Determination of Forward and Futures Prices

1. Determination of Forward and Futures Prices Chapter 5

2. The Goals of Chapter 5 • 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 • Forward and futures values • 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

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. 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

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

7. 5.2 Futures Prices for Investment Assets

8. Theoretical Futures Price for Investment Assets • The effect of the daily settlement of futures is ignored and suppose the interest rate is constant • Under this simplified assumption, the forward and futures prices are identical and used interchangeably • Suppose there is no income or storage costs for the underlying asset of futures • The spot price today is , and the futures price today for delivery in years is . Chapter 1 shows , where is the risk-free interest rate expressed with annual compounding

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) sellthe gold spot and take a long position of the 1-year futures on gold

10. When Interest Rates are Measured with Continuous Compounding • The theoretical futures price expressed with continuous compounding is , where is the risk-free zero rate, with continuous compounding, for the time to maturity • This equation holds for any investment asset that provides no income and has no storage costs, e.g., non-dividend-paying stocks • In this course, we always use the formulae expressed with continuous compounding

11. Consider a Known Dollar Income of Investment Assets • When an investment asset provides a known dollar income at time point , then , where is the present value of the income • If there are multiple dollar incomes in , is the sum of the present values of them • An intuitive way to understand this formula • You can treat as the stock price and as the cash dividend payment at time • It is known that after the payment of the cash dividend at , an identical amount is deducted from the stock price • To reflect the above situation today, the PV of the cash dividend payment , i.e., , should be deducted from the current stock price

12. Consider a Known Dollar Income of Investment Assets • Suppose = $900, an income of $40 occurs at 4 months, and 4-month and 9-month rates are 3% and 4% per annum. If the 9-month futures price is $910 (or $870), is there any arbitrage opportunity? • The PV of the income at 4 months is • The theoretical futures price is • As long as the futures price deviates from this theoretical price, there is an arbitrage opportunity

13. Consider a Known Dollar Income of Investment Assets • For , which is overvalued than its theoretical value • At • Borrow $900: $39.6 for 4 months and $860.4 for 9 months • Buy one unit of asset at = $900 • Enter into a short position of the 9-month futures () • At 4 months • Receive $40 of income from the asset • Use this $40 (= $39.6) to repay the first loan • At 9 months • Sell the asset through the futures for $910 • Use $860.4$886.6 to repay the second loan • Profit realized = $910 – $886.6 = $23.4

14. Consider a Known Dollar Income of Investment Assets • For , which is undervalued than its theoretical value • At • Short sell one unit of asset at = $900 • Invest $39.6 for 4 months and $860.4 for 9 months • Enter into a long position of the 9-month futures () • At 4 months • Receive $39.6$40 from the 4-month investment • Use this $40 to pay the lender of the asset • At 9 months • Receive $860.4$886.6 from 9-month investment • Buy the asset through the futures for $870 • Return the asset to the lender • Profit realized = $886.6 – $870 = $16.6

15. Consider a Known Yield Income of Investment Assets • When an investment asset provides a known yield income (with continuous compounding) in the period , then • An intuitive way to understand the formula • A yield income means that the income is expressed as a percent of the asset’s price at the time the income is paid • Similar to the dollar income, whenever the yield income is paid to the asset holder, there is a negative impact on the asset price • Suppose the asset price is today, the expected annual growth rate of the asset price is , and the asset provides an annual yield of

16. Consider a Known Yield Income of Investment Assets • Annual compounding: • Semiannual compounding: • When the compounding frequency approaches infinity • Thus, the term reflects the negative impact of the yield income on the asset price for a year • If , , which means an amount of is paid to the asset holder • Based on the original formula , if the negative impact of the yield income is considered, the formula should be adjusted as • Note that the role of is similar to the role of in the futures price formula on Slide 5.11

17. Consider a Known Yield Income of Investment Assets • The no-arbitrage argument for the formula of • If • Buy units of the asset at today by borrowing dollars • Invest the continuously generated income in the same asset, and thus by time , it is expected to have units of the asset • Enter into a futures to sell units at • The final sales proceeds from the futures position, , minus the repayment amount of the debt, , can generate a positive payoff, i.e.,

