Question

0. After a detailed analysis on the risks and returns of the stock market and Stock XYZ (current price=€50), you conclude that the price of XYZ is unlikely to change a lot during the course of the next 3 months. You decide to bet on this analysis and establish a short straddle position, by simultaneously writing 100 puts and 100 calls with maturity of 3 months and a strike price of €50. The premium you receive for writing each put/call is €2.50.

   •  What is the maximum profit of your position? What should the stock price be, in order to realize this maximum gain?

   •  Report the lower and higher bounds for XYZ’s price 3 months from now that provide you with a non-negative return on your position.

   •  Draw a profit diagram separately for your a) short position on puts, b) short position on calls, and c) total position

Answer 1

Lets first look at a few data points of the short straddle:

Short Call			Short Put
Spot price	Exercise price	Premium	Spot price	Exercise price	Premium
	50	2.5		50	2.5
	Payoff	Profit		Payoff	Profit	Total Profit
39	0	2.5	39	-11	-8.5	-6
40	0	2.5	40	-10	-7.5	-5
41	0	2.5	41	-9	-6.5	-4
42	0	2.5	42	-8	-5.5	-3
43	0	2.5	43	-7	-4.5	-2
44	0	2.5	44	-6	-3.5	-1
45	0	2.5	45	-5	-2.5	0
46	0	2.5	46	-4	-1.5	1
47	0	2.5	47	-3	-0.5	2
48	0	2.5	48	-2	0.5	3
49	0	2.5	49	-1	1.5	4
50	0	2.5	50	0	2.5	5
51	-1	1.5	51	0	2.5	4
52	-2	0.5	52	0	2.5	3
53	-3	-0.5	53	0	2.5	2
54	-4	-1.5	54	0	2.5	1
55	-5	-2.5	55	0	2.5	0
56	-6	-3.5	56	0	2.5	-1
57	-7	-4.5	57	0	2.5	-2
58	-8	-5.5	58	0	2.5	-3
59	-9	-6.5	59	0	2.5	-4
60	-10	-7.5	60	0	2.5	-5

And below is the payoff diagram for this position:

• We know that if the price of the underlying goes above 50, then the call will be exercised. but below 50 it wont be exercised.
  • When price is above 50, the loss can be infinite because price can go as high as possible
  • Loss is reduced by the amount of premium received for writing the option
  • When the price is 50 or less, the profit is fixed at 2.5 or the amount of premium received
• We also know that if the price of the underlying goes below 50, then the put will be exercised. but above 50 it wont be exercised.
  • When price is below 50, the price can go as low as 0 and therefore the loss can at max be -50+2.5 = -47.5
  • Loss is reduced by the amount of premium received for writing the option
  • When the price is above 50 , the profit is fixed at 2.5 or the amount of premium received
• Using the above data sets, and the analysis, we can see that the maximum profit from the position is achieved when the price is 50 and the maximum profit is $5 or the total premium received
• From the above data set, the non negative return or profit can be achieved in the price range of 45 to 55, including both.
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