1. Determination of Forward and Futures Prices
Chapter 5
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

2. Consumption vs Investment Assets
Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: stock, bonds, gold, silver)
Consumption assets are assets held primarily for consumption (Examples: copper, oil)
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

3. Short Selling (Page )
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 6th Edition, Copyright © John C. Hull 2005

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 6th Edition, Copyright © John C. Hull 2005

5. Example
Consider the position of an investor who shorts 500 IBM shares in April, when the price is $120 per share
He closes out the position by buying them back in July when the price per share is $100
A dividend of $1 per share is paid in May
The net gain of the investor is 500*$ *$1-500*$100=$9.500
The investor is required to maintain a margin account with the broker.
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6. Assumptions 1
The market participants are subject to no transaction costs when they trade 2
The market participants are subject to the same tax rate on all net trading profits 3
The market participants can borrow and lend at the same risk-free rate of interest 4
The market participants take advantage of arbitrage opportunities as they occur
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

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 6th Edition, Copyright © John C. Hull 2005

8. 1. Gold: An Arbitrage Opportunity?
Suppose that:
The spot price of gold is US$390
The quoted 1-year forward price of gold is US$425
The 1-year US$ interest rate is 5% per annum
No income or storage costs for gold
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18

9. 2. Gold: Another Arbitrage Opportunity?
Suppose that:
The spot price of gold is US$390
The quoted 1-year forward price of gold is US$390
The 1-year US$ interest rate is 5% per annum
No income or storage costs for gold
Is there an arbitrage opportunity?
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10. The Forward Price of Gold
If the spot price of gold is S and the futures price for a contract deliverable in T years is F, then
F = S (1+r )T
where r is the 1-year (domestic currency) risk-free rate of interest.
In our examples, S=390, T=1, and r=0.05 so that
F = 390(1+0.05) =
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11. Arbitrage Position 1: Arbitrageurs can adopt the following strategy:
Borrow $390 at 5% for one year
Buy 1 ounce of gold
Enter into a short forward contract to the gold for $425 in one year
In one-year 390(1+0.05) = is required to repay the loan.
The arbitrageur can shell the gold for $425. The result is a riskless profit of $15.5.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19

12. Arbitrage Position 2: Arbitrageurs can adopt the following strategy:
Short 1 ounce of gold
Invest the proceeds (390) at 5% for 1-year
Enter into a long forward contract to repurchase the gold for $390 in 1-year
In 1-year the $390 investment grows to 390(1+0.05) =
The gold is purchased for 390 and the short position is closed out. The result is a riskless profit of $19.50.
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13. When Interest Rates are Measured with Continuous Compounding
F0 = S0erT
This equation relates the forward price and the spot price for any investment asset that provides no income and has no storage costs
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

14. Another Example Consider a stock that is currently selling for $30
The quoted 2-years forward price of the stock is $35
The 2-years US$ interest rate is 5% per annum
This means that the stock can be bought or sold for 35 with delivery in two years
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18

15. Arbitrage Positions: Arbitrageurs can adopt the following strategy:
Borrow 3000 at 5% for two years
Buy 100 shares of the stock
Enter into a short forward contract to sell 100 shares for 3500 in two years
In two-years 3000e0.05*2=3316 is required to repay the loan.
The arbitrageur can shell the shares for The result is a riskless profit of 184.
It is easy to see that any forward price above gives rise to this type of riskless arbitrage profit
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16. Continued.. Consider a stock that is currently selling for $30
The quoted 2-years forward price of the stock is $31
The 2-years US$ interest rate is 5% per annum
This means that the stock can be bought or sold for $31 with delivery in two years
Is there an arbitrage opportunity?
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18

17. Arbitrage Positions: Arbitrageurs can adopt the following strategy:
Short 100 shares of the stock
Invest the proceeds (3000) at 5% for two-years
Enter into a long forward contract to repurchase 100 shares for 3100 in two-years
In two-years the 3000 investment grows to 3000e0.05*2=3316.
100 shares are purchased for 3100 and the short position is closed out. The result is a riskless profit of 216.
This riskless arbitrage opportunity is available whenever the forward price is below 33.16
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19

18. Arbitrage
Riskless arbitrage opportunities, if they are observed at all, rarely last for long
As traders take advantage of these arbitrage opportunities, the forward price will be driven up or down and the arbitrage opportunity will disappear.
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19. Generalization: The relationship between F0 and S0 is: F0 = S0erT
When F0 > S0erT an arbitrageur can buy the asset and short forward (futures) contacts on the asset
When F0 < S0erT an arbitrageur can short the asset and buy forward (futures) contracts on it
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

20. Example
Consider a four-month forward contract to buy a zero-coupon bond that will mature one year from today
The current price of the bond is 930
We can regard the contract as on an eight-month zero-coupon bond
The 4-months US$ interest rate is 6% per annum
Then the forward price is obtained by
F0= P0er*T = 930e0.06*4/12 =948.79
This would be the delivery price in a contract negotiated today
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21. When an Investment Asset Provides a Known Dollar Income (page 105, equation 5.2)
F0 = (S0 – I )erT
where I is the present value of the income during life of forward contract
Examples are stocks paying dividends
and coupon bearing bonds.
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

