QUESTION 4 (DISCRETE RANDOM VARIABLE) An option to buy a stock is priced at $200

Question

QUESTION 4 (DISCRETE RANDOM VARIABLE)

An option to buy a stock is priced at $200. If the stock closes above 30, the option will be worth $1000. If it closes below 20, the option will be worth nothing. If it closes between 20 and 30 (inclusive), then the option will be worth $200. A trader thinks there is 50% chance that the stock will close in the 20 - 30 range and a 30% chance that it will close below 20.

1) How much does the trader expect to gain?

2) What is the standard deviation of her gain?

3) Do you think the trader should buy the stock option? Explain your choice in two sentences.

Explanation / Answer

given that stick price = 200

If the stock closes above 30 the option will be worth $1000.

If it closes below 20, the option will be worth nothing

If it closes between 20 and 30 (inclusive), then the option will be worth $200

probability that stock will close in the 20 - 30 range according to trader = 0.5 or 50%

probability that stock will close below 20 according to trader = 0.3 or 30%

probability that stock will close above 30 according to trader = 1 - probability that stock will close in the 20 - 30 range according to trader - probability that stock will close below 20 according to trader

= 1 - 0.5 - 0.3 = 0.2

1)

Possible gains of trader are (1000-200), (0-200), (200-200) i.e 800, -200,0

Probability for gain to be 800 = probability that stock will close above 30 according to trader = 0.2

Probability for gain to be -200 = probability that stock will close below 20 according to trader = 0.3

Probability for gain to be 0= probability that stock will close in the 20 - 30 range according to trader = 0.5

expected profits = 800*Probability for gain to be 800 + (-200)*Probability for gain to be -200 + 0*Probability for gain to be 0

= 800*0.2 + (-200)*0.3+ 0*0.5 = 100

2)

variance

= (800-mean)2*Probability for gain to be 800 + (-200-mean)2*Probability for gain to be -200 + (0-mean)2*Probability for gain to be 0

= (800-100)2*0.2 + (-200-100)2*0.3 + (0-100)2*0.5 = 130000

standard deviation of her gain = varinace = 130000 = 360.555

3)

yes the trader should buy the stock option because expected profit is 100 which is positive

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