Futures Pricing

Basis and Spreads Basis= Spot price – futures price Basis should converge to zero—i.e spot price is the same as the futures price at expiration. If not arbitrage opportunities appear.

Implied Repo rate The repo rate is the best estimate of financing costs, it is equal to the implicit interest rate embedded in a repurchase agreement—a repurchase agreement is a contract in which a trader sells a security today with the commitment to repurchase it at a later date. If Then

Futures-Spot Arbitrage Decision: If the implied repo > actual repo ratesell futures and buy the spot=Cash and carry arbitrage If the implied repo < actual repo rate buy futures and sell the spot = reverse Cash and carry arbitrage

Example Spot price for silver is 4.65 per troy ounce 1 year futures is 5.2. The repo rate is 8 % (you can borrow at 8%) Trade Today 1-year Borrow So +4.65 -4.65*1.08=-5.02 Short futures 5.2-St Buy silver -4.65 St Total 0.18

Example 2 Spot price for silver is 4.65 per troy ounce 1 year futures is 4.8. The repo rate is 8 % (you can borrow at 8%) Build the right arbitrage strategy

Futures – Futures Arbitrage Relationship between 2 futures contracts with different maturities. Implied repo rate

Arbitrage rule If the implied repo rate > actual repo rate  F(0, t + dt) is overpriced. Then, Sell F(0, t + dt) ; buy F(0, t); borrow F(t, t) at repo rate and buy S(t,t) with proceed = this a forward cash and carry arbitrage If the implied repo rate < actual repo rate  F(0, t + dt) is underpriced. Then, buy F(0, t + dt); sell F(0, t); invest F(t, t) at repo rate and sell short S(t,t) = this a reverse forward cash and carry arbitrage

Example 1 year futures price of silver is 5.2 1year repo rate is 8% and the two year repo rate is 9% Trade today 1year 2 year Buy the 1y Futures S1-5.2 sell the 2y Futures 5.85-S2 Borrow in 1 year at forward rate 5.20 -5.2 x 1.1= -5.72 Buy silver in 1 year -S1 S2 Total 0.13

Example 1 year futures price of silver is 5.2 1year repo rate is 8% and the two year repo rate is 9% Use the right arbitrage strategy

Imperfect markets and arbitrage Transaction costs exist: let’s call them T (it is a rate). Different rates of borrowing and lending: Lets call them R(B) and R(L). Although let’s switch to discrete time value of money (it is easier to understand…)

Cash and Carry Arbitrage with T Theoretically (R per period), F(0,t) = S(0,t) x (1+R) In CCA, you sell F, buy S Transaction costs exist (T) and increase the cost of spot purchase. Thus, F(0,t) < S(0,t) x (1+R) x (1+T) RCCA, you buy F and sell S (proceed from short sale reduced by T) Thus, F(0,t) > S(0,t) x (1+R) x (1-T) In sum RCCA CCA S(0,t) x (1+R) x (1-T)< F(0,t) < S(0,t) x (1+R) x (1+T)

Cash and Carry Arbitrage with T and Different Borrowing and investing rates Same idea as before CCA borrow at RB to buy S RCCAinvest at RL the proceed from the short-sale of S Short sell restriction “you can only reinvest f% of proceed from short sell of S) RCCA CCA S(0,t) x (1+ f x RL) x (1-T)< F(0,t) < S(0,t) x (1+RB) x (1+T)

Example Silver spot is 4.65 1-year futures is 5.2 RL is 7.9% RB is 8.1% T is 1% F is 80% (short sellers have only access to 80% of proceed from short sales) Show that an arbitrage opportunity exists 4.89<F(0,t)<5.08

Arbitrage With T-Bill Futures If an arbitrageur can discover a disparity between the implied financing rate and the available repo rate, there is an opportunity for riskless profit If the implied financing rate is greater than the borrowing rate, then he/she could borrow, buy T-bills, and sell futures If the implied financing rate is lower than the borrowing rate, he/she could borrow, buy T-bills, and buy futures

Example We are on Aug. 1st ; you observe the following spot and futures interest rates. Propose an arbitrage strategy… Effective Yield Sept contract (Sep 21=52 days) 10% 52 days Tbill 3% 142 days Tbill 8%

Step 1 Prices FV=PV x (1+R)^t or PV=FV/(1+R)^t Your futures contract has 90 days to maturity from the date of the futures contract maturity. PV=100/(1+10%)^(90/360)=97.6454 52-days T-Bill is PV=100/(1+3%)^(52/360)=99.5739 142-days T-Bill is PV=100/(1+8%)^(142/360)=97.0099

Step 2Look for arbitrage Opportunity The 142-day Tbill will be a 90-day Tbill in 52 days! F=S x (1+R)^t R=repo rate=(F/S)^(360/52)-1 R=(97.6454/ 97.0099)^(360/52)-1=4.624% This is greater that the actual 3% rate on a 52-days Tbill! F is overpriced, then use a cash and carry arbitrage, where the futures is sold and the spot is purchased.

Step 3: design the strategy Today aug 1st Sep. 21st Borrow So 97.0099 -97.0099 x 1.03^52/360 =-97.425 Buy 142 Tbill -97.0099 To be delivered or St Sell futures 97.6454-St Total =.2204

Spreading