Chapter 10: Futures Arbitrage Strategies

Transcription

1 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 agreementis the sale of securitiestogether 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 1

2 Using Eurodollar futures with spot to earn an arbitrage profit. Sept day Eurodollar 8.25% 180-day Eurodollar 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) = f: f = 100 (8.63)(90/360) = (price/contract=978,425) r = (f/s) 1/T -1 = ( /95.625) 1/(90/365) 1 = % 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: 3 rd bond (6.875%, 08/15, 2025). On 11/13, f=116, delivery date =3/11, reinvestment rate =1% (Accrued Interest: AI t =1.6814) (AIT=0.4558) Invoice Price = f (CF) + AI T = 116 x ( ) + [(6.875/2) x (24/181)] = = 2

3 184 days 181 days 24 days 8/15 2/15 3/11 8/15 90 days 118 days AI= AIT = Forward Price (cost of cash bond) = (S + AI t ) (1+r) T FV (CI) AI t = [(6.875/2) x (90/184)] = FV(CI) = (6.875/2) (1.01) 24/365 = (S + AI t ) (1+r) T FV (CI) = ( ) (1.01) 118/ = Invoice Price forward price = = The cheapest-to-deliver bond is the 6.75%, 08/15/2026 bond. Using Excel to calculate the Cost of carry Intermediate cash 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. 3

4 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 (8 th - 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) + AI T Cash Bond Cost (forward price) = (S + AI t )(1+r) T FV(CI) No-Arbitrage Condition: f (CF) + AI T = (S + AI t ) (1+r) T - FV(CI) r 1 / T f ( C F ) + A I T + F V ( C I ) = 1 = S + A I t R Arbitrage is profitable if r > R 4

5 4. T-Bond Arbitrage Example On 11/13: Cash Bond S = , AI t = , FV(CI)= /11 T-bond futures f = 116, AI T = o.4558 CF = r r r 1 / T f ( C F ) + A I T + F V ( C I ) = 1 S + A I t / ( ) = = % 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 5

6 3. Example S = 1,305, d = 3%, R = 5.2% f = 1,316.30, T = (40/365) = r = [(ln(f) ln(s))/t] + d = [(ln(1,316.3) ln(1,305)) / ] = Implied repo > actual repo, arbitrage is profitable. Spot Futures /8 Borrow 5.2% Short 61 1,316.3 Long $20M of stocks 12/18 S T = Sell stocks: ( /1305)x 20M = 19,928, (20M) / ( x 250) = 61 (HR=1) Spot Futures Sell stocks: $19,928, Close out futures: Repay Loan: (20M) (1.052) (40/365) = ($20,111,428) Dividends: (20M) (0.03) (40/365) π = - (S T f) (61) π = -[( ) (250) (61)] π = 243,085 = $65, Gain/Loss (116,784.69) 243,085 Problems: 8, 9, 10 6

7 Date Cash Market Futures Market 09/16 Borrow R=8.25% Short Dec for 3 months Buy 6-month /16 Repaid 3-month loan & interest: (1.0825) (90/365) = month Eurodollar has 3- Deliver 3-month Euro into month remaining maturity, futures & receive Deliver into futures Arbitrage profit = f S(1+R) T = = = $3,295/contract 7

8 8.63% Loss = 16,750 8