Derivatives Problem set

“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
Group: Bárbara Bello, Kejsi Hasanaj and Dziugas Staniulis
Problem Set 1
1. (3 points) On 17th, March, 2029, an investor takes a long position at 9am in a futures
contract (on ABC shares) with the following characteristics:
• Contract maturity: 19th, March, 2029.
• Contract Size:
100 shares.
• (Initial) futures price: $49.44 / share.
• Initial margin:
$600 / contract.
• Maintenance level: $460 / contract.
• The futures market opens at 9am and closes at 5pm.
• This underlying asset will not pay dividends.
The next table provides the evolution of spot prices for the ABC share:
Day
17 – March
18 – March
19 – March
9am
48
46.5
48
5pm
47.5
47
49
Assume that the daily risk-free interest rate is 1% (discretely compounded). Then, complete
(and explain briefly) the following table.
Day
Future
Settlement
Price
17 – March 47.5
18 – March 47
19 – March 49
Daily
Profit /
Loss
Accumulated
Profit / Loss
Margin
Account
Balance
-194
-50
+200
-194
-244
-44
406
550
550
Additional
Margin
194
-
From the table, we can see that the buyer (long position) is obligated to perform one margin
call (March 17th) since the price of ABC shares decreases incurring losses of $194 and
lowering the margin account under the maintenance margin barrier. In the maturity day of the
contract, the price increases reducing the daily losses incurred by $200. At the end of the
contract, the buyer has lost $44 and gets back $550 from the margin account, totaling a loss of
$94.
2. (2 points) Assume the annual risk-free rate is 2%, the annual dividend yield on the S&P500
is 3%, and the price of a 1-year futures contract is 1,000.
a) What is the implied fair value of the index?
b) What is the value of a 2-year futures contract on the index?
a) Implied fair value of the index is $990.049.
1,000=S_oe^(2%-3%)1
S_o=990.049
b) The value of the 2-year futures contract on the index is $1010.050.
F(0,2)=990.049e^(2%-3%)2
F(0,2)=1010.050
Problem Set, “Derivatives and Risk Management”
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“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
3. (2 points) The two-month interest rate in Switzerland and the United States are 3% and 8%
per annum, respectively, with continuous compounding. The spot price of the Swiss franc is
$0.6500. The futures price for a contract deliverable in two months is $0.6600. Explain
carefully the arbitrage opportunity and the final outcome.
Non-arbitrage future contract’s price = 0.6500e^((8%-5%)*2/12)) = 0.653258
Since 0.653258 < 0.66000, the arbitrage strategy would be buying Swiss francs and short
sale the futures of the Swiss francs.
Problem Set, “Derivatives and Risk Management”
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“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
4. (2 points) Fill in the following payoff table (as a function of the stock price in 1 year) of a
portfolio that is (i) long one share of stock, (ii) long one 1-year put with an exercise price of
$50, (iii) short one 1-year call with an exercise price of $60, (iv) short one 1-year call with
an exercise price of $80, and (v) short a 1-year zero-coupon bond with a face amount of
$40. Graph the payoff as a function of the stock price in 1 year. (Note: Payoff not profit,
ignore the cost of these securities.)
Price
Long Stock
Long Put (X=50)
Short Call (X=60)
Short Call (X=80)
Short Bond (F=40)
Total Payoff
$30 $40 $50 $60 $70 $80 $90 $100 $110
$30
$20
$0
$0
-$40
$10
$40 $50 $60 $70 $80 $90 $100
$10 $0
$0
$0
$0
$0
$0
$0
$0
$0
-$10 -$20 -$30 -$40
$0
$0
$0
$0
$0
-$10 -$20
-$40 -$40 -$40 -$40 -$40 -$40 -$40
$10 $10 $20 $20 $20 $10 $0
$110
$0
-$50
-$30
-$40
-$10
5. (2 points) A four-month European call option on a dividend-paying stock is currently selling
for $5. The stock price is 64$, the strike price is $60, and a dividend of $0.80 is expected in
one month. The risk-free interest rate is 12% per annum for all maturities. Explain carefully
the arbitrage opportunity and the final outcome.
In order to form a conclusion for the arbitrage opportunities, we need to value the present values
of both the price and the dividend.
The PV of the strike price is 60e^(-0.12x4/12) = 57.648
On the other hand the PV of the dividend is 0.80e^(-0.12x1/12)=0.789
In this situation since 5<6.56 (65 – 57.648-0.789), an investor willing to do arbitrage should buy
the option and, after that, short the stock making a profit of $59 ($64-$5). On the other hand he
would invest 0.79 at the rate of 12% to pay dividend of $0.80. What is left which is $58.21 is
invested for four months at 12%
6. (2 points) A one-month European put option on a non-dividend-paying stock is currently
selling for $2.50. The stock price is $47, the strike price is $50, and the risk-free interest rate
is 6% per annum for all maturities. Explain carefully the arbitrage opportunity and the final
outcome.
