Arbitrage

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Arbitrage

• Arbitrage is a concept that means capitalizing on
  discrepancies in quoted prices.
• All major investment banks and hedge funds have
  an arbitrage department.
• It is difficult to make a killing but it is
  possible to make money over a long period with
  complicated models and great timing.

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Example 2 Can you make money?

• YES!
• You buy DM from Bank A at .500 /DM
• You sell DM to Bank B at .505 /DM.
• You make .005/DM
• A 100,000 round trip yields 1000 in seconds.

• Bank quotes

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Triangular Arbitrage

• Suppose that the British pound is worth 1.50 and
  the DM is worth .50. Then the synthetic rate
  for pounds in DM is .

• The units we want are DM/GBP.
• (1.50 USD/GBP)/ (.50 USD/DM) 3 DM/GBP

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Can you Make Money?

• The synthetic cross rate for the pound in terms
  of francs is (2 USD/GBP)/(.20 USD/FRF) 10
  FRF/GBP. This is cheaper than the cross rate.

• Bank quotes 2USD/GBP and 11 FRF/GBP and .20
  USD/FRF
• You have 10,000
• Can you make money?

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How to capitalize on the discrepancy between the
rates

• Use 10,000 to buy 5,000 GBP.
• Convert to FRF to yield 5,000 GBP x 11 FRF/GBP
  55,000 FRF
• Sell FRF for dollars to yield 55,000 FRF x .20
  USD/FRF to yield 11,000.
• Make a 1000 profit!

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MORAL

• Locational and triangular arbitrage force the
  structure of rates into equilibrium.
• Dealers can quickly detect misalignments and buy
  and sell accordingly.

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Covered Interest Arbitrage

• YES!
• Buy pounds spot to obtain 1m/2 .5 million GBP
• Invest at .04 to obtain 1.04 x 500,000 520,000
  GBP
• Sell forward for 520,000 x 2 1,040,000

• S 2 USD/BRP
• F 2 USD/BRP
• rUS90 .02
• rUK90 .04
• You have 1,000,000 to invest.
• Can you make money?

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What will happen to the forward rate?

• Currently the forward premium on the pound is 0
• People will rush to buy pounds spot and sell
  pounds forward
• Other things equal, the forward rate will fall
  and the spot rate will rise, decreasing the
  forward premium.

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Whats the Intuition?

• If theres no difference in the spot and the
  forward rate, money will flow to the country with
  the higher interest rate.
• Other things equal, the forward rate (HC/FC) must
  fall to adjust for the rate differential and make
  foreign investments less attractive.

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Deriving Bid Ask Spreads

• S 23.00-05 LUF/AUD
• rLUF180 10.00-25 p.a.
• rAUD180 20.00-25 p.a.
• What are the bounds on the forward bid/ask
  dictated by arbitrage, other things equal?

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Three Conditions that Must Hold

• Fbid lt Fask
• No money machines
• Least cost dealing

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No Money Machine Step 1

• Borrow LUF at home and invest abroad
• It must not be possible to make money

• Remember you borrow at the ASK rate (the
  higher)
• You lend at the BID rate (the lower)

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The Mechanics

• Suppose that you borrow the present value of 1
  LUF now, buy AUD spot, invest in AUD assets, and
  sell AUD forward.
• How much will you have?
• In equilibrium, it must be less than or equal to
  1 LUF

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Deriving the Upper Bound

• The proceeds of the loan are 1/ rLUF ASK180
  1/(1(.1025/2)) .9512 LUF
• Buy AUD spot and get .9512/23.05 .0413 AUD and
  invest to get .0413 AUD (1 .2/2) .0454 AUD
• Sell AUD forward at Fbid for .0454 AUD Fbid
  (LUF/AUD)
• .0454 Fbidlt 1 or Fbidlt 22.05 LUF/AUD

