Exercise $5$

A company wishes to hedge its exposure to a new fuel whose price changes have a $0\mathrm{.}6$ correlation with gasoline futures price changes. The company will lose $$1$ million for each $1$ cent increase in the price per gallon of the new fuel over the next three months. The new fuel's price changes have a standard deviation that is $50\%$ greater than price changes in gasoline futures prices. If gasoline futures are used to hedge the exposure, what should the hedge ratio be$?$ What is the company's exposure measured in gallons of the new fuel? What position, measured in gallons, should the company take in gasoline futures? How many gasoline futures contracts should be traded? Each contract is on $42,000$ gallons.

Exercise $6$

It is now October $2014.$ A company anticipates that it will purchase $1$ million pounds of copper in each of February $2015,$ August $2015,$ February $2016,$ and August $2016.$ The company has decided to use the futures contracts traded in the COMEX division of the CME Group to hedge its risk. One contract is for the delivery of $25,000$ pounds of copper. Devise a hedging strategy for the company.

Assume the market prices $($in cents per pound$)$ today and at future dates are as in the following table. What is the impact of the strategy you propose on the price the company pays for copper?

$\backslash$table$[[$Date$,$October $2014,$February $2015,$August $2015,$February $2016,$August $2016],[$Spot Price,$372\mathrm{.}00,369\mathrm{.}00,365\mathrm{.}00,377\mathrm{.}00,388\mathrm{.}00],[$March $2015$ Futures Price,$372\mathrm{.}30,369\mathrm{.}10,,,],[$September $2015$ Futures Price,$372\mathrm{.}80,370\mathrm{.}20,364\mathrm{.}80,,],[$March $2016$ Futures Price,,$370\mathrm{.}70,364\mathrm{.}30,376\mathrm{.}70,],[$September $2016$ Futures Prices,,,$364\mathrm{.}20,376\mathrm{.}50,388\mathrm{.}20]]$

Exercise $7$

An index is $1,200.$ The three$-$month risk$-$free rate is $3\%$ per annum and the dividend yield over the next three months is $1\mathrm{.}2\%$ per annum. The six$-$month risk$-$free rate is $3\mathrm{.}5\%$ per annum and the dividend yield over the next six months is $1\%$ per annum. Estimate the futures price of the index for threemonth and six$-$month contracts. All interest rates and dividend yields are continuously compounded.

Exercise $8$

A one$-$year long forward contract on a non$-$dividend$-$paying stock is entered into when the stock price is $$40$ and the riskfree rate of interest is $10\%$ per annum with continuous compounding. The forward price is $$40.$

a$)$ Is there an arbitrage possibility here? Why?

b$)$ How can you make a profit out of this arbitrage strategy?