Classification of Calls and Puts: Practice Problems and Solutions

This is Section 7 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.

This is the generalized formula for currency options - relating calls denominated in one currency (say, the dollar or any other currency A of our choice) to puts denominated in another currency (say, the euro or any other currency B of our choice). We will use this formula in our exploration of how puts and calls are classified with relation to our choice of the underlying asset and the strike asset.

C_A(x₀, K, T) = x₀ K P_B(1/x₀, 1/K, T)

Explanation of variables:

K = strike price, denominated in currency A.

T = time to expiration of the options.

x₀ = exchange rate in terms of [currency A/currency B].

C_A(x₀, K, T) = price of A-denominated call option on B (in units of currency A).

P_B(1/x₀, 1/K, T) = price of B-denominated put option on A (in units of currency B).

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 9, p. 292.

Problem CCP1. Bard Co. stock and Drab Co. stock are used as underlying or strike assets in writing options contracts. Geoffrey owns an option contract with the following payoff: max(0, S_Bard - S_Drab). Which of the following are correct ways of describing this contract? More than one answer may be correct.

(a) A call option on Bard stock with Drab stock as the strike asset.

(b) A call option on Drab stock with Bard stock as the strike asset.

(c) A put option on Bard stock with Drab stock as the strike asset.

(d) A put option on Drab stock with Bard stock as the strike asset.

(e) A call option on both Bard and Drab stock.

(f) A put option on both Bard and Drab stock.

Solution CCP1.

For Geoffrey, if Bard is the underlying asset, then Geoffrey can gain at expiration if the price of Bard exceeds the price of Drab - so this option functions as a Bard call.

Someone aiming for the price of Drab to decline relative to the price of Bard can gain at expiration if the price of Drab is less than the price of Bard. So this option can also be interpreted as a put option on Drab stock. All that differs between the interpretations is the labeling of assets as underlying assets or strike assets. So (a) and (d) are both correct answers.

Problem CCP2. In this list, some two of the descriptions are equivalent, as are the remaining two descriptions. Find the pairs of equivalent descriptions.

(a) A put option on dollars, with XYQ stock as the strike asset.

(b) A call option on dollars, with XYQ stock as the strike asset.

(c) A call option on XYQ stock, with dollars as the strike asset.

(d) A put option on XYQ stock, with dollars as the strike asset.

Solution CCP2.

Both (b) and (d) refer to transactions in which you buy X dollars in exchange for one share of XYQ stock. So (b) and (d) are the same.

Both (a) and (c) refer to transactions in which you sell X dollars in exchange for one share of XYQ stock. So (a) and (c) are the same.

Problem CCP3. A 1-year dollar-denominated call option on euros with a strike price of $1.50 has a payoff of max(0, R - 1.50), where R is the exchange rate in dollars per euro one year from now. Determine the payoff of a 1-year euro-denominated put option on dollars.

Solution CCP3. Conceptually, a 1-year euro-denominated put option on dollars is identical to a 1-year dollar-denominated call option on euros; the only difference is the scale of the payoff for one contract in terms of the euro. The exchange rate one year from now in terms of euros per dollar will be 1/R, and the strike price will be 1/1.50 euros = 2/3 euros. Furthermore, on the put option, we gain whenever the exchange rate in terms of euros per dollar is less than the strike price.

Thus, the 1-year euro-denominated put option on dollars has a payoff of

max(0, 2/3 - 1/R)

Problem CCP4. A 1-year dollar-denominated call option on euros with a strike price of $1.50 has a payoff of max(0, R - 1.50), where R is the exchange rate in dollars per euro one year from now. The current exchange rate is $1.52 per euro. The premium on one dollar-denominated call option on euros is $0.04. Calculate the premium in euros on one euro-denominated put option on dollars.

Solution CCP4. We first determine the cost of buying 1/1.5 1-year dollar-denominated call options, which is (1/1.5)*0.04 = $0.02666666666667. We need to convert this to euros - and the result will be the cost of buying one 1-year euro-denominated put options.

$0.02666666666667*(1/1.52)euros per dollar = the premium in euros on one euro-denominated put option on dollars = 0.0175438596 euros

Problem CCP5. A 1-year call option on Terlefian nivands (TNV), denominated in Roblanian dufords (RD), costs 34 RD. This option has a strike price of 356 RD. The exchange rate between the currencies is 350 RD/TNV. Find the price in terms of TNV of one 1-year put option on RD, denominated in TNV.

Solution CCP5. We use the formula C_A(x₀, K, T) = x₀ K P_B(1/x₀, 1/K, T), with A = RD and B = TNV. Here, C_A(x₀, K, T) = 34, x₀ = 350, and K = 356.

So 34 = 350*356* P_B(1/x₀, 1/K, T) and P_B(1/x₀, 1/K, T) = 34/(350*356) = 0.0002728731942 TNV

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