Chapter 10: Futures Arbitrage Strategies
We use a number of tools to manage our investment portfolio
for the highest return, while minimizing the risk. Some of
these tools happen to be called 'derivatives.'
Anonymous respondent quoted in
1996 Capital Access Survey on Derivatives
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 1
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Important Concepts in Chapter 10




Futures spread and arbitrage strategies
Cheapest-to-deliver bond
Delivery options
Use of futures in market timing, alpha capture, and asset
allocation
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 2
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Short-Term Interest Rate Arbitrage

Cash and Carry/Implied Repo
 Cash and carry transaction means to buy asset and sell
futures
 Repurchase agreement/repo to obtain funding
 Overnight vs. term repo
 Cost of carry pricing model: f0(t) = S0 + q
 Implied repo rate:
1/t
 f 0 (t) 

r̂  
 S 
0 

Chance/Brooks
1
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 3
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Short-Term Interest Rate Arbitrage
(continued)


Cash and Carry/Implied Repo Rate
 Also equivalent to buying longer term and converting it
to shorter term.
 Example. See Table 10.1.
Eurodollar Arbitrage
 Using Eurodollar futures with spot to earn an arbitrage
profit.
 See Table 10.2.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 4
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Intermediate and Long-Term Interest Rate
Arbitrage
 Recall
the option to deliver any T-bond with at least 15
years to maturity or first call.
 Adjustment to futures price using conversion factor,
which is the price per $1.00 par of a 6% bond delivered
on a particular expiration.
 Invoice price = (Settlement price on position
day)(Conversion factor) + Accrued interest
 Example: Delivery on March 2009 contract.
Settlement price is 112 ($112,000) on position day.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 5
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)
 You
plan to deliver the 5 1/2 of 2028 on March 8. CF
= 0.9389. Coupon dates of February 15 and August 15.
Last coupon on February 15, 2009. Days from 2/15 to
3/13 is 26. Days from 2/15 to 8/15 is 181. Accrued
interest
 $100,000(0.055/2)(26/181) = $395.03
 Invoice price:
 $112,000(0.9433) + $395.03 = $106,044.63
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 6
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)
 Next
day, Notice of Intention Day, Thursday, March 9,
the short invoices the long $106,044.63. The long pays
for and receives the bond on Friday, March 10.
 Table 10.3 shows CFs and invoice prices for other
deliverable bonds on the March 2009 contract.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 7
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract
 Recall the option to deliver any T-bond with at least 15
years to maturity or first call.
 Example: Delivery on March 2009 contract of 6 1/4s
of May 15, 2030.
 Cost of delivering bond
 f0(T)(CF) + AIT - [(B + AIt)(1+r)(T-t) – CIt,T]
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 8
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract (continued)
 Example: On 11/13/08, plan to deliver the 6 1/4s of
5/15/30 on the March 2009 contract on March 11.
f0(T) = 116 , CF = 1.0296, AIt = 3.09, AIT = 2.00
(deliver on March 11), B = 122.75. 118 days between
November 13 and March 11. Reinvestment rate =
1.0%.
 Invoice price
 116(1.0296) + 2.00 = 121.43
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 9
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract (continued)
 There are no interim coupons paid
 Forward price of deliverable bond
 (122.75 + 3.09)(1.01)118/365
+ 3.125(1.01)118/365 = 123.11
 So the bond would cost 1.68 (=123.11 – 121.43) more
than it would return.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 10
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract (continued)
 All we can do, however, is compare this result with that
for another bond. For the 6 3/4s of August 15, 2026
with CF = 1.0798 and price of 127 3/32, we have
accrued interest of 1.65 on November 13 and 0.45 on
March 11. Coupon of 3.375 on February 15 is
reinvested at 1.0% for 24 days and grows to
3.375(1.01)24/365 = 3.38.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 11
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract (continued)
 Forward price is, therefore,
 (127.09375 + 1.65)(1.01)118/365 – 3.38 = 125.78
 Invoice price is
 116(1.0798) + 0.45 = 125.71.
 Thus, this bond would produce -0.07
(= 125.71 – 125.78). So the 6 3/4 bond is better than
the 6 1/4 bond.
 Table 10.4 shows these calculations for all deliverable
bonds. See Ctd8e.xls for these calculations.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 12
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Determining the Cheapest-to-Deliver Bond on the
Treasury Bond Futures Contract (continued)
 Why identifying the cheapest-to-deliver bond is
important:
 Identifying the true spot price
 Calculating the correct hedge ratio
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 13
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options
 The Wild Card Option
 Futures market closes at 3:00 pm while spot market
stays open until at least 5:00 pm.
 This allows the holder of a short futures contract
during the delivery month to potentially profit from
a decline in the price of a deliverable bond during
that two hour period in the expiration month.
 Illustration: f3 = futures price at 3:00 pm, B3 = spot
price at 3:00 pm. CF = conversion factor
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 14
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options (continued)
 The Wild Card Option (continued)
 Let the short own 1/CF bonds (CF must be > 1.0, so
coupon must be > 6 percent). This is less than one
bond per contract so additional bonds, called “the
tail,” will have to be purchased in order to make
delivery.
 At 5:00 pm, the spot price is B5. It is profitable to
purchase these bonds at 5:00 pm if B5 < f3(CF).
 This holds because the invoice price is locked in but
the spot price of the bonds can potentially fall.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 15
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options (continued)
 The Wild Card Option (continued)
 If the spot price does not fall sufficiently, then the
short simply waits until the next day. By the last
eligible delivery day, the short would have to make
delivery.
 This is a potentially valuable option granted by the
long to the short and its value would have to be
reflected in a lower futures price at 3:00 pm.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 16
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options (continued)
 The Quality Option
 Also called the switching option, it gives the short
the right to change deliverable bonds if another
becomes more attractive. This right also exists in
various other futures markets.
 Similar to this is the location option, which is the
right to choose from among several eligible delivery
locations. This can be valuable when the underlying
is a storable commodity.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 17
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options (continued)
 The End-of-the-Month Option
 The right to make delivery any of the business days
at the end of the month after the futures contract has
stopped trading, around the third week of the month.
 Similar to the wild card option because the invoice
price is locked in when the futures stops trading.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 18
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Delivery Options (continued)
 The Timing Option
 The right to deliver on any eligible day of the
delivery month.
 Delivery will be made early in the month if the bond
earns a coupon that is less than the cost of financing.
 Delivery will be made late in the month if the bond
earns a coupon that exceeds the cost of financing.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 19
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Implied Repo/Cost of Carry
 Buy spot T-bond, sell futures.
 This will produce a return (implied repo rate) of
 (CF)f 0 (T)  AI T  CI0,T 
r̂  

