Q1. A stock price is currently $100. Every month, it is expected to go up by 3%

Question

Q1. A stock price is currently $100. Every month, it is expected to go up by 3% or down by 3%. So, if the price goes up 3 consecutive months, the price will become $100 * (1+3%) * (1+3%) * (1+3%). The risk-free interest rate is 1% per annum.

What is dt, the length of one (not the whole) period?

What is u, the up factor?

What is d, the down factor?

Calculate p, the risk-neutral probability that the stock price will go up next period.

For Q2 and Q3, use the following information.

A non-dividend paying stock is currently trading at $100 and its volatility is 40%. Consider a put option on this stock, with a strike price of $110, expiring in 6 months. The current risk-free rate is 1% per annum. We will price the put option with a 3-step binomial tree (the number of steps = 3).

Q2. First, calculate the European put option price in a spreadsheet. Then use Derivagem to price it. Confirm these 2 prices match. Include the Derivagem output (screenshot or copy paste).

Hint: The put pricing exercise in class was a 2-step tree. You can extend 1 more step to that example to create a 3-step tree.

Q3. Calculate the American put option price in a spreadsheet. Then use Derivagem to price it and confirm. These 2 prices must match but will be higher than Q3 answer. Include the Derivagem output.

Derivagem:

http://www-2.rotman.utoronto.
ca/~hull/software/

Explanation / Answer

Q1)

The risk neutral probability of an up move = (1+Rf - D)/(U - D)

Where; Rf = Risk free rate

U = size of an up-move = 1*103% = 1.03

D = Size of a down move = 1-1*3% = 0.97

So applying the formula, risk-neutral probability that the stock price will increase each period

= (1+0.01 - 0.97)/(1.03-0.97)

= 0.04/0.06 = 0.6667

= 66.67%

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