Name:
M339D=M389D Introduction to Actuarial Financial Mathematics
University of Texas at Austin
Mock Exam II
Instructor: Milica Čudina
Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points.
Time: 75 minutes
MULTIPLE CHOICE
TRUE/FALSE
1.3 (2)
TRUE
FALSE
?? (5)
a
b
c
d
e
1.8 (5)
a
b
c
d
e
1.9 (5)
a
b
c
d
e
FOR THE GRADER’S USE ONLY:
Def’n
T/F
??
1.4
??
M.C.
Σ
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1.1. DEFINITIONS.
Problem 1.1. (5 points) Write down the definition of what it means for a derivative security to be
short with respect to its underlying asset.
Solution: A derivative security is said to be short with respect to an underlying asset if its payoff is
a nonincreasing/decreasing function of the final price of that asset.
1.2. TRUE/FALSE QUESTIONS. Please, circle the correct answer on the front page of this exam.
Problem 1.2. (2 pts) A non-dividend-paying stock sells today for $100 per share. The yearly effective
interest rate is 0.21. Then, F0,1/2 (S) > 110.
Solution: FALSE
In fact, F0,1/2 (S) = 100(1.21)1/2 = 100 × 1.1 = 110.
Problem 1.3. (2 pts) A (long) put is a short position with respect to the underlying asset price.
Solution: TRUE
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1.3. FREE-RESPONSE PROBLEMS.
Please, record your final numerical answers in the boxes at the bottom of every page with
a free-response problem. Single-word or one-number answers without a justification will
receive zero credit.
Problem 1.4. (25 points) You produce tiramisu cakes. You plan to sell 1,000 cakes in a month.
Your (unhedged) payoff will be $10, 000 − S(1), where S(1) denotes the price of the amount of belgian
chocolate required to dust the 1,000 cakes.
Assume that the continuously compounded annual risk-free interest rate equals 6%.
Your hedge consists of the following two components:
(1) one long one-month, $9,000-strike call option on the amount of chocolate you need; it’s premium is VC (0) = $60.00,
(2) one written one-month, $8,500-strike put option on the amount of chocolate you need; it’s
premium is VP (0) = $200.00.
Calculate the profit of the hedged portoflio if the final price of the amount of chocolate you need
turns out to be $8, 800.
For the final price of chocolate equal to $8,800, the profit of the hedged portfolio is:
Solution: The hedged portfolio consists of the following components:
(1) the payoff from the tiramisu cake sales,
(2) one long one-year, $9,000-strike call option on the chocolate whose premium was $60.00,
(3) one written one-year, $8,500-strike put option on the chocolate whose premium was $200.00.
The initial cost for this portfolio is the cost of hedging (all other accumulated production costs are
incorporated in the revenue expression 10, 000 − S(1)). Their future value is
(60 − 200)e0.005 ≈ −140.70.
As usual, the negative initial cost signifies an initial influx of money for the principal character (in
this case, you: the producer of tiramisu cakes).
The profit for S(1) = 8, 800 is
10, 000 − 8, 800 + 140.70 = 1, 340.70.
Problem 1.5. (10 points)Suppose that the current price of a dividend-paying stock equals $100. Let
r = 0.05 and δ = 0.03. You notice that a forward price for delivery of this stock in two-years equals
F = $110. You suspect that this forward price creates an arbitrage opportunity.
Construct an arbitrage portfolio which consists of taking a position in the forward contract, purchase
or short-sale of the underlying asset, and borrowing/lending at the risk-free interest rate.
Solution: The forward price based on the initial stock price, r and δ equals
F0,T (S) = S(0)e(r−δ)T = 100e(0.05−0.03)·2 ≈ 104.08 < F = 110.
The conclusion is that the observed forward price is “too high”. One way to exploit this arbitrage
opportunity would be to do the following:
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(1) engage in the short forward contract,
(2) borrow S(0)e−δT at the risk free rate to be repaid at time T ,
(3) use the above loan to buy e−δT shares of stock.
So, the initial cost of this portfolio is zero.
During the time period (0, T ], all of the continuously paid dividends are automatically reinvested
in the asset S. So, at the end, one share of stock is owned. If we want to, we can deliver it to fulfill
the forward contract. Thus, at time−T , the payoff is
−(S(T ) − F ) + S(T ) − e(r−δ)T S(0) > 0.
The portfolio we constructed is, indeed, an arbitrage portfolio.
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1.4. MULTIPLE CHOICE QUESTIONS.
Please, circle the correct answer on the front page of this exam.
Problem 1.6. (5 points) A stock’s price today is $1000 and the annual effective interest rate is given to
be 5%. You write a one-year, $1,050-strike call option for a premium of $10 while you simulataneously
buy the stock. What is your profit if the stock’s spot price in one year equals $1,200?
(a) $150.00
(b) $139.90
(c) $10.50
(d) - $39.00
(e) None of the above.
Solution: (c)
S(T ) − 1000(1.05) − (S(T ) − K)+ + 10(1.05) = 1050 − 990(1.05) = 10.50.
Problem 1.7. (5 points) The S&P 500 Index is priced at $950.46. The annualized dividend yield on
the index is 1.40%. The continuously compounded annual interest rate is 8.40%. What is the forward
price for a forward contract with delivery date 9 months from today?
