1. Pricing Futures
By Ryota Kasama

2. Outline Why futures price is important?
How is the futures price decided?
FT= S0 (1+rf)T
Arbitrage
Why does this formula always work?
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

3. Why futures pricing is important?
The Exchange
The price of wheat may go down…
The price of wheat may go up….
Wheat Farmer
Baker
Agrees to sell 2 tons of wheat to baker at $200/ton 3 months later.
Futures Contract
Agrees to buy 2 tons of wheat from wheat farmer at $200/ ton 3 months later.
Sell Futures Contract
Buy Futures Contract

4. Arbitrage Possibility of a risk-free profit at zero cost New York
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
New York
$1= ¥101
Buy $1,000
 Payment ¥100,000
Sell $1,000
 Receive ¥101,000

5. 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

6. 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 = $420
Suppose the future price is $430 or $410?
Price Inefficiency
Violations of the formula: Arbitrage opportunity

7. What should the arbitrage profit be when futures price is $420??
FT =S0 (1 + rf)1 = $420
Spot gold price: $400/oz.
The interest rate is 5%
Arbitrageurs sell  The price goes down
The price goes up  Arbitrageurs buy
What if the actual futures price of gold for delivery one year is $430?
That is, FT > ST (1+rf)T
What should the arbitrage profit be when futures price is $420??
What if the actual futures price of gold for delivery one year is $410?
That is, FT < ST (1+rf)T
What would you do??
What would you do??
Sell gold at $400
Invest $400 for the gold
Strategy-2
Strategy-1
Borrow $400
Buy the gold at $400
 -400(1+0.05)=- $420
400(1+0.05)=+$420
Sell the Futures Contract at $430 after a year
Buy the Futures Contract at $410 after a year
+$430
-$410
Arbitrage profit= $430- $420 =$10/oz.
“Cash and Carry Arbitrage”
Arbitrage profit= $420- $410 =$10/oz.
“Reverse Cash and Carry Arbitrage”

8. So, Futures price is decided in order to eliminate profits.
FT =S0 (1 + rf)1 = $420
Spot gold price: $400/oz
The interest rate is 5%
Consider second strategy
‘Reverse Cash and carry arbitrage’
Sell the gold for $400
Invest $400 for the gold
Buy the futures contract at $420 after a year
Consider first strategy ‘Cash and carry arbitrage’
Borrow $400
Buy the gold at $400
Sell the futures contract at $420 after a year
 -400(1+0.05)=- $420
400(1+0.05)=+$420
-$420
+$420
Arbitrage profit= $420- $420 =$0/oz.
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.

9. Commodities and Financial Assets
Commodities Assets: Wheat, coffee, Corn, gold etc…
Financial Assets:
T-bills, stock, and bond etc…

10. 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” -y
y: Dividend yield
FT= S0(1+rf )T

11. Future Prices –Commodity
FT= S0 (1+rf)T :Today’s spot rate and risk-free interest rate
The difference between “ Buy for immediate delivery at the spot price” and “Buy for future delivery at the futures price”
In future contracts,
You can earn interest rate on the purchase price.
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

12. 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
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

13. Thank you. Questions?