In: Finance

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.

(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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