Determination of Forward and Futures Prices Chapter 3

1. Determination of Forward and Futures PricesChapter 3

2. Determinants of Forward and Futures Prices • Examine forward prices first and then show how forward and futures prices are related • Forward prices on investment assets • that provide no income • e.g. non-dividend paying stock • that provide a known cash income • e.g. coupon bond or • that provide a known dividend yield • Forward prices on consumption assets

3. Consumption vs Investment Assets • Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold, silver) • Consumption assets are assets held primarily for consumption (Examples: copper, oil)

4. Short Selling (Page 40-42) • Short selling involves selling securities you do not own • Your broker borrows the securities from another client and sells them in the market in the usual way

5. Short Selling(continued) • At some stage you must buy the securities back so they can be replaced in the account of the client • You must pay dividends and other benefits the owner of the securities receives

6. Some Preliminaries: Measuring Interest Rates • The compounding frequency used for an interest rate is the unit of measurement • The difference between quarterly and annual compounding is analogous to the difference between miles and kilometers

7. Some Preliminaries: Continuous Compounding(Page 43) • In the limit as we compound more and more frequently we obtain continuously compounded interest rates • $100 grows to $100eRTwhen invested at a continuously compounded rateRfor time T • $100 received at time Tdiscounts to $100e-RTat time zero when the continuously compounded discount rate is R

8. Conversion Formulas(Page 43) Define Rc : continuously compounded rate Rm: same rate with compounding m times per year

9. Notation

10. Gold Example (From Slide 1.26) • For gold F0 = S0(1 + r )T (assuming no storage costs) • If ris compounded continuously instead of annually F0 = S0erT

11. Extension of the Gold Example(Page 46, equation 3.5) • For any investment asset that provides no income and has no storage costs F0 = S0erT

12. When an Investment Asset Provides a Known Dollar Income (page 49, equation 3.6) F0= (S0– I )erT where I is the present value of the income

13. When an Investment Asset Provides a Known Yield (Page 51, equation 3.7) F0 = S0e(r–q )T where q is the average yield during the life of the contract (expressed with continuous compounding)

14. Valuing a Forward ContractPage 51 • The value of a forward contract at the time when it is first entered into is zero. • Value of forward contract after the initiation of the contract • Suppose that K is delivery price in a forward contract & F0is forward price that would apply to the contract today • The value of a long forward contract, ƒ, is ƒ = (F0 – K )e–rT • Similarly, the value of a short forward contract is (K – F0)e–rT

15. Forward prices:Forwards as a delayed sale • An investor is considering the sale of a stock three months from today: • Immediate sale: sell the stock today at the current spot price for the asset, S0 (e.g. S0 = 61.00). • deposit S0 and earn the interest rate r per year with continuous compounding (e.g. r = 6.71%) • Delayed sale: enter into a forward contract today to sell the stock in three months for F0. • What is the least that the investor is willing to receive for the stock in three months?

16. Forward prices:Forwards as a delayed purchase • An investor is considering the purchase of a stock three months from today: • Immediate purchase: buy the stock today at the current spot price for the asset, S0 (e.g. 61.00) • Delayed purchase: Take the current price of the stock today, S0, and deposit it and earn the interest rate r per year with continuous compounding (e.g. r = 6.71%) • What is the most that the investor is willing to pay for the stock in three months?

17. Forward prices:Forwards as a delayed sale or purchase • Putting it together: If there are to be no arbitrage opportunities in the market, what is the forward price, F, for a contract to buy or sell this stock three months from today? • Suppose the stock pays income to an investor between today and the maturity of the contract? • How should we, if at all, adjust the price? • Think about the buyer • Think about the seller • Think about whether the asset is a financial asset or a commodity for consumption

18. Financial Investments Investments that provide no income Investments that provide a known cash income Investments that provide a known dividend yield Commodities for investment with storage costs (e.g., gold and silver) Consumption Investments Commodities with a known cash storage cost Commodities with a known continuous storage cost Convenience yields Forward contract on Types of Assets:Each asset has an associated cost of carrying the asset from the spot to forward market

19. No Arbitrage Pricing • Consider a six-month forward contract on a dividend paying stock with a price of $34.375 per share. • Dividends of $0.20 per share are expected after three and six months. • The three-month and six-month risk-free rates of interest are, respectively, 7.875% and 8.00% per year with continuous compounding. • The quoted forward price for a contract maturing in six-months is $35.80. • Does an arbitrage opportunity exist?

