Outline
1. Why futures price is important?
2. How is the futures price decided?
 FT= S0 (1+rf)T
 Arbitrage
3. Why does this formula always work?
4. Futures prices of Financial assets
 FT= S0 (1+rf--y)T
5. Futures prices of Commodity assets
 FT= S0 (1+rf + storage cost –convenience yield )T
Why futures pricing is important?
The price of
wheat may
go down…
The Exchange
Wheat Farmer
Agrees to sell 2 tons of
wheat to baker at $200/ton
3 months later.
Sell Futures Contract
The price of
wheat may
go up….
Baker
Futures
Contract
Agrees to buy 2 tons of
wheat from wheat farmer at
$200/ ton 3 months later.
Buy Futures Contract
Arbitrage
Possibility of a risk-free profit at zero cost
 By Buying the cheap and Selling the expensive
 Market (Price) inefficiency
 Arbitrage opportunity is eliminated in a second
Profit
¥101,000 - ¥100,000= ¥1,000
*Risk-free / Zero cost
Tokyo
$1=¥100
Buy $1,000  Payment ¥100,000
New York
$1= ¥101
Sell $1,000  Receive ¥101,000
What should the futures price be?
Pricing is determined by the spot price and interest rate.
You don’t pay up front, so you can earn interest on the purchase price.
Violation of this formula gives Arbitrage opportunity
FT = S0 (1 + rf)T
FT = Futures Price lasting T period
S0 = Today’s Spot Price
r f = Risk free Interest rate
Simple Example
Today, Spot price of gold: $400/oz
The one year interest rate: 5%
For there to be no arbitrage, the future price of gold for delivery one year
should be:
FT = S0 (1 + rf)T
= 400(1+ 0.05)1 = $420
Suppose the future price is $430 or $410?
 Price Inefficiency
 Violations of the formula: Arbitrage opportunity
FT =S0 (1 + rf )1 = $420
Arbitrageurs sell  The price goes down
The price goes up  Arbitrageurs buy
Spot gold price: $400/oz.
The interest rate is 5%
• What if the actual futures price of gold for
• What if the actual futures price of gold for delivery
What should the arbitrage
profit
beiswhen
delivery
one
year
$410?
one year is $430?
That is, FT > ST (1+rf)T futures price is $420??That is, FT < ST (1+rf)T
What would you do??
What would you do??
Strategy-1
Strategy-2
1.
2.
3.
Borrow $400  -400(1+0.05)=- $420
Buy the gold at $400
Sell the Futures Contract at $430 after a year
+$430
1.
2.
3.
Sell gold at $400
Invest $400 for the gold 400(1+0.05)=+$420
Buy the Futures Contract at $410 after a year
-$410
Arbitrage profit= $430- $420 =$10/oz.
Arbitrage profit= $420- $410 =$10/oz.
“Cash and Carry Arbitrage”
“Reverse Cash and Carry Arbitrage”
FT =S0 (1 + rf )1 = $420
Spot gold price: $400/oz
The interest rate is 5%
• Consider first strategy ‘Cash and carry arbitrage’
1.
2.
3.
Borrow $400  -400(1+0.05)=- $420
Buy the gold at $400
Sell the futures contract at $420 after a year
+$420
Arbitrage profit= $420- $420 =$0/oz.
• Consider second strategy
‘Reverse Cash and carry arbitrage’
1. Sell the gold for $400
2. Invest $400 for the gold 400(1+0.05)=+$420
3. Buy the futures contract at $420 after a year -$420
Arbitrage profit= $420- $420 =$0/oz.
When the futures price is $420/oz , the arbitrage profit has disappeared.
So, Futures price is decided in order to eliminate profits.
Commodities and Financial Assets
Commodities Assets:
Wheat, coffee, Corn, gold etc…
Financial Assets:
T-bills, stock, and bond etc…
Futures Prices- Financial Assets
FT= S0 (1+rf)T : Today’s spot rate and risk-free interest rate
Consider again the difference between “ Buy for immediate delivery at the
spot price” and “Buy for future delivery at the futures price”
FT= S0(1+rf -y)T
y: Dividend yield
Future Prices –Commodity
0 FT= S0 (1+rf)T :Today’s spot rate and risk-free interest rate
0 The difference between “ Buy for immediate delivery at the spot price” and
“Buy for future delivery at the futures price”
In future contracts,
1. You can earn interest rate on the purchase price.
2. You don’t need to store commodities Save warehouse costs
3. No Convenience Yield: the benefit associated with holding an physical good
FT= S0 (1+ rf+ storage costs- convenience yield)T
Summary
 Futures pricing is important
 FT= S0 (1+rf)T
 No arbitrage opportunity and profit
 FT> S0 (1+rf)T or FT< S0 (1+rf)T
 Arbitrage opportunity
 FT= S0 (1+rf)T
Futures prices of Financial assets
 FT= S0 (1+rf--y)T
Futures prices of Commodity assets
 FT= S0 (1+rf + storage cost –convenience yield )T
Thank you. Questions?