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# 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

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