Pricing FuturesBy Ryota Kasama

Outline1. 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

The price of wheat may go down…

Wheat Farmer

Futures Contract

The price of wheat may

go up….

Agrees to sell 2 tons of wheat to baker at $200/ton

3 months later.

Agrees to buy 2 tons of wheat from wheat farmer at $200/ ton 3 months later.

Why futures pricing is important?

The Exchange

Sell Futures Contract Buy Futures Contract

Baker

ArbitragePossibility of a risk-free profit at zero cost

By Buying the cheap and Selling the expensiveMarket (Price) inefficiency Arbitrage opportunity is eliminated in a second

New York $1= ¥101

Tokyo $1=¥100 Buy $1,000 Sell $1,000

　　　　　 Profit¥101,000 - ¥100,000= ¥1,000 *Risk-free / Zero cost　　

Payment ¥100,000 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 periodS0 = Today’s Spot Pricer f = Risk free Interest rate

Simple ExampleToday, 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

Spot gold price: $400/oz. The interest rate is 5%

• 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??

400(1+0.05)=+$420

-$4103. Buy the Futures Contract at $410 after a year

Arbitrage profit= $420- $410 =$10/oz.

“Reverse Cash and Carry Arbitrage”

1. Sell gold at $4002. Invest $400 for the gold

Strategy-2

• What if the actual futures price of gold for delivery one year is $430?

That is, FT > ST (1+rf)T

What would you do??

Strategy-11. Borrow $400 2. Buy the gold at $4003. Sell the Futures Contract at $430 after a year

Arbitrage profit= $430- $420 =$10/oz.

“Cash and Carry Arbitrage”

-400(1+0.05)=- $420

+$430

Arbitrageurs sell The price goes down The price goes up Arbitrageurs buy

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%

400(1+0.05)=+$420 -$420

Arbitrage profit= $420- $420 =$0/oz.

• Consider first strategy ‘Cash and carry arbitrage’

1. Borrow $4002. Buy the gold at $4003. Sell the futures contract at $420 after a year

Arbitrage profit= $420- $420 =$0/oz.

-400(1+0.05)=- $420

+$420

• Consider second strategy ‘Reverse Cash and carry arbitrage’

1. Sell the gold for $4002. Invest $400 for the gold3. Buy the futures contract at $420 after a year

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 )T -y y: Dividend yield

Future Prices –Commodity

0FT= S0 (1+rf)T :Today’s spot rate and risk-free interest rate

0The 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 costs3. 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 assetsFT= S0 (1+rf--y)T

Futures prices of Commodity assets FT= S0 (1+rf + storage cost –convenience yield )T

Thank you. Questions?