1. Determination of Forward and Futures Prices
Chapter 5
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

2. 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)
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

3. Short Selling 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

4. 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

5. Short Selling: Example
An investor shorts 500 shares in April when the price is $120 and close out position by buying themback in July when price is $100. Supose a dividend of $1 is paid in May.
Short sale of share:
Apr: Borrow 500 shares and sell at $120:+$60,000
May: Pay dividend : $- 500
July: Buy back 500 shares at $ : -$50,000
Net Profit : $9,500

6. Exp. cont. Purchase of shares
Apr: Purchase 500 shares for $120: -$60,000
May: Receive dividend : +$500
July: Sell 500 shares for $100 : +$50,000
Net Profit (Loss) : -$9,500

7. 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

8. 1. 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?
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 18

9. Yes Now; In 3 months; Borrow $40 at 5% for 3 months
Buy 1 unit of asset
Enter into forward contract to sell in 3 months for $43.
In 3 months;
Sell asset for $43
Repay loan with interest: 40e0.05x3/12 = $40.50
Profit = $43-$40.50 = $2.5.

10. 2. Another Arbitrage Opportunity?
Suppose that:
The spot price of nondividend-paying stock is $43
The 3-month forward price is US$39
The 1-year US$ interest rate is 5% per annum
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 19

11. Yes Now; In 3 months; Short 1 unit of asset to realize $40
Invest $40 at 5% for 3 months
Enter into foeward contract to buy asset in 3 months for $39.
In 3 months;
Buy asset for $39
Close out position
Receive 40e0.05x3/12 = $40.50
Profit = = $1.5

12. 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 1-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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008 20

13. If F0 > S0erT : Arbitrageurs can buy the asset and short forward contracts.
1. Borrow S0 at r for T years
2. Buy the asset
3. Short a forward contract on the asset.
Profit= F0 - S0erT
If F0 < S0erT : Arbitrageurs can short the asset and enter into long forward contracts on it.
1. Sell the asset for S0
2. Invest the proceed at r for T years
3. Take a long position in a forward contract
Profit= S0erT - F0

14. 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

15. When an Investment Asset Provides a Known Dollar Income
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

16. Exp. 5.2 (Pg 110)
Consider a 10-month forward contract on a stock priice is $50. We assume that the risk-free rate of interest continuously compounded is 8% per annnum for all maturities. We also assume that dividends of $0.75 per share are expected after three months, six months, and nine months. Find the forward price.

17. When an Investment Asset Provides a Known Yield
F0 = S0 e(r–q )T
where q is the average yield during the life of the contract (expressed with continuous compounding)
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

18. Exp 5.3 (Pg.111)
Consider a six-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once durng a six-month period. The risk-free rate of interest with continuous compounding is 10% per annum. The price is $25. Find the forward price.

19. Valuing a Forward Contract
Suppose that
K is delivery price in a forward contract and
F0 is 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

20. Exp 5.4 (Pg 112)
A long forward contract on a non-dividend paying stock was entered into some time ago. It currently has six months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is $25, and the delivery price is $24. Find the value of the forward contract.

21. 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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

22. Stock Index F0 = S0 e(r–q )T
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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

23. Stock Index (continued)
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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

24. Exp. 5.5 (Pg. 115)
Consider a three-month futures contract on the S&P 500. Suppose that the stocks undelying the index provide a dividend yield of 1% per annum, that the current value of the index is 1,300, and the continuously compounded risk-free interest rate is 5% per annum. Find the index futures price.

25. Index Arbitrage
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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

26. Index Arbitrage (continued)
Index arbitrage involves simultaneous trades in futures and many different stocks
Very often a computer is used to generate the trades
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

27. Futures and Forwards on Currencies
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 rf is the foreign risk-free interest rate
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

28. Exp. 5.6 (Pg. 118)
Suppose that the two-year interest rates in Australia and the US are 5% and 7% respectively, and the spot exchange rate between the AUD and the USD is o.62 USD PER AUD.
Suppose that the two-year forward exchange rate is 0.63
Suppose that the two-year forward exchange rate is 0.66
Define the arbitrage strategies.

29. Futures on Consumption Assets
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.
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008

30. Exp. 5.8 (Pg. 121)
Consider a one-year futures contract on gold. We assume no income and tht it costs $2 per ounce per year to store gold,with the payment being made at the end of the year. The spot price is $600 and the risk-free rate is 5% per annum for all maturities.
Suppose the actual gold futures price is $700
Suppose the actual gold futures price is $610.
Define the arbitrage opportunities.

31. The Cost of Carry
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
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull 2008