Question

A stock price follows geometric Brownian motion with an annual expected return of 6% and a volatility of 35%. The current price is $38.

a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in 12 months will be exercised?

b) What is the probability that the holder of the European call option on the stock will profit $4.5 upon exercising the option?

Answer 1

a)

The probability that the European call option with exercise price of $40 will be exercised is the probability of stock price being more than $40.

P(St>K) = N (ln (St/K) + (T-t)*(r-?²/2))

                        ?*(T-t)^.5

z = ln (St/K) + (T-t)*(r-?²/2))

                        ?*(T-t)^.5

St	38
K	40
T-t	1
r	0.06
std dev	0.35
z	-0.14834
Required probability	0.441038

b)

To profit $4.5 from option, the stock price must be 44.5 or higher.

z = ln (St/K) + (T-t)*(r-?²/2))

                        ?*(T-t)^.5

St	38
K	44.5
T-t	1
r	0.06
std dev	0.35
z	-0.45294
Required probability	0.325297