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No Money Machine Step 2

• Borrow AUD abroad and invest at home
• It must not be possible to make money

• Remember you borrow at the ASK rate (the
  higher)
• You lend at the BID rate (the lower)

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The Mechanics

• Suppose that you borrow the present value of 1
  AUD now, buy LUF spot, invest in LUF assets, and
  sell LUF forward.
• How much will you have?
• In equilibrium, it must be less than or equal to
  1 AUD

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Deriving the Lower Bound

• The proceeds of a loan requiring a payment of 1
  AUD in 180 days is 1/(1 (rAUD ASK180/2)
  1/(1.10125) .9080 AUD
• Sell AUD, yielding .9080Sbid .908023 20.89
  LUF
• Invest to get 20.89(1 rLUF Bid180/2)21.93
• Buy AUD forward at Fask to get 21.93/ Fask lt 1
  AUD
• Fask gt 21.93

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Summary of General Arbitrage Bounds
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Condition 3 Least Cost Dealing

• It is possible to to obtain domestic and foreign
  currency at time T without going through the
  forward market
• You can replicate the transaction in the money
  markets and spot markets
• Bank quotes must be competitive

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Intuition Behind Replication

• Suppose a Luxembourg investor needs 1 AUD in 6
  months
• She could borrow LUF now, exchange at the spot
  rate, invest in AUD, and have AUD in 6 months
• Whats the cost? Repayment of the LUF loan
• If this is less expensive than the forward ask,
  why do business with the bank?

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Is this realistic?

• In the real world, if the direct rate is not
  within the spread of the synthetic rates, there
  will be no dealing in the direct market
• In the presence of transactions costs, the cost
  of making a particular transaction depends on the
  route chosen, so investors will always compare
  the direct with the synthetic rate
• A forward hedge and a money market hedge will not
  always produce the same result, so there is a
  small cost from choosing the wrong hedge.

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Calculating the Synthetic Forward Bid

• Forward bid is the price at which the investor
  exchanges AUD for LUF in the future
• How else could I get LUF at time T?
• By borrowing AUD today, selling it spot,
  investing it in LUF assets

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Calculating the Synthetic Forward Bid

• If you borrow the present value of a unit of
  foreign currency, you will have 1/(1 rask/2)
  AUD to work with
• Buy LUF spot and you have Sbid 1/(1 rask/2) )
• Invest in LUF assets to get
• Sbid (1 rbid/2)/(1 rask/2)
• This is what you get in exchange for paying back
  1 AUD at time T
• Sbid (1 r/2)/(1 rask/2) lt Fbid

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Calculating the Synthetic Forward Ask

• You want to receive 1 AUD at time T
• Invest its present value now 1/(1rbid/2)
• Borrow the LUF equivalent Sask/(1rbid/2)
• Youll pay back Sask(1 rask/2)/(1rbid/2) at
  time T
• This is the synthetic forward ask (the rate at
  which you can buy AUD forward synthetically)
• Sask(1 rask/2)/(1rbid/2) gt Fask

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Summary of General Least Cost Dealing Bounds
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MORAL

• The results themselves are not important
• What IS important is the logic arbitrage and
  least cost dealing rules give us RANGES for
  forward quotes rather than the exact formula for
  covered interest parity
• Still, simple covered interest parity is a good
  approximation

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Does CIP Hold?

• Empirical tests of the BOUNDS of IRP hold
  PERFECTLY if Euro rates are used
• Eurocurrency markets are markets for time
  deposits and loans in countries outside the
  country of the currency in which the credit
  instrument is denominated.
• Euromarkets dominate money markets.
• CIP holds perfectly in these markets.

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What about simple CIP?

• Take a look at the Excel sheet

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What about simple CIP?

• Take a look at the Excel sheet

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What about simple CIP?

• Interest rate parity does not always hold when
  comparing domestic as opposed to Eurocurrency
  rates because of the existence of capital
  controls which prevent arbitrage/
• The covered interest differential is a widely
  used measure of the effectiveness of capital
  controls.