Maverick Manufacturing, Inc., must purchase gold in three months for use in its
ID: 2741231 • Letter: M
Question
Maverick Manufacturing, Inc., must purchase gold in three months for use in its operations. Maverick’s management has estimated that if the price of gold were to rise above $1,550 per ounce, the firm would go bankrupt. The current price of gold is $1,470 per ounce. The firm’s chief financial officer believes that the price of gold will either rise to $1,645 per ounce or fall to $1,360 per ounce over the next three months. Management wishes to eliminate any risk of the firm going bankrupt. Maverick can borrow and lend at the risk-free EAR of 5.0 percent. a-1. Should the company buy a call option or a put option on gold? Call option Put option a-2 What strike price would the company like this option to have? (Do not round intermediate calculations.) Strike price b. How much should such an option sell for in the open market? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Option price $ c. Suppose no options currently trade on gold. What are the transactions needed to create a synthetic option with identical payoffs to a traded option? (Do not round intermediate calculations and round your final answers to 2 decimal places. (e.g., 32.16)) c-1. $ shares of stock c-2. Amount to $ d. How much does the synthetic option cost? (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) Synthetic option $
Explanation / Answer
Part a-1)
The company should buy the call option having a strike price of $1,550 per ounce and 3 months expiration. This would help the company to cover the risk associated with an increase in the price of gold above the strike price of the option.
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Part a-2)
The company would like to have a strike price of $1,550 per ounce. It is so, because any increase in the price of gold above the strike price would provide a payoff to the company (difference between price of gold on expiry and strike price) and in case of a decrease in the price of gold, the company can let the option expire, resulting in no loss.
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Part b)
The selling price of the option is calculated as follows:
Return if the Price of Gold Rises to $1,645 = (1,645 - 1,470)/1,470*100 = 11.90%
Return if the Price of Gold Falls to $1,360 = (1,360 - 1,470)/1,470*100 = -7.48%
Now, we will have to calculate the risk-neutral probability of a rise and decline in the price of gold with the use of following formula:
Risk Free Rate = (Probability of Rise in the Price of Gold)*(Return if the Price of Gold Rises) + (1-Probability of Rise in the Price of Gold)*(Return if the Price of Gold Falls)
where Risk Free Rate over Next Three Months = (1+5%)^(1/4) - 1 = 1.23% or .0123
Substituting the values calculated above, we get,
.0123 = (Probability of Rise in the Price of Gold)*(.1190) + (1-Probability of Rise in the Price of Gold)*(-.0748)
Rearranging Values, we get,
.0123 = (Probability of Rise in the Price of Gold)*(.1190) - .0748 + (.0748*Probability of Rise in the Price of Gold)
Probability of Rise in the Price of Gold = .0871/.1938 = 44.94%
Probability of Fall in the Price of Gold = 1-Probability of Rise in the Price of Gold = (1-44.94%) = 55.06%
With these value of Risk-Neutral Probabilities, we will have to calculate value of expected payoff at the date of expiration as follows:
Expected Payoff at Expiration = (Probability of Rise in the Price of Gold)*(Expected Rise in Price of Gold - Strike Price of Call Option) + (Probability of Fall in the Price of Gold)*(0)
Expected Payoff at Expiration = (.4494)*(1,645 - 1,550) + (.5506)*(0) = 42.693
To determine the current selling price, we need to calculate the present value of Expected Payoff at Expiration as follows:
Selling Price of Option (PV of Expected Payoff at Expiration) = (Expected Payoff at Expiration/(1+Risk Free Rate)^(1/4) = 42.693/(1+5%)^(1/4) = $42.175 or $42.18
Notes:
There can be a slight variation in final answer on account of rounding off values.
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Part c-1)
The synthetic call option with identical payoffs can be created by buying gold and borrowing money at the risk free rate. The quantity (shares of stock) of gold that has to be bought can be calculated with the use of following formula for delta of the option:
Delta = (Swing of Option or Expected Payoff at Expiration)/(Swing of Price of Gold) = (1,645 - 1,550)/(1,645 - 1,360) = .3333
To create a synthetic call option, the company will have to buy .33 ounce of gold (shares of stock).
Notes:
There can be a slight variation in final answer on account of rounding off values.
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Part c-2)
To determine the amount to be borrowed, we will have to perform a comparision of the payoffs that would result from an actual call option and payoff from delta shares at expiration. The comparision is given as follows:
Call Option:
Payoff if the Price of Gold Rises to $1,645 = 1,645 - 1,550 = $95
Payoff if the Price of Gold Falls to $1,360 = 0 (as the option will not be exercised)
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Delta Shares:
Payoff if the Price of Gold Rises to $1,645 = (.3333)*(1,645) = $548.28
Payoff if the Price of Gold Falls to $1,360 = (.3333)*(1,360) = $453.29
The payoff of the actual call option is required to be the same as that of a synthetic all option. As can be observed, that when the company buys delta shares, it will have a minimum payoff of $453.29 irrespective of the fact whether there is an increase in the price of gold or not. Therefore, the total payoff will have to be reduced by $453.29 to match the payoff from actual call option. This can be achieved by borrowing $453.29 at today's value which is calculated as follows:
Amount to be Borrowed = 453.29/(1+5%)^(1/4) = $447.79
Notes:
There can be a slight variation in final answer on account of rounding off values.
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Part d)
The cost of synthetic option is calculated as follows:
Cost of Synthetic Option = Amount Paid to Purchase Gold - Amount Borrowed = .3333*1,470 - 447.79 = $42.16
Notes:
There can be a slight variation in final answer on account of rounding off values.