1Arbitrage Analysis of Forwards and Futures

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1Arbitrage Analysis of Forwards and Futures

• Arbitrage is about identifying risk-free yet
  profitable trading strategies.
• Arbitrage strategies involve a generalization of
  a simple trading strategy.
• Arbitrage can be used to value a contract because
  if markets are efficient there should be no such
  thing as a risk-free economic profit.
• First, we consider short selling as a trading
  technique.

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2Short Selling

• Hull, Slide 3.5
• Hull, Slide 3.6
• Now we make the following assumptions about
  arbitrage activities
• there are no transactions costs
• all trading profits and losses are subject to the
  same tax rate ? ? (1 - ?)?, ? constant
• borrowing and lending takes place at the same
  rate, r
• the market is efficient in that arbitrage
  opportunities are taken as they arise so that
  they do not persist for long
• short selling is allowed.

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3Compounding and interest rate Issues

• Hull, Slides 3.2, 3.3, and 3.4
• Hull, Slide 3.12
• For further details, see Cox, Ingersoll and Ross,
  Journal of Financial Economics (Dec., 1981).
• For random interest rates, the idea is that when
  you make a profit (loss) the rate of return on
  the money you have to invest is high (low).

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4Spot-Forward Arbitrage-Enforced Relations

• Let S(t) be the time t spot price on an
  investment asset, and F(T,t) the forward/futures
  price. Interest rate, r is constant.
• Hull, Slide 3.7
• Hull, Slide 3.8
• These relationships do not depend on the ability
  to sell short. They just require that a
  significant fraction of asset holders do so for
  purely investment purposes..

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5Intuition for Arbitrage-Enforcement

• Suppose that relation (3.5) in Hull were not
  true. If F(T,t) gt S(t)er(T-t). Then trade as
  follows
• At time t, borrow S(t) for duration T - t at
  rate r.
• Simultaneously, buy the asset and sell forward at
  price F(T,t).
• At time T, deliver the asset and receive F(T,t).
• Simultaneously, use S(t)er(T-t) to repay the
  loan.
• The profit is F(T,t) - S(t)er(T-t) gt 0.
• If F(T,t) lt S(t)er(T-t), then reverse the
  strategy.

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6When there is Investment Income

• Hull, Slide 3.9
• If the forward price is relatively too high, buy
  the asset, borrow enough to pay I when it comes
  due, and sell forward the asset. If the forward
  price is relatively too low, reverse your
  positions. If short sales are not allowed,
  investors who own the asset will sell it to avail
  of the arbitrage opportunity, and will buy
  forward.
• Hull, Slide 3.10. Here, the rate of dividend
  payout on the security is q S(t).

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7Valuing a Futures/Forward Contract already
Written

• Now lets move away from writing the contract.
  Assume that it has already been written, i.e.,
  that K has been set at a time t before t.
• Prices then evolve, and the contract assumes
  positive or negative value. We wish to value
  that contract. That is, what would one
  pay/accept to get into/out of the contract?
• Hull, Slide 3.11. To obtain some intuition for
  eqn. (3.8), suppose that the contract were
  written at time t. Then F(T, t) would be set
  so that f 0.

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8Intuition for Eqn. (3.8)

• For a forward, if you go long at time t you
  agree to take delivery at time T for price K.
• At time t (t gt t), you could cancel your
  position by going short at price F(T, t).
• Wait until T when you buy at K and sell at F(T,
  t). You make/lose the non-random amount F(T,t)
  - K.
• Discount back to time t to obtain eqn. (3.8). If
  you go short at time t, then reverse the
  arguments.
• When dealing with futures, there is the issue of
  marking to market. But it is irrelevant if
  interest rates are constant.

