Pricing Futures By Ryota Kasama

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

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

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

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

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 3. Sell the Futures Contract at $430 after a year 3. 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”

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.

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” -y y: Dividend yield FT= S0(1+rf )T

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

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

Thank you. Questions?