20. Forward vs Futures Prices • Forward and futures prices are usually assumed to be the same. When interest rates are uncertain they are, in theory, slightly different: • A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward price • A strong negative correlation implies the reverse

21. Stock Index • Stock index tracks the change in the value of a hypothetical portfolio of stocks. • S&P 500 Index is based on a portfolio of 500 stocks (400 industrial, 40 utilities, 20 transportation, and 40 financial companies.) • Nikkei 225 Stock Average is based on a portfolio of 225 of the largest stocks trading on the Tokyo Stock Exchange • Stock index futures: futures contract on the stock index • S&P 500 Futures is on 250 times the index • Nikkei 225 Stock Average is on 5 times the index

22. Stock Index continued • Can be viewed as an investment asset paying a dividend yield • The futures price and spot price relationship is therefore F0 = S0e(r–q )T where q is the dividend yield on the portfolio represented by the index • For the formula to be true it is important that the index represent an investment asset • In other words, changes in the index must correspond to changes in the value of a tradable portfolio • The Nikkei index viewed as a dollar number does not represent an investment asset

23. Stock Index Arbitrage • The purchase or sale of a portfolio of stocks that replicates a stock index and the sale or purchase of a futures contract on the index • Occurs when the F0 does not equal S0e(r-q)T • When F0>S0e(r-q)Tan arbitrageur buys the stocks underlying the index and sells futures • When F0<S0e(r-q)Tan arbitrageur buys futures and shorts or sells the stocks underlying the index • Often implemented through program trading • execution on a stock market of a large number of simultaneous buy or sell orders. • Triggered by a computer program that detects an arbitrage opportunity or suggests some other reason for quickly establishing a large portfolio of stocks.

24. Index Arbitrage(continued) • Index arbitrage involves simultaneous trades in futures and many different stocks • Very often a computer is used to generate the trades • Occasionally (e.g., on Black Monday) simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold

25. Stock Index Arbitrage • On November 8, the S&P 500 Index is at $1150.50, the dividend yield is 3% and the risk-free rate is 5%. The December futures contract is priced at $1,160.46 and expires on December 18. T = 40/365 = 0.1096 • What is the theoretical price of the December futures contract? • What type of an arbitrage can be executed using $20 million. • On December 18 the S&P 500 Index is at 1,146.41.

26. Futures and Forwards on Currencies (Page 57-59) • A foreign currency is analogous to a security providing a dividend yield • The continuous dividend yield is the foreign risk-free interest rate • It follows that if rfis the foreign risk-free interest rate

27. Futures on Consumption Assets(Page 62) F0S0e(r+u )T where u is the storage cost per unit time as a percent of the asset value. Alternatively, F0(S0+U )erT where U is the present value of the storage costs.

28. Convenience Yields • Benefits to users of commodity that are not obtained by holders of futures contracts • Ability to keep a production process running • Ability to profit from temporary local shortages • The convenience yield, y, is defined so that • If storage costs are a known cash amount F0eyT = (S0+U)erT • If storage costs are proportional to the commodity price F0 = S0e(r+u-y)T

29. The Cost of Carry (Page 63) • The cost of carry, c, is the storage cost plus the interest costs less the income earned • For an investment asset F0 = S0ecT • For a consumption asset F0S0ecT • The convenience yield on the consumption asset, y, is defined so that F0 = S0 e(c–y )T

30. Cost of Carry: Summarizes relation between futures and spot price

31. Cost of Carry Pricing

32. Futures Prices & Expected Future Spot Prices (Page 64) • Suppose k is the expected return required by investors on an asset • We can invest F0e–r Tnow to get STback at maturity of the futures contract • This shows that F0 = E (ST )e(r–k )T

33. Futures Prices & Future Spot Prices (continued) • If the asset has • nosystematic risk, then k = r andF0 is an unbiased estimate of ST • positive systematic risk, then k> r and F0 < E (ST ) • negative systematic risk, then k< r and F0> E (ST )

34. Convergence of Futures Price to Spot Priceand Backwardation and Contango (a) (b) Futures Price Spot Price Futures Price Spot Price Time Time