Quiz: Option Basics

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5 questions to check your understanding of the chapter “Option Basics”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.

1. Question 1

   As a hypothetical example, suppose the underlying asset is priced at 20,000 yen today. A put option (put) with a strike price of 21,000 yen has an option premium of 1,300 yen. What is the time value of this put?

   1. A.1,300 yen
   2. B.1,000 yen
   3. C.300 yen
   4. D.2,300 yen
   Answer and explanation

   Answer: C

   The premium of an option splits into intrinsic value and time value. Intrinsic value is what you would get by exercising now, and time value is the part of the premium above the intrinsic value. The intrinsic value of a put is "strike price − price of the underlying asset," so it is 21,000 − 20,000 = 1,000 yen. The time value is the premium minus the intrinsic value: 1,300 − 1,000 = 300 yen. Time value reflects the chance that the price of the underlying asset moves in your favor before expiry, and it shrinks as expiry approaches.

   The 1,300 yen choice comes from using the formula for a call option (call) and computing the intrinsic value as "price of the underlying asset − strike price." Since 20,000 − 21,000 is negative, you would set the intrinsic value to zero and treat the whole premium as time value. A put has intrinsic value when the strike price is above the price of the underlying asset, as in this example. 1,000 yen is the intrinsic value itself, and 2,300 yen is the premium plus the intrinsic value (1,300 + 1,000).

   Learn more: Option Basics: Calls and Puts

2. Question 2

   As a hypothetical example, suppose you sell one unit of a right to sell the underlying asset at a strike price of 200 yen (a put) for a premium of 8 yen. Ignore interest between receiving the premium and expiry, as well as trading fees and taxes. If the price of the underlying asset at expiry is 180 yen, what is the seller's profit or loss at expiry?

   1. A.A loss of 12 yen
   2. B.A profit of 12 yen
   3. C.A loss of 20 yen
   4. D.A profit of 8 yen
   Answer and explanation

   Answer: A

   When the price at expiry falls below the strike price of 200 yen, the put is exercised and the seller must buy the underlying asset at 200 yen. The profit or loss is therefore "price at expiry − strike price + premium," or 180 − 200 + 8 = −12 yen, a loss of 12 yen. The break-even point is 200 − 8 = 192 yen: below 192 yen the seller loses, and above it the seller gains.

   If the price at expiry is 200 yen or higher, the put is not exercised and the 8 yen received is the seller's profit. That 8 yen is the most the seller can earn. The loss, on the other hand, grows as the price falls. Assuming the price of the underlying asset cannot go below 0 yen, the largest loss is 192 yen, at a price of 0 yen.

   A profit of 12 yen is the buyer's result on the same put. The buyer exercises and receives a payoff of 200 − 180 = 20 yen, and after subtracting the 8 yen premium paid, makes a profit of 12 yen. For the same option, the buyer's and seller's results have opposite signs and add up to 0. A loss of 20 yen comes from forgetting to add the 8 yen premium received. A profit of 8 yen is the result when the put is not exercised (a price at expiry of 200 yen or higher).

   Learn more: Option Payoffs and Profit/Loss

3. Question 3

   With an American option, which can be exercised at any time until expiry, the holder can choose to exercise before expiry or wait until expiry. Assume interest rates are positive. For which of the following option holders can exercising before expiry be the better choice?

   1. A.Holding an American call on a stock that pays no dividends, when the stock price is well above the strike price
   2. B.Holding an American put on a stock that pays no dividends, after the stock price has fallen sharply
   3. C.Holding an American call on a stock that pays no dividends, when the market price of the call is above its intrinsic value
   Answer and explanation

   Answer: B

   What a put can pay is capped at the strike price, so after the stock price has fallen sharply, there is little left to gain by waiting. As a hypothetical example, with a put at a strike price of 200 yen and a stock price of 10 yen, exercising now gives you 190 yen. Waiting until expiry gives at most 200 yen, so the room to gain is only up to 10 yen. On the other hand, the 190 yen received now earns interest until expiry. So even for a stock without dividends, exercising early can be better.

