932N1 – Financial Derivatives – Workshop 3
1. (JC 5.14) The following option prices are given for Sunstar Inc., whose stock price equals
$50.00:
Strike Price
45
50
55
Call Price
5.50
1.50
1.00
Put Price
1.00
1.50
5.55
Compute the intrinsic value of each of these options and identify whether they are in-themoney, at-the-money, or out-of-the-money.
2. (JC 15.6) The price of a put option with strike K1 = 20 is $0.75 and the price of a put
option with strike K2 = 25 is $3.50.
(a) What is a bullish vertical spread
(b) Draw a bullish vertical spread by trading put options with strike prices K2 = 20 and
K4 = 25.
Solution:
(a) A vertical spread is an option trading strategy that involves buying an option and
simultaneously selling an option of the same type (so that the trade involves both
calls or both puts) but with a different strike price, where both options have the
same underlying security and the same date of maturity. The strategy gives a flat
profit (or loss) line for stock prices at expiration that are above the higher strike
price or below the lower strike price. A bullish vertical spread gives a flat profit line
above the higher strike price and a flat loss line below the lower strike price. Traders
set up bullish spreads when they are optimistic about the underlying stock.
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932N1
Financial Derivatives
Workshop #3
(b) See diagram:
3. (JC 15.14) Use the following data for options expiring on the same date. Draw profit diagram
and identify the stock price corresponding to zero profit and the maximum profit and loss
for a long butterfly spread made up of put options. Remember that a long butterfly spread
is something that makes a small profit, if the volatility is low.
Stock
29
Strike Price
25
30
35
Call Price
5
2
1
Put Price
1
3
6
4. (JC 15.18)
(a) What is a collar in the options market?
(b) How would you create a zero-cost collar?
(c) Why might a copper manufacturer find it useful to employ this strategy?
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932N1
Financial Derivatives
Workshop #3
Solution:
(a) A collar on a long stock position is created by adding a long put/s and short call/s.
(b) A zero cost collar is a collar where the cost of purchasing a put/s is paid for by
writing call/s so that no net inflow or outflow of cash occurs.
(c) A copper manufacturer may employ a collar to hedge output price risk to protect
against a decline in copper prices in the market. The company surrenders some of
the potential upside gains from an increase in copper prices due to the written calls.
5. (JC 16.1) Use the following data for European options: Call price = $5, risk-free compounded
interest rate r = 5% per year, stock price S = $55, strike price K = $55, time to maturity
T = 1 month.
If the quoted put price is pQ = $9, show how to capture arbitrage profits in this market.
Solution: The price of a zero-coupon bond maturing in one month is e−rT = e−0.05×1/12 =
$0.995842. By Result 16.1 put-call parity (PCP) for European options, the arbitrage-free
put price
pAF = cQ + BK − S = 5 + 0.9958 × 55 − 55 = $4.77.
This is significantly lower than the quoted put price pQ = $9, suggesting that the quoted
put price is inconsistent with PCP and hence an arbitrage opportunity.
Arbitrage profits are made by selling the relatively overpriced market-quoted put (pQ )
and by buying the underpriced synthetic put (by buying quoted call, buying zero-coupon
bonds equal to present value of the strike, and short-selling the stock). This gives
pQ − pAF = pQ − (cQ + BK − S) = 9 − 5 − 0.9958 × 55 + 55 = $4.23
as immediate arbitrage profits. And, there are two possibilities on the expiration date:
• If the stock price at expiration S(T ) is less than or equal to $55, then the put is
in-the-money and short put has a payoff of −[55 − S(T )]. The $55 available from
maturing zero-coupon bonds is used to pay off the $55, and the stock available
from the put option is used to meet the short stock obligation. This gives a zero
net payoff.
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932N1
Financial Derivatives
Workshop #3
• If S(T ) is greater than $55, then the put expires worthless. However the long
call trade has a payoff of [S(T ) − 55]; from this, the stock is used to meet the
short stock obligation, and the $55 liability is met with the $55 available from the
maturing zero-coupon bonds. This also gives a zero net payoff.
Thus, the portfolio we have created gives an immediate arbitrage profit of $4.23 and has
zero payoffs on other dates under all possible stock prices.
6. (JC 16.6) Using put-call parity, given c = $2, PV(Div)= $1, p = $1, S = K = $100,
r = 0.05 per year and T = 0.25 years, can you make arbitrage profits? Explain.
Solution: The price of a zero-coupon bond maturing in 0.25 years is e−rT = e−0.05×0.25 =
$0.9876. By put-call parity for European options (adjusted for dividends), the arbitragefree put price
pAF = cQ + BK − S + P V (Div) = 2 + 0.9876 × 100 − 100 + 1 = $1.76.
This is higher than the quoted put price pQ = $1, suggesting that the quoted put price is
inconsistent with PCP and hence an arbitrage opportunity. We can set up a strategy by
buying the relatively underpriced market-quoted put and selling the overpriced synthetic
put (by selling quoted call, selling zero-coupon bonds equal to present value of the strike,
buying the stock, and selling zero-coupon bond equal to present value of dividends). This
gives
−pQ +pAF = −pQ +[cQ +BK −S +P V (Div)] = 1+2+0.9876×100−100+1 = $0.76
in immediate arbitrage profit. And it can be shown that the portfolio has zero payoff at
all future dates (on the ex-dividend date and on the expiration date, under all stock price
possibilities).
7. (JC 16.12) It it true that the lower the exercise price, the more valuable the call? Explain
your answer.
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932N1
Financial Derivatives
Workshop #3
Solution: This is true. The lower the strike price, the more chance that the option will
be in-the-money since the stock price has a lower bound to surpass. The lower the strike
price, the more valuable the European call. This result holds for American options as
well.
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