(a) $937.48
(b) $942.66
(c) $984.36
(d) $1001.69
(e) None of the above.
Solution: (d)
In our usual notation,
F0,T (S) = S(0)e(r−δ)T = 950.46e(0.084−0.014)0.75 ≈ 1001.69.
Problem 1.8. (5 points) A non-dividend-paying stock sells for $100 per share today. The one-year
forward price is $110. You short sell the stock and close the short sale in exactly one year. Find your
profit if the stock’s spot price in one year equals $130 per share.
(a) 20 loss
(b) 20 gain
(c) 30 loss
(d) 30 gain
(e) None of the above.
Solution: (a)
Because the stock pays no dividends, we have that F V0,T (S(0)) = F0,T (S). So, the profit equals
−S(1) − F V0,1 (−S(0)) = −S(1) + F0,1 (S) = −130 + 110 = −20.
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Problem 1.9. (5 points) To plant and harvest 20,000 bushels of corn, Farmer Jayne incurs costs
totaling $33,000. The current spot price of corn is $1.80 per bushel. What is the profit if the spot
price is $1.90 per bushel when she harvests and sells her corn?
(a) About $3,000 gain
(b) About $3,000 loss
(c) About $5,000 loss
(d) About $5,000 gain
(e) None of the above
Solution: (d)
1.90 · 20, 000 − 33, 000 = 5, 000
Problem 1.10. Farmer Jayne bought a $1.70 strike put option for $0.11 and sold a $1.75 strike call
option for a premium of $0.14. Both options expire in six months. Her total costs of producing the
corn are $1.65 per bushel. She will sell the 20, 000–bushel corn crop in six months. Assume that the
effective interest rates for a six month period are 4.0% What is the minimum per-bushel profit in her
strategy?
(a) $624
(b) $1,624
(c) $2,624
(d) $3,624
(e) None of the above.
Solution: (b)
Per bushel, Farmer Jayne’s profit is
(K1 − ST )+ − (ST − K2 )+ − 1.65 + ST − (P − C)(1 + i)
where ST denotes the price of a bushel of corn in six months, i.e., at time T = 0.5, and where
K1 = 1.70, K2 = 1.75, C = 0.14, P = 0.11 and i = 0.04. If you draw the graph of the profit above, you
will notice that it is nondecreasing in ST . This means that the minimum profit occurs for ST = 0. In
that case, the profit for the 20,000 bushels of corn is
20, 000(1.70 − 1.65 − (−0.03)(1.04)) = 1, 624.
Problem 1.11. (5 points) The initial price of the market index is $900. After 3 months the market
index is priced at $915. The nominal rate of interest convertible monthly is 4.8%.
The premium on the put, with a strike price of $930, is $8.00. What is the profit or loss at expiration
for a short put?
(a) $15.00 loss
(b) $6.90 loss
(c) $6.90 gain
(d) $15.00 gain
(e) None of the above.
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Solution: (b)
The profit for a short put is
0.048 3
−(K − S(T ))+ + VP (0) 1 +
= −15 + 8 · 1.0043 = −6.90.
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Problem 1.12. The initial price of the market index is $900. After 3 months the market index is
priced at $920. The nominal rate of interest convertible monthly is 4.8%.
The premium on the long call, with a strike price of $930, is $2.00. What is the profit or loss at
expiration for this long call?
(a) $2.00 loss
(b) $2.02 loss
(c) $2.02 gain
(d) $2.00 gain
(e) None of the above.
Solution: (b)
In our usual notation, the profit is
(ST − K)+ − C · (1 + j)3
with C denoting the price of the call and j the effective monthly interest rate. We get
(920 − 930)+ − 2 · 1.043 ≈ −2.02.
Problem 1.13. Assume that you open a 100−share short position in a common stock S when the bidask is $100.00-$101.00. When you close your position the bid-ask prices are $99.50-$100.00. Assume
that you pay a commission rate of 1.00%. Calculate your (roundtrip) gain or loss on this short
investment (assume r = 0)?
(a) The investor breaks even; i.e., the gain/loss is 0.
(b) About $200 loss
(c) About $132.50 loss
(d) About $200 gain
(e) None of the above
Solution: (b)
The commission needs to be paid for both transactions, so the total outcome for the short-seller is
100(100 · 0.99 − 100 · 1.01) = −200.
Problem 1.14. A market index is currently trading at $1,000. Which of the following options is/are
in the money? More than one answer can be true. You get the credit if you circled all acceptable
answers and none of the incorrect ones.
(a) $1,500-strike put
(b) $900-strike put
(c) $1,250 strike call
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(d) $950 strike call
(e) None of the above.
Solution: (a) and (d)
SHORT-ANSWER QUESTIONS
Problem 1.15. (10 points) Believe it or not, we all enter forward contracts on a daily basis. Explain,
in eight lines or fewer, how ordering a pizza is a forward contract. Ignore time-limits on when the
pizza should be delivered. Imagine that the pizza is to be delivered in 30 minutes exactly.
Solution: The quantity and type of pizza is specified at order time. So is the agreed upon price and
the delivery time. The logistics are also explained (you give your address and contact info). The pizza
parlor will confirm that you, indeed, ordered your pizza. While there are no formal steps to aleviate
“credit risk” (you decide you do not want the pizza, even though you did order it, and you refuse to
pay), you might get “blacklisted” by a particular pizza parlor.