# BFF5915: The Black-Scholes-Merton option pricing model assumes a risk-free rate as the expected rate of return on the asset: OPTIONS, FUTURES AND RISK MANAGEMENT Assignment, MUM, Malaysia

School

###### Monash University Malaysia (MUM)

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Assignment Type

Subject

Date

###### 11/11/2022

Question 1

For each of the FIVE (5) statements below, state whether it is true or false (2 marks), and explain the reason (3 marks).

(1) The Black-Scholes-Merton option pricing model assumes a risk-free rate as the expected rate of return on the asset. This assumption makes the model impractical.

(2) The seller of a European put option will benefit if the underlying asset price goes up in value.

(3) Consider a \$200 million interest rate swap with semi-annual payment that has a remaining life of 3 months. The fixed rate associated with the swap is 4% per annum and the floating rate is LIBOR + 2% per annum. Both rates are compounded semiannually. If the LIBOR rate was 3% per annum three months ago, the floating rate payer will, in three months, receive \$1 million on a net basis.

(4) Consider a Credit Default Swap (CDS) on the principal of \$100 million with recovery rate of 60%. The payment by the CDS seller, excluding the premium paid by the buyer, to the buyer is \$60 million in the event of default.

(5) Suppose that the risk-free zero curve is flat at 7% per annum with continuous compounding and that defaults can occur half way through each year in a new fiveyear CDS. Suppose that the recovery rate is 30% and the default probabilities each year conditional on no earlier default are 3%. The price of this CDS is the present value of its expected future payoff.

Question 2

The spot exchange rate between the Australian dollars (AUD) and U.S. dollar (USD) is 0.7512 (USD per 1 AUD). Three-month interest rates in the U.S. and Australia are 1% and 1.5% per annum respectively, with continuous compounding. The three-month forward exchange rate is 0.7495 (USD per 1 AUD).

(a) What arbitrage strategy is possible? Calculate arbitrage profits on AUD10 million.

(b) List out issues that can make this arbitrage profit unachievable.

Question 3

The December Eurodollar futures contract is quoted as 98.20 and a company plans to borrow USD8 million for three months starting in December at LIBOR plus 0.5% per annum.

(a) What rate can the company lock in by using the Eurodollar futures contract?

(b) What position and how many should the company take in the contracts?

(c) If the actual three-month rate turns out to be 1.4% per annum, what is the final settlement price on the futures contracts?

(d) Based on the actual rate in (c), what is the effective borrowing cost for the company if it hedges using the Eurodollar futures contract? Is it a perfect hedge? Explain.

Question 4

(a) Discuss the three possible ways in which an option position can be terminated. Would your answer be different if the option was created in an OTC market?

(b) A one-month European put option on a non-dividend-paying stock is selling for \$2.50. The stock price is \$47, the strike price is \$50, and the risk-free interest rate is 6 percent per annum with continuous compounding. Suggest an arbitrage strategy. Provide full details of the arbitrage transactions.

Question 5

Consider an option on a non-dividend-paying stock where the stock price is \$60, the exercise or strike price is \$54, the risk-free interest rate is 1% per annum with continuous compounding, the volatility is 35% per annum, and the time to maturity is 6 months. Assume that the stock price follows a Geometric Brownian Motion.

(a) Using the Black-Scholes-Merton model, compute the price of the option specified above if it is a European put.

(b) Without using the Black-Scholes-Merton model, find the price of the corresponding European call option with the same strike price and maturity.

(c) If the actual price of the put option in the market is \$0.10 higher than the value you computed in (a), what can you say about the implied volatility in relation to the 35% volatility stated in the question? Is the implied volatility higher than, lower than or equal to 35%? Briefly explain your answer.

Question 6

Big Fund is an active market maker in the gold options market. They aim to maintain a deltaneutral options portfolio at all times. A trader at Big Fund has just purchased a gold call option. The option has a face value of 1000 ounces and a delta of 0.30 (per ounce).

(a) Specify the trade required to delta-hedge this option.

(b) Sometime later, the delta of this option moves to 0.10. Specify the action required to maintain delta neutrality of this option.

(c) Explain why it is common practice for trading entities, such as Big Fund, to maintain delta neutrality of their portfolio. When would they choose NOT to be delta neutral?

(d) List two assumptions used in the derivation of the Black-Scholes-Merton model (when used to price European style options). Discuss the validity of these assumptions.

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