Post on 23-Dec-2015
8-1
Chapter 8
Arbitrage
8-2
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.
8-3
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.
NYSE PACSell Buy
+$53 -$50 = $3.
8-4
Assumptions for Arbitrage
No transactions costs.No default.No collateral.The ability to shortsell securities and
use the proceeds from the shortsale. This is called unrestricted shortselling.
8-5
Lender of certificates
SellerShortseller buys
Lender of certificates
IOU Certificate
$
Purchase of certificate
CertificateReturnIOU
Short Position Is Established
Short Position Is Closed
Shortseller BuyerSale of certificate
$
8-6
Shortseller must buy back at some future.
Profit: Shortsale price > Purchase price.Loss: Shortsale price < Purchase price.
Potential shortsale losses have no upper bound, implying shortselling is very risky.
8-7
Not an issue for bonds because of daily accrued interest.
$
TimeEx-dividend
point
After-tax value of dividends
For stocks, shortsellers must pay dividends to lender of certificates.
8-8
Shortselling a Bond Equals Borrowing
Points in Time
0 1 2
Cash flows +$82.64 0 -$100
8-9
Hypothetical Strips Prices
Points in time
0 1 2
$70 $100 –
$80 $100
8-10
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
Arbitrage Cash Flows
8-11
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.
8-12
Points in time
0 1 2
$88 $100 –
$80 $100
8-13
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
A Non-arbitrage Position
8-14
Arbitrage and Bond Coupons
8-15
Points in Time
0 1 2
Bond G -$100 +$6 +$106
Bond H -$100 +$8 +$108
Two-period Bonds
8-16
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
Arbitrage for Two-period Bonds
8-17
Action Points in Time
0 1 2
Shortsell Bond G +$100 -$6 -$106
Buy Bond H -$103.60 +$8 +$108
Net cash flows -$3.60 +$2 +$2
Cumulative net cash flows -$3.60 -$1.60 +$.40
Two-period Bonds: No Arbitrage Profit
8-18
Cash FlowsPoints in Time
210
Bond G
Bond H
100
106
106
108
6
8
8-19
ArbitragePoints in Time
210
Buy G
Short H
-100
+106
+106
-108
+6
-8
Net
Cumulative Net
+6
+6
-2
+2
-2
+4
8-20
Price
Coupon
Arbitrage
P = c[PVA] + PAR[PV]
Arbitrage
0 6 8
104
100
S
8-21
Points in Time
0 1 2
Bond G $100 $6 $106
Bond H $102 $8 $108
Suppose
8-22
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
There is an arbitrage profit as follows
8-23
The forward interest rate is negative
%.38.106
1
12.088.1
2 ,0
2 ,0
f
f
8-24
210
PG
PH
6
108
106
1168
112
Total Future Inflows
8-25
.5714.103P
,100P If
035714.1PP
112
116PP
H
G
GH
GH
To avoid arbitrage (a negative forward rate)
8-26
124
120
Total Future Inflows
Bonds of Different Maturities
100
100
6
104
106
4
210 3 4 5
Bond H
Bond G 66
4 4 4
8-27
-100
+100
+6
-104
+106
-4
Arbitrage
210 3 4 5
Short Bond H
Buy Bond G +6+6
-4 -4 -4
Net
Cum Net
+20
0
+2 +2 +102
+2 +4 +6 +108
-104
+4
8-28
General Case
4CG + PAR
5CH +
PAR
Total Future Inflows
PG
PH
CG
CH +
PAR
CG +
PARCH
210 3 4 5
CGCG
CH CH CH
Arbitrage unless
PARC5PARC4
PPH
GHG
8-29
(94.34)(.06) + (1.06)(85.73) = $96.53
210
94.34
85.73
100
100
100 1066
Replicating Portfolio
8-30
Action Points in Time
0 1 2
Two-period bond $100 $6 $106
One-period strip $94.34(6%) $100(6%)
Two-period strip $85.73(106%) $100(106%)
Arbitrage between Coupon-bearing Bonds and Strips
(94.34)(.06) + (1.06)(85.73) = $96.53
8-31
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
Arbitrage between Coupon-bearing Bonds and Strips
8-32
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
Cash Flows in Equilibrium When Price of Two-period Strip is $89
8-33
210
S2S1
• Individual bonds can be stripped and reconstituted.• Principal (Par) strips from the specific bond must
be used to reconstitute the bond.• Coupon strips from any bond can be used to
reconstitute the coupons.
U.S. Treasury Strips
n3
C
. . .
C C C + PAR
S3 Sn + Sp,n
8-34
5-Year Bond
4-Year Bond
C1 C5 +
PAR5C4 +
PAR4
C1
210 3 4 5
C3C2
C2 C3
C4
• To reconstitute a 4-year bond, coupon strips from either bond can be used for the coupons at times 1, 2, 3, 4.• Only Par4 can be used to reconstitute the 4-year
par value.
8-35
• In reconstituting a bond, the principal strip coming from the original bond must be used.
• Principal strips with the appropriate maturing may be used to reconstitute any bond.
Principal Strips vs. Coupon Strips
8-36
In practice prices of principal and coupon strips with the same maturity may be different
Time
Sp
SHORT
LONG
Sc
Maturity
Par
8-37
• Arbitrage positions between principal and coupon strips in practice require collateral.
• The differences in price must be big enough to justify investing this collateral.
• Price differences may get larger over time and more collateral may be required.
8-38
Creating Forward Contracts from Spot
Securities
8-39
Points in Time
0 1 2
Long forward 0 -Forward +Par
Long Forward Position
8-40
210
Spot
Strips
85.73 = S2
96.15 = S1
100
100
Long Forward
0 +100-F
= = 0.8916.85.7396.15
S2
S1
Numerical Example
8-41
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
8-42
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
8-43
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
Creating a Long Forward Position
8-44
2,01
2
f1PAR
SSF
8-45
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%.
1215.1100
F = = 89.16.
8-46
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
8-47
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
Borrow forward at 5% +$89.16 -$100
________________________________________________
Net 0 +$2.20 0
8-48
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
8-49
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
Lend forward at 12.15% -$89.16 +$100
________________________________________________
Net 0 +$6.08 0
8-50
Price
Coupon
P2High
C1
P3
P2
P1
C2 C3
P2Low
Arbitrage if P2High > P2 rr if P2
Low < P2