Question

In: Finance

The price of a stock is $40. The price of a one-year put with strike price...

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.

Solutions

Expert Solution

(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.


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