FIN 413 RISK MANAGEMENT

1
FIN 413 RISK MANAGEMENT

• Forward and Futures Prices

2
Topics to be covered

• Compounding frequency
• Assumptions and notation
• Forward prices
• Futures prices
• Cost of carry
• Delivery options

3
Suggested questions from Hull

• 6th edition 4.4, 4.10, 5.2, 5.5, 5.6, 5.14
• 5th edition 4.4, 4.9, 5.2, 5.5, 5.6, 5.14

4
Compounding frequency

• Interest can be compounded with varying
  frequencies.
• We will often assume that interest is compounded
  continuously.
• Two rates of interest are said to be equivalent
  if for any amount of money invested for any
  length of time, the two rates lead to identical
  future values.

5
Annual compounding

• The interest earned on an investment in any one
  year is reinvested to earn additional interest in
  succeeding years.
• R EAR, effective annual rate
• FV A(1R)n
• PV A(1R)-n

A
A(1R)-n
0
n
6
Compounding m times per year

• The year is divided into m compounding periods.
  Interest earned in any compounding period is
  reinvested to earn additional interest in
  succeeding periods.
• Rm the annual (or nominal) rate of interest
  compounded m times per year
• Rm/m the effective rate of interest for each
  mth of a year

7
Compounding m times per year

• FV A(1Rm/m)mn
• PV A(1Rm/m)-mn

A(1Rm /m)mn
A
0
n
A
A(1Rm /m)-mn
0
n
8
Continuous compounding
R8 the annual rate of interest compounded
continuously

• FV lim A(1Rm/m)mn m?8 AeR8n
• PV lim A(1Rm/m)-mn m?8 Ae-R8n

9
Eulers number

• 2 lt e lt 3
• e 2.71828183

10
Conversion formulas
11
Conversion formulas
12
Natural log function

• Properties
• -8ltln(x)lt8, for 0ltxlt8
• ln(x)lt0, for 0ltxlt1
• ln(1) 0
• ln(x)gt0, for xgt1
• ln(ax) ln(a) ln(x)
• ln(a/x) ln(a) - ln(x)
• ln(ax) xln(a)
• ln(ex) xln(e) x

13
Exponential function

• Properties
• exgt0, for -8ltxlt8
• 0ltexlt1, for xlt0
• e0 1
• exgt1, for xgt0
• e-x 1/ex
• exey exy
• (ex)y exy
• eln(x) x

14
Example

• Consider an interest rate that is quoted at 10
  per annum with monthly compounding. What is the
  equivalent rate with continuous compounding?

15
Short selling in the spot market

• Involves selling securities that you do not own
  and buying them back later.
• When you initiate a short sale, your broker
  borrows the securities from another client and
  sells them on your behalf in the spot market.
  You receive the proceeds of the sale.
• Through your broker, you must pay the client any
  income received on the securities.
• At some later stage, you must buy the securities,
  close your short position, and return the
  securities to the client from whom you borrowed.
• Ignoring the income foregone, short selling
  yields a profit if the price of the security
  falls.

Buy
Sell
16
Example

• Suppose you short sell IBM stock for 90 days.
  The cash flow are

Note Short selling is the opposite of buying.
17
Analysis forward prices

• Forward contracts are easier to analyze than
  futures contracts.
• We begin our analysis with them.
• We will consider forward contracts on the
  following underlying assets
• Assets that provide no income.
• Assets that provide a known cash income.
• Assets that provide a known yield.
• Commodities
• Later we will consider futures contracts.

18
Assumptions

• There are some market participants (such as large
  financial institutions) that
• - pay no transactions costs (brokerage fees,
  bid-ask spreads) when they trade.
• - are subject to the same tax rate on all
  profits.
• - can borrow or lend at the risk-free rate of
  interest.
• - exploit arbitrage opportunities as they arise.
• Note The quality of any theory is a direct
  result of the quality of the underlying
  assumptions. The assumptions determine the
  degree to which the theory matches reality.

19
Notation

• T the time (in years) until the delivery date
  of a forward contract
• S (or S0) the current spot price of the asset
  underlying a forward contract
• K the delivery price specified in a forward
  contract
• F (or F0) the current forward price
• f the current value of a forward contract to
  the long
• -f the current value of a forward contract to
  the short
• r the risk-free interest rate (expressed as an
  annual, continuously compounded rate) for an
  investment maturing in T years
• Note In practice, r is set equal to the LIBOR
  with a maturity of T years.

