Pricing Forwards and Futures

1
FINA 4327Professor Andrew Chen

• Pricing Forwards and Futures
• Lecture Note 9

2
Outline

• Determination of Forward/Futures Prices
• Forward/Futures prices on stocks(with or without
  dividends), stock indices, and foreign
  currencies.
• Commodity Forward/Futures prices with storage
  costs and convenience yield
• Valuation of Previously Issued Forward Contracts
• Relation between Forward and Futures Prices
• Summary of Costs of Carry

3
Determination of Forward/Futures Prices

• Cost of Carry Models
• Notations
• T time until delivery (in years)
• S price of underlying asset today (spot price
  now)
• ST price of underlying asset at time T (spot
  price at T)
• F forward price today
• K delivery price in the forward contract
• f value of forward contract today
• c cost of carry per year, with continuous
  compounding
• r risk-free rate of interest per year, with
  continuous compounding
• Note when a contract is initiated, FK and f0

4
Determination of Forward/Futures Prices

• I. Perfect Markets
• Forward Prices for an Investment Asset That
  Provides No Income
• Consider the following investment strategy
• Buy 1 unit of the asset in the spot market for S
• Short one forward contract with zero initial cost
• Under this investment strategy, the forward
  contract requires you to sell the asset for F at
  time T, thus, we know that ST (F ST) F.

5
Determination of Forward/Futures Prices

• Therefore,
• F SerT
• Or
• S Fe-rT
• The forward price, F, must be the future value to
  which S grows at the risk-free interest rate for
  a time T or equivalently, the current value of
  the stock, S, should be equal to the discounted
  forward price.

(2.1)
(2.2)
6
Example

• Consider the following facts
• 6 month forward contract on ZIX stock
• Assume the current price of one share is 15.00
• The continuously compounded annual risk-free
  interest rate is 4. What must the forward price
  of the stock be?
• F SerT 15e0.04x0.5 15.30
• This would be the delivery price, K, in the
  contract negotiated today.
• Why must the forward price equal to 15.30?

7
What if Forward Prices were 16.50/share

• Use cash carry C C
• Note F SerT 16.5 15e0.04 x 0.5 1.20

8
What if Forward Prices were 14/share

• Use reverse cash carry RC C
• Note SerT F 15e0.04 x 0.5 - 14 1.30
• In a perfect market, the cost of carry equals the
  risk-free interest rate, i.e., c r.

9
CC Gold Arbitrage Transactions

• Prices for Analysis
• Spot price of Gold 900
• Forward price of Gold (for delivery in 1
  year) 950
• Annual continuous compounded interest rate 5
• CC if forward price is higher than the
  equilibrium value

10
CC Gold Arbitrage Transactions
11
CC Gold Arbitrage Transactions

• Rule 1 To prevent CC

FT lt SecT SerT
(2.3)
12
Reverse CC Gold Arbitrage Transactions

• Prices for Analysis
• Spot price of Gold 920
• Forward price of Gold (for delivery in 1
  year) 950
• Annual continuous compounded interest rate 5

13
RCC Gold Arbitrage Transactions
14
RCC Gold Arbitrage Transactions

• Rule 2 To prevent RCC
• Rule 3 To prevent arbitrage opportunity
• Rule 4 Implied Repo Rate
• Note In a perfect market, the implied repo rate
  should be equal to the actual repo rate

FT gt SecT SerT
(2.4)
FT SecT SerT same as (2.1)
(2.5)
(2.6)
c ln(FT/S) / T
15
Gold Forward CC Arbitrage

• Prices for Analysis
• Futures price for gold expiring in 1 year 900
• Futures price for gold expiring in 2 years 950
• Annual continuous compounded interest rate 5
• from years 1 - 2

16
Gold Forward CC Arbitrage
17
Gold Forward CC Arbitrage

• Rule 5 To prevent Forward CC

F2 lt F1ecT F1erT
(2.7)
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Gold Forward Reverse CC Arbitrage

• Prices for Analysis
• Futures price for gold expiring in 1 year 940
• Futures price for gold expiring in 2 years 950
• Annual continuous compounded interest rate 5
• from years 1 - 2

19
Gold Forward Reverse CC Arbitrage
20
Gold Forward Reverse CC Arbitrage
21
Gold Forward Reverse CC Arbitrage

• Rule 6 To prevent Forward RCC
• Rule 7 No Forward Arbitrage opportunity
• Rule 8 Implied Forward Repo Rate (year 1 to year
  2)

F2 gt F1ecT F1erT
(2.8)
(2.9)
F2 F1ecT F1erT
(2.10)
c lnF2/F1 /T
22
Example

• Consider the following March 2009 spot and
  futures prices of the silver contracts
• Assuming that risk-free interest rates are
  continuously compounding and that there is no
  storage cost, what are the implied annual spot
  repo rates?
• With the same assumptions, what are the implied
  annual July forward repo rates?