B0  AI 0


Chance/Brooks
(1/T)
1
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 20
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Implied Repo/Cost of Carry (continued)
 Example: On November 13, 2008, CTD bond on March
contract is 6 3/4s maturing on August 15, 2026. Spot
price is 127 3/32, accrued interest is 1.65, CF = 1.0798
and futures price is 116. From November 13 to March
11 is 118 days so T = 118/365 = 0.3233. There one
coupon payment made with a future value of 3.38
(3.375(1.01)(24/365). Accrued interest on March 11th,
• AIT = 0.45
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 21
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Implied Repo/Cost of Carry (continued)
 Implied repo rate is, therefore,
116(1.0798 )  0.45  3.38 
r̂  

127.09375  1.65


1/0.3233
 1  0.008
 If
the bond can be financed in the repo market for less
than this rate, then the arbitrage would be profitable.
Obviously that is not the case here.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 22
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Treasury Bond Spread/Implied Repo Rate
 Let time t be expiration of nearby futures and T be
expiration of deferred futures.
 Go long the nearby and short the deferred.
 When nearby expires, take delivery and hold until
expiration of deferred. This creates a forward
transaction beginning at t and ending at T
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 23
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Treasury Bond Spread/Implied Repo Rate (continued)
 Implied repo rate
 (CF(T))f 0 (T)  AIT  CI t,T 
r̂  

