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The pricing of forward and futures contracts

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Outline Spot and futures prices for non-dividend paying investment assets Spot and futures prices for investment assets paying a known income Spot and futures prices for investment assets paying a known yield/return Spot and futures prices for commodities with storage costs Spot and futures prices for consumption commodities with storage costs The cost of carry The valuation of forward contracts

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1a: Non-dividend paying investment asset The forward price of a contract expiring in three months is $43. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected Arbitrage profit at expiration : $2.50

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Case 1a: Implications Eventually, investors would bid up the stock price, and drive down the forward price

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Case 1b: Non-dividend paying investment asset The forward price of a contract expiring in three months is $39. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1b: Non-dividend paying investment asset The forward price of a contract expiring in three months is $39. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1b: Non-dividend paying investment asset The forward price of a contract expiring in three months is $39. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected

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Case 1b: Non-dividend paying investment asset The forward price of a contract expiring in three months is $39. The three-month annualized interest rate is 5%, and the current price of the underlying asset is $40/share. No dividend is expected Arbitrage profit at expiration : $1.50

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Case 1b: Implications Eventually, investors would drive down the stock price, and bid up the forward price

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Relationship between spot and forward/futures prices for a non-dividend paying investment asset F 0 = S 0 e rT F 0 = forward/futures price today S 0 = underlying asset spot price today r = risk-free rate T = time to expiration

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2a: Asset with a known income A bond has the one-year forward price of $930. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%. Arbitrage profit at expiration : $17.61

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Case 2a: Implication Eventually, investors would drive down the forward price, and bid up the spot price of the bond

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%.

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Case 2b: Asset with a known income A bond has the one-year forward price of $905. The current spot price of the bond is $900. Coupon payments are $40 every six months. The six-month risk-free rate is 9%/year and the one-year risk-free rate is 10%. Arbitrage profit at expiration : $ $40 - $905 = $7.39

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Case 2b: Implication Eventually, investors would drive down the spot price, and bid up the forward price of the bond

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Relationship between spot and forward/futures prices for an investment asset providing a known income F 0 = (S 0 - PV income )e rT In our example: PV income = $40e -(0.09)(0.5) + $40e -(0.1)

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63.

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Case 2c: Asset providing a known yield/return Assume two-year rates in the US and Canada are 7% and 5% respectively. The spot rate of the C$ is US$0.62. The two-year forward rate US$0.63. Arbitrage profit = US$16.91

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Case 2c: Implication Eventually, investors would drive down the forward price and bid up the spot price of the US$

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Relationship between spot and forward/futures prices for an investment asset providing a known yield/return F 0 = S 0 e (r-q)T Where q is the known yield/return provided by the investment asset In case 2c, q is the interest rate on the foreign currency.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year.

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Case 3a: Commodities The one-year futures price of gold is $500 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year. Arbitrage profit = $1,537

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Case 3a: Implications In the long run, investors would bid up the spot price of gold and drive down its futures price.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears.

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Example 3b: Commodities The one-year futures price of gold is $470 per troy once. The spot price is $450 per troy once and the risk-free rate is 7%/year. The storage cost of gold is $2 per troy once per year, payable in arrears. Arbitrage profit = $1,463

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Case 3b: Implications In the long run, investors would bid up the futures price of gold and drive down the spot price

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Relationship between spot and forward/futures prices for an investment commodity with storage costs F 0 = (S 0 + U)e rT Where U = PV of storage costs (negative income). If the storage cost is proportional to the price of the asset, storage costs can be viewed as a negative yield: F 0 = S 0 e (r + u)T

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What if the commodity is held for consumption only? In example 3b, one would might not want to sell the gold and engage in arbitrage. Hence, F 0 = < (S 0 + U)e rT F 0 = < S 0 e (r + u)T

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Relationship between spot and forward/futures prices for a consumption commodity with storage costs F 0 = (S 0 + U)e (r - y)T or F 0 = S 0 e (r + u - y)T Where y is a fudge factor called convenience yield

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The cost of carry It measures the interest paid to finance the asset plus storage costs less income earned on the asset. For an investment asset paying no dividend cc = r For a stock index cc = q For a foreign currency cc = r f For a commodity with storage costs proportional to price cc = r + u

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The valuation of forward contracts What is the value of a forward contract at inception? Zero

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The valuation of forward contracts: Investment asset paying no dividend/income What is the value of a forward contract between inception and maturity? Long contract f = (F 0 - F)e -rT f = S 0 - Fe -rT Short contract f = (F - F 0 )e -rT f = Fe -rT - S 0 Where F is the current forward price of the contract.

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The valuation of futures contracts What is the value of a futures contract between inception and maturity? At the end of each trading day, the value of futures is set back to zero as a result of marking-to-market.

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Spot and forward/futures prices: A summary

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