# Binomial Trees and Properties of Options

## Problem A3: Binomial Trees and Properties of Options (17 marks)

In this question, you need to price options with binomial trees. You will consider puts and calls on a share with spot price of \$30. Strike price is \$34. Furthermore, assume that over each of the next two four-month periods, the share price is expected to go up by 11% or down by 10%. The risk-free interest rate is 6% per annum with continuous compounding.

a. Use a two-step binomial tree to calculate the value of an eight-month European call option using the no-arbitrage approach. [3 marks]

b. Use a two-step binomial tree to calculate the value of an eight-month European put option using the no-arbitrage approach. [3 marks]

c. Show whether the put-call-parity holds for the European call and the European put prices you calculated in a. and b. [1 mark]

d. Use a two step-binomial tree to calculate the value of an eight-month European call option using risk-neutral valuation. [1 mark]

e. Use a two step-binomial tree to calculate the value of an eight-month European put option using risk-neutral valuation. [1 mark]

f. Verify whether the no-arbitrage approach and the risk-neutral valuation lead to the same results. [1 mark]

g. Use a two-step binomial tree to calculate the value of an eight-month American put option. [1 mark]

h. Without calculations: What is the value of an eight-month American call option with a strike price of \$34? Why? [2 marks]

i. Without calculations: Consider an at-the-money European put on the same share and a time-to-maturity of 8 months. Will the price of the at-the-money put be higher or lower compared to the put price you calculated in e.? Why? [2 marks]

j. Without calculations: What would happen to the option prices you calculated in d. and e. if the interest rate drops to 4%? Why? [2 marks]

Note: When you use no-arbitrage arguments, you need to show in detail how to set up the riskless portfolios at the different nodes of the binomial tree.

## Part B: Risk Management Case Study (30 marks)

Assume today is Sept 1, 2021. Company IMineGold is a gold mining company producing gold with plans to increase its size and valuation over the next few years. The company has invested in developing goldfields and commenced extracting gold and gold production from its underground mines. The company is confident that it will have an output of 100,000 ounces of gold on 1 March 2022 and is willing to sell this output amount on that day.

The company’s revenue is subject to changes of the gold price. As the market price of gold is quite volatile, the company is considering using some strategies to manage its risk exposure.

Following further information is available:

• Profit margin is defined as (Revenue – Cost of Goods Sold) / Revenue.

• The cost of mining one ounce of gold is \$1,420.

• The company has a target profit margin of 18%.

• The minimum acceptable profit margin below which the company will have difficulties servicing its debt is 14%.

• The spot price of gold on 1 Sept 2021 is \$1,800.

• The interest rate curve is flat and the risk-free interest rate for all maturities is 2% (cont. comp.).

• The annual gold price volatility is estimated to be 15%.

The company regularly transacts with the ABC Bank which is available as a counterparty of forward and option contracts with gold as an underlying asset.

Based on this information and the knowledge you have accumulated while taking this Derivatives course, respond to the questions below.

A. What would be the profit margin of IMineGold if the current spot price is used? (1 mark)

B. What is the gold price at which IMineGold achieves exactly its target rate? (1 mark)

C. What is the critical gold price for IMineGold? (1 mark)

D. What is the forward price for a contract expiring on 1 March 2022? (1 mark)

E. Use the Black-Scholes model to price following options on gold: (5 marks)

• Option 1: A call option with strike price 1680 expiring on 1 March 2022.

• Option 2: A put option with strike price 1680 expiring on 1 March 2022.

• Option 3: A call option with strike price 1850 expiring on 1 March 2022.

• Option 4: A put option with strike price 1850 expiring on 1 March 2022.

Does the put-call-parity hold for Options 1 and 2?

F. IMineGold considers the following five hedging strategies for managing commodity price risk:

Strategy I: No hedge at all.

Strategy II: Hedging 100% of the mining output with a forward contract.

Strategy III: Hedging 50% of the mining output with a forward contract and leaving the remaining 50% unhedged.

Strategy IV: A strategy using one of the options from sub-question E. to meet the target rate and benefit from favourable movements in the gold price.

Strategy V: A strategy using one of the options from sub-question E. for worst-case protection only (meet critical rate).

For each of these strategies:

• Calculate the profit margin for the following two scenarios:

a) The gold spot price on 1 March 2022 is \$1,990 per ounce.

b) The gold spot price on 1 March 2022 is \$1,610 per ounce.

• If a strategy includes options, state clearly which option contract should be used, whether as a long or short position, and why.

Sub-question F is for a total of 19 marks.

G. As an alternative hedging strategy, IMineGold might consider using gold futures contracts traded on the CME, which is a world leading derivatives marketplace

Without any calculations, discuss advantages and disadvantages of a hedging strategy using CME futures as compared to using forward contracts with the ABC bank. (2 marks)

Notes:

1. Assume that gold doesn’t cause storage costs.

2. For the sake of simplicity, assume that all amounts are given in USD and the company is not concerned about currency risk.

3. If a hedging strategy uses options, the option premium needs to be considered as an expense (added to Cost of Goods Sold) for profit margin calculations.

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