Question

You own Macquarie Group Shares (MQG) currently valued at $170 per share, and believe the recent weakness in the banking system may present some downside risks. As a result, you decide to purchase a 2-month 'at-the-money' Put Option on MQG shares to protect your risk. The volatility of MQG shares is 25%, and the Shares pay a dividend of $5 at the end of month 1. The risk free rate is 4.5% for all maturities. All variables are required to be calculated using continuous compounding (recall use of e^rt). Show all workings and calculate:

The up movement factor and down movement factor when a one-month time step is used?
The up movement probability (Pu) and down movement probability (Pd) when a one-month time step is used?

Answer 1

To find the up and down movement factors for Macquarie Group Shares (MQG), we use the binomial model formulas and the given volatility of 25%. The risk-neutral probabilities are then calculated using these factors and the continuous compounding risk-free rate of 4.5%. This approach is used to price a protective put option on MQG shares at $170 per share.

Step-by-step explanation:

To calculate the up and down movement factors for Macquarie Group Shares (MQG) using a one-month time step, we will use the binomial options pricing model, which considers the volatility and the time step. The annual volatility given is 25%, and with one month being 1/12 of a year, the monthly volatility (sigma) can be calculated as 0.25 / √12. For a binomial model, we typically use the following formulas to calculate the movement factors:

Up movement factor (u) = e^(sigma * √Δt)
Down movement factor (d) = 1 / u

Here, Δt is the length of the time step, which is 1/12 for a month. Plugging in the monthly volatility:
u = e^(0.25/√12 * √(1/12))
d = 1 / u

Once we have the up and down factors, the risk-neutral probabilities can be computed using the following formulas, taking into account the risk-free rate (rf), which is continuous compounded:

Up movement probability (Pu) = (e^(rf * Δt) - d) / (u - d)
Down movement probability (Pd) = 1 - Pu

Substitute the continuous compounding risk-free rate of 4.5% and the values for u and d to find Pu and Pd:

Pu = (e^(0.045 * (1/12)) - d) / (u - d)

Pd = 1 - Pu

These calculations will determine the required movement factors and probabilities to price the put option on MQG shares valued at $170 per share.