23
Example

• Spot repo rates
• May/March
• July/March
• September/March
• December/March

F SecT,
cT ln(F/S),
c ln(F/S)/T.
c ln(17.41/17.38)/.167 1.03
c ln(17.45/17.38)/.333 1.21
c ln(17.49/17.38)/.5 1.26
c ln(17.55/17.38)/.75 1.29
24
Example

• July Forward Repo Rates
• September/July
• December/July

c ln(17.49/17.45)/.167 1.37
c ln(17.55/17.45)/.417 1.37
25
The Cost-of Carry Model in Imperfect Market

• Attempted Cash-and-Carry Gold Arbitrage
  Transactions
• Prices for the Analysis
• Spot price of Gold 900
• Futures price of Gold (for delivery in 1
  year) 950
• Interest Rate 5
• Transaction cost (Q) 4

26
The Cost-of Carry Model in Imperfect Market
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The Cost-of Carry Model in Imperfect Market

• Rule 9 To prevent CC with transaction cost

(2.11)
FT lt S(1 Q)erT
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The Cost-of Carry Model in Imperfect Market

• Attempted Reverse CC Gold Arbitrage Transactions
• Prices for the Analysis
• Spot price of Gold 920
• Futures price of Gold (for delivery in 1
  year) 950
• Interest Rate 5
• Transaction cost (Q) 4

29
The Cost-of Carry Model in Imperfect Market
30
The Cost-of Carry Model in Imperfect Market

• Rule 10 To prevent Reverse CC with transaction
  cost
• Combinging (2.11) and (2.12), we know that the
  conditions to
• prevent arbitrage opportunity with transaction
  costs are as follows
• Rule 11

FT gt S(1 - Q)erT
(2.12)
(2.13)
S(1 Q)erT FT S(1 Q)erT
31
Fwd Prices for Asset That Provides Known Cash
Incomes

• Consider a T period forward contract on a stock
  that is certain to pay a dividend of D at time t,
  
• where 0 lt t lt T.
• Consider the following investment strategy
• Buy one unit of the asset in the spot market.
• Short one forward contract.
• Upfront cost
• S the initial cost of taking a position in the
  forward contract is zero and one unit of the
  asset costs S

32
Fwd Prices for Asset That Provides Known Cash
Incomes

• Strategy is worth F Der(T-t) at time T
• The forward contract requires you to sell the
  asset for F at time T and the dividend received
  at time t can be invested at r over the period
  from t to T
• Equating the initial outflow with the PV of the
  cash inflow yields
• or
• where PV(D) stands for present value of dividend.

(2.14)
(2.15)
33
Fwd Prices for Asset That Provides Known Cash
Incomes

• Simplify the notation by denoting I for PV(D),
  then (2.15) can be written as
• Where I is the present value of the income
  received from the underlying asset during the
  life of the contract.

(2.16)
34
Example (Dividend Paying Stock)

• Consider the following facts
• 5-month forward contract with GM
• Current price is 25
• Dividend of 0.50 is expected at the end of 4
  months
• Risk-free interest rate is 5 (continuously
  compounded)
• What is the current forward price?
• F25-.5e-0.05x(4/12)e0.05x(5/12) 25.02
• There will be arbitrage opportunities is the
  forward price is above or below 25.02.