20
LIBOR

• LIBOR London Interbank Offer rate
• The rate at which large international banks are
  willing to lend to other large international
  banks for a specified period.
• The rate at which large international banks fund
  most of their activities.
• A variable interest rate.
• A commercial lending rate, higher than
  corresponding Treasury rates.

21
Analysis

• Objective to derive formulas for F and f.
• We will use arbitrage pricing methods.
• Note The basis of any arbitrage is to sell what
  is relatively overvalued and to buy what is
  relatively undervalued.

22
Forward contract UA provides no income

• Examples forward contracts on non-dividend-paying
  stocks and zero-coupon bonds.
• Proposition F SerT, in the absence of
  arbitrage opportunities
• Note F SerT gt S

23
Forward contract UA provides no income

• Proposition F SerT, in the absence of
  arbitrage opportunities
• Proof Suppose F gt SerT.
• Arbitrage strategy (to be implemented today)
• Buy one unit of the UA in the spot market by
  borrowing S dollars for T years at rate r.
• Short a forward contract on one unit if the UA.
• At time T
• Sell the UA for F dollars under the terms of the
  forward contract.
• Repay the bank SerT dollars.
• Arbitrage profit per unit of UA F SerT gt
  0.
• S is bid up and F is bid down.

24
Forward contract UA provides no income

• Suppose F lt SerT.
• Arbitrage strategy (to be implemented today)
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This
  leads to a cash inflow of S dollars. Invest this
  for T years at rate r.
• At time T
• The proceeds from the sale/short sale have grown
  to SerT dollars.
• Buy the UA for F dollars under the terms of the
  forward contract.
• Return the UA to your portfolio or to the client
  from whom it was borrowed.
• Arbitrage profit per unit of UA SerT F gt
  0.
• F is bid up and S is bid down.
• Thus F SerT

25
Alternative derivation of formula

• Spot transaction
• Price agreed to.
• Price paid/received.
• Item exchanged.
• Prepaid forward contract
• Price agreed to.
• Price paid/received.
• Item exchanged in T years.
• Forward contract
• Price agreed to
• Price paid/received in T years.
• Item exchanged in T years.

26
Alternative derivation of formula

• Underlying asset provides no income
• FP S
• Explanation With a prepaid forward contract, as
  compared to a spot transaction, physical exchange
  of the asset is delayed T years. But since the
  asset, by assumption, pays no income to the
  holder, the holder neither receives nor foregoes
  income due to the delay.
• F FP erT SerT
• Explanation The forward contract allows the long
  to delay payment for T years and requires the
  short to delay receipt. The long can earn
  interest on the cash that would otherwise have
  been paid. The short foregoes this interest.
  The forward price (which is arrived at by
  multiplying the prepaid forward price, equal to
  S, by erT) compensates the short for the delay.

27
Forward contract UA provides no income

• Proposition f S Ke-rT
• Proof
• In general f (F K )e-rT
• We derived F SerT
• Thus f (SerT K )e-rT S Ke-rT
• Also -f -(F K )e-rT (K F )e-rT Ke-rT -
  S

28
Forward contract UA provides no income

• We derived f S Ke-rT
• Thus f gt 0 iff S gt Ke-rT

K
0
T
29
Forward contract UA provides no income

• We derived -f Ke-rT S
• Thus -f gt 0 iff Ke-rT gt S

K
0
T
30
Example 5.9, page 121

• T 1 year
• S 40
• r 10
• (a) F SerT 40e(0.101) 44.21
• f S Ke-rT
• 40 44.21e-(0.101)
• 0

0
1
31
Example (continued)

• (b) T ½ year
• S 45
• r 10
• F S erT 45e(0.100.5) 47.31
• f S Ke-rT
• 45 44.21e-(0.100.5)
• 2.95

0
1
0.5
32
Creating a forward contract synthetically

• A security is created synthetically by
  assembling a portfolio of traded assets that
  replicates the payoff to the security.
• A long position in a forward contract can be
  created synthetically by
• Buying the UA with borrowed funds.
• Buying a call option and writing a put option.

33
Creating a forward contract synthetically

• Method 1
• Consider a forward contract on a stock with a
  delivery date in T years. The stock will pay no
  dividends during the next T years.
• The forward contract can be created synthetically
  by buying the stock with borrowed funds.
• r the annual, continuously compounded rate at
  which funds can be borrowed.
• S0 the current price of the stock.

34
Creating a forward contract synthetically
35
Creating a forward contract synthetically

• Value at time T of a long position in a forward
  contract fT FT - K ST K ST S0erT

Value at time T of replicating portfolio
fT
ST
36
Forward contract UA provides a known cash income

• Examples forward contracts on dividend-paying
  stocks and coupon bonds.
• I the present value of the income to be
  received over the remaining life of the forward
  contract
• Proposition F (S I )erT, in the absence of
  arbitrage opportunities

37
Forward contract UA provides a known cash income

• Note F (S I )erT lt SerT
• This price is lower than if the asset didnt pay
  income.

