Observe the following options prices for February 26, 2001:
Call Call Call Put Put Put S X MAR APR JUL MAR APR JUL $29.50 $25.00 4.88 5.30 7.00 0.44 1.00 2.08 $29.50 $27.50 2.63 3.63 - 1.00 1.94 - $29.50 $30.00 1.56 2.44 - 2.00 3.00 - $29.50 $32.50 0.53 1.83 - 3.50 4.50 - $29.50 $35.00 0.31 0.81 2.19 5.75 6.25 7.38 $29.50 $37.50 - 0.50 - - 9.00 - $29.50 $40.00 0.08 0.31 1.25 11.13 11.25 11.50
In solving all the problems below, assume no transaction cost, and frictionless instant trading.
The expiration dates are: March 16(18 days); April 20 (53 days) and July 20 (114 days).
The respective risk-free rate are: 5%, 4.85% and 4.52%.Q1. Use the above data to: a. verify the put-call parity for European options on a non-dividend paying stock for the JUL, $35 options.
-C + P + S
= Xe-r(T-t) -2.19 + 7.38 + 29.50 = 35(2.71828)-(0.0452)(114/365) 34.69
= 34.50 .19
= 0
b. If the result does not hold, indicate how would you make arbitrage profit. (I.C.F. = Initial Cash Flow)
At Expiration Strategy
I.C.F.
ST < 35 ST > 35 Short JUL 35 Call
-2.19
0
ST - 35
Long JUL 35 Put
7.38
ST - 35
0
Issue debt @ 4.52%
-34.50
35
35
Long Stock
29.50
-ST
-ST
Total
.19
0
0
P/L
. .19
.19
Q2. a.Your broker calls you and tells you that even though the MAR, $27.50 call is listed for $2.63/share, she can get it for you for $1.50/share. The option is an American call. Explain to your broker how to make immediate arbitrage profit, without having to wait till the option’s expiration.
Buy the call and immediately exercise and sell the stock.
b. Your broker calls you again and inform you that actually, this call is European. Explain to her how you can still, make arbitrage profit, although this time, the strategy requires you to wait until the option's expiration.
Buy the call. Hold until just before expiration, exercise, and sell the stock.
Q3. You are interested in the JUL, $40 put. Your broker informs you that the actual price is not $11.50/share, as is shown in the table; instead, it is $11.15/share. She claims that you can make arbitrage profit if you act on fast, before the price changes. If you agree, explain how to do it. If you do not agree, explain to her why there is no arbitrage opportunity even with the price at $11.15/share.
Pt
> MAX(0, X - ST) Pt
> MAX(0, 40 - 29.50) 40 - 25.5
> 10.50 11.50
> 10.50 In this case the value of the JUL 40 put is greater than the exercise price minus the current stock price. This follows the rule that before expiration an American put must be worth at least the exercise price minus the stock price.
Q4. Your broker calls yet, once more, suggesting to open a Box spread with the APR, $30 and the APR, $35 exercise prices.
a. Use a table of cash flows to verify that this strategy guarantees you $5 at expiration.
At Expiration Strategy
I.C.F.
ST < 30 30< ST < 35 ST > 35 Long APR 30 call
-2.44
0
ST - 30
ST - 30
Short APR 35 call
0.81
0
0
35 - ST
Long APR 35 put
-6.25
35 - ST
35 - ST
0
Short APR 30 put
3.00
ST - 30
0
0
Total
-4.88
5
5
5
P/L
. 0.12
0.12
0.12
b. Calculate the risk-free rate you earn from this strategy.
FV
= PVert 5.00
= 4.88er(53/365) 1.0246
= er(53/365) 0.0243 = r(0.1452) 0.1673 = r
c. Compare the risk-free rate you calculated in b. with the market risk-free rate. If they are not equal, explain how you can make arbitrage profit.
The risk-free rate associated with the "box" strategy calculated above (16.73%) is higher than the risk-free rate associated with the April options (4.85%). Arbitrage profit can be made by borrowing to pay for the "box" spread to earn an overall return of 11.88%. The table is shown below.
PVert
= FV 4.88e(0.0485)(53/365)
= FV 4.88(1.00707)
= FV 4.9145
= FV
Strategy |
I.C.F. |
ST < 30 | 30< ST< 35 | ST > 35 |
Long APR 30 call |
-2.44 |
0 |
ST - 30 |
ST - 30 |
Short APR 35 call |
0.81 |
0 |
0 |
35 - ST |
Long APR 35 put |
-6.25 |
35 - ST |
35 - ST |
0 |
Short APR 30 put |
3.00 |
ST - 30 |
0 |
0 |
Issue debt @ 4.85% |
4.88 |
4.914 |
4.914 |
4.914 |
Total |
0 |
.086 |
.086 |
.086 |
P/L |
. |
.086 |
.086 |
.086 |