Financial Derivatives
Spring 2016
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Problem Set 3: Option Pricing and Hedging
1.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation (ignoring transactions costs).
• Digital Datawhack stock is selling for $95 per share.
• Call options on Digital Datawhack at 90, with August 2016 expiration, are
selling for $12 per share.
• Put options on Digital Datawhack at 90, with August 2016 expiration, are selling
for $2.50 per share.
• At the current T-Bill rate, $86 invested today would grow to $90 at the expiration
date of the options.
2.
The following prices are observed. Formulate an arbitrage strategy to profit from the
situation (ignoring transactions costs).
• Arcott Labs stock is selling for $50 per share.
• Call options on Arcott Labs at 45, with expiration in nine months, are selling for
$11 per share.
• Call options on Arcott Labs at 50, with expiration in nine months, are selling for
$2 per share.
• Put options on Arcott Labs at 45, with expiration in nine months, are selling for
$3 per share.
• Put options on Arcott Labs at 50, with expiration in nine months, are selling for
$5 per share.
• At the current T-Bill rate, $42.70 invested today would grow to $45 (and $47.44
would grow to $50) at the expiration date of the options.
3.
The following prices are observed. Show how to borrow money at the risk-free rate,
using only option contracts.
• Morgenstern Labs stock is selling for $91.50 per share.
• Call options on Morgenstern Labs at 85, with expiration in nine months, are
selling for $14.50 per share.
• Call options on Morgenstern Labs at 90, with expiration in nine months, are
selling for $11.875 per share.
• Put options on Morgenstern Labs at 85, with expiration in nine months, are
selling for $3.75 per share.
• Put options on Morgenstern Labs at 90, with expiration in nine months, are
selling for $5.875 per share.
• At the current T-Bill rate, $85.50 invested today would grow to $90 (and $80.75
would grow to $85) at the expiration date of the options.
Financial Derivatives
Spring 2016
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4.
Digital Datawhack stock rose by $2, from $60 to $62. “Crunch” Murdoch owns call
options on Datawhack, which he bought two months ago for $5 per share. They
expire in seven months, and have an exercise price of $55. Which of the following is
a theoretically possible response of the option value (on a per-share basis), to the
recent move of the stock?
a. rise by $2.10
b. rise by $1.80
c. fall by $2.05
d. fall by $1.75
e. None of the above
5.
Option A is a call option written on Digital Datawhack stock, with expiration in one
month. Option B is a call on Datawhack with expiration in six months. Both options
have an exercise price of $50 per share. Which of the following is a theoretically
correct relationship?
a. Option A is worth more than Option B.
b. Option B is worth more than Option A.
c. Option B is worth the same as Option A.
6.
Option A is a call option written on Digital Datawhack, with exercise price of $50.
Option B is also a call option on Datawhack, with exercise price of $55. Both
options expire six months from now. Which of the following is a theoretically
correct relationship?
a. Option A is worth more than Option B.
b. Option B is worth more than Option A.
c. Option B is worth the same as Option A.
7.
Suppose Treasury Bill rates have risen, but nothing else has changed. What would
be the theoretical response of call option values (for calls written on U.S. stocks)?
a. rise
b. fall
c. remain unchanged
8.
Suppose the exercise price of an option is $50, the Treasury Bill rate is 5%, and there
are 179 days remaining until expiration. Discounted continuously, what is the
present value of the exercise price?
9.
Suppose the stock price is $56 and the exercise price of a call option is $50.
Volatility of the stock is a standard deviation of 20%. The information in problem 8
holds for this call. Do the following calculations:
a. Find d1 and d2
b. Using the Normal Distribution table, estimate N(d1) and N(d2)
c. Use these estimates to calculate the value of the call option
d. Use the option calculator to find the value of this call. Explain why the
computer’s answer differs from your hand calculations.
e. What is Delta for this option?
f. Find the value of a put with the same exercise price and expiration.
