Only B There are two apple orchards, in Chicago and NYC. Driving between the apple orchards takes 8 hours. For simplicity, suppose the risk-free rate is 0. There are liquid futures contracts for delivery of a ton of apples at Chicago and NYC in 1 month’s time. For notational clarity, let fC and fN represent the prices of Chicago/NYC futures contracts in the first period, and let pC and pN represent the ultimate prices of a ton of apples in Chicago and NYC in the second period (one month later, when the future contracts expire). a). Suppose you think there is going to be excess demand for apples in NYC relative to Chicago, but you are not sure what the level of apple prices will be. You would then enter into a spread contract: you would long an NYC futures contract, and short a Chicago futures contract. If you enter into this contract, what is your payoff in one month when the contract expires? Write an expression for the payoff of the spread contract, in terms of fC, fN, pC, pN. (Hint: this looks complicated, but the easy way to do it is to start by thinking about the payoff of an individual futures position). b). Suppose you are not sure whether there will be excess demand in NYC or Chicago. However, you believe that the absolute value of the difference between the Chicago and NYC prices, pC − pN, will be large (either very positive or very negative). You want to construct a bet that will pay you the absolute value of the difference between prices, |pC − pN|, at expiration. You can construct these bets using call and put options on the spread. For example, a call option on the spread at strike price 1 would pay you: (pC − pN) − 1 pC − pN ⩾1 0 pC − pN < 1 That is, the payoff is the difference between (pC − pN) and 1 if the difference pC − pN is greater than 1, so the call option is in the money. The payoff is 0 otherwise. Suppose you can trade call and put options on the spread between pC and pN at any strike price. How do you construct a bet whose payoff (ignoring the upfront price you pay for it) is equal to |pC − pN|? Describe the set of call and put options that you must buy. c). It costs x dollars, paid upfront in the first period, to reserve an 8-hour truck rental (well, let’s say 12 hours, so you have time to make the trip and then return the truck) which can hold one ton of apples. Using your answer to 2., suppose the upfront price of the bet which pays |pC − pN|is s dollars in the first period. For some values of x and s, there is a way to construct an arbitrage trade, combining the truck rental and the option position, in a way that guarantees that you will make money. What is the condition on x and s under which an arbitrage trade is possible? Describe how you would do this arbitrage trade. Explain precisely why the trade is riskless: that is, explain the payoffs of each part of the trade and how they contribute to creating a riskless profit.
Only B
There are two apple orchards, in Chicago and NYC. Driving between the apple orchards takes 8
hours. For simplicity, suppose the risk-free rate is 0. There are liquid futures contracts for delivery of a ton
of apples at Chicago and NYC in 1 month’s time. For notational clarity, let fC and fN represent the prices
of Chicago/NYC futures contracts in the first period, and let pC and pN represent the ultimate prices of a
ton of apples in Chicago and NYC in the second period (one month later, when the future contracts expire).
a). Suppose you think there is going to be excess demand for apples in NYC relative to
Chicago, but you are not sure what the level of apple prices will be. You would then enter into a spread
contract: you would long an NYC futures contract, and short a Chicago futures contract. If you enter
into this contract, what is your payoff in one month when the contract expires? Write an expression
for the payoff of the spread contract, in terms of fC, fN, pC, pN. (Hint: this looks complicated, but the
easy way to do it is to start by thinking about the payoff of an individual futures position).
b). Suppose you are not sure whether there will be excess demand in NYC
or Chicago. However, you believe that the absolute value of the difference between the Chicago and
NYC prices, pC − pN, will be large (either very positive or very negative). You want to construct a bet
that will pay you the absolute value of the difference between prices, |pC − pN|, at expiration. You can
construct these bets using call and put options on the spread. For example, a call option on the spread
at strike price 1 would pay you:
(pC − pN) − 1 pC − pN ⩾1
0 pC − pN < 1
That is, the payoff is the difference between (pC − pN) and 1 if the difference pC − pN is greater than
1, so the call option is in the money. The payoff is 0 otherwise. Suppose you can trade call and put
options on the spread between pC and pN at any strike price. How do you construct a bet whose
payoff (ignoring the upfront price you pay for it) is equal to |pC − pN|? Describe the set of call and
put options that you must buy.
c). It costs x dollars, paid upfront in the first period, to reserve an 8-hour
truck rental (well, let’s say 12 hours, so you have time to make the trip and then return the truck)
which can hold one ton of apples. Using your answer to 2., suppose the upfront price of the bet which
pays |pC − pN|is s dollars in the first period. For some values of x and s, there is a way to construct
an arbitrage trade, combining the truck rental and the option position, in a way that guarantees that
you will make money. What is the condition on x and s under which an arbitrage trade is possible?
Describe how you would do this arbitrage trade. Explain precisely why the trade is riskless: that is,
explain the payoffs of each part of the trade and how they contribute to creating a riskless profit.
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