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The pricing of forward and futures contracts

The pricing of forward and futures contracts. 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

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The pricing of forward and futures contracts

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  1. The pricing of forward and futures contracts

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

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

  10. 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

  11. 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

  12. 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

  13. 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

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

  15. Relationship between spot and forward/futures prices for a non-dividend paying investment asset F0 = S0erT F0 = forward/futures price today S0 = underlying asset spot price today r = risk-free rate T = time to expiration

  16. 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%.

  17. 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%.

  18. 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%.

  19. 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%.

  20. 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%.

  21. 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%.

  22. 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

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

  24. 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%.

  25. 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%.

  26. 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%.

  27. 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%.

  28. 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%.

  29. 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 : $952.39 - $40 - $905 = $7.39

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

  31. Relationship between spot and forward/futures prices for an investment asset providing a known income F0 = (S0 - PVincome)erT In our example: PVincome = $40e-(0.09)(0.5) + $40e-(0.1)

  32. 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.

  33. 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.

  34. 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.

  35. 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.

  36. 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.

  37. 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.

  38. 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

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

  40. Relationship between spot and forward/futures prices for an investment asset providing a known yield/return F0 = S0e(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.

  41. 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.

  42. 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.

  43. 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.

  44. 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.

  45. 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.

  46. 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

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

  48. 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.

  49. 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.

  50. 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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