Arbitrage

1
Chapter 8

• Arbitrage

2

• Suppose that a particular stock is selling for
  53 on the New York Stock Exchange and
  simultaneously selling for 50 on the Pacific
  Coast stock exchange.
• On arbitrageur can simultaneously buy on the
  Pacific Coast exchange for 50 and sell on the
  New York stock exchange for 53.

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NYSE PAC Sell Buy 53 -50 3.

• The arbitrageur makes an instant, risk-free
  profit of three dollars. The ability to
  repeatedly carry out this transaction will force
  the prices to be the same in equilibrium.

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Assumptions for Arbitrage

• No transactions costs.
• No default.
• The ability to shortsell securities and use the
  proceeds from the shortsale. This is called
  unrestricted shortselling.

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Short Position Is Established
Sale of certificate
Shortseller
Buyer

IOU
Certificate
Lender of certificates
Short Position Is Closed
Purchase of certificate
Seller
Shortseller buys

Certificate
ReturnIOU
Lender of certificates
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Shortseller must buy back at some future.

• Profit Shortsale price gt Purchase price.
• Loss Shortsale price lt Purchase price.

Potential shortsale losses have no upper bound,
implying shortselling is very risky.
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For stocks, shortsellers must pay dividends to
lender of certificates.

After-tax value of dividends
Time
Ex-dividend point
Not an issue for bonds because of daily accrued
interest.
8
Shortselling a Bond Equals Borrowing
Points in Time 0 1 2 Cash
flows 82.64 0 -100
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Hypothetical Strips Prices
Points in time 0 1 2 70 100 80 100
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Arbitrage Cash Flows
Action Points in Time 0 1 2 Buy
one-period strip -70 100 0 Shortsell
two-period strip 80 0 -100 Net cash
flows 10 100 -100 Cumulative net cash
flows 10 110 10
11

• In a multi-period context, a sufficient condition
  for arbitrage is for the cumulative cash flows to
  never be negative and have the possibility of
  being positive at a future point in time.

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Points in time 0 1 2 88 100 80 100
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A Non-arbitrage Position
Action Points in Time 0 1 2 Shortsell
one-period strip 88 -100 0 Buy two-period
strip -80 0 100 Net cash
flows 8 -100 100 Cumulative net cash
flows 8 -92 8
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• Arbitrage and Bond Coupons

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Two-period Bonds
Points in Time 0 1 2 Bond G
-100 6 106 Bond H -100 8 108
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Arbitrage for Two-period Bonds
Action Points in Time 0 1 2 Shortsell
Bond G 100 -6 -106 Buy Bond H -100
8 108 Net cash flows 0 2.00
2.00 Cumulative net cash flows
0 2.00 4.00
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Two-period Bonds No Arbitrage Profit
Action Points in Time 0 1 2 Shortsell
Bond G 100 -6 -106 Buy Bond H -103.90
8 108 Net cash flows -3.90
2 2 Cumulative net cash flows -3.90
-1.90 .10
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Cash Flows
Points in Time
2
1
0
Bond G
100
106
6
Bond H
106
108
8
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Arbitrage
Points in Time
2
1
0
Buy G
-100
106
6
Short H
106
-108
-8
Net
6
-2
-2
Cumulative Net
6
2
4
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Price
Arbitrage
104 100 S
?
P cPVA PARPV
?
Arbitrage
Coupon
0 6 8
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Replicating Portfolio
(94.34)(.06) (1.06)(85.73) 96.53
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Arbitrage between Coupon-bearing Bonds and Strips
Action Points in Time 0 1 2 Short
two-period bond 100 -6 -106 Buy 6 of a
one-period strip -5.66 6 ? Buy 106 of a
two-period strip -90.87 ? 106 Net cash
flows 3.47 0 0 Cumulative net cash
flows 3.47 3.47 3.47
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Cash Flows in Equilibrium When Price of
Two-period Strip is 89
Action Points in Time 0 1 2 Short
two-period bond 100 -6 -106 Buy 6 of a
one-period strip -5.66 6 ? Buy 106 of a
two-period strip -94.34 ? 106 Net cash
flows 0 0 0 Cumulative net cash
flows 0 0 0
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• Creating Forward Contracts from Spot Securities

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Long Forward Position
Points in Time 0 1 2 Long
forward 0 -Forward Par
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Numerical Example
2
1
0
Spot
85.73 S2
100
Strips
96.15 S1
100
Long Forward
0
100
-F
85.73 96.15
0.8916.
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A Numerical Example ofCreating a Long Forward
Position
Action (at time 0) Points in Time 0 1
2 Long two-period strip -85.73 ?
100 Short 0.8573/0.9615 one-period bonds
85.73 -89.16 ? Net Long
forward 0 -89.16 100
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A Numerical Example of Creating a Short Forward
(Borrowing) Position
Action (at time 0) Points in Time 0 1
2 Short 1 two-period strip 85.73
? -100 Long 0.8573/0.9615 one-period
bonds -85.73 89.16 ? Net
Short forward 0 89.16 -100
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Creating a Long Forward Position
Action (at time 0) Points in Time 0 1
2 Long 1 two-period strip -S2 ?
100 Short S2/S1 one-period bonds S1/(S2/S1)
-1(S2/S1) ? Net Long forward 0
-(S2/S1) 100
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(No Transcript)
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Arbitrage and Forward Interest Rates
Suppose that R0,1 4, R0,2 8, implying that
a forward loan can be created with an interest
rate of 12.15.
F 89.16.
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Suppose the actual forward rate is 15, while the
rate implied by strips is 12.15.
Action (at time 0) Points in Time 0 1
2 Lend forward at 15 0 -100/1.15
100
-86.96 Short 1
two-period strip 85.73 ?
-100 Long 0.8573/0.9615 -85.73
89.16 ? one-period strips Net
0 2.20 0
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Suppose the actual forward rate is 5 and the
implied forward rate is 12.15.
Action (at time 0) Points in Time 0 1
2 Borrow forward at 5 0 100/1.05
-100
95.24 Long 1 two-period
strip -85.73 ? 100 Short
0.8573/0.9615 85.73 -89.16
? one-period strips Net 0
6.08 0
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Suppose
Points in Time 0 1 2 Bond G
100 6 106 Bond H
102 8 108
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There is an arbitrage profit as follows
Points in Time 0 1 2 Short 1.02 units
Bond G 102 -6.12 -108.12
Buy Bond H -102
8 108 Net 0 1.88
-0.12 Cumulative Net
0 1.88 1.76
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The forward interest rate is negative
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Price
P2High
P3
?
P2
P2Low
P1
?
Coupon
C1
C2
C3
Arbitrage if P2High gt P2 rr if P2Low lt P2