As a hypothetical example, take a call option with a strike price of 100 and a term of 1 year. The price of the underlying asset is observed four times, once every 3 months (the last observation is on the expiry date), and the observed prices are 105, 117, 99, and 111, in that order. An Asian option pays the average of the four observations minus the strike price. What is its payoff?
Quiz: Types of Exotic Options
Last reviewed: September 25, 2026
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7 questions to check your understanding of the chapter “Types of Exotic Options”. 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.
Question 1
Answer and explanation
Answer: B
An Asian option is a path-dependent option whose payoff is set by the average price over the period. The average of the four observations is (105 + 117 + 99 + 111) ÷ 4 = 108. The payoff is the average minus the strike price: 108 − 100 = 8.
The 11 is the price at expiry, 111, minus the strike price. A standard call whose payoff depends only on the price at expiry (a vanilla option) would pay 11. The 17 is the highest price, 117, minus the strike price, which has the form of the payoff of a lookback option, which is set by the highest price. The 7 comes from leaving the observation on the expiry date out of the average and using (105 + 117 + 99) ÷ 3 = 107. An Asian option, set by the average, and a lookback option, set by the highest price, are both path-dependent types of exotic option. For this price path, the payoff changes to 8, 17, or 11 depending on whether it is set by the average, the highest price, or the price at expiry.
Learn more: Overview of Exotic Options
Question 2
Take an up-and-out call (knock-out) and an up-and-in call (knock-in), both barrier options that can be exercised only at expiry. They have the same strike price, barrier, and expiry, and there is no rebate. If the monitoring changes from continuous (checking the barrier at all times until expiry) to discrete (checking only at set times, such as each day's closing price), how do the two prices change?
Answer and explanation
Answer: A
Under discrete monitoring, a path that crosses the barrier and comes back between monitoring times counts as not having touched it. The barrier is therefore touched less often than under continuous monitoring. The knock-out, whose right is less likely to disappear, is priced higher, and the knock-in, whose right is less likely to come into existence, is priced lower.
"The knock-out falls" comes from the mistaken idea that discrete monitoring makes the barrier easier to touch. "The knock-out does not change and the knock-in falls" does not fit the fact that the sum of the two prices does not change. With no rebate in this example, the sum equals the price of a call with the same strike price and expiry and no barrier (a vanilla call). A vanilla call does not look at any barrier, so changing the monitoring method does not change its price. If the knock-in falls, the knock-out must rise to keep the sum. "Neither changes" comes from assuming the monitoring method has no effect on the price. The prices of both the knock-out and the knock-in change with the monitoring method.
Learn more: Barrier Options
Question 3
As a hypothetical example, a buyer pays a premium of 9 yen for a call on a cash-or-nothing digital option (a digital call) with a strike price of 200 yen and a payout of 20 yen. Ignore interest on the premium, trading fees, and taxes. If the price of the underlying asset at expiry is 260 yen, what is the buyer's profit or loss at expiry?
Answer and explanation
Answer: C
A cash-or-nothing digital call pays the set amount if the price at expiry is above the strike price, and nothing if it is not. The amount received does not grow with how far the price ends above the strike price, so even at a price of 260 yen at expiry, the payoff is 20 yen. Subtracting the premium paid up front gives 20 − 9 = 11 yen, a profit. If the price at expiry is 200 yen or less, there is no payoff, and the 9 yen premium paid is the loss.
A profit of 51 yen treats the option as paying like a standard call with the same strike price (a vanilla call). The payoff would then be 260 − 200 = 60 yen, and subtracting the premium gives 60 − 9 = 51 yen. The payoff of a vanilla call grows with the price at expiry, but a digital call jumps from 0 to 20 yen at the strike price and does not grow beyond that. A profit of 20 yen comes from forgetting to subtract the premium. A loss of 9 yen is the result if the price at expiry does not end above the strike price.
Learn more: Digital Options
Question 4
As a hypothetical example, take a call on an average-price Asian option. The price of the underlying asset is observed 6 times, at the end of each month, and the 6th observation is the expiry. The average is an arithmetic average, and it is above the strike price. Once the 4th observation has been made, the 2 prices not yet observed each rise by 1 yen. About how much does the payoff of the call increase?
Answer and explanation
Answer: D
The 4 prices already observed will not change, so only the 2 prices not yet observed can move the average. The average is the sum of the 6 prices divided by 6. If both of the 2 prices rise by 1 yen, the sum rises by 2 yen. The average therefore rises by 2 ÷ 6 = about 0.33 yen, and because the average is above the strike price, the payoff rises by the same amount. This 0.33 yen is the upper limit of how strongly the payoff responds to a move in the price of the underlying asset (the counterpart of delta). The limit falls as the observations progress.