18. Consider a Known Yield Income of Investment Assets • If • Short sell units of the asset at today and deposit the proceeds in a bank to earn the interest rate • Enter into a futures to buy units at • When continuously generated income is paid on the asset, the arbitrageur owes more on the short position. As a result, the short selling position grows at the rate and thus the arbitrageur needs to return units at time • The interest and principal of the deposit, , minus the fund to buy units at , , can generate a positive payoff, i.e., • To eliminate the above two arbitrage opportunities, we can derive theoretically

19. Stock Index Futures • The underlying asset of stock index futures is a stock index level, which can be viewed as an investment asset paying a yield income • A stock index reflects the performance of a virtual portfolio of stocks • It is infeasible to take the PVs of all cash dividends of all stocks in this portfolio into account • In practice, the continuous compounding dividend yield is estimated for this virtual portfolio • The futures and spot prices of a stock index futures follows , where is the continuous compounding dividend yields on the stock index portfolio during the life of the futures contract

20. Index Arbitrage • When , an arbitrageur buys the portfolio of stocks underlying the index and takes a short position of stock index futures • When , an arbitrageur takes a long position of futures and (short) sells the portfolio of stocks underlying the index • The above two strategies are known as index arbitrage and the details are similar to the trading strategies introduced on Slides 5.17 and 5.18

21. Index Arbitrage • Index arbitrage involves simultaneous trades in futures and many different stocks • Very often a computer program is used to generate the trades, which is known as program trading • During a financial crisis, simultaneous trades could be not possible and the no-arbitrage relationship between and does not hold • On Oct. 19, 1987 (Black Monday), the S&P 500 index was 225.06 (down 57.88 on that day) and the futures price for the Dec. S&P 500 index futures was 201.5 (down 80.75 on that day) • The overloaded system on exchanges delays the execution of orders and thus the index arbitrage becomes infeasible

22. Stock Index Futures • The futures contract on the Nikkei 225 Index in CME views 5 times the Nikkei 225 Index, which is measured in yen, as a dollar number • Suppose you take a long position of the Nikki 225 index futures with to be 1000, and on the delivery date, the Nikki 225 index is 1100 • Your payoff is USD$5×(1100 – 1000) = USD$500 • Note that traders cannot trade the stock index portfolio underlying the Nikkei 225 Index in USD • The formula of cannot apply to Nikkei 225 index futures, which is a “quanto” futures (匯率連動期貨) where the underlying asset is measured in one currency and the payoff is in another currency

23. Futures and Forwards on Currencies • A foreign currency is analogous to a security providing a yield income • The foreign risk-free interest rate is the yield an investor can earn if he holds that currency • It follows that if the dividend yield is replaced with the foreign risk-free interest rate, we can derive the futures price as • () is the spot (futures) price of the foreign currency in terms of the domestic currency, i.e., the current exchange rate (the exchange rate applied on the delivery date) • The and are the domestic and foreign risk-free interest rates, respectively

24. Why the Relation Must Be True (US$ is the Domestic Currency) Enter into a foreign currency futures to sell units of foreign currency at  • If , an arbitrage opportunity occurs • To eliminate all arbitrage opportunities, we can derive

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

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

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

28. Valuing a Futures or a Forward Contract • Suppose that is the delivery price specified in a futures contract and at time point , is the current futures price that would apply to the futures contracts • The value of a long futures contract at is , and the value of a short futures contract at is , where is the risk-free interest rate corresponding to the maturity of • Note that the values of the long and short positions are with the same magnitude but with opposite signs

29. Valuing a Futures or a Forward Contract (for a long position) • Suppose that at , , , , and • In practice, when a futures is initiated, the delivery price is set to the current futures price and thus the initial value of the futures equals 0 • Suppose that at (after half a year), , , , and • With the passage of time, as long as the futures price (determined by the demand and supply of futures) does not equal the delivery price, the value of the futures emerges

