BE332 Options and Futures Workshop 4
QUESTION 1
Explain carefully the meaning of the terms convenience yield and cost of carry. What is the
relationship between futures price, spot price, convenience yield, and cost of carry?
Convenience yield measures the extent to which there are benefits obtained from ownership of the
physical asset that are not obtained by owners of long futures contracts. The cost of carry is the
interest cost plus storage cost less the income earned. The futures price, , and spot price, , are
related by
where is the cost of carry, is the convenience yield, and is the time to maturity of the futures
contract.
QUESTION 2
The spot price of silver is $9 per ounce. The storage costs are $0.24 per ounce per year payable
quarterly (i.e. $0.06 per quarter) in advance. Assuming the interest rates are 8% per annum for all
maturities, what should the futures price of silver be for delivery in 9 months? Explain the arbitrage
strategy that could be used if the current 9-month futures price was instead $9.25.
! = (! + )"# = )9 + 0.06$!.!&×! + 0.06$!.!&× !"# + 0.06$!.!&× $"#. !.!&× %"# = (9 + 0.06 + 0.059 + 0.058)!.!&× %"# = 9.74
Note: remember that any number raised to the power of 0 equals 1 e.g. ! = 1
If current 9-month futures price is $9.25, then it is low relative to the spot rate. The basic arbitrage
strategy that could be used is therefore to short sell (if it is not already owned) silver in the spot
market, and enter a long futures position to buy silver in the futures market. In more detail:
1. Short sell the commodity and save the storage costs (! + ) = 9 + 0.06 + 0.059 +0.058 = $9.177
2. Invest (! + ) = $9.177 at the risk-free rate of 8%, for 9 months
3. Take long position in futures contract to buy one ounce of silver at price of $9.25 in 9
months
After 6 months:
1. Investment has grown in value to 9.177!.!&× %"# = $9.74
2. Pay $9.25 for one ounce of silver from long futures position and use to close out short spot
position
Net profit at maturity of $9.74 - $9.25 = $0.49 per ounce of silver
0F 0S
( )
0 0
c y TF S e -=
c y T
QUESTION 3
The two-month interest rates in Switzerland and the United States are 2% and 5% per annum,
respectively, with continuous compounding. The spot price of the Swiss franc is $0.8000. The
futures price for a contract deliverable in two months is $0.8100. What arbitrage opportunities
does this create?
The theoretical no-arbitrage futures rate is:
& = &'()(!*+ = 0.08000(&.&.)&.&/)× "#" = 0.8040
If the actual futures rate/price is $0.8100, then it is too high relative to the current spot rate. This
suggests that an arbitrageur should:
1. Buy Swiss francs in the spot market
2. Sell Swiss francs via a short futures position
Today:
1. Borrow USD ! = $0.8000 at the US risk-free rate of 2% per annum
for two months. (Cash flow +$0.8000)
2. Use money to buy 1 Swiss franc, and invest at Swiss (i.e. foreign) risk-
free rate of 5% p.a. (Cash flow -$0.8000)
3. Take short futures position to sell erfT Swiss Francs at price of $0.8100
in two months time (Cash flow 0)
In 2 months:
1. The 1 SF invested has grown into "2# = !.!(× #"# = 1.033 SF,
withdraw this amount of foreign currency
2. Sell the 1.033 swiss francs via the short forward position at price of ! = $0.8100 per SF, for total cash flow amount of 1.033 ×$0.8100 = $0.8127
3. Repay the USD loan - including interest this is 0.800!.!)× #"# =0.8067 USD (cash flow of -$0.8067)
Net profit: 0.8127 - 0.8067 = 0.0060