The PV of the strike price is 50e^(-0.06x1/12)= 49.751
Since we have 2.5 < 2.75 (49.75-47), the investor should buy the stock and the put option by
borrowing the total amount invested $49.50 at the 6% to take advantage of arbitrage opportunity.
1. If the stock price is more than $50 the option expires without any worth whatsoever, but
the stock can be sold for a minimum of fifty. $50 that we take in one month has a present
value of $49.75 today. The profit we make is definitely the difference of 0.25.
2. If the stock price is less than $50, the option is exercised, and the stock will be sold for
exactly $50. We generate a profit of $0.25 in present value.
7. (3 points) The price of a European call that expires in six months and has a strike price of
$30 is $2. The underlying stock price is $29 and a dividend of $0.50 is expected in two months
and again in five months. The term structure is flat, with all risk-free interest rates being 10%.
Problem Set, “Derivatives and Risk Management”
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“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
a) What is the price of a European put option that expires in six months and has a strike
price of $30?
C+Xe^(-rT)+D=p+S – This is the main form of the equation
p= 2 + 30e-0.1*0.6+ (0.5e-0.1*2/12+0.5e-0.1*5/12) – 29= 2.3487  $2.35
b) Explain carefully the arbitrage opportunity and the final outcome if the European put
price is $3.
If the price is 3, then we can simply buy the call and short both put and stock and thus
we will get at 0.65 of profit (the difference).
8. (2 points) A bond issue by Standard Oil worked as follows. The holder received no interest.
At the bond's maturity the company promised to pay $1,000 plus an additional amount based
on the price of oil at that time. The additional amount was equal to the product of 170 and the
excess (if any) of the price of a barrel of oil at maturity over $25. The maximum additional
amount paid was $2,550 (which corresponds to a price of $40 per barrel). (Intuition: If the
price of the commodity went up, the company was in good position to provide the bondholder
with the additional payment).
Show how to design this bond, that is, describe in detail the combination of assets and
derivatives that replicates the pay-off of this bond.
We can see that the price is a combination of a long call, a short call, and a regular bond
If we call “P” the price that it is mentioned in the exercise ( the oil price ) than by what we
understand from the content of what it is given, we can write the below equations :
If P < 25 we get nothing
If P > 25 and smaller than 40 than the equation stands at 170(P-25)
If P > 40 than we what we get is 2250$ (as given)
We can interpret this as a bull spread which is very evident actually because this bond is the same
with a regular bond adding a long position in 170 call options of oil having the strike price that
we mentioned as $25 adding here obviously a short position in 170 call options on oil with a
strike price of now 40$.
9. (2 points) Consider European calls and puts on a non-dividend paying stock index. Case
assumptions are S=100, X=100, T=1, rf = 2% (continuous compounding), = 40%. Using the
Black-Scholes-Merton model, compute premiums, deltas, and betas for calls and puts.
Using the formulas, we obtain:
D1 = ( (ln(100/100) + (0.02+(0.5*0.4^2)*1) / (0.4*sqtr1) = 0.1
D2= D1 – 0.4sqrt1= -0.3
Problem Set, “Derivatives and Risk Management”
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“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
Call premium = (100*0.1)-100e^(-0.02*1)*-0.3
Call premium =10 – (-29.4) = $39.4
Put option premium= 100e^(-0.02*1)*-(-0.3) – (100*(-0.1))
Put option premium= 29.4 – (-10) = $39.4
We don’t know how to compute the betas and deltas. 
Problem Set, “Derivatives and Risk Management”
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“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
Problem Set 2
1. (1 point) Suppose that you enter into a short future contract to sell July silver for $5.20 per
ounce on the New York Commodity Exchange. The size of the contract is 5,000 ounces. The
initial margin is $4,000 and the maintenance margin is $3,000. What change in the futures
price will lead to a margin call? What happens if you do not meet the margin call?
A loss of $1,000 ($4,000 - $3,000 = $1,000) will lead to a margin call. Therefore, an
increase of more than $0.20 ($1,000 = 5,000 * X; X= 0,2) in the price of silver per ounce
would lead to a margin call on the futures’ contract.
In the case this happens, the seller will receive a margin call to deliver additional
money/fixed income asset to the clearinghouse of a value of $1,000 or more depending on
the magnitude of the futures’ price increase. In the case the seller does not fulfill this
delivery, the position will be liquidated.
2. (1 point) An investor enters into two long futures contracts on frozen orange juice. Each
contract is for the delivery of 15,000 pounds. The current futures price is 160 cents per pound,
the initial margin is $6,000 per contract, and the maintenance margin is $4,500 per contract.
What price change would lead to a margin call? Under what circumstances $2,000 could be
withdrawn from the margin account?
A margin call will happen when the investor losses $1,500 (initial margin ($6,000) –
maintenance margin ($4,5000) = $1,500), and this will happen if the price per pound lowers
under 150 cent per pound ( $1,500 = 15,000 * X, X= 10 cents, 160 cents -10 cents = 150
cents).
