Please show your work. The following quotes were observed for options on a given
ID: 2780530 • Letter: P
Question
Please show your work. The following quotes were observed for options on a given stock on November 1 of a given year. These are American calls except where indicated. Use the information to answer questions 7 through 20. Strike: 105, 110, 115 Call Nov: 8.40, 4.40, 1.50 Call Dec: 10, 7.10, 3.90 Call Jan: 11.50, 8.30, 5.30 Put Nov: 5.30, 0.90, 2.80 Put Dec: 1.30, 2.50, 4.80 Put Jan: 2.00, 3.80, 4.80 The stock price was 113.25. The risk-free rates were 7.30 percent (November), 7.50 percent (December) and 7.62 percent (January). The times to expiration were 0.0384 (November), 0.1342 (December), and 0.211 (January). Assume no dividends unless indicated. Show your work. 15. What is the European lower bound of the December 105 call? 16. What is the European lower bound of the November 115 call? 17. From American put-call parity, what are the minimum and maximum values that the sum of the stock price and December 110 put price can be? 18. The maximum difference between the January 105 and 110 calls is which of the following? 19. Suppose you knew that the January 115 options were correctly priced but suspected that the stock was mispriced. Using put-call parity, what would you expect the stock price to be? For this problem, treat the options as if they were European. 20. Suppose the stock is about to go ex-dividend in one day. The dividend will be $4.00. Which of the following calls will you consider for exercise? Show your work
Explanation / Answer
The current stock price (S) is 113.25. The below tables summarizes all the call put options given in the question:
CALL
Strike(X)
Nov
Dec
Jan
105
8.4
10
11.5
110
4.4
7.1
8.3
115
1.5
3.9
5.3
PUT
Strike(X)
Nov
Dec
Jan
105
5.3
1.3
2
110
0.9
2.5
3.8
115
2.8
4.8
4.8
15) Without getting into the derivation the minimum value of a European Call option (C) is given by:
C=Max (0, S-Xe-rt)
Where S= Stock Price=113.25
X= Strike Price=105
R= Risk free rate =7.50% (December)
t= Time to expiry =0.1342 (December)
C=Max (0,113.25-105e-0.0750×0.1342)
=Max (0, 113.25-103.9484757)
=Max(0,9.05)
=9.05
Conclusion: Thus, the minimum value of 105 December European call option is $ 9.05.
16) Using the same logic as question the lower bound of European 115 November call can be found as:
C=Max (0, S-Xe-rt)
Where S= Stock Price=113.25
X= Strike Price=115
R= Risk free rate =7.30% (November)
t= Time to expiry =0.0384 (November)
C=Max (0,113.25-115e-0.0730×0.0384)
=Max (0, 113.25-114.67808)
=Max (0,-1.4280)
=0
Conclusion: Thus, the minimum value of 115 November European call option is $ 0.
17) For American options Put-Call Parity is given by the inequality:
S-X C-P S-Xe-rt
Where S=Stock Price
X= Strike Price
r= Risk free rate
t=time to maturity
C=Call price
P= Put Price
Now the question asks us find the minimum and maximum of the sum of stock price and put price (December 110) using the above inequality. So we will just rearrange the above inequality in terms of S+P
C+Xe-rt S+P C+X
C=7.10 (table in question 1)
X=110
r=7.50%
t=0.1342
S=113.25
P= 2.5 (table in question 1)
Therefore,
7.10+110e-0.075×0.1342113.25+2.57.10+110
7.10+108.90 113.25+2.5 7.10+110
116115.75117.10
Conclusion: The minimum value is 116 while maximum value is 117.10.
18) The maximum value for both European and American call option is:
C=Max (0, S-X)
Therefore, max value of January 105 call is:
C=Max (0,113.25-105)
=Max(0,8.25)
=8.25
Therefore, max value of January 110 call is:
C=Max (0,113.25-110)
=Max (0,3.25)
=3.25
Conclusion: Thus, the maximum difference between Jan 105 and Jan 100 call is 8.25-3.25=$ 5
The current stock price (S) is 113.25. The below tables summarizes all the call put options given in the question:
CALL
Strike(X)
Nov
Dec
Jan
105
8.4
10
11.5
110
4.4
7.1
8.3
115
1.5
3.9
5.3
PUT
Strike(X)
Nov
Dec
Jan
105
5.3
1.3
2
110
0.9
2.5
3.8
115
2.8
4.8
4.8
15) Without getting into the derivation the minimum value of a European Call option (C) is given by:
C=Max (0, S-Xe-rt)
Where S= Stock Price=113.25
X= Strike Price=105
R= Risk free rate =7.50% (December)
t= Time to expiry =0.1342 (December)
C=Max (0,113.25-105e-0.0750×0.1342)
=Max (0, 113.25-103.9484757)
=Max(0,9.05)
=9.05
Conclusion: Thus, the minimum value of 105 December European call option is $ 9.05.
16) Using the same logic as question 15 the lower bound of European 115 November call can be found as:
C=Max (0, S-Xe-rt)
Where S= Stock Price=113.25
X= Strike Price=115
R= Risk free rate =7.30% (November)
t= Time to expiry =0.0384 (November)
C=Max (0,113.25-115e-0.0730×0.0384)
=Max (0, 113.25-114.67808)
=Max (0,-1.4280)
=0
Conclusion: Thus, the minimum value of 115 November European call option is $ 0.
17) For American options Put-Call Parity is given by the inequality:
S-X C-P S-Xe-rt
Where S=Stock Price
X= Strike Price
r= Risk free rate
t=time to maturity
C=Call price
P= Put Price
Now the question asks us find the minimum and maximum of the sum of stock price and put price (December 110) using the above inequality. So we will just rearrange the above inequality in terms of S+P
C+Xe-rt S+P C+X
C=7.10 (table in question 15)
X=110
r=7.50%
t=0.1342
S=113.25
P= 2.5 (table in question 15)
Therefore,
7.10+110e-0.075×0.1342113.25+2.57.10+110
7.10+108.90 113.25+2.5 7.10+110
116115.75117.10
Conclusion: The minimum value is 116 while maximum value is 117.10.
18) The maximum value for both European and American call option is:
C=Max (0, S-X)
Therefore, max value of January 105 call is:
C=Max (0,113.25-105)
=Max(0,8.25)
=8.25
Therefore, max value of January 110 call is:
C=Max (0,113.25-110)
=Max (0,3.25)
=3.25
Conclusion: Thus, the maximum difference between Jan 105 and Jan 100 call is 8.25-3.25=$ 5
CALL
Strike(X)
Nov
Dec
Jan
105
8.4
10
11.5
110
4.4
7.1
8.3
115
1.5
3.9
5.3