Question

1. The current spot price for a stock is $100, using a binomial model, in every period it has been determined that the probability for this stock to go up is 70%, in this case the stock will increase in value a 12 %. If the stock goes down, the value will decrease 13%. For a call option with strike price of $186  and after 12 periods:

            1) Calculate the values of the factor "u" and "d".

            2) Show a diagram with the binomial development of the price for 3 periods.

            3) Calculate theoretically the minimum number of times the stock should go up in order to exercise the call option.

            4)Calculate the probability of exercising the call option.

Answer 1

1.

u = 1.12 (increase factor)

d= 0.87 (decrease factor)

2. The binomial development of the price is as shown in the stock price lattice (for 3 periods)

			140.4928
		125.44	109.1328
	112	97.44	84.7728
100	87	75.69	65.8503
t=0	t=1	t=2	t=3

3. For the option to be exercised, the price > 186

So, no. of upmoves (n) is given as

100*1.12^n > 186

n> 5.475 or n=6

So, the stock has to go up at least 6 times before the call option is exercised

(Please note that x upmoves and y downmoves are equivalent to x-y upmoves)

4. Probability that the option will be exercised

= probability of 12 upmoves and 0 downmoves + probability of 11 upmoves and 1 downmoves + probability of 10 upmoves and 2 downmoves +probability of 9 upmoves and 3 downmoves

=12C0*0.7^12*0.3^0 + 12C1*0.7^11*0.3^1+12C2*0.7^10*0.3^2+12C3*0.7^9*0.3^3

= 0.7^12+12*0.7^11*0.3+66*0.7^10*0.3^2+220*0.7^9*0.3^3   

=0.492516