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9Forwards/Futures on Stock Indices

• Under consideration here are SP500, NASDAQ, Dow,
  etc., indices. The formula must accommodate
  dividend payments by underlying stocks.
• Hull, Slide 3.13.
• Hull, Slide 3.14. The problem is that the Nikkei
  Index is not something anyone in the U.S. an
  invest in. In Japan it is possible to replicate
  via holding a basket of stocks. In the U.S., one
  can only invest in the exchange rate times the
  replicated index. The straight Nikkei index can
  not be arbitraged, and so eqn. (3.12) can not be
  arrived at. There are alternative means of
  valuation.

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10Index Arbitrage Activities

• Hull, Slide 3.15.
• Hull, Slide 3.16.
• On Black Monday (10/19/87), and at other times
  the index futures market was effectively
  inoperative. The system was overloaded and
  either futures did not keep up with stocks or
  vice versa. Therefore, it was not possible to
  take risk-free arbitrage.
• It is a continual battle for exchanges to keep
  computer capacity up with the ever-expanding
  volume of trade.

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11Uses of Stock Index Futures

• If one fully hedges a fairly well-diversified
  stock portfolio by use of the appropriate index,
  then what should your rate of return be? By
  CAPM, it should be circa r. Why not just invest
  in an interest-bearing account in the first
  place?
• Possible reason You might be speculating that
  the portfolio you have chosen (in say oil stocks
  or large stocks) will outperform the overall
  index, but you dont know where the market itself
  will go. The index will hedge away the systemic
  risk, leaving only the idiosyncratic risk. With
  this hedge, the market could fall and you could
  still win (or vice versa).

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12Uses of Stock Index Futures, Contd

• Possible Reason You may wish to be exposed to
  market risk over the long run, but you may not
  want to be exposed at a particular point in time
  (year end, pre-election, pre-Fed meeting, triple
  witching). Index hedging is cheaper than
  unwinding and rewinding positions.
• Hull, Slide 3.17. The number of futures
  contracts required to immunize your portfolio
  against market risk depends on how your portfolio
  tends to move against the index, i.e., your
  portfolio ?.
• Hull, Slide 3.18. Index futures can also modify
  portfolio ?.

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13Other Arbitrage Relations

• Forwards/Futures on currencies Hull, Slide 3.19.
• Forwards/Futures on investment assets with
  storage costs. In this case (some precious
  metals), the holder of a long actuals position
  will have to be compensated for storage costs.
• If the storage costs are of a fixed per unit
  nature then the usual no-arbitrage arguments give
  Hull eqn. (3.15), F(T,t) S(t) Uer(T-t).
  
  
  Here, U is the
  present value of all storage costs over time
  interval (t, T. See Hull, eqn. (3.6) and Slide
  3.9.

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14Investment Assets with Storage Costs

• If storage costs are proportional to stock value
  (e.g., insurance), then no-arbitrage gives Hull
  eqn. (3.16), F(T,t) S(t)e(r u)(T-t).
  
  Here, u is the continuously compounded rate of
  storage costs.
• Some mixture of the storage cost methods might be
  most appropriate.

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15Arbitrage for Consumption Assets

• Now lets turn to why it was necessary to assume
  that the assets in question were primarily
  investment in nature, as distinct from
  consumption. Assets such as corn, wheat, lumber,
  copper, etc., are not covered.
• Hull, Slide 3.20. Handout 8.
• The reason for the inequality is because people
  want the stock on hand for purposes other than
  investment. The magnitude of the inequality can
  be related to the desire to have stocks on hand.
  Convenience yield measures it.

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16Cost of Carry

• Hull, Slide 3.21.
• Apply to an index with continuous dividends,
  foreign exchange, etc.

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17Futures Price vs. Expected Future Spot Price

• For an investment asset, we know that
  F(T,t) S(t) er(T-t).
  
  ()
• If CAPM gives an expected rate of return on a
  security of k, then we have
  
  EtS(T) S(t)ek(T-t) or
  S(t)
  EtS(T)e-k(T-t) .
  ()
• Substitute () into () to obtain Hull, Eqn.
  (3.25). See Handout 9.
• Hull, Slide 3.22. Hull, Slide 3.23.