σ - 1/20σ + 2σ = -1/40σ - 4σ. Thus, d₂ decreased by 1/40σ + 4σ.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.