Question

As a financial analyst at Citibank Derivative Trading desk, you have collected data for a power option. A power call option pays off (max(ST-X, 0))2 at time T, where ST is the stock price at time T and X is the exercise price. A stock price is currently $60. It is known that at the end of one year it will be either $66 or $54. The risk-free rate of interest with continuous compounding is 5% per annum. Calculate the value of a one year power call option with an exercise price of $60.

What is the delta of the option ?

What is the risk neutral probability of up move ?

What is the value of the option ?

Answer 1

x = $60

so = $60 = p = 0.75    us=$66 Cu =66-60 = 6

                   1-p =0.25 ds =54 Cd = 0

a) That is delta of the option

      = Cu - Cd / Us -ds = 6 - 0 / 66 - 54

           6 / 12 = 0.5

R = 5% continuous compounding

    i.e 1.05

T = 1 Year

U = 66 / 60 = 1.1

d = 54 / 60 = 0.9

b) Risk neutral probability of up move

i.e p = R-d / u-d = 1.05 - 0.9 / 1.1 - 0.9 = 0.75

c) Value of the option

p * Cu + (1 - p) * Cd / e^{r t}

= 0.75 * 6 + 0.25 * 0 / e ^0.05*1 = 4.5 / 1.0513 = $ 4.28