Financial Derivatives
Spring 2016
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10. Use the option calculator spreadsheet to find Gamma for the option in problem 9.
Suppose there is also an option with the same expiration and an exercise price of
$55. Formulate a Gamma hedge for the option in problem 9. (See the discussion of
Gamma hedging in the text.)
11. Suppose the option in problem 9 has a market premium of $822.80 for a 100-share
contract ($8.228 per share). Use the option calculator spreadsheet or one of the
programs provided with the text to find the implied volatility by trial and error.
12. Explain each of the following concepts as they relate to call options. What roles do
they play in hedges?
a. Delta
b. Gamma
c. Rho
d. Vega
e. Theta
13. Explain what we mean when we say that the binomial model is a discrete time model
and the Black-Scholes model is a continuous time model.
14. Be prepared to explain Brownian Motion in class, giving examples. What makes us
think stock price changes follow this pattern?
15. Suppose you subscribe to a service that gives you estimates of the theoretically
correct volatilities of stocks. You note that the implied volatility of a particular
option is substantially higher than the reported volatility. Assuming you trust the
accuracy of the estimates you receive, what action would you consider?
16. Answer the following questions as they relate to implied volatilities.
a. Can implied volatilities be expected to vary for options on the same stock with
the same exercise prices but different expirations?
b. Can implied volatilities be expected to vary for options on the same stock with
the same expiration but different exercise prices?
17. Is it practical to expect a delta hedge to be risk free? Explain.
18. Is it practical to expect a gamma hedge to be risk free? Explain.
Financial Derivatives
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2.
3.
Solutions: Problem Set 3
This is a violation of put-call parity,
which offers a nice opportunity for
arbitrage. To take advantage of it, buy
stock and puts for $97.50 per share.
Sell calls and bonds (that is, write a call
and borrow the present value of the
exercise price) and receive $98.00 per
share. Having bought a put and written
a call with exercise price of $90 per
share, you are committed to sell your
stock for $90, which will be just enough
money to pay off the loan. Your future
obligations are covered, and you can
still pocket a profit of $.50 per share. If
you can do this in a large volume, say
100 contracts at 100 shares per contract,
you'll make an arbitrage profit of
$5,000.
This is a double violation of put-call
parity, which offers an opportunity to
take advantage of a box spread. At the
$45 exercise price, you would buy stock
and puts while selling calls and bonds.
At the $50 exercise price, you would
sell stock and puts while buying calls
and bonds. The stock transactions
cancel each other, and the bond
transactions leave you with a net
purchase of $5 face value. You are thus
committed to buy a share of stock for
$50 and sell it for $45 (having bought a
call and written a put with exercise
price of $50 per share, and also having
bought a put and written a call with
exercise price of $45)—so you break
even, after you collect the $5 from the
bond investment. Your future
obligations are completely covered, and
you can still pocket a profit of $6.26 per
share.
Contract to buy stock at $90 per share
and sell it at $85, thus creating an
obligation to pay $5 on the expiration
date (this is the loan repayment). In
order to do this, create a box spread in
which you buy a call and write a put at
the $90 exercise price, while selling a
Spring 2016
call and buying a put at the $85 exercise
price.
4.
B.
5.
B. Option B has longer to expiration.
6.
The lower the exercise price, the higher
the value of the call option. Prior to
expiration, A is worth more than B.
After expiration, both are worthless.
7.
A
8.
Enter .05 * 179/365. Change the sign to
negative, then press the ex button (exp
on some calculators). Multiply the
result times $50. The answer is $48.79.
Save this in the calculator’s memory so
you have the full precision for use in
later problems.
9.
d1 is 1.0543 and d2 is 0.9142. Delta is
0.8541, and the model premium is
$7.84. Remaining questions are for
class discussion.
10. For class discussion
11. For class discussion
12. For class discussion
13. For class discussion
14. For class discussion
15. For class discussion
16. For class discussion
17. For class discussion
18. For class discussion