38
Forward contract UA provides a known cash income

• Proposition F (S I )erT, in the absence of
  arbitrage opportunities
• Proof Suppose F gt (S I )erT.
• Arbitrage strategy (to be implemented today)
• Buy one unit of the UA in the spot market by
  borrowing S dollars for T years at rate r.
• Short a forward contract on one unit if the UA.
• Use the income from the asset to repay the loan.
• At time T
• Sell the UA for F dollars under the terms of the
  forward contract.
• Repay the bank (S I )erT dollars.
• Arbitrage profit per unit of UA F (S I
  )erT gt 0.
• S is bid up and F is bid down.

39
Forward contract UA provides a known cash income

• Suppose F lt (S I )erT.
• Arbitrage strategy (to be implemented today)
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This
  leads to a cash inflow of S dollars. Invest this
  for T years at rate r.
• At time T
• The proceeds from the sale/short sale have grown
  to (S I )erT dollars.
• Buy the UA for F dollars under the terms of the
  forward contract.
• Return the UA to your portfolio or to the client
  from whom it was borrowed.
• Arbitrage profit per unit of UA (S I )erT
  F gt 0.
• F is bid up and S is bid down.
• Thus F (S I )erT

40
Alternative derivation of formula

• Spot transaction
• Price agreed to.
• Price paid/received.
• Item exchanged.
• Prepaid forward contract
• Price agreed to.
• Price paid/received.
• Item exchanged in T years.
• Forward contract
• Price agreed to
• Price paid/received in T years.
• Item exchanged in T years.

41
Alternative derivation of formula

• Underlying asset provides a known cash income
• FP S - I
• Explanation With a prepaid forward contract, as
  compared to a spot transaction, physical exchange
  of the asset is delayed T years. As a result of
  the delay, the long foregoes income with present
  value I and the short receives this income.
  Thus, the price paid by the long and received by
  the short is reduced by amount I.
• F FP erT (S I )erT
• Explanation The forward contract allows the long
  to delay payment for T years and requires the
  short to delay receipt. The long can earn
  interest on the cash that would otherwise have
  been paid. The short foregoes this interest.
  The forward price (which is arrived at by
  multiplying the prepaid forward price, equal to S
  - I, by erT) compensates the short for the delay.

42
Forward contract UA provides a known cash income

• Proposition f S I Ke-rT
• Proof
• In general f (F K )e-rT
• We derived F (S I )erT
• Thus f (S I )erT Ke-rT (S I ) Ke-rT
  
• Also -f Ke-rT (S I )

43
Forward contract UA provides a known cash income

• We derived f S I Ke-rT
• Thus f gt 0 iff S gt Ke-rT I

K
0
T
44
Forward contract UA provides a known cash income

• We derived -f Ke-rT (S I)
• Thus -f gt 0 iff Ke-rT I gt S

K
0
T
45
Example 5.23, page 123

• S 50
• r 8
• T 6/12
• (a) I 1e-(0.082/12) 1e-(0.085/12)
  1.9540
• F (S I )erT (50 1.9540)e(0.086/12)
  50.0068
• -f -(S I Ke-rT)
• -(50 1.9540 50.0068e-(0.086/12)) 0

1
1
6/12
0
2/12
5/12
46
Example (continued)

• (b) S 48
• r 8
• T 3/12
• I 1e-(0.082/12) 0.9868
• F (S I)erT (48 0.9868)e(0.083/12)
  47.9629
• -f -(S I Ke-rT)
• -(48 0.9868 50.0068e-(0.083/12)) 2.00

1
1
6/12
0
2/12
5/12
3/12
47
Example (continued)

• S 50
• T 6/12
• (a) I 1e-(0.0782/12) 1e-(0.0825/12)
  1.9535

1
1
6/12
0
2/12
5/12
Term structure of interest rates
48
Forward contract UA provides a known yield

• Examples forward contracts on stock portfolios
  and currencies.
• q the average yield per annum expressed as a
  continuously compounded rate
• Proposition F Se(r-q)T, in the absence of
  arbitrage opportunities

49
Forward contract UA provides a known yield

• Note F Se(r-q)T lt SerT
• This price is lower than if the asset didnt pay
  income.