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Forward Price is 26.00

• F (S PV(D))erT 26 25.02 0.98

36
Example (Bonds with Coupon Interest)

• Case 1 (Forward price is too high use Cash and
  Carry)
• Forward price of a one year bond is 930
• Current spot price is 900
• Coupon payments of 40 are expected in 6 months
  and 1 year
• 6 month risk free interest rate is 9
• 1 year risk free interest rate is 10
• Strategy
• Borrow 900 to buy one bond spot
• Short one forward contract on one bond

37
Example (Bonds with Coupon Interest)

• We know from (2.13) that
• F (S I)erT
• I 40e-.09(.5) 40e-.09(1) 74.433
• Thus, the equilibrium forward price should be
• F (900 74.433)e.10(1) 912.39, and the
  arbitrage profit is 17.61.

38
Example (Bonds with Coupon Interest)

• Case 2 (Forward price is too low use Reverse
  CC)
• Forward price of bond maturing in one year 905
• Current spot price of bond 900
• Coupon payments due in 6 months and 1 year 40
• 6 month risk-free rate 9
• 1 year risk-free rate 10
• Strategy
• Short one bond.
• Enter into a long forward contract to repurchase
  the bond in one year.
• Arbitrage profit 912.39 905 7.39

39
Fwd Prices for Inv That Provides Known Div Yield

• When pricing stock index forward/futures (or
  options on stock indices and index futures),
  standard practice is to assume that dividends on
  the underlying index of stocks are proportional
  to the value of the index and paid continuously.
• Assume that the dividend yield is paid
  continuously at an annual rate of q.

40
Fwd Prices for Inv That Provides Known Div Yield

• Example
• Current value of the SP 500 index 1,300
• Current dividend yield (compounded 3 per year
• continuously compounded) q 0.03.
• This means that dividends on the index in the
  next small interval of time are paid at the rate
  of 39 (0.03)(1,300) per year.

41
Fwd Prices for Inv That Provides Known Div Yield

• If you hold one share of the index (valued at
  1,000) and reinvest the dividends received in
  additional shares, at the end of 6 months you
  will own
• At the end of 1 year, you will own

1 x eqxT 1 x e0.03x0.5 1.015 shares
1 x eqxT 1 x e0.03x1.0 1.031 shares
42
Fwd Prices for Inv That Provides Known Div Yield

• Thus, if you want to set up a strategy where you
  own one share of the index at time T, you must
  purchase e-qT of a share of the index today.
• To see this, note that

(1 x e-qT) x eqT 1 share
43
Fwd Prices for Inv That Provides Known Div Yield

• Consider the following strategy
• Buy e-qT units of the index in the spot market.
  The current spot price (value) of one unit of the
  index is denoted as S.
• Short one forward contract on the index.

44
Fwd Prices for Inv That Provides Known Div Yield

• Upfront cost of strategy
• Value of strategy at time T
• Equating the initial outflow with the present
  value of the cash inflow yields
• or

Se-qT
STe-qTeqT (F ST) F
Se-qT Fe-rT
(2.17)
F Se(r-q)T
(2.18)
45
Fwd Prices for Inv That Provides Known Div Yield

• Example
• Consider a 6-month forward (futures) contract on
  the SP 500 index.
• Current value of index (S) 1,300
• Dividend yield (q) 3
• Risk-free rate of interest (r) 4
• What is the correct index forward (futures)
  price?
• If the price is different from 1,306.50, index
  arbitrage is possible. Note that index arbitrage
  is implemented through program trading, with a
  computer system used to generate the trades.
•  

F Se(r-q)T (1,300)e(0.04-0.03)(0.5)
1,306.50
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Example (continued)

• Arbitrage opportunity if SP 500 index forward
  (futures) price is 1,318?

47
Example (continued)

• What should you do to take advantage of the
  arbitrage opportunity if the SP 500 index
  forward (futures) price is 1,250?

48
Forward/Future Prices of Currencies

• The pricing of foreign currency forwards/futures
  is very similar to pricing index futures
• A unit of foreign currency can be thought of as a
  stock with a continuous dividend yield that is
  equal to the foreign interest rate.

49
Forward/Future Prices of Currencies

• Denote S as the current spot price (in dollars)
  of one unit of the foreign currency.
• Note that the currency of a given country can be
  deposited in a money market account earning that
  countrys risk-free interest rate or can be
  invested in that countrys currency-denominated
  government bonds.
• Define rf as the foreign risk-free interest rate
  per year with continuous compounding.

