1. Options, Futures, and Other Derivatives
Tenth Edition
Chapter 5
Determination of Forward and Futures Prices
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2. Chapter Agenda
Deriving relationship between forward (or futures) prices and spot prices
Examine relationship between futures prices and spot prices for contracts on stock indices, foreign exchange and commodities

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

4. Short Selling (Page 108--109) (1 of 2)
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 (2 of 2)
At some stage you must buy the securities so they can be replaced in the account of the client
You must pay dividends and other benefits the owner of the securities receives
There may be a small fee for borrowing the securities

6. Example
You short 100 shares when the price is $100 and close out the short position three months later when the price is $90
During the three months a dividend of $3 per share is paid
What is your profit?
What would be your loss if you had bought 100 shares?

7. Assumptions
The market participants are subject to no transaction costs when they trade
The market participants are subject to the same tax rate on all net trading profits
The market participants can borrow money at the same risk-free rate of interest as they can lend money
The market participants take advantage of arbitrage opportunities as they occur

8. Notation for Valuing Futures and Forward Contracts
Spot price today
F0:
Futures or forward price today T:
Time until delivery date r:
Risk-free interest rate for maturity T

9. An Arbitrage Opportunity?
Suppose that:
The spot price of a non-dividend-paying stock is $40
The 3-month forward price is $43
The 3-month US$ interest rate is 5% per annum
Is there an arbitrage opportunity?

10. Another Arbitrage Opportunity?
Suppose that:
The spot price of nondividend-paying stock is $40
The 3-month forward price is US$39
The 1-year US$ interest rate is 5% per annum (continuously compounded)
Is there an arbitrage opportunity?

11. The Forward Price
If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then
F0 = S0erT
where r is the T-year risk-free rate of interest.
In our examples, S0 =40, T = 0.25, and r = 0.05 so that
F0 = 40e0.05×0.25 = 40.50

12. Example
What should be the 4-month forward price of the zero-coupon bond?

13. If Short Sales Are Not Possible…
Formula still works for an investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low

14. When an Investment Asset Provides a Known Income (page 113, equation 5
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
Is there an arbitrage opportunity?

15. When an Investment Asset Provides a Known Income (page 113, equation 5
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
Is there an arbitrage opportunity?

16. When an Investment Asset Provides a Known Income (page 113, equation 5
What should be the 10 months forward price of the stock?

17. When an Investment Asset Provides a Known Yield (Page 115, equation 5
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
Following formula is used to convert non-annual compounding rate to continuous compounding rate
What should be the 6-months forward price of the asset?

18. Valuing a Forward Contract
A forward contract is worth zero (except for bid-offer spread effects) when it is first negotiated
Later it may have a positive or negative value
Suppose that K is the delivery price and F0 is the forward price for a contract that would be negotiated today

19. Valuing a Forward Contract (pages 115-117)
By considering the difference between a contract with delivery price K and a contract with delivery price F0 we can deduce that:
the value of a long forward contract is (F0 – K )e–rT
the value of a short forward contract is
(K – F0 )e–rT

20. Valuing a Forward Contract (pages 115-117)
What should be the 6 months forward price of the stock?
What would be the value of the forward contract?

21. Valuing a Forward Contract (pages 115-117)

22. Forward vs Futures Prices
When the maturity and asset price are the same, forward and futures prices are usually assumed to be equal.
In theory, when interest rates are uncertain, they are 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

23. Stock Index (Page 118-120) (1 of 2)
Can be viewed as an investment asset paying a dividend yield
The futures price and spot price relationship is therefore
F0 = S0 e(r–q )T
where q is the average dividend yield on the portfolio represented by the index during life of contract
What should be the futures price of the index?

24. Stock Index (2 of 2)
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

25. Index Arbitrage (1 of 2)
When F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futures
When F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

26. Index Arbitrage (2 of 2)
Index arbitrage involves simultaneous trades in futures and many different stocks
Very often a computer is used to generate the trades
Occasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold

27. Futures and Forwards on Currencies (Page 120-123)
A foreign currency is analogous to a security providing a yield
The yield is the foreign risk-free interest rate
It follows that if rf is the foreign risk-free interest rate

28. Figure 5.1: Explanation of the Relationship Between Spot and Forward

29. Futures and Forwards on Currencies

30. Consumption Assets: Storage is Negative Income
F0 ≥ S0 e(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.

31. Consumption Assets: Storage is Negative Income
Required: What should be the theoretical futures price?

32. The Cost of Carry (Page 126-127)

33. The Cost of Carry (Page 126-127)
The cost of carry, c, is the storage cost plus the interest costs that is paid to finance the asset 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

34. Futures Prices & Expected Future Spot Prices (Page 127--129) (1 of 2)
Suppose k is the expected return required by investors in an asset
We can invest F0e–r T at the risk-free rate and enter into a long futures contract to create a cash inflow of ST at maturity
This shows that

35. Futures Prices & Future Spot Prices (2 of 2)
No Systematic Risk
k = r
F0 = E(ST)
Positive Systematic Risk
k > r
F0 < E(ST)
Negative Systematic Risk
k < r
F0 > E(ST)
Positive systematic risk: stock indices Negative systematic risk: gold (at least for some periods)

36. Practice Questions
Problem 5.3.
Suppose that you enter into a six-month forward contract on a non-dividend-paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 5% per annum. What is the forward price?
Problem 5.4.
A stock index currently stands at 350. The risk-free interest rate is 4% per annum (with continuous compounding) and the dividend yield on the index is 3% per annum. What should the futures price for a four-month contract be?

37. Practice Questions
Problem 5.9.
A one-year long forward contract on a non-dividend-paying stock is entered into when the stock price is $40 and the risk-free rate of interest is 5% per annum with continuous compounding.
What are the forward price and the initial value of the forward contract?
Six months later, the price of the stock is $45 and the risk-free interest rate is still 5%. What are the forward price and the value of the forward contract?
Problem 5.10.
The risk-free rate of interest is 7% per annum with continuous compounding, and the dividend yield on a stock index is 3.2% per annum. The current value of the index is 150. What is the six-month futures price?

38. Practice Questions
Problem 5.11.
Assume that the risk-free interest rate is 4% per annum with continuous compounding and that the dividend yield on a stock index varies throughout the year. In February, May, August, and November, dividends are paid at a rate of 5% per annum. In other months, dividends are paid at a rate of 2% per annum. Suppose that the value of the index on July 31 is 1,300. What is the futures price for a contract deliverable on December 31 of the same year?
Problem 5.12.
Suppose that the risk-free interest rate is 6% per annum with continuous compounding and that the dividend yield on a stock index is 4% per annum. The index is standing at 400, and the futures price for a contract deliverable in four months is 405. What arbitrage opportunities does this create?

39. Practice Questions
Problem 5.29.
The spot price of oil is $50 per barrel and the cost of storing a barrel of oil for one year is $3, payable at the end of the year. The risk-free interest rate is 5% per annum, continuously compounded. What is an upper bound for the one-year futures price of oil?

40. Copyright