Chapter 2
Additional Questions and Problems
1.
The price of XYZ stock is $52 per share. Six-month European call and put
options, both with a striking price of $45, are traded on the stock. The present
value of the striking price, discounted at the current market rate of interest for six
months, is $43. The market price of the call option is $11 per share. What should
be the theoretical market price of the put?
2.
Consider a one-year European call option with a striking price of $100. The price
of the underlying stock is $108, and the risk-free rate of interest is 5 percent per
year compounded annually. (In other words, $1 invested in a safe asset should be
worth $1.05 after one year.) The stock pays no dividends. What is the lowest
possible value for this call option, assuming the market for the option is efficient?
3.
Consider the following information relevant to the pricing of one-year European
put and call options on XYZ stock.
Stock Price
Striking price for put and call
Market price of put
Market price of call
Risk-free interest rate
$53.00
$50.00
$ 2.00
$ 8.00
5 percent per year compounded annually
Determine whether the put and call options are priced correctly, given the pricing
information shown in the table above. If you determine that the options are not
priced correctly, describe a transaction that should provide you with an immediate
arbitrage profit while guaranteeing that you will have a net payoff of zero one
year from now at the time the options mature.
Project
Testing Put-Call Parity
On Monday, January 7, 2002, options on Microsoft with the maturity dates shown below
were available for trading. Also shown are risk-free rates for investing and borrowing
through each date as of January 7.
Days
Risk-Free
Risk-Free
Actual
until
Investing
Borrowing
Option
Maturity Date Maturity
Rate (%)
Rate (%)
Jan 2002 January 18
11
1.73
2.03
Feb 2002 February 15
39
1.70
2.00
Apr 2002 April 19
102
1.71
2.01
Jul 2002
July 19
193
1.81
2.11
Jan 2003 January 17
375
2.40
2.70
Jan 2004 January 16
739
3.32
3.62
All interest rates are annualized, assuming annual compounding.
Example present value calculation: Present value of $1 to be
received from investing through July 19, 2002:
1
= 0.99056
1.0181193/ 365
On the web site for the textbook, http://www.rendleman.com/book, you should download
the Excel file MICROSOFT PUT_CALL_JAN 7, 2002.xls to use in connection with this
project. The file contains price data for all exchange-traded options in Microsoft as of
10:01 A.M., January 7, 2002. A portion of the data from the file is shown below.
Stock
Bid
69.70
Bid
02 Jan 30.00 (MQF AF-E)
02 Jan 35.00 (MQF AG-E)
02 Jan 40.00 (MQF AH-E)
02 Jan 45.00 (MQF AI-E)
02 Jan 50.00 (MSQ AJ-E)
02 Jan 55.00 (MSQ AK-E)
02 Jan 60.00 (MSQ AL-E)
02 Jan 65.00 (MSQ AM-E)
02 Jan 70.00 (MSQ AN-E)
Ask
Volume
69.71
6,326,600
Calls
Ask
Volume
Open Int
Bid
Puts
Ask
Volume
Open Int
39.5
39.9
0
1616
0
0.05
0
9697
34.5
34.9
0
1421
0
0.05
0
12829
29.6
29.9
0
4243
0
0.05
0
12677
24.6
24.9
0
11804
0
0.05
0
26208
19.6
19.9
5
19175
0
0.1
0
72228
14.6
14.9
20
25584
0
0.1
0
63059
64539
9.7
10
6
54899
0.05
0.15
10
5.1
5.3
123
68097
0.35
0.5
71
70630
1.55
1.7
457
124510
1.8
2
121
114868
For each option, a bid and ask price is given along with the volume of trading as of 10:01
and the open interest. Open interest represents the number of option contracts
outstanding.
For each maturity and striking price combination, compute the up-front cost of
establishing a position involving stock, a call, a put and safe asset that will have a payoff
of zero at maturity based on the following two put-call-parity-based pricing relationships:
0 = stock + put − call − pv ( strike )
0 = call + pv ( strike ) − stock − put
The first equation assumes you buy the stock, buy the put, write the call and borrow the
present value of the striking price. The second equation assumes the opposite; you buy
the call, buy a safe asset that will pay the striking price when the options mature, write
the put and short the stock. When computing up-front costs assume all security
purchases are made at ask prices and that all sales, writing positions, and borrowing is
done at bid prices.
If you find any arbitrage opportunities, see if you can come up with rational explanations.