1. Determination of Forward and Futures Prices Chapter 5 5.1

2. The Goals of Chapter 5 5.2 Background knowledge –Investment vs. consumption assets, short selling ( 賣空 ), and assumptions for market participants Futures prices for investment assets –Adjustment for known dollar incomes or yields –Futures on stock indices and foreign currencies –Futures vs. forward prices –Valuing forward or futures contracts Futures prices for consumption assets –Convenience yield ( 便利殖利率 ) and cost of carry theory ( 持有成本理論 ) Futures price vs. expected spot price

3. 5.1 Background Knowledge 5.3

4. Consumption vs. Investment Assets Investment assets are assets held for investment purposes, e.g., stock shares, bonds, currencies, gold, silver Consumption assets are assets held for consumption, e.g., copper, oil, pork, corn ※ The no-arbitrage argument can (cannot) be used to fully determine the forward and futures prices of investment (consumption) assets ※ Some investment assets, like gold or silver, have a number of industrial uses and thus can be consumed, so they are consumption assets as well 5.4

5. Short Selling Short selling ( 賣空 ) involves selling securities you do not own –Your broker borrows securities from another clients and sells them in a market on behalf of you –Earn positive payoffs if the security price declines –At some stage you must buy the securities and return them back to the accounts of the clients who lend you these securities –You must pay dividends and any incomes that the owners of the securities should receive in this short selling period (The security owners feel as if they continuously held these securities) –There may be a small fee for borrowing securities 5.5

6. Assumptions for Market Participants Four assumptions associated with market participants –They are subject to no transaction costs when they trade –They are subject to the same tax rate on their net trading profits –They can borrow or lend money at the risk-free rate with unlimited amount –They take advantage of any arbitrage opportunity as it occurs 5.6

7. 5.2 Futures Prices for Investment Assets 5.7

8. 5.8 Theoretical Futures Price for Investment Assets

9. 5.9 Arbitrage Example for Gold Futures Suppose that –Spot price of gold today is $1000 –1-year gold futures price today is $1100 ($990) –The interest rate is 5% per annum –No income or storage costs for gold The theoretical value of the futures price on gold is $1,000×(1+5%)=$1,050 –Futures price > $1050  Buy the gold spot and take a short position of the 1-year futures on gold –Futures price < $1050  (Short) sell the gold spot and take a long position of the 1-year futures on gold

10. When Interest Rates are Measured with Continuous Compounding 5.10

11. Consider a Known Dollar Income of Investment Assets 5.11

12. Consider a Known Dollar Income of Investment Assets 5.12

13. Consider a Known Dollar Income of Investment Assets 5.13

14. Consider a Known Dollar Income of Investment Assets 5.14

15. Consider a Known Yield Income of Investment Assets 5.15

16. Consider a Known Yield Income of Investment Assets 5.16

17. Consider a Known Yield Income of Investment Assets 5.17

18. Consider a Known Yield Income of Investment Assets 5.18

19. Stock Index Futures 5.19

20. Index Arbitrage 5.20

21. Index Arbitrage 5.21

22. 5.22 Stock Index Futures

23. Futures and Forwards on Currencies 5.23

24. Why the Relation Must Be True (US$ is the Domestic Currency) 5.24 

25. Forward vs. Futures Prices Forward and futures prices are usually assumed to be the same When interest rates are stochastic ( 隨機 ), forward and futures prices could be different due to the enforcement of daily settlement for trading futures –The difference between forward and futures prices can be significant if there exists a relationship between the interest rate and the underlying variable, e.g., Eurodollar futures introduced in Ch. 6 5.25

26. Forward vs. Futures Prices –A positive correlation between interest rates and the asset price With the increase of the asset price, the futures holder can earn doubly from the increase of the balance of the margin account and the higher interest rate With the decrease of the asset price, the futures holder losses, but the opportunity cost of fund for these losses is low due to the lower interest rate Thus, the futures contract is more attractive and demanded so that futures price > forward price 5.26

27. Forward vs. Futures Prices –A negative correlation between interest rates and the asset price With the increase of the asset price, the benefit of the increase of the balance of the margin account will be offset by the lower interest rate With the decrease of the asset price, the balance of the margin account decreases such that the futures holder cannot fully enjoy the higher interest rate Thus, the futures contract is relatively not attractive so that futures price < forward price 5.27

28. Valuing a Futures or a Forward Contract 5.28

29. 5.29 Valuing a Futures or a Forward Contract (for a long position)

30. 5.30 Theoretical Futures/Forward Prices and Futures Values Asset Without any income

31. 5.3 Futures Prices for Consumption Assets 5.31

32. Futures Prices for Consumption Assets 5.32

33. Futures Prices for Consumption Assets 5.33

34. Futures on Consumption Assets 5.34

35. Convenience Yield and Cost of Carry 5.35

36. Convenience Yield and Cost of Carry 5.36

37. 5.4 Futures Price vs. Expected Spot Price 5.37

38. Relationship Between Futures Prices and Expected Spot Prices 5.38

39. 5.39 Relationship Between Futures Prices and Expected Spot Prices

40. 5.40 Relationship Between Futures Prices and Expected Spot Prices