Question

The price of a stock is $40. The price of a one-year put with strike price $30 is $0.70 and a call with the same time to maturity and a strike of $50 costs $0.50. Both options are European.

(a) An investor buys one share, shorts one call and buys one put. Draw and comment upon the payoff of this portfolio at maturity as a function of the underlying price.

(b) How would your answer to (a) change if the investor buys one share, shorts two calls and buys two puts instead.

Answer 1

(a)

Stock price	Long stock payoff	Payoff short call strike $50	Payoff long put strike $30	Combined position
20	-20	0.5	9.3	-10.2
21	-19	0.5	8.3	-10.2
22	-18	0.5	7.3	-10.2
23	-17	0.5	6.3	-10.2
24	-16	0.5	5.3	-10.2
25	-15	0.5	4.3	-10.2
26	-14	0.5	3.3	-10.2
27	-13	0.5	2.3	-10.2
28	-12	0.5	1.3	-10.2
29	-11	0.5	0.3	-10.2
30	-10	0.5	-0.7	-10.2
31	-9	0.5	-0.7	-9.2
32	-8	0.5	-0.7	-8.2
33	-7	0.5	-0.7	-7.2
34	-6	0.5	-0.7	-6.2
35	-5	0.5	-0.7	-5.2
36	-4	0.5	-0.7	-4.2
37	-3	0.5	-0.7	-3.2
38	-2	0.5	-0.7	-2.2
39	-1	0.5	-0.7	-1.2
40	0	0.5	-0.7	-0.2
41	1	0.5	-0.7	0.8
42	2	0.5	-0.7	1.8
43	3	0.5	-0.7	2.8
44	4	0.5	-0.7	3.8
45	5	0.5	-0.7	4.8
46	6	0.5	-0.7	5.8
47	7	0.5	-0.7	6.8
48	8	0.5	-0.7	7.8
49	9	0.5	-0.7	8.8
50	10	0.5	-0.7	9.8
51	11	-0.5	-0.7	9.8
52	12	-1.5	-0.7	9.8
53	13	-2.5	-0.7	9.8
54	14	-3.5	-0.7	9.8
55	15	-4.5	-0.7	9.8
56	16	-5.5	-0.7	9.8
57	17	-6.5	-0.7	9.8
58	18	-7.5	-0.7	9.8
59	19	-8.5	-0.7	9.8
60	20	-9.5	-0.7	9.8

Here, from the combined position payoff, we observe that when the stock price at maturity is lesser than 40, the investor faces a loss and if the stock price at maturity is greater than 40, the investor has a profit. The maximum loss of the investor is -10.2 when the stock price is less than $30 and the maximum profit is 9.8 when the stock price crosses $20. Hence, both the upper as well as the lower limit of the investor position is capped.

Inshort, this payoff replicates a bull call spread.

b)

Stock price	Long stock payoff	Payoff 2 short call strike $50	Payoff 2 long put strike $30	Combined position
0	-40	1	58.6	19.6
1	-39	1	56.6	18.6
2	-38	1	54.6	17.6
3	-37	1	52.6	16.6
4	-36	1	50.6	15.6
5	-35	1	48.6	14.6
6	-34	1	46.6	13.6
7	-33	1	44.6	12.6
8	-32	1	42.6	11.6
9	-31	1	40.6	10.6
10	-30	1	38.6	9.6
11	-29	1	36.6	8.6
12	-28	1	34.6	7.6
13	-27	1	32.6	6.6
14	-26	1	30.6	5.6
15	-25	1	28.6	4.6
16	-24	1	26.6	3.6
17	-23	1	24.6	2.6
18	-22	1	22.6	1.6
19	-21	1	20.6	0.6
20	-20	1	18.6	-0.4
21	-19	1	16.6	-1.4
22	-18	1	14.6	-2.4
23	-17	1	12.6	-3.4
24	-16	1	10.6	-4.4
25	-15	1	8.6	-5.4
26	-14	1	6.6	-6.4
27	-13	1	4.6	-7.4
28	-12	1	2.6	-8.4
29	-11	1	0.6	-9.4
30	-10	1	-1.4	-10.4
31	-9	1	-1.4	-9.4
32	-8	1	-1.4	-8.4
33	-7	1	-1.4	-7.4
34	-6	1	-1.4	-6.4
35	-5	1	-1.4	-5.4
36	-4	1	-1.4	-4.4
37	-3	1	-1.4	-3.4
38	-2	1	-1.4	-2.4
39	-1	1	-1.4	-1.4
40	0	1	-1.4	-0.4
41	1	1	-1.4	0.6
42	2	1	-1.4	1.6
43	3	1	-1.4	2.6
44	4	1	-1.4	3.6
45	5	1	-1.4	4.6
46	6	1	-1.4	5.6
47	7	1	-1.4	6.6
48	8	1	-1.4	7.6
49	9	1	-1.4	8.6
50	10	1	-1.4	9.6
51	11	-1	-1.4	8.6
52	12	-3	-1.4	7.6
53	13	-5	-1.4	6.6
54	14	-7	-1.4	5.6
55	15	-9	-1.4	4.6
56	16	-11	-1.4	3.6
57	17	-13	-1.4	2.6
58	18	-15	-1.4	1.6
59	19	-17	-1.4	0.6
60	20	-19	-1.4	-0.4

Here, we observe that payoff is asymetrical and the maximum profit occours when the stock price at maturity is $50. However, the downside is unlimited as if the stock price ends below $60, the payoff is negative. Higher the stock goes above $60, more negative the payoff is and higher the losses for the investor.