50
Forward contract UA provides a known yield

• Proposition F Se(r-q)T, in the absence of
  arbitrage opportunities
• Proof Suppose F gt Se(r-q)T.
• Arbitrage strategy (to be implemented today)
• Buy one unit of the UA in the spot market by
  borrowing S dollars for T years at rate r.
• Short a forward contract on one unit if the UA.
• Use the income from the asset to repay the loan.
• At time T
• Sell the UA for F dollars under the terms of the
  forward contract.
• Repay the bank Se(r-q)T dollars.
• Arbitrage profit per unit of UA F Se(r-q)T
  gt 0.
• S is bid up and F is bid down.

51
Forward contract UA provides a known yield

• Suppose F lt Se(r-q)T.
• Arbitrage strategy (to be implemented today)
• Go long a forward contract on one unit if the UA.
• Sell or short sell one unit of the UA. This
  leads to a cash inflow of S dollars. Invest this
  for T years at rate r.
• At time T
• The proceeds from the sale/short sale have grown
  to Se(r-q)T dollars.
• Buy the UA for F dollars under the terms of the
  forward contract.
• Return the UA to your portfolio or to the client
  from whom it was borrowed.
• Arbitrage profit per unit of UA Se(r-q)T F
  gt 0.
• F is bid up and S is bid down.
• Thus F Se(r-q)T

52
Alternative derivation of formula

• Spot transaction
• Price agreed to.
• Price paid/received.
• Item exchanged.
• Prepaid forward contract
• Price agreed to.
• Price paid/received.
• Item exchanged in T years.
• Forward contract
• Price agreed to
• Price paid/received in T years.
• Item exchanged in T years.

53
Alternative derivation of formula

• Underlying asset provides a known yield
• FP Se-qT
• Explanation FP equals the investment required in
  the asset today that will yield one unit of the
  asset in T years when physical delivery occurs.
  e-qT units of the asset will grow to e-qT eqT
  1 unit of the asset in T years, assuming that
  the income provided by the asset is reinvested in
  the asset. e-qT units of the asset cost Se-qT
  today. Therefore, FP Se-qT .
• F FP erT Se-qTerT Se(r-q)T
• Explanation The forward contract allows the long
  to delay payment for T years and requires the
  short to delay receipt. The long can earn
  interest on the cash that would otherwise have
  been paid. The short foregoes this interest.
  The forward price (which is arrived at by
  multiplying the prepaid forward price, equal to
  Se-qT, by erT) compensates the short for the
  delay.

54
Forward contract UA provides a known yield

• Proposition f Se-qT Ke-rT
• Proof
• In general f (F K )e-rT
• We derived F Se (r-q)T
• Thus f Se(r-q)T K e-rT Se(r-q)T/erT
  Ke-rT
• Se-qT Ke-rT
• Also -f Ke-rT Se-qT

55
Example 5.11, page 122

• r 9
• S 300
• T 5/12
• q (5 2 2 5 2)/5 3.2
• F Se (r-q)T 300e((0.09-0.032)5/12) 307.34

56
Forward prices futures prices

• Like forward contracts, futures contracts are
  contracts for deferred delivery.
• But, unlike forward contracts, futures contracts
  are marked to market daily.
• Consider corresponding forward and futures
  contracts
• Same underlying asset.
• Delivery date in two days.
• The contracts are identical except
• Forward contract is settled at maturity.
• Futures contract is settled daily.
• Ignore taxes, transaction costs, and the
  treatment of margins.
• F the forward price
• G the futures price

57
Forward prices futures prices
Day 0F00
Day 1F10
Day 2F2 S2F2 K S2 F0
Forward pricePayoff to buyer
Day 0G00
Day 1G1G1 G0
Day 2G2 S2G2 G1 S2 G1
Futures pricePayoff to buyer
58
Forward prices futures prices

• Example Suppose we have
• Day 0 G0 2
• Day 1 G1 1 with a 50 probability
• 3 with a 50 probability
• Day 2 G2 S2 since the futures contract
  terminates.

59
Example (continued)

• Suppose that the interest rate is a constant 10
  (effective per day).
• On day 1, if G1 1 the futures buyer has a
  loss (G0 G1) 1. S/he would borrow this
  amount at r 10 and have to repay 1.10 on
  day 2.
• On day 1, if G1 3 the futures buyer has a
  gain (G1 G0) 1. S/he would invest this
  amount at r 10 and have 1.10 on day 2.
• Since there is a 50 chance of paying interest of
  0.10 and a 50 chance of earning interest of
  0.10, there is no expected benefit from marking
  to market on day 1.
• Since the futures contract offers no benefit as
  compared to the forward contract, G0 F0.

