Determination of Forward and Futures Prices Chapter 5

1
Determination of Forward and Futures
PricesChapter 5
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)

3
Continuous Compounding

• In the limit as we compound more and more
  frequently we obtain continuously compounded
  interest rates
• 100 grows to 100eRT when invested at a
  continuously compounded rate R for time T
• 100 received at time T discounts to 100e-RT at
  time zero when the continuously compounded
  discount rate is R

4
Conversion Formulas

• Define
• r continuously compounded rate
• R same rate with compounding m times per year

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Examples

• Bank account pays 4 compounded monthly
• What is the (annual) continuously compounded
  returns?
• r 12ln(1.04/12) 3.993
• What is the future value of 100 dollars in 15
  months?
• FV 100exp.03993x15/12 105.12
• or
• FV 100(1.04/12)15 105.12

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Notation
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Gold Example

• For gold
• F0 S0(1 r )T
• (assuming no storage costs)
• If r is compounded continuously instead of
  annually
• F0 S0erT

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Extension of the Gold Example

• For any investment asset that provides no income
  and has no storage costs
• F0 S0erT

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Suppose F gt SerT

• T 0 CF0
• sell forward 0
• buy spot -S
• borrow S _at_ r S
• Net cash flow 0
• Arbitrage portfolio

• T1 CF1
• Deliver F
• Payoff debt -SerT
• Profit F - SerT

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When an Investment Asset Provides a Known Dollar
Income

• F0 (S0 I )erT
• where I is the present value of the income

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Suppose F gt (S I)erT

• T 0 CF0
• sell forward 0
• buy spot -S
• Sell Income I
• borrow S _at_ r (S-I)
• Net cash flow 0
• Arbitrage portfolio

• T1 CF1
• Deliver F
• Payoff debt -(S-I)erT
• Profit F (S-I)erT

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When an Investment Asset Provides a Known Yield

• F0 S0 e(rq )T
• where q is the average yield during the life
  of the contract (expressed with continuous
  compounding)

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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

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Stock Index

• Can be viewed as an investment asset paying a
  dividend yield
• The futures price and spot price relationship is
  therefore
• F0 S0 e(rq )T
• where q is the dividend yield on the portfolio
  represented by the index

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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
• The Nikkei index viewed as a dollar number does
  not represent an investment asset

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Index Arbitrage

• When F0gtS0e(r-q)T an arbitrageur buys the stocks
  underlying the index and sells futures
• When F0ltS0e(r-q)T an arbitrageur buys futures and
  shorts or sells the stocks underlying the index

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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

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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

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Suppose F gt Se(r-rf)T

• Let S be the spot exchange rate in /
• Then S dollars buys 1 pound
• Alternative to forward is to convert dollars to
  pounds and then invest in Britain
• Therefore, carry cost in domestic currency is
  offset by foreign currency rate
• q rf

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repay
SerT
Borrow S
exchange
/ Se(r-rf)T
1
erfT
invest
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Futures on Consumption Assets

• F0 ? S0 e(ru )T
• where u is the storage cost per unit time as a
  percent of the asset value.
• Alternatively,
• F0 ? (S0U )erT
• where U is the present value of the storage
  costs.

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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(cy )T