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Chapter 10: Futures Arbitrage Strategies
� I. Short-Term Interest Rate Arbitrage
� 1. Cash and Carry/Implied Repo
� Cash and carry transaction means to buy asset and sell futures
� Use repurchase agreement/repo to obtain funding
� A repurchase agreement is the sale of securities together with an agreement for the seller to buy back the securities at a later date.
� Repo Rate: financing rate (overnight vs. term repo)
� The repurchase price should be greater than the original sale price, the difference effectively representing interest, is called the repo rate.
� Implied Repo Rate
� The financing rate that produces no arbitrage profit
� Cost of carry pricing model: f= S + θ� π = f – S - θ = 0
= f – S(1+r)T = 0� r = (f / S)1/T– 1 � r is the equilibrium rate� Arbitrage will be profitable if
implied repo rate (r) > actual repo rate (R)That is, f is over-priced
2. Eurodollar Arbitrage� Eurodollar Futures as a synthetic loan: Buyer of Eurodollar futures agrees to “lend” (e.g., buy $1,000,000 Eurodollar TD), while seller agrees to “borrow”
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� Using Eurodollar futures with spot to earn an arbitrage profit.
� Sept. 16� 90-day Eurodollar discount@ 8.25% � 180-day Eurodollar discount@ 8.75% � December Eurodollar Futures: IMM=91.37 � yield 8.63%� Repo rate (R) = 8.25%
� Is Arbitrage profitable?� S: 180-day Eurodollar price S=100-(8.75)(180/360) = 95.625� f: f = 100 – (8.63)(90/360) = 97.8425 (price/contract=978,425)� r = (f/S)1/T -1 = (97.8425/95.625)1/(90/365) – 1 = 9.7445%� Since r > R, arbitrage is profitable
� Arbitrage examples� Long Eurodollar, short Eurodollar futures� Table 10.2, p. 332
II. T-Bond Arbitrage – Short futures, long bond
1. Determine Cheapest-to-Deliver (CTD) Bond
� The bond that maximizes (invoice price – forward price )
� shows these calculations for all deliverable bonds. Example: 3rd bond (6.875%, 08/15, 2025). On 11/13, f=116, delivery date =3/11, reinvestment rate =1%
(Accrued Interest: AIt=1.6814) (AIT=0.4558)
� Invoice Price = f • (CF) + AIT
= 116 x ( ) + [(6.875/2) x (24/181)] = 126.4284+0.4558 =
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� Forward Price (cost of cash bond) = (S + AIt) (1+r)T – FV (CI)
� AI t = [(6.875/2) x (90/184)] = 1.6814
� FV(CI) = (6.875/2) (1.01) 24/365= 3.4397
� (S + AIt) (1+r)T – FV (CI)
= (128.469 + 1.6814) (1.01)118/365– 3.4397 = 127.1297
→ Invoice Price – forward price = 126.884 – 127.13 =
�The cheapest-to-deliver bond is the 6.75%, 08/15/2026 bond.
� Using Excel to calculate the
Cost of carryIntermediate cash
184 days
181 days
8/15 2/15 3/11 8/15
90 days 118 days
24 days
AI=1.6814
AIT = 0.4558
� 2. Delivery Options
� A. The Quality Option – The short has the right to deliver any of a number of acceptable bonds.
� The quality option makes it difficult to undertake “long arbitrage”, because the arbitrageur does not know which cash bond will be delivered.
� B. The Timing (Accrued Interest) Option - The short has the option of choosing when to deliver the cash bond during the futures expiration month.
� If R (borrowing repo rate) > coupon interest, the short has incentive to deliver early.
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� C. The Wild Card Option – The short can announce delivery until 9:00 p.m. (EST).If delivery is undertaken, then the 3:00 p.m. futures closing
price is employed to calculate the invoice price, while spot market operates until 5:00 p.m. If cash bond price drops, the short can buy bond at lower price to deliver.
� D. The End-of-the-Month Option – The invoice price can be set based on the settlement price on the futures final trading day (8th-to-last business day of the delivery month). The short can wait for the spot price to fall and deliver during the remaining business days (the remaining 7 days).
� 3. T-Bond Implied Repo Rate
� Invoice Price = f • (CF) + AIT� Cash Bond Cost (forward price) = (S + AIt)(1+r)T – FV(CI)� No-Arbitrage Condition:
f • (CF) + AIT = (S + AIt) (1+r)T - FV(CI)
• Arbitrage is profitable if r > R
1 /( ) ( )
1T
T
t
f C F A I F V C Ir R
S A I
• + += − = +
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� 4. T-Bond Arbitrage Example_______________________________________________________
On 11/13:
Cash Bond S = 128.469, AIt = 1.6814, FV(CI)=3.4397
03/11 T-bond futures f = 116, AIT = o.4558
CF = 1.0899
-------------------------------------------------------------------------------------
1 /
3 6 5 /1 1 8
( ) ( )1
1 1 6 (1 . 0 8 9 9 ) 0 . 4 5 5 8 3 . 4 3 9 71
1 2 8 . 4 6 9 1 . 6 8 1 4
0 . 4 1 %
T
T
t
f C F A I F V C Ir
S A I
r
r
• + += − +
• + + = − +
=
Arbitrage will be profitable if the cost of financing is < 0.41%. The repo rate in this example is 1%, hence no arbitrage.
III. Stock Index Futures Arbitrage� 1. Compare cost-of-carry model price with market price
� 2. Calculate implied repo rate
� Since f = Se (r-d)T,
� Implied repo rate = r = [(ln(f) – ln(S))/T] + d
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� 3. Example___________________________________________________________
S = 1,305, d = 3%, R = 5.2%
f = 1,316.30, T = (40/365) = 0.1096
-----------------------------------------------------------------------------------------
� r = [(ln(f) – ln(S))/T] + d
= [(ln(1,316.3) – ln(1,305)) / 0.1096 ] + 0.03
= 0.1087
� Implied repo > actual repo, arbitrage is profitable.___________________________________________________________
Spot Futures-----------------------------------------------------------------------------------------11/8 Borrow $20M @ 5.2%
Long $20M of stocks
Short 61 futures @ 1,316.3
(20M) / (1316.3 x 250) = 61 (HR=1)
12/18 ST = 1300.36
Sell stocks: (1300.36/1305)x 20M
= 19,928,889.89
___________________________________________________________Spot Futures
-----------------------------------------------------------------------------------------
Repay Loan: (20M) (1.052)(40/365)
= ($20,111,428)
Dividends: (20M) (0.03) (40/365)
= $65,753.42
Sell stocks: $19,928,889.89 Close out futures:
π = - (ST – f) (61)
π = -[(1300.36 – 1316.3) (250) (61)]
π = 243,085
___________________________________________________________
Gain/Loss (116,784.69) 243,085
___________________________________________________________
Problems: 8, 9, 10
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Date Cash Market Futures Market
Borrow 95.625 @ R=8.25% for 3 months
Short Dec. futures @ 97.8425 09/16
Buy 6-month Eurodollar @ 95.625
Repaid 3-month loan & interest: 95.625 (1.0825)(90/365) = 97.513
12/16
6-month Eurodollar has 3-month remaining maturity,
Deliver into futures →
Deliver 3-month Euro into futures & receive 97.8425
Arbitrage profit = f – S(1+R)T = 97.8425 – 97.513 = 0.3295 = $3,295/contract
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Loss = 16,750
8.63%
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