60
Example (continued)

• Now suppose that the interest rate is not
  constant. Suppose that r 12 on day 1 if G1
  3 and r 8 on day 1 if G1 1.
• On day 1, if G1 1 the futures buyer has a
  loss (G0 G1) 1. S/he would borrow this
  amount at r 8 and have to repay 1.08 on day
  2.
• On day 1, if G1 3 the futures buyer has a
  gain (G1 G0) 1. S/he would invest this
  amount at r 12 and have 1.12 on day 2.
• Now there is an expected benefit from marking to
  market (50 0.12 50 0.08) 0.02.
• Since the futures contract offers a benefit as
  compared to the forward contract, G0 must exceed
  F0.

61
Example (continued)

• Now suppose that the interest rate is not
  constant. Suppose that r 8 on day 1 if G1
  3 and r 12 at day 1 if G1 1.
• On day 1, if G1 1 the futures buyer has a
  loss (G0 G1) 1. S/he would borrow this
  amount at r 12 and have to repay 1.12 on
  day 2.
• On day 1, if G1 3 the futures buyer has a
  gain (G1 G0) 1. S/he would invest this
  amount at r 8 and have 1.08 on day 2.
• Now the expected gain from marking of market
  (50 0.08 50 0.12) -0.02.
• Since the forward contract offers a benefit as
  compared to the futures contract, F0 must exceed
  G0.

62
Forward prices futures prices

• With this reasoning
• - G0 F0 when interest rates are uncorrelated
  with the futures price.
• - G0 gt F0 when interest rates are positively
  correlated with the futures price.
• - F0 gt G0 when interest rates are negatively
  correlated with the futures price.
• Empirical evidence
• - Differences between the forward and futures
  prices are usually trivial once factors such as
  taxes, transaction costs, and the treatment of
  margin are controlled for.
• - Exceptions
• . Contracts on fixed income instruments, like
  T-bills. The prices of T-bills are highly
  negatively correlated with interest rates. F0 gt
  G0
• . Long-lived contracts.
• Formulas for F use to calculate both forward
  prices and futures prices.

63
Stock index futures contracts

• Heavily traded. See National Post website.
• Stock index a weighted average of the prices of
  a selected number of stocks.
• Underlying the portfolio of stocks comprising
  the index.
• Examples of stock indices (futures exchanges)
• SP/TSX Canada 60 Index (ME)
• SP 500 Composite Index (CME)
• NYSE Composite Index (NYFE)

64
Stock index futures contracts

• A futures contract on an asset that provides
  income.
• Formulas
• F Se(r-q)T
• F (S I )erT
• S denotes the current value of the index.
• Index arbitrage what kind of trader might engage
  in this arbitrage?
• F gt Se(r-q)T
• F lt Se(r-q)T

65
Stock index futures contracts

• Cash-settled contracts.
• More likely to lead to delivery.
• On the last trading day, the settlement price is
  set equal to the closing value of the index.
• Multiplier (m)
• SP 500 composite index futures, m 250
• SP/TSX Canada 60 index futures, m 200
• The long gains if F2 gt F1. The short gains if F2
  lt F1
• F1 the futures price at the time the position is
  initiated.
• F2 the futures price at the time the position is
  terminated.

66
Example

• On May 20, 2005, you go long two March 2006
  futures contracts on the SP 500 Composite Index.
  The contract is trading at 1206.60. Suppose you
  hold the contract to expiration and the index is
  at 1193.50 at that time. What is your gain/loss?
• Solution
• F1 1206.60
• F2 1193.50
• Your loss ((F1 F2)2502) ((1206.60
  1193.50)2502) 6,550
• Note
• If you had shorted the contracts, you would have
  gained 6,550.
• If m 1, your loss would have equaled 26.20.

67
Stock index futures contracts

• SP 500 composite index futures m 250
• Mini SP 500 futures m 50
• Both of these contracts trade on CME.
• See www.cme.com
• Question Who trades the mini? Designed for
  individual investors, rather than professional
  portfolio managers.

68
Forward and futures contracts on currencies

• See National Post website.
• Foreign currency a security that provides a
  known yield at rate q rf
• Our earlier formula, F Se(r-q)T, becomes F
  Se(r-rf )T
• Notation
• r the domestic risk-free interest rate
• rf the foreign risk-free interest rate
• S the spot price of the foreign currency (or
  spot exchange rate) expressed in units of the
  domestic currency, e.g., 1 CAD 0.9270 USD
• F the forward or futures price of the foreign
  currency expressed in units of the domestic
  currency, e.g., 1 CAD 0.9342 USD (1-year
  forward)

69
Forward and futures contracts on currencies

• Proposition F Se(r-rf )T, in the absence of
  arbitrage opportunities
• Proof Suppose F gt Se(r-rf )T.
• Arbitrage strategy (to be implemented today)
• Buy one in the spot market by borrowing S
  dollars for T years at rate r.
• Short a forward contract on one .
• Use the income from the invested to repay the
  loan.
• At time T
• Sell the for F dollars under the terms of the
  forward contract.
• Repay the bank Se(r-rf )T dollars.
• Arbitrage profit per F Se(r-rf)T gt 0.
• S is bid up and F is bid down.