1 yen is how a standard call (a vanilla option), whose payoff depends only on the price at expiry, behaves. When it is above the strike price, a 1 yen move in the price at expiry moves its payoff by 1 yen. About 0.83 yen is how an average-strike option behaves when only the last observed price rises by 1 yen. Its payoff is the price at expiry minus the average, and if only the last observed price rises by 1 yen, the payoff moves by 5 ÷ 6 = about 0.83 yen. About 0.17 yen is 1 ÷ 6, which counts only 1 of the 2 prices as having risen.
Because the response is small, for the buyer the payoff grows only slowly when the price of the underlying asset rises just before expiry. Likewise, it falls only slowly if the price drops just before expiry.
Learn more: Asian Options
Question 5
For a call on a floating-strike lookback option, how is the payoff at expiry set? Consider the standard form, in which the prices on the start date and at expiry are also observed.
Answer and explanation
Answer: B
A call on a floating-strike lookback option is a right to buy the underlying asset at the lowest price during the period, and its payoff is the price at expiry minus the lowest price. The price at expiry is also a candidate for the lowest price, so this amount is never negative. The price on the start date is also a candidate, so the payoff does not fall below that of a standard call (a vanilla option) whose strike price is the price on the start date.
The highest price during the period minus the price at expiry is the payoff of a floating-strike put option (put), which mixes up the call and the put. The lowest price minus the price at expiry reverses the direction of the subtraction in the correct answer and is 0 or less. The price at expiry minus the price on the start date is the payoff of a standard call whose strike price is the price on the start date (0 if negative), a form that does not look at prices during the period.
The deeper the price of the underlying asset falls along the way, the lower the lowest price and the larger the payoff. The premium the buyer pays up front is correspondingly higher than for a standard call. If the payoff is below the premium, the buyer loses.
Learn more: Lookback Options
Question 6
As a hypothetical example, take a European call on a basket option whose underlying asset is the basket value, the weighted sum of the prices of stocks X and Y (weight 0.5 each). The strike price is 100 yen, and at expiry the price of X is 140 yen and the price of Y is 70 yen. What is the payoff at expiry per unit?
Answer and explanation
Answer: C
The basket value is the weighted sum of the prices of the stocks. At expiry it is 0.5 × 140 + 0.5 × 70 = 105 yen. The payoff is the amount by which the basket value exceeds the strike price, which is 105 − 100 = 5 yen. The rise in X, 0.5 × 40 = 20 yen, and the fall in Y, 0.5 × 30 = 15 yen, are added together before the payoff is set. The net is 20 − 15 = 5 yen.
The 20 yen answer is the payoff if you buy calls on X and Y separately, 0.5 unit each. It is 0.5 × (140 − 100) + 0.5 × 0 = 20 yen. Each separate call pays only for the rise and does not use the fall to offset it. In this example, with positive weights and a strike price of 100 yen for each stock, the payoff of the basket call never exceeds the payoff of buying calls on X and Y, 0.5 unit each. The 40 yen answer looks only at the 40 yen rise in X and ignores the fall in Y. The 0 yen answer assumes that because Y is below the strike price, the whole payoff disappears. If the basket value is above 100 yen, there is a payoff.
If the basket value at expiry is 100 yen or less, the payoff is 0 yen and the buyer loses the premium paid. For example, if X is 80 yen and Y is 90 yen, the basket value is 85 yen.
Learn more: Basket Options
Question 7
A Japanese company bought a call on a quanto option whose underlying asset is a US-dollar-denominated stock. At expiry the stock price is above the strike price. Which rate is used to convert the dollar-denominated payoff into yen?
Answer and explanation
Answer: A
A quanto option converts the price of an underlying asset quoted in a foreign currency into yen at the fixed rate set at the start of the trade and sets the payoff from that. The exchange rate at expiry does not enter the payoff.
As a hypothetical example, take a strike price of USD 100, a size of 100 shares, a fixed rate of 150 yen per dollar, and a stock price of USD 120 at expiry. The payoff is USD 20 × 100 shares × 150 yen = 300,000 yen. This amount does not change whether the exchange rate at expiry is 160 yen per dollar (a stronger dollar) or 140 yen per dollar (a weaker dollar). An ordinary foreign-currency option converted at the exchange rate at expiry would pay 320,000 yen at 160 yen per dollar and 280,000 yen at 140 yen per dollar, so its payoff changes.
"The exchange rate on the expiry date" is how that ordinary foreign-currency option converts. "Whichever is better for the buyer" comes from the mistaken idea that the exchange rate at expiry is used when the dollar is stronger and the fixed rate when the dollar is weaker. A quanto converts at the fixed rate in both cases. "The average of the daily exchange rates" mixes this up with a type that sets the rate by an average. If the stock price at expiry is at or below the strike price, the payoff is 0 yen at any rate, and the buyer loses only the premium.
Learn more: Quanto Options