Parity of Options on Currencies: Practice Problems and Solutions

This is Section 4 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here.

The formula for parity of options on currencies is

C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT

Explanation of Variables:

K = strike price of the options.

T = time to expiration of the options.

C(K, T) = price of a European call with strike price K and time to expiration T.

P(K, T) = price of a European put with strike price K and time to expiration T.

e^-rT*K= prepaid forward price for the strike.

r = the continuously compounded interest rate, denominated in dollars or the currency of one's choice

u = the continuously compounded interest rate, denominated in euros or some other currency of one's choice

x₀ = the current exchange rate, denominated in dollars/euro or in [currency 1]/[currency 2].

F_0,T = forward price for the underlying asset.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 286.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem POC1. One euro currently trades for 1.4831 U.S. dollars. The annual dollar-denominated interest rate is 0.01% (due to the Federal Reserve's incessant artificial credit expansions and manipulations of the money supply), while the annual euro-denominated interest rate is 3%. Both interest rates are on a continuously compounded basis. A euro call option with strike price $1.34 has price $0.34. What is the price a euro put option with the same strike price? Both options expire in one year.

Solution POC1. We use the equation C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT

and rearrange it as follows: P(K, T) = C(K, T) - x₀e^-uT + Ke^-rT

Here, C(K, T) = 0.34, K = 1.34, r = 0.0001, u = 0.03, x₀ = 1.4831, T = 1.

Thus, P(K, T) = 0.34 - 1.4831*e^-0.03 + 1.34e^-0.0001 = P(K, T) = 0.2405982359

Problem POC2. One Terlefian nivand (TNV) currently trades for 89 Roblanian dufords (RD) (Yes, the names are randomly generated.). A call option on nivands with strike price of 100 RD is currently worth 8.93 RD, while a TNV put option sells for 4.35 RD. The options expire in six months. The annual continuously compounded TNV-denominated interest rate is 0.044. What is the annual continuously compounded RD-denominated interest rate?

Solution POC2. One of the purposes of this problem is to show that the currencies in question need not be dollars or euros; the formula still applies, but we need to be careful about identifying which rate is u and which rate is r. Here, the TNV is the asset behind the call and options, whose prices are expressed in RD. So, RD is the currency analogous to the dollar. Thus, r is the rate we seek. We know that u = 0.044, C(K, T) = 8.93, P(K, T) = 4.35, x₀ = 89, K = 100.

We use the equation C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT

and rearrange it as follows: P(K, T) - C(K, T) + x₀e^-uT = Ke^-rT

Thus, 4.35 - 8.93 + 89e^-0.044/2 = 82.48338092 = Ke^-rT.

82.48338092 = 100e^-r/2

0.8248338092 = e^-r/2

-2ln(0.8248338092) = r = 0.3851467127 (either the Roblanian government is hyperinflating the duford, or the Roblanians tend to have a really high rate of time preference. In either case, you would probably be better off holding Terlefian nivands.)

Problem POC3. In the aftermath of Robert Mugabe's government-induced hyperinflation, One U. S. dollar (USD) currently trades for 30000 Zimbabwe dollars (ZWD). An oppressed Zimbabwean citizen wishes to get rid of his Zimbabwe dollars and trade them for the safer U. S. dollars. However, he does not yet know the annual continuously compounded USD-denominated interest rate. The annual effective ZWD interest rate is 1004. (No, this is not a typo. The inflation rate alone in Zimbabwe has exceeded 100000%-- the highest in history!) Help the oppressed Zimbabwean figure out the annual continuously compounded USD-denominated interest rate using the following information. The price of a dollar call with a strike price of 834000 ZWD is 28000 ZWD, while the price of a dollar put with the same strike price is 2 ZWD. The options expire in one year.

Solution POC3. This is partly a trick question. Even though the rate we want to find is US- dollar-denominated, it is actually u in our formula, since the USD is the asset behind the options.

We use the equation C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT and rearrange it as follows:

x₀e^-uT = C(K, T) - P(K, T) + K/(1+i), where i = 1004 is the annual effective ZWD-denominated interest rate.

Here, C(K, T) = 28000 and P(K, T) = 2. K = 834000 and x₀ = 30000. T = 1.

Thus, 30000e^-u = 28000 - 2 + 834000/1005.

30000e^-u = 28828.85075

e^-u = 0.9609616915

-ln(0.9609616915) = u = 0.0398207339

Problem POC4. You want to exchange fonbats for lomteds, but you do not know the exchange rate between them. You do know that a lomted call costs 34 fonbats, and a lomted put costs 35 fonbats. Both options have a strike price of 650 and expire three months from now. The lomted-denominated annual effective interest rate is 0.04, while the fonbat-denominated annual continuously compounded interest is 0.06. What is the exchange rate in fonbats per lomted?

Solution POC4.

In this case, we are given the continuously compounded interest rate r, but instead of u, we have an annual effective interest rate i.

We use the equation C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT and rearrange it as follows:

x₀(1+i)^-T = C(K, T) - P(K, T) + Ke^-rT

Here, C(K, T) = 34; P(K, T) = 35; K = 650; T = ¼; r = 0.06; i = 0.04. Thus,

x₀(1.04)^-1/4 = 34 - 35 + 650e^-0.06/4

x₀(1.04)^-1/4 = 639.3227607

639.3227607/(1.04)^-1/4 = x₀ = 645.6222678 fonbats/lomted

Problem POC5. Tabmids are exchanged for fillomtams at a rate of 0.03 tabmids per fillomtam. The prepaid forward price for the strike on fillomtam options is 0.02. The semiannual effective fillomtam-denominated interest rate is 0.34. A fillomtam put expiring 45 months from now with the strike price in question has price 0.02. Find the price of a fillomtam call with the same strike price and time to expiration.

Solution POC5. We use the equation C(K, T) - P(K, T) = x₀e^-uT - Ke^-rT and rearrange it as follows: C(K, T) = x₀e^-uT - Ke^-rT + P(K, T).

Here, Ke^-rT = 0.02; P(K, T) = 0.02; T = 45/12; x₀ = 0.03. We multiply x₀ by (1.34)^2(-45/12) to get the appropriate present value factor.

Thus, C(K, T) = (1.34)^2(-45/12)0.03 - 0.02 + 0.02 = C(K, T) = 0.003340683

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