Barrier Options: Practice Problems and Solutions

This is Section 53 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. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here. See Section 49 here. See Section 50 here. See Section 51 here. See Section 52 here.

The payoff for a barrier option depends on whether the underlying asset price ever reaches a specified level - or barrier - during the life of the option. Barrier puts or calls are always no more expensive than otherwise equivalent regular puts or calls, because the payoff on the former can never be larger than the payoff on the latter.

Three types of barrier options are

Knock-out options: Options go out of existence when the asset price reaches the barrier. These options are called down-and-out when the price has to decline to reach the barrier. They are called up-and-out when the price has to increase to reach the barrier.

Knock-in options: Options come into existence when asset price reaches the barrier. These options are called down-and-in when the price has to decline to reach the barrier. They are called up-and-in when the price has to increase to reach the barrier.

Rebate options: Options make a fixed payment if the asset price reaches the barrier. The payment can be made at the time the barrier is reached or at the time the option expires. If the latter is true, then the option is deferred rebate. Up rebate options occur when the barrier is above the current price. Down rebate options occur when the barrier is below the current price.

For barrier options, the following parity relationship holds:

Knock-in option + Knock-out option = Ordinary option

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 14, pp. 449-451.

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

Problem BO1. The stock of Tradable Co. once traded for $100 per share. Several barrier option contracts were then written on the stock. Suddenly, the stock price increased to $130 per share - which is the barrier for the options. Which of these barrier options would have a positive payoff in this case? More than one answer may be possible.

(a) Up-and-out put with a strike price of $120
(b) Up-and-out call with a strike price of $120
(c) Up-and-in put with a strike price of $120
(d) Up-and-in call with a strike price of $120
(e) Rebate option that pays a rebate of $12

Solution BO1. Since the barrier has been reached, the properties of all these options have been triggered. The up-and-out options go out of existence when the barrier is reached, so (a) and (b) are now worthless. The up-and-in put in (c) went into existence when the barrier was reached, but the stock price is above its strike price, so it is also worthless. The up-and-in call in (d), however, now has a payoff of 130-120 = $10. The rebate option in (e) has also been triggered and will now pay $12 to the holder. Thus, (d) and (e) will have positive payoffs.

Problem BO2. An ordinary call option on Lucrative Co. stock with a strike price of $50 and time to expiration of 1 year trades for $4. An otherwise identical up-and-in call option on Lucrative Co. stock with a barrier of $55 trades for $2.77. Find the price of an up-and-out call option on Lucrative Co. stock with a barrier of $55, a strike price of $50, and time to expiration of 1 year.

Solution BO2. We use the formula

Knock-in option + Knock-out option = Ordinary option

We are given that

Ordinary option = 4 and Knock-in option = 2.77. Thus,

Knock-out option = 4 - 2.77 = $1.23

Problem BO3. You own a portfolio of the following options on the stock of Imperious LLC:

- An up-and-in call with strike = $63 and barrier = $66

- An up-and-out put with strike = $78 and barrier = $65

- An up rebate option with rebate = $14 and barrier = $61

- An up-and-out call with strike = $35 and barrier = $61

Originally, the stock traded at $59 per share. Right before the options expired, the stock began to trade at $63 per share, its record high. What is your total payoff on this portfolio?

Solution BO3. The barrier for the up-and-in call with strike = $63 and barrier = $66 was not hit, so this option is worthless. The barrier for the up-and-out put with strike = $78 and barrier = $65 was not hit, so this option is still in existence and has the payoff of a regular put: 78 - 65 = 13. The barrier for the up rebate option was reached, so this option pays its rebate of $14. The barrier for the up-and-out call with strike = $35 and barrier = $61 was reached, so this option is worthless. Thus, the total payoff on this portfolio is

13 + 14 = $27.

Problem BO4. Hildebrand owns a portfolio of the following options on the stock of Imperious LLC:

- An up-and-in call with strike = $63 and barrier = $66

- An up-and-out put with strike = $78 and barrier = $65

- An up rebate option with rebate = $14 and barrier = $61

- An up-and-out call with strike = $35 and barrier = $61

Originally, the stock traded at $59 per share. Right before the options expired, the stock began to trade at $67 per share, its record high. What is Hildebrand's total payoff on this portfolio?

Solution BO4. The barrier for the up-and-in call with strike = $63 and barrier = $66 was reached, so this call now has a payoff of 67 - 63 = 4. The barrier for the up-and-out put with strike = $78 and barrier = $65 has been reached, so this option is now worthless. The barrier for the up rebate option with rebate = $14 and barrier = $61 has been reached, so the option pays its rebate of $14. The barrier for the up-and-out call with strike = $35 and barrier = $61 has been reached, so the option goes out of existence and is worthless. Thus, Hildebrand's total payoff is 4 + 14 = $18

Problem BO5. The stock of Tractable Co. pays no dividends. The stock currently trades at $54 per share. An up-and-in call with strike = $55 and barrier = $60 has a price of $3.04, and an up-and-out call with strike = $55 and barrier = $60 has a price of $1.32. The options expire in 2 months, and the annual continuously compounded interest rate is 0.03. Find the price of one ordinary put option on the stock of Tractable Co. with strike price of $55 and time to expiration of 2 months.

Solution BO5. First we use the formula

Knock-in option + Knock-out option = Ordinary option, from which we find that the price of the ordinary call for this stock is 3.04 + 1.32 = 4.36

Now we use put-call parity: C - P = S - Ke^-rT, where r = 0.03, T = 1/6, K = 55, C = 4.36, and S = 54. Thus, P = C - S + Ke^-rT = 4.36 - 54 + 55e^-0.03/6 = P = $5.085686356

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