70
Forward and futures contracts on currencies

• Suppose F lt Se(r-rf)T.
• Arbitrage strategy (to be implemented today)
• Go long a forward contract on one .
• Sell one . This leads to a cash inflow of S
  dollars. Invest this for T years at rate r.
• At time T
• The proceeds from the sale have grown to Se(r-rf
  )T dollars.
• Buy one for F dollars under the terms of the
  forward contract.
• Return the to your portfolio.
• Arbitrage profit per Se(r-rf)T F gt 0.
• F is bid up and S is bid down.
• Thus F Se(r-rf )T

71
Forward contract UA is a foreign currency

• For an asset that provides a known yield, we had
  
• f Se-qT Ke-rT
• -f Ke-rT Se-qT
• Foreign currency a security that provides a
  known yield at rate q rf
• Thus, for a forward contract on a foreign
  currency, we have
• f Se-rfT Ke-rT
• -f Ke-rT Se-rfT

72
Futures on commodities

• Commodity bulky, entails storage costs if held
• Types
• Investment commodity held primarily for
  investment purposes, e.g., gold, silver
• Consumption commodity held primarily to be used,
  e.g., oil, copper, canola

73
Investment commodities

• Examples gold, silver
• Ignoring storage costs, these are assets that pay
  no income. Thus F SerT.
• But storage costs can be treated as negative
  income.
• Letting U the present value of the storage
  costs incurred during the life of a
  forward/futures contract
• F (S I )erT (S (U ))erT (S U )erT

74
Investment commodities

• Proposition F (S U )erT, in the absence of
  arbitrage opportunities
• Proof Suppose F gt (S U )erT.
• Arbitrage strategy (to be implemented today)
• Buy one ounce of gold in the spot market, and
  arrange to store it, by borrowing (SU ) dollars
  for T years at rate r.
• Short a forward contract on one ounce of gold.
• At time T
• Sell the ounce for F dollars under the terms of
  the forward contract.
• Repay the bank (SU )erT dollars.
• Arbitrage profit per ounce F (SU )erT gt
  0.
• S is bid up and F is bid down.

75
Investment commodities

• Suppose F lt (SU )erT.
• Arbitrage strategy (to be implemented today)
• Go long a forward contract on one ounce of gold.
• Sell one ounce of gold and forego storage costs.
  This leads to a cash inflow of (SU ) dollars.
  Invest this for T years at rate r.
• At time T
• The proceeds from the sale have grown to (SU
  )erT dollars.
• Buy one ounce for F dollars under the terms of
  the forward contract.
• Return the ounce to your portfolio.
• Arbitrage profit per ounce (SU )erT F gt
  0.
• F is bid up and S is bid down.
• Thus F (SU )erT

76
Alternative derivation of formula

• Spot transaction
• Price agreed to.
• Price paid/received.
• Item exchanged.
• Prepaid forward contract
• Price agreed to.
• Price paid/received.
• Item exchanged in T years.
• Forward contract
• Price agreed to
• Price paid/received in T years.
• Item exchanged in T years.

77
Alternative derivation of formula

• Underlying requires the payment of storage costs
  (expressed in present value dollar terms)
• FP S U
• Explanation With a prepaid forward contract, as
  compared to a spot transaction, physical exchange
  of the asset is delayed T years. As a result,
  the long forgoes storage costs with present value
  U and the short has to pay these costs. Thus, the
  price paid by the long and received by the short
  is increased by amount U.
• F FP erT (S U )erT
• Explanation The forward contract allows the long
  to delay payment for T years and requires the
  short to delay receipt. The long can earn
  interest on the cash that would otherwise have
  been paid. The short foregoes this interest.
  The forward price (which is arrived at by
  multiplying the prepaid forward price, equal to S
  U, by erT) compensates the short for the delay.