   For a call on a stock without dividends, selling the option in the market gives you more than exercising before expiry. As a hypothetical example, take a strike price of 200 yen, a stock price of 230 yen, and a market price for the call of 34 yen. Exercising now gives you the intrinsic value (the value of exercising now) of 230 − 200 = 30 yen, while selling gives you 34 yen. This is because, with positive interest rates and no dividends, the price of a call before expiry is higher than its intrinsic value. The 4 yen difference is the time value, which you give up by exercising before expiry. When the price of the call is above its intrinsic value (choice C), selling is better for the same reason.

   A European option, which can be exercised only on the expiry date, cannot be exercised before expiry. For a call on a stock without dividends there is no reason to exercise early, so the American option is worth the same as the European option. Exercising early when the price of the underlying asset is well above the strike price can be better for a call on an underlying asset that pays dividends or interest. This holds only when the dividends or interest received exceed the interest lost by paying the strike price early, and it does not apply to a call on a stock without dividends.

   Learn more: Exercise Styles: European, American and Bermudan

4. Question 4

   As a hypothetical example, suppose an underlying asset that pays no dividends is priced at 200 today, expiry is one year away, and the one-year discount factor is 0.97. A European put on this underlying asset with strike price 200 and expiry in one year is priced at 10. Using put-call parity, what is the price of the call with the same underlying asset, strike price, and expiry?

   1. A.4
   2. B.10
   3. C.6
   4. D.16
   Answer and explanation

   Answer: D

   For European options with the same underlying asset, strike price, and expiry, put-call parity links the call price $C$, the put price $P$, the current price of the underlying asset $S$, the strike price $K$, and the discount factor to expiry $DF$ as follows.

   $$C - P = S - K \times DF$$

   The present value of the strike price is 200 × 0.97 = 194, so the call is 200 − 194 = 6 more expensive than the put, or 10 + 6 = 16. The relationship comes from comparing two combinations. One is buying a call and selling a put. The other is buying the underlying asset and borrowing money on a promise to repay $K$ at expiry. The amount you can borrow today is the present value of $K$, $K \times DF$, so the money needed to set up the second combination today is $S - K \times DF$. Both pay "price at expiry − $K$" at expiry, so their values today are also equal.

   6 is the difference between the call and put prices; it comes from forgetting to add the put price. 4 reverses the direction of the difference and subtracts 6 from the put price. 10 is what you get by ignoring discounting (a discount factor of 1). With zero interest, the call and put with strike price 200 would have the same price, but the discount factor in this example is 0.97.

   When the price difference departs from the formula, you can lock in the gap through arbitrage by selling the expensive side and buying the cheap side, apart from costs such as trading fees.

   Learn more: Put-Call Parity

5. Question 5

   As a hypothetical example, suppose you sell both a call and a put on a stock priced at 200 yen today, with the same expiry and a strike price of 200 yen, receiving a premium of 9 yen for each (selling a straddle). Which statement about the profit or loss of this combination at expiry is correct?

   1. A.The loss is largest when the stock price at expiry is exactly 200 yen
   2. B.The loss has no upper limit as the stock price at expiry rises
   3. C.There is a profit when the stock price at expiry is below 182 yen
   4. D.There is a loss when the stock price at expiry is between 182 yen and 218 yen
   Answer and explanation

   Answer: B

   The profit or loss of a combination at expiry is the sum of the results of each trade in it. The profit-and-loss shape of selling a straddle is that of buying the same call and put (buying a straddle), flipped upside down. The premiums received total 9 + 9 = 18 yen. With a stock price of 230 yen at expiry, the call you sold is exercised for a loss of 230 − 200 = 30 yen, so the result is 18 − 30 = −12 yen, a loss of 12 yen. As the stock price rises, the loss on the call you sold grows without limit.

   On the downside, the loss on the put you sold grows. However, the largest loss is 200 − 18 = 182 yen, when the stock price reaches 0. So the loss has no upper limit only on the upside, and below 182 yen the result is not a profit but a growing loss.

   The break-even points are "strike price ± total premium received," or 182 yen and 218 yen, and a profit remains when the stock price is between them. At a stock price of exactly 200 yen, neither option is exercised and the 18 yen received is all profit, and this is the largest profit. "The loss is largest at 200 yen," "a loss between 182 yen and 218 yen," and "a profit below 182 yen" describe the shape of buying a straddle.

   Learn more: Option Strategies