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Pricing Futures By Ryota Kasama

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Outline 1. Why futures price is important? 2. How is the futures price decided? F T = S 0 (1+r f ) T Arbitrage 3. Why does this formula always work? 4. Futures prices of Financial assets F T = S 0 (1+r f- -y) T 5. Futures prices of Commodity assets F T = S 0 (1+r f + storage cost –convenience yield ) T

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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

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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 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

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What should the futures price be? Pricing is determined by the spot price and interest rate. You dont pay up front, so you can earn interest on the purchase price. Violation of this formula gives Arbitrage opportunity F T = S 0 (1 + r f ) T F T = Futures Price lasting T period S 0 = Todays Spot Price r f = Risk free Interest rate

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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: F T = S 0 (1 + r f ) T = 400( ) 1 = $420 Suppose the future price is $430 or $410? Price Inefficiency Violations of the formula: Arbitrage opportunity

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F T =S 0 (1 + r f ) 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, F T < S T (1+r f ) T What would you do?? 400(1+0.05)=+$420 -$ Buy the Futures Contract at $410 after a year Arbitrage profit= $420- $410 =$10/oz. Reverse Cash and Carry Arbitrage 1.Sell gold at $400 2.Invest $400 for the gold Strategy-2 What if the actual futures price of gold for delivery one year is $430 ? That is, F T > S T (1+r f ) T What would you do?? Strategy-1 1.Borrow $400 2.Buy the gold at $ 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??

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F T =S 0 (1 + r f ) 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 $400 2.Buy the gold at $400 3.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 $400 2.Invest $400 for the gold 3.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.

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

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Futures Prices- Financial Assets F T = S 0 (1+r f ) T : Todays 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 F T = S 0 (1+r f ) T -y y: Dividend yield

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Future Prices –Commodity 0 F T = S 0 (1+r f ) T :Todays 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 dont need to store commodities Save warehouse costs 3. No Convenience Yield: the benefit associated with holding an physical good F T = S 0 (1+ r f + storage costs- convenience yield) T

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Summary Futures pricing is important F T = S 0 (1+r f ) T No arbitrage opportunity and profit F T > S 0 (1+r f ) T or F T < S 0 (1+r f ) T Arbitrage opportunity F T = S 0 (1+r f ) T Futures prices of Financial assets F T = S 0 (1+r f- -y) T Futures prices of Commodity assets F T = S 0 (1+r f + storage cost –convenience yield ) T

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Thank you. Questions?

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