50
Forward/Future Prices of Currencies

• Consider the following investment strategy 
• Buy e-rfT units of the foreign currency.
• Short a forward contract on one unit of the
  foreign currency.
• The up front cost of the strategy is
• The investment strategy is worth F at time T
•  

51
Forward/Future Prices of Currencies

• Equating the initial outflow with the PV of Cash
  inflow yields
• or

(2.19)
(2.20)
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Example

• Assume the following facts
• Current / exchange rate 1.925/
• U.S. interest rate (Annualized) 4
• British interest rate (continuously
  compounded) 6
• What is the forward price of GBP() for a 6-month
  forward contract?
• Arbitrage opportunity exists if this relationship
  is not satisfied

53
Example (Arbitrage Opportunity)

• Forward price is 1.918/

54
Pricing Commodity Forward/Futures with Storage
Costs and Convenience Yield

• What are storage costs
• Higher storage cost increases the forward/futures
  price relative to the spot price of the
  commodity.
• PV of storage costs
• The cost of storage can be treated like a
  negative dividend yield. In this case, the
  forward/futures price is given by

(2.21)
(2.22)
55
Example

• Consider the following facts
• 1 year forward contract on platinum
• Storage costs are paid upfront at 1.50/6-months
• Current spot price of platinum is 2,120/oz.
• Risk-free interest rate (continuously
  compounded) 4.5

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Example (continued)

• PV of storage costs
• Forward/futures price

57
Pricing Commodity Forward/Futures with Storage
Costs and Convenience Yield

• Riskless arbitrage opportunities
• Takes place if F gt (S U)erT or F lt (S U)erT.
• F gt (S U)erT
• Buy the platinum and short the forward/futures
  contract
• F lt (S U)erT
• Sell the platinum and long the forward/futures
  contract

58
Pricing Commodity Forward/Futures with Storage
Costs and Convenience Yield

• Using that the storage costs are expressed as a
  proportion of the commodity price, the
  convenience yield, y, is defined to be the fudge
  factor that makes the relation F Se(ru)T an
  equality
• Therefore, the convenience yield can be found as
  follows
• y r u ln(F/S)/T

(2.23)
(2.24)
(2.24A)
59
MV of Previously Issued Forward Contracts

• Although the forward price is initially set to
  make the market value of the contract equal to
  zero, as time passes and the forward prices for
  contemporary contracts change, the market value
  of the previously issued contract, f, may become
  greater or less than zero.
• Consider a long forward contract that was issued
  in the past with a delivery price of K.
  (Remember that the delivery price K is the
  forward price that sets the initial value of the
  contract equal to zero.)

60
Example

• Consider the following facts
• Current forward price for an identical contract
  is F.
• Identical contracts are contracts on the same
  underlying asset and having the same maturity T
  as the previously issued contract.
• The difference in value between the previously
  issued contract and the identical contract
• f 0 f
• Since the identical contract which is written at
  the current forward price of F has a zero initial
  value.

61
Example (continued)

• At the maturity of the contracts, T, the
  difference in the payoffs is
• Since the time T difference is a constant (the
  difference between the two known delivery prices,
  (F K), and the current time difference is f, it
  follows that the value of the previously issued
  forward contract, f, is the discounted value of F
  K at the riskless rate of interest

(2.25)
62
MV of Previously Issued Forward Contracts

• For a long forward contract on an investment
  asset that provides no income
• For a long forward contract on an investment
  asset that provides a known income with present
  value I
• For a long forward contract on an index that
  provides a known dividend yield at the rate q

(2.26)
(2.27)
(2.28)
63
MV of Previously Issued Forward Contracts

• For a long forward contract on a currency
• For a long forward contract on an investment
  asset with present value storage costs of U
• For a long forward contract on an investment
  asset with proportional storage costs u

(2.29)
(2.30)
(2.31)
64
Example

• Consider the following facts
• Long forward on a Non-dividend paying stock
• Remaining maturity is 10 months
• Risk-free rate (continuously compounded) is 5
• Current stock price is 40
• Delivery Price is 42
• What is the current value of this long forward
  contract?

65
Example (continued)

• Facts
• T 10/12, r 0.05, S 40, K 42
• or

66
Example

• Consider the following facts
• 6-month long forward contract in gold
• Current spot price is 980/oz.
• Risk-free rate (continuously compounded) is 4.5
• Delivery Price is 960
• Storage costs are 1/oz. for every 3 months,
  payable in advance
• What is the current value of this forward
  contract?

