Solutions to Assignment 1 (Financial Math. 1, Spring 2017)
Problem 1.30
The price of gold is currently $1,000 per ounce. The forward price for delivery in one year is $1,200. An arbitrageur can borrow money at 10% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no income.
The arbitrageur should borrow money to buy a certain number of ounces of gold today and short forward contracts on the same number of ounces of gold for delivery in one year. This means that gold is purchased for $1000 per ounce and sold for $1200 per ounce. Assuming the cost of borrowed funds is less than 20% per annum this generates a riskless profit.
Problem 1.31
The current price of a stock is $94, and three-month European call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase is trying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?
The investment in call options entails higher risks but can lead to higher returns. If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who buys call options gains
An investor who buys shares gains
The strategies are equally profitable if the stock price rises to a level, S, where
or
The option strategy is therefore more profitable if the stock price rises above $100.
Problem 2.30
Suppose that there are no storage costs for crude oil and the interest rate for borrowing or lending is 5% per annum. How could you make money on May 26, 2010 by trading July 2010 and December 2010 contracts on crude oil? Use Table 2.2.
The July 2010 settlement price for oil is $71.51 per barrel. The December 2010 settlement price for oil is $75.23 per barrel. You could go long one July 2010 oil contract and short one December 2010 contract. In July 2010 you take delivery of the oil borrowing $71.51 per barrel at 5% to meet cash outflows. The interest accumulated in five months is about 71.51×0.05×5/12 or $1.49. In December the oil is sold for $75.23 per barrel which is more than the amount that has to be repaid on the loan. The strategy therefore leads to a profit. Note that this profit is independent of the actual price of oil in June 2010 or December 2010. It will be slightly affected by the daily settlement procedures.
Problem 2.31
What position is equivalent to a long forward contract to buy an asset at on a certain date and a put option to sell it foron that date?
The long forward contract provides a payoff of ST− K where ST is the asset price on the date and K is the delivery price. The put option provides a payoff of max (K−ST, 0). If STK the sum of the two payoffs is ST– K. If STK the sum of the two payoffs is 0. The combined payoff is therefore max (ST– K, 0). This is the payoff from a call option. The equivalent position is therefore a call option.
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