Question

A stock price follows geometric Brownian motion with an expected return of 16% 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 6 months will be exercised? (b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?

Answer 1

(a) What is the probability that a European call option on the stock with an exercise price of $40 and a maturity date in 6 months will be exercised?

European call option on the stock with an exercise price of $40 will be exercised if the stock price will be above $40 in six months, therefore we have to calculate the probability of the stock price being above $40 in six months. Let’s assume that the stock price in six months is X, therefore the probability distribution of ln X is

{ln 38 + ( 16% - 35%^2/2) * 0.5, 35%^2 * 0.5}

Or {3.6376 + (0.16 – 0.35^2/2) * 0.5, 0.35^2 * 0.5}

Or {3.6376 + 0.0494, 0.06125}

Or {3.687, 0.06125}

But ln 40 = 3.689

Therefore the required probability

= 1-N {(3.689 – 3.687)/ √0.06125}

=1-N {(0.002)/ 0.247}

= 1 – N (0.008)

Now refer normal distribution table, we get N (0.008) = 0.5032

Therefore the required probability = 1- 0.5032 = 0.4968

(b) What is the probability that a European put option on the stock with the same exercise price and maturity will be exercised?

European put option on the stock with an exercise price of $40 will be exercised if the stock price will be less than $40 in six months; therefore we have to calculate the probability of the stock price being below $40 in six months. That is-

= 1- the probability of the stock price being above $40 in six months

= 1-0.4968

= 0.5032