30. Theoretical Futures/Forward Prices and Futures Values • The current time point is , the maturity time point is , and is the delivery price of futures/forwards • If is 0 (at the beginning of a contract), the theoretical futures/forward prices () are identical to those introduced on Slides 5.10, 5.11, and 5.15 • The formulae of the theoretical values of the futures/forward can be obtained by replacing the futures/forward prices with the formulae in the middle column

31. 5.3 Futures Prices for Consumption Assets

32. Futures Prices for Consumption Assets • Commodities that are consumption assets rather than investment assets usually provide no income, but are subject to significant storage costs • The first way to model the storage cost: , where is the storage cost per unit time as a percent of the asset value (The arbitrage strategies on gold leading to this formula are on Slides 1.34 to 1.36) • Alternative way (for the one-time payment of storage costs): , where is the present value of the storage costs

33. Futures Prices for Consumption Assets • If • Borrow at the risk-free rate and use it to purchase one unit of the commodity and to pay storage costs • Short a futures contract on one unit of the commodity • Lead to a positive payoff of  cannot hold • If • Sell the commodity at , save the PV of the storage costs, , and invest the proceeds to earn the risk-free rate • Take a long position in a futures contract • Lead to a positive payoff of • Owners of a commodity are reluctant to do so because they can consume the commodity but cannot consume the long position of a futures or forward contract

34. Futures on Consumption Assets • Due to the concern for the need of using or consuming commodities, the relationship between the futures and spot prices of a consumption commodity is , where is the storage cost per unit time as a percent of the asset value, or is , where is the present value of the storage costs

35. Convenience Yield and Cost of Carry • The benefits from holding the physical asset are referred to as the convenience yield (便利殖利率) provided by the commodity. Denote the convenience yield as , then we can derive • The convenience yield reflects the concern of the future availability of the commodity • The greater the possibility that shortages will occur, the higher the convenience yield • Use to explain normal and inverted markets on Slides 2.28 and 2.29 • If  increases with  normal market • If  decreases with  inverted market

36. Convenience Yield and Cost of Carry • The relationship between futures and spot prices can be unified in terms of cost of carry (持有成本) • The cost of carry, , is the interest costs plus the storage cost less the income earned, i.e., • For an investment asset, • For a consumption asset, • The convenience yield on the consumption asset, , is defined so that • Note that for investment assets, the convenience yield must be zero to eliminate arbitrage opportunities

37. 5.4 Futures Price vs. Expected Spot Price

38. Relationship Between Futures Prices and Expected Spot Prices • Is the futures price an unbiased estimate of the expected spot prices on the delivery date? • Keynes and Hicks: Hedgers prepare to lose, but speculators hope to make money • If hedgers hold short positions and speculators hold long positions, then • Speculators buy undervalued futures, i.e., , and hedgers would sacrifice (to accept undervalued futures price) in order to reduce risks • If hedgers hold long positions and speculators hold short positions, then • Speculators sell overvalued futures, i.e., , and hedgers would sacrifice (to accept overvalued futures price) in order to reduce risks

39. Relationship Between Futures Prices and Expected Spot Prices • Analysis of the systematic risk based on the CAPM • Suppose is the expected return required by investors on an asset with the price • We can invest the amount of now (to earn the risk-free rate to get back at maturity via a futures contract with the futures price • The present value of (discounted by ) should be the initial investment • According to the CAPM, if the asset has • nosystematic risk (), then and is an unbiased estimate of • positive systematic risk (), then and • negative systematic risk (), then and

40. Relationship Between Futures Prices and Expected Spot Prices • (Normal) backwardation (逆價差)is the market condition where the price of a forward or futures contract is trading below the expected spot price, i.e., (Backwardation is the normal case in practice since for most assets, they have positive systematic risk) • Contango(正價差)is the market condition where the price of a forward or futures contract is trading above the expected spot price, i.e., • Note that in practice, the backwardation and contango are sometimes used to refer to whether the futures prices is below or above the current spot price, i.e., the backwardation is for and the contago is for