Since the investor contracted two futures contract, he/she will have to earn $1,000 ($2,000 /
2 = $1,000) to be able to withdraw the $2,000 from the margin account. This situation will
happen if the price of the orange juice pound increases by 6.67 cents ($1,000 = $15,000 *
X; X = $1,000/$15,000 ; X = $0.0667). Therefore, if the price of the orange juice per pound
reaches 166.7 cents, the investor will be able to withdraw $2,000 from the margin account.
3. (1 point) At the end of one day a Clearinghouse member is long 100 contracts and the
settlement price is $50,000 per contract. The original margin is $2,000 per contract. On the
following day the member becomes responsible for clearing an additional 20 long contracts,
entered into at a price of $51,000 per contract. The settlement price at the end of this day is
$50,200. How much does the member have to add to its margin account with the exchange
Clearinghouse?
You initial margin account has $200,000 from the 100 long contracts at $50,000 each. Once
you enter into 20 more long contracts, you need to add $40,000 more.
At the end of the day, settlement price is $50,200 per contract, therefore you have:
a. Gains from the 100 contracts at $50,000: $20,000 ( ($50,200 - $50,000)* 100 =
$20,000)
b. Losses from the 20 contracts at $51,000: $16,000 ( ( $50,200 - $51,000)*20 = $16,000)
Therefore, at the end, the member has to add a margin of $56,000 for the 20 new contracts
($40,000 (initial margin) + $16,000 (loss in the daily operation)), but since he/she has a gain
in the 100 contracts from the beginning of the day of $20,000, at the end of the day he/she
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Problem Set, “Derivatives and Risk Management”
“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
will have to add $36,000 ($56,000 - $20,000) to its margin account with the exchange
Clearinghouse.
4. (1 point) Assume that the risk-free interest rate is 9% per annum with continuous
compounding and that the dividend yield on a stock index varies throughout the year. In
February, May, August, and November, dividends are paid at a rate of 5% per annum. In other
months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on
July 31, 2002, is 300. What is the futures price for a contract deliverable on December, 31,
2002?
Since we are at July 31 with index value of 300:
Average dividend yield= (5%+2%+2%+5%+2%)/5 = 3.2%
December futures’ price = 300e^((9%-3.2%)*5/12)= $307.338
5. (2 points) Suppose that the risk-free interest rate is 10% per annum with continuous
compounding and that the dividend yield on a stock index is 4% per annum. The index is
standing at 400 and the futures price for a contract deliverable in four months is 405. Describe
the arbitrage opportunity that exists and detail the final outcome of the arbitrage strategy that
can be implemented.
Futures’ price w/o arbitrage opportunity = 400e^((10%-4%)*4/12)=$408.08
The potential arbitrage opportunity, since F(0,T)<S_oe^(r-q)T, would be to buy a futures’
contract at 405 and short the stocks of the index. In four months, you will use the money
from shorting to pay 405 of the futures
6. (3 points) Explain in detail if the following statements are true or false:
• (1 point) “A short hedger's position improves when the basis strengthens
unexpectedly and worsens when the basis weakens unexpectedly.”
True. A short hedger’s position has a long position on the underlying asset and a
short position on the respective futures’ contract. Being the basis as the difference
between the spot price and the futures price, its position benefits from an increased
basis, and worsens from a decreased basis.
• (1 point) ”If the minimum variance hedge ratio is calculated as 1.0, the hedge must
be perfect".
False. Even if the MVHR is equal to 1, the correlation coefficient (p) can be just 0.5
and Std Dev S/ Std Dev F= 2 leading to a MVHR of 1.0 when their correlation is
just 0.5 whereas a perfect hedge needs p=1
• (1 point) “If there is no basis risk, the minimum variance hedge ratio is always 1.0".
True. If there is no basis risk, it means that the hedge is perfect since the correlation
between the underlying asset and the futures’ asset is 1 (implying it is the same asset).
7. (1 point) The standard deviation of monthly changes in the spot price of live cattle is 1.2 (in
cents per pound). The standard deviation of monthly changes in the futures price of live cattle
for the closest contract is1.4. The correlation between the futures price changes and the spot
price changes is 0.7. It is now October 15. A beef producer is committed to purchasing
200,000 pounds of live cattle on November 15. The producer wants to use the December live
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Problem Set, “Derivatives and Risk Management”
“Derivatives and Risk Management”
Master Sc. in Banking and Finance, Barcelona School of Management - UPF
Professor: Manuel Moreno. [email protected]
cattle futures contract to hedge its risk. Each contract is for the delivery of 40,000 pounds of
cattle. What strategy should the beef producer follow?
The beef producer, in order to hedge for the risk of the live cattle’s price to increase, should enter
a long position on live cattle’s futures.
HR = 0.7 (1.2/1.4) = 0.6 is the optimal hedge ratio.
200,000/40,000 * 0.6 = 3 futures contracts would be the optimal hedge ratio
Therefore, the beef producer should long 3 December futures contracts of 40,000 pounds of live
cattle each to offset the risk of buying 200,000 pounds of live cattle on November 15.
Problem Set, “Derivatives and Risk Management”
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