Question

Consider a call option with strike price of 2.5.

Underlying stock is expected to follow the distribution:

Price Prob

1 0.05

2 0.20

3 0.25

4 0.25

5 0.20

6 0.05

1. When stock price is above the strike price of 2.5, what is the average value of the stock?

(hint: first find conditional probabilities and then do a weighted average)

2. What is the average payment from the call option when the call option is in the money (ie stock price is above strike price of 2.5)?

(Hint: two ways to do this. a. weighted avg pmt of call option using conditional probabilities; b. just take the difference b/w conditional mean price of the stock and the strike price)

3. how much should the call be priced today (hint: this is asking for the unconditional mean of call option payment)

Answer 1

Probability that S > 2.5 = P(S = 3) + P(S = 4) + P(S = 5) + P(S = 6) = 0.25 + 0.25 + 0.20 + 0.05 = 0.75

Conditional Probabilities:

P(S = 3 | S > 2.5) = P(S = 3) / P(S > 2.5) = 0.25 / 0.75 = 1/3;

P(S = 4 | S > 2.5) = P(S = 4) / P(S > 2.5) = 0.25 / 0.75 = 1/3;

P(S = 5 | S > 2.5) = P(S = 5) / P(S > 2.5) = 0.20 / 0.75 = 4/15

P(S = 6 | S > 2.5) = P(S = 6) / P(S > 2.5) = 0.05 / 0.75 = 1/15

1. When stock price is above the strike price of 2.5, what is the average value of the stock?

Average value of the stock, S_avg = Sum (Conditional probabilities) x Value = 1/3 x 3 + 1/3 x 4 + 4/15 x 5 + 1/15 x 6 = 4.07

2. What is the average payment from the call option when the call option is in the money (ie stock price is above strike price of 2.5)?

Average payment = max (S_avg - K, 0) = max (4.07 - 2.5, 0) = 1.57

3. how much should the call be priced today (hint: this is asking for the unconditional mean of call option payment)

S	p	Call payment	p x V
		V = max (S - 2.5, 0)
1	0.050	-	-
2	0.200	-	-
3	0.250	0.500	0.125
4	0.250	1.500	0.375
5	0.200	2.500	0.500
6	0.050	3.500	0.175
		Total	1.175

Hence, the price of the call today = 1.175