67
Example (continued)

• or

68
Example

• Consider the following facts
• 6-month forward contract on the GBP
• Current spot exchange rate is 1.9708/
• U.S. risk-free rate (continuously compounded) is
  4.32
• U.K. risk-free rate (continuously compounded) is
  5.25
• Delivery Price is 1.9825/
• What is the current value of this long forward
  contract?

69
Example (continued)

• or

70
Relation between Forward and Futures Prices

• Major difference between forwards and futures
  contract is the timing of cash flows.
• Consider a long forward contract and a long
  futures contract written on the same underlying
  asset and having the same maturity
• The payoff on the forward contract occurs at
  maturity and is equal to the difference between
  the spot price at maturity and the delivery
  price.
• There are no cash flows on a forward contract
  prior to maturity.
• By contrast, the futures contract is marked to
  market daily.
• Your account is credited or debited daily to
  reflect the daily change in the futures price.
• In effect, the futures contract is rewritten at
  the end of every day to have a zero market value.

71
Relation between Forward and Futures Prices

• Does the cash flow timing difference between
  forward and futures contracts create a difference
  between forward and futures prices on otherwise
  identical contracts (i.e., contracts written on
  the same asset and having the same maturity)?
• The answer is maybe it depends on interest
  rates.
• The key point is that credits to a futures
  account can be invested to earn a rate of return.
  The same is not true of a forward contract,
  because the investor does not receive a payoff
  until the maturity of the contract.

72
Relation between Forward and Futures Prices

• Consider the following points
• If interest rates are zero, forward and futures
  prices will be equal
• Consider a long futures contract and a long
  forward contract on the same spot security, both
  initiated on day 0 with the same delivery price
  340 and 5 days to delivery.

73
Relation between Forward and Futures Prices

• If interest rates are not zero, forward and
  futures prices may not be equal.
• However, in the special cases where the risk-free
  interest rate is a constant and the same for all
  maturities or where the risk-free interest rate
  is a known function of time, then it is possible
  to prove that forward and futures prices will be
  equal.
• Given that these special conditions are unlikely
  to hold in practice (especially for contracts
  with longer maturities), forward and futures
  prices on otherwise identical contracts may not
  be equal.

74
Relation between Forward and Futures Prices

• When interest rates are uncertain, it can be
  shown that futures prices contain an additional
  term that is related to the correlation between
  the price of the underlying asset and interest
  rates.

75
Relation between Forward and Futures Prices

• Suppose the correlation is positive.
• Asset price and interest rates change
• Receive credits to your account as you are MTM
• You can reinvest these credits at high rates of
  interest, and you accumulate wealth rapidly. As
  the asset price falls, your account is debited,
  yet the forgone rate of return on reinvestments
  is relatively low.
• Thus, when the correlation between the underlying
  asset price and interest rates is positive,
  investors will bid up futures prices relative to
  forward prices on otherwise identical contracts.
  
• Using the same reasoning, when the correlation is
  negative, futures prices will tend to be less
  than forward prices.

76
Example

• Treasury bond futures
• Underlying asset in Treasury bond futures
  contract is a bond
• Since bond prices are inversely related to
  interest rates, we would expect T-bond futures
  prices to be less than corresponding forward
  prices
• When contracts have long maturities or when the
  underlying asset is highly correlated with
  interest rates, differences between forward and
  futures prices can be economically significant
• Nevertheless, we will typically assume that
  forward and futures prices are equal F will be
  used to represent both the futures price and the
  forward price of an asset

77
Example (continued)

• Important differences between forward and futures
  contracts
• Futures contracts can create a short-term cash
  flow problem when they are used to hedge an
  existing position that has a payoff at T only.
• A hedger may face substantial margin calls if
  futures prices move against her position prior to
  the maturity of the contract.
• Note that this risk is not present with forward
  contracts since they are settled at T.
• The troubles at Metallgesellschaft A.G. in 1993
  and 1994 illustrate this problem with futures
  contracts.

78
Summary of Costs of Carry

• Cost of carry relates to the cost of holding the
  underlying asset relative to using a forward or
  futures contract to purchase the underlying asset
  in the future.
• Thus, the higher is the cost of carry, the higher
  is the forward/futures price relative to spot.

79
Summary of Costs of Carry