78
Investment commodities

• As an alternative, storage costs can be expressed
  as a proportion or percentage of the current spot
  price of the commodity.
• Storage costs can then be treated as a negative
  yield.
• Letting u storage costs per annum as a
  proportion or percentage of the spot price
• F Se(r-q)T Se(r-(-u))T Se(ru)T

79
Alternative derivation of formula

• Spot transaction
• Price agreed to.
• Price paid/received.
• Item exchanged.
• Prepaid forward contract
• Price agreed to.
• Price paid/received.
• Item exchanged in T years.
• Forward contract
• Price agreed to
• Price paid/received in T years.
• Item exchanged in T years.

80
Alternative derivation of formula

• Underlying requires the payment of storage costs
  (expressed as a percentage of the spot price)
• FP SeuT
• Explanation FP equals the investment required in
  the asset today that will yield one unit of the
  asset in T years when physical delivery occurs.
  euT units of the asset will grow to euT e-uT
  1 unit of the asset in T years, taking into
  consideration the storage costs that must be
  paid. euT units of the asset cost SeuT.
  Therefore, FP SeuT.
• F FP erT SeuTerT Se(ru)T
• Explanation The forward contract allows the long
  to delay payment for T years and requires the
  short to delay receipt. The long can earn
  interest on the cash that would otherwise have
  been paid. The short foregoes this interest.
  The forward price (which is arrived at by
  multiplying the prepaid forward price, equal to
  SeuT, by erT) compensates the short for the delay.

81
Consumption commodities

• Examples copper, oil, canola
• Proposition F (S U )erT
• F Se(ru)T

• F gt (S U )erT (F gt Se(ru)T)

• F lt (S U )erT (F lt Se(ru)T)

Buy
Sell
Buy
Sell
Traders will respond. S will be bid up and F
will be bid down.
Traders may not respond. If they dont, S will
not be bid down and F will not be bid up.
82
Consumption commodities

• Note We can convert the inequalities to
  equalities by using the concept of convenience
  yield a measure of the benefits of holding the
  physical commodity.
• Letting y the convenience yield, expressed as
  an annual, continuously compounded rate
• F (S U )e(r-y )T
• F Se(ru-y )T

83
Estimating convenience yield

• Provide an estimate of the convenience yield of
  oil
• It is May 2007.
• Current spot price (WTI) 64.35
• The August 2007 contract (NYMEX) is trading at
  66.52.
• Let u 10.
• There are 3 months to maturity of the contract.
• 3-month LIBOR 5.32
• F Se(ru-y)T
• 66.52 64.35e(0.05320.10-y)(3/12)
• y 2.0537

84
Example 5.15, page 122

• S 9
• Storage costs 0.24 per year payable quarterly
  in advance
• r 10
• T 9/12
• U (0.24/4) (0.24/4)e-(0.103/12)
  (0.24/4)e-(0.106/12) 0.1756
• F (S U )erT (9 0.1756)e(0.109/12)
  9.89

0.24/4
0.24/4
0.24/4
0
3/12
6/12
9/12
85
No-Arbitrage Bounds

• The analysis has ignored transaction costs
  trading fees, bid-ask spreads, different interest
  rates for borrowing and lending, and the
  possibility that buying or selling in large
  quantities will cause prices to change.
• With transaction costs, there is not a single
  no-arbitrage price but rather a no-arbitrage
  region.

86
Example

• A trader owns silver as part of a long-term
  investment portfolio. There is a bid-offer
  spread in the market for silver. The trader can
  buy silver for 12.02 per troy ounce and sell for
  11.97 per troy ounce. The six-month risk-free
  interest rate is 5.52 per annum compounded
  continuously. For what range of six-month
  forward prices of silver does the trader have an
  arbitrage opportunity?
• Solution For silver F (S U )erT
• F Se(ru)T
• Assume U u 0 since we are given no
  information on storage costs.
• Thus, F SerT in the absence of arbitrage
  opportunities.

87
Example (continued)

• There is an arbitrage opportunity if
• 1) F gt SerT 12.02e(0.05526/12) 12.36
• 2) F lt SerT 11.97e(0.05526/12) 12.31
• The trader has an arbitrage opportunity for F gt
  12.36 and F lt 12.31. There is no arbitrage
  opportunity for 12.31 F 12.36.

88
Example (continued)

• Now suppose that the trader must pay a 0.10
  transaction fee per ounce of silver.
• There is an arbitrage opportunity if
• 1) F gt SerT (12.02 0.10)e(0.05526/12)
  12.46
• 2) F lt SerT (11.97 - 0.10)e(0.05526/12)
  12.20
• The trader has an arbitrage opportunity for F gt
  12.46 and F lt 12.20. There is no arbitrage
  opportunity for 12.20 F 12.46.

