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 six 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? Note: leave answers as N(x).

Answer 1

Step 1

The required probability is the probability of stock price to be higher than the strike price at the time of expiration in case of the call option. This is because a call could only be profitable to exercise if the price of the underlying stock is greater than the exercise price at the time of exercise of the option.

This is computed with the help of a probability distribution equation or a table that links every result of statistical research with its probability of occurrence.

Step 2

European options:

A European option is an option where the trader can exercise the option only at the time of maturity, on the date of expiration. European options sometimes trade at discount values because they compete with the American options, that allow more opportunities for the investors to exercise the contract.

Step 3

Put options

Step 4

A put option gives the option holder the right to sell the asset/security at a certain fixed price, where the buyer expects the stock prices to go down.

Step 5

Call options

Step 6

A call option gives the option holder the right to purchase the security at a certain fixed price, where the buyer expects the stock prices to go up.

Step 7

Calculate the probability distribution of the European call option with an exercise price of $40 and maturity 6 months.

The stock is normally distributed with a mean and standard deviation and has a lognormal distribution. The formula is as follow:

Here,

The stock price at future timeis .

The stock price at time 0 is .

The expected rate of return is .

The volatility of the stock is .

The future time period .

The stock is distributed normally and has a lognormal distribution.

Step 8

Note: The call option could only be exercised if the price of the underlying stock is more than the strike price of the option.

Let’s assume that the price of the stock in 6-months will be x.

Compute the probability distribution if the stock price is x in 6-months as follows:

Note: Result is rounded up to 6 decimal places.

Step 9

Now, find the ln40 using a table or spreadsheet ln function.

The.

Note: Result is rounded up to 6 decimal places.

Step 10

The requirement is to find the probability of.

Formula for calculating the probability of the European call option is given below.

Here,

The normal distribution is denoted by N

A) Substitute, the value of probability distribution at is 3.688879, the probability distribution at is 3.686961, and 0.06125 (computed in previous step) in denominator in the above stated probability formula.

Compute the probability as follows:

Thus, the probability of the European call option to be exercised is .

Here, the value of N is computed using the NORMSDIST function of spreadsheet.

B)

Note: The put option could only be exercised if the price of underlying stock is lesser than the strike price at the time of expiration.

Therefore, the requirement is to find the probability of stock price to be lesser than $40 in 6-months.

Compute the probability to exercise the put option as follows:

Thus, the probability to exercise the put option is.

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