(CF(t))f
(t)

AI
0
t


1/(T t)
1
 Example:
On November 13, 2008 CTD was 6 3/4s
maturing in 2026. Examine the March-June spread.
March priced at f0(t) = 116 with CF(t) = 1.0798. June
priced at f0(T) = 115 with CF(T) = 1.0792.
AIt (March 13) = 0.45 and AIT (June 11) = 2.16. No
coupons in the interim so CIt,T = 0. From March 13 to
June 11 is 90 days.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 24
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Intermediate and Long-Term Interest Rate
Arbitrage (continued)

Treasury Bond Spread/Implied Repo Rate (continued)
 Implied repo rate
115(1.0792 )  2.16 
r̂  

116(1.0798
)

0.40


365/90
 1  0.0199
 Compare
to actual repo rate and note that this is a
forward rate.
 Note the turtle trade: Implied repo rate on T-bond
spread to Fed funds futures rate
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 25
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Stock Index Arbitrage

Stock Index Arbitrage
 Recall the stock index futures pricing model
f 0 (T)  S0 e (rc  c )T


Example: Let S&P 500 = 1305.00, risk-free rate is 5.2%, dividend
yield is 3% and time to expiration is 40 days so
T = 40/365 = 0.1096. Futures should be at
 1305e(0.052 - 0.03)(.1096) = 1308.15
Now let the actual futures price be 1309.66. This is too high so
sell the futures and buy the index. Hold until expiration. Sell the
stocks and buy back the futures.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 26
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Stock Index Arbitrage (continued)

Stock Index Arbitrage (continued)
 Now find the implied repo rate. Let f0(T) be the actual
futures price. Then
^
r
 In
ln(f 0 (T)/S 0 )
c
T
our example, this is
^
r
ln(1309.66 /1305)
 0.03  0.0625
0.1096
 So
if you could get financing at less than this rate, the
arbitrage would be worth doing.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 27
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Stock Index Arbitrage (continued)

Stock Index Arbitrage (continued)
 Some practical considerations
 buying and selling all stocks simultaneously
 buying fractional contracts
 transaction costs of about 0.005 % of spot value.
 Program trading.
 See Table 10.5 for stock index arbitrage example.
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 28
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Summary
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 29
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Appendix 10: Determining the CBOT
Treasury Bond Conversion Factor

Determine maturity in years (YRS), months (MOS) and
days as of first date of expiration month. Use first call date
if callable. Ignore days. Let c be coupon rate. Round
months down to 0, 3, 6, or 9. Call this MOS*.
 If MOS* = 0,
c 1  (1.03) 2*YRS 
 2*YRS
CF0  

(1.03)

2
.03

Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 30
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Appendix 10: Determining the CBOT
Treasury Bond Conversion Factor (continued)
 If
MOS* = 3,
CF3  (CF0  c/2)(1.03)0.5  c/4
 If
MOS* = 6,
c 1  (1.03)  ( 2*YRS1) 
 ( 2*YRS1)
CF6  

(1.03)

2
.03

 If
MOS* = 9,
CF9  (CF6  c/2)(1.03)0.5  c/4
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 31
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Appendix 10: Determining the CBOT
Treasury Bond Conversion Factor (continued)

Example: 5 1/4s of February 15, 2029 delivered on March
2006 contract. On March 1, 2006 remaining life is 22
years, 11 months, 14 days. YRS = 22, MOS = 11. Round
down so that MOS* = 9. Find CF6:
0.0525 1  (1.03)  ( 2*(22)1) 
 ( 2*(22)1)
CF6 

(1.03)
 0.908055


2 
0.03


Then CF9 is
CF9  (0.908055  0.0525/2)(1 .03) 0.5  0.0525/4  0.9075
Chance/Brooks
An Introduction to Derivatives and Risk Management, 8th ed.
Ch. 10: 32
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