22. Example
Consider a one-year forward contract to purchase a coupon-bearing bond whose current price is 900, that will mature in five years
Coupon payments of 40 are expected after 6 and 12 months
Six-month and one-year riskless rates are 9% and 10% respectively
So, S0=900, I=40e-0.09* e-0.1*1 =74.433, and
F0 = (S0-I)er*T=( )e0.1*1 =912.39
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23. Arbitrage
If F0>(S0-I)er*T, an arbitrageur can lock in a profit by buying the asset and shorting a forward contract on the asset.
If F0<(S0-I)er*T, an arbitrageur can lock in a profit by shorting the asset and taking a long position in a forward contract
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 18

24. Example
Consider a 10-month forward contract on a stock with price of 50
The riskless rate is 8% per annum for all maturities
Dividends of 0.75 per share are expected after three, six and nine months
The present value of the dividends, I, is given by I=0.75e-0.08*3/ e-0.08*6/ e-0.08*9/12 =2.162
The forward price is given by
F0 = (S0-I)er*T=( )e0.08*10/12 =51.14
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25. When an Investment Asset Provides a Known Yield (Page 107, equation 5
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 6th Edition, Copyright © John C. Hull 2005

26. Example
Consider a 6-month forward contract on a stock with price of 25
The riskless rate is 10% per annum
The stock is expected to provide income equal to 2% of the stock price in 6 months.
The yield is 3.96% per annum with continuous compounding.
The forward price is given by
F0 = S0e(r-q)*T=25e( )*0.5 =25.77
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27. Valuing a Forward Contract Page 108
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 6th Edition, Copyright © John C. Hull 2005

28. f = (F0-K)e-r*T = S0-Ke-r*T=25-24e-0.1*0.5=2.17
Example
Consider a six-month long forward contract on a non-dividend paying stock, that was entered into some time ago
The stock price is 25, and the delivery price is 24
The risk-free rate of interest is 10% per annum
Then, the forward price, F0, is given by
F0 = S0er*T= 25e0.1*0.5=26.28
The value of the forward contract is
f = (F0-K)e-r*T = S0-Ke-r*T=25-24e-0.1*0.5=2.17
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29. 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 6th Edition, Copyright © John C. Hull 2005

30. F0 = S0 e(r–q )T Stock Index (Page 110-112)
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 6th Edition, Copyright © John C. Hull 2005

31. Example Consider a 3-month futures contract on the S&P 500
The riskless rate is 6% per annum
The stocks underlying the index provide a dividend yield of 1% per annum.
The current value of the index is 800
The futures price is given by
F0 = S0e(r-q)*T=800e( )*0.25 =810.06
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32. 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 6th Edition, Copyright © John C. Hull 2005

33. 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
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

34. Futures and Forwards on Currencies (Page 112-115)
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 6th Edition, Copyright © John C. Hull 2005

35. Why the Relation Must Be True Figure 5.1, page 113
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

36. Example
Suppose the 2-year interest rate in Australia and the US are 5% and 7% respectively
The exchange rate is 0.62 USD for 1 AUD
The 2-year forward exchange rate should be
F0 = S0e(r-rf)*T= 0.62e( )*2 =0.6453
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37. Arbitrage Position 1:
Suppose the 2-year forward exchange rate is less than this, say An arbitrageur can:
Borrow 1000 AUD at 5% per annum for 2-years, convert to 620 USD and invest the USD at 7%
Enter into a forward contract to buy 1, AUD for 1,105,17*0.63= USD
The 620 USD grow to 620e0.07*2= USD.
USD are used to purchase 1,105,17 AUD under the terms of the forward
Repay the principal and interest on the 1000 AUD that are borrowed (1000e0.05*2=1,105.17)
This riskless profit is =16.91 USD
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005 19

38. Arbitrage Position 2:
Suppose the 2-year forward exchange rate is greater than this, say An arbitrageur can:
Borrow 1000 USD at 7% per annum for 2-years, convert to 1000/0.62=1, AUD and invest the AUD at 5%
Enter into a forward contract to sell 1, AUD for 1,782.53*0.66=1, USD
The 1, AUD grow to 1,612.90e0.05*2=1, AUD.
The forward contract has the effect of converting this to 1, USD
Repay the principal and interest on the 1000 USD that are borrowed (1000e0.07*2=1,150.27)
This riskless profit is 1, ,150.27=26.20 USD
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39. Futures on Commodities (Page 117)
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 6th Edition, Copyright © John C. Hull 2005

40. Arbitrage When F0 > (S0+U)erT an arbitrageur can:
Borrow S0+U at the risk free rate and use it to purchase one unit of commodity and pay storage cost
Short a forward contract on one unit of the commodity
When F0 < (S0+U)erT an arbitrageur can:
Sell the commodity, save the storage costs and invest the proceeds at r
Take a long position in a forward
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

41. Futures on Consumption Assets (Page 117-118)
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.
Convenience yield y:the benefit from holding the physical asset:
F0eyT = (S0+U )erT
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

42. The Cost of Carry (Page 118-119)
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 6th Edition, Copyright © John C. Hull 2005

43. Futures Prices & Expected Future Spot Prices (Page 119-121)
Suppose k is the expected return required by investors on an asset
We can invest F0e–r T at the risk-free rate and enter into a long futures contract so that there is a cash inflow of ST at maturity
This shows that
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005

44. Futures Prices & Future Spot Prices (continued)
If the asset has
no systematic risk, then k = r and F0 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 )
Options, Futures, and Other Derivatives 6th Edition, Copyright © John C. Hull 2005