89
Forward and futures contracts on currencies

• If interest rates are expressed as annual rates
  compounded continuously
• If interest rates are expressed as equivalent
  effective annual rates

90
Cost of carry

• Cost of carry (c) the cost of holding an asset,
  including the interest paid to finance purchase
  of the asset plus storage costs minus income
  earned on the asset.
• c can be positive, zero, or negative.
• The concept allows us to express our formulas for
  F in a more general way
• Investment asset F SecT
• Consumption asset F Se(c-y)T

91
Cost of carry
IA F SecT
CA F Se(c-y)T
c r
c r q
c r rf
c r u
c r u
92
Cost of carry

• Investment asset F SecT
• Consumption asset F Se(c-y)T
• T 0 implies F Se0 S
• That is, the forward/futures price of an asset
  equals its spot price at the time the contract
  expires.

93
Cost of carry

• Investment asset F SecT
• Consumption asset F Se(c-y)T
• ?F/?S the amount by which the forward (futures)
  price changes in response to an infinitesimal
  change in the spot price, ceteris paribus
• Investment asset ?F/?S ecT gt 0
• Consumption asset ?F/?S e(c-y)T gt 0
• F and S are positively correlated.

94
Cost of carry
Investment asset F SecT

• c gt 0 implies ecT gt 1 and F gt S
• Normal, contango market

• c lt 0 implies 0 lt ecT lt 1 and F lt S
• Inverted market, backwardation

95
Cost of carry
Consumptiom asset F Se (c-y)T

• c gt y implies e (c-y)T gt 1 and F gt S
• Normal, contango market

• c lt y implies 0 lt e (c-y)T lt 1 and F lt S
• Inverted market, backwardation

96
Cost of carry

• Investment asset F SecT
• Consumption asset F Se(c-y)T
• ?F/?T the amount by which the forward (futures)
  price changes in response to an infinitesimal
  change in the time to expiration of the contract,
  ceteris paribus
• Investment asset ?F/?T SecT c
• c gt 0 implies ?F/?T gt 0 normal or contango
  market
• c lt 0 implies ?F/?T lt 0 inverted market,
  backwardation
• Consumption asset ?F/?T Se(c-y)T (c y)
• c gt y implies ?F/?T gt 0 normal or contango
  market
• c lt y implies ?F/?T lt 0 inverted market,
  backwardation

97
Normal market

• c gt 0 (investment asset)
• c gt y (consumption asset)
• F gt S
• ?F/?T gt 0, that is, forward (futures) contracts
  with longer times to expiration trade at higher
  prices than forward (futures) contracts with
  shorter times to expiration.

98
Inverted market

• c lt 0 (investment asset)
• c lt y (consumption asset)
• F lt S
• ?F/?T lt 0, that is, forward (futures) contracts
  with longer times to expiration trade at lower
  prices than forward (futures) contracts with
  shorter times to expiration.

99
Amaranth Advisors LLC

• The Connecticut-based hedge fund lost about 6
  billion (40 of its value) in September 2006
  trading natural gas derivatives.
• GM, September 22, 2006 The problem with oil
  and gas these days is that the market is morphing
  from backwardation, when spot prices are higher
  than prices for delivery in the future, to
  contango, when futures prices are higher than
  spot.

100
Spread trades

• A spread trade provides exposure to the
  difference between two prices.
• It is a long-short futures position.
• Example
• Calendar spread go long long-term contract and
  short short-term contract on the same underlying
  asset, or vice versa.
• Intercommodity spread go long futures on
  commodity A and short futures on commodity B
• Geographical spread go long NYMEX oil futures
  and go short Londons ICE Brent oil futures
• For speculators, it offers reduced risk.

101
Amaranth Advisors LLC

• Traders expected the market to be in
  backwardation but it has moved into contango.
• They implemented spread trades based on this
  expectation.

Traders at Amaranth were betting .
Expectation of a cold winter, active hurricane
season, instability in oil and gas producing
countries
What has happened .Winter of 2005-2006 was
warm, the hurricane season was benign, supply of
oil and gas was relatively high
F
F
T
T
102
Delivery options for a futures contract

• T the time to expiration of a forward or
  futures contract.
• Forward contract we know T.
• Futures contract we must estimate T.
• Question When during the delivery period of a
  futures contract will the short choose to make
  delivery?

103
Delivery options for a futures contract

• Normal Market
• Investment asset, c gt 0
• Consumption asset, c gt y
• Short should deliver as soon as possible.

104
Delivery options for a futures contract

• Inverted Market
• Investment asset, c lt 0
• Consumption asset, c lt y
• Short should deliver as late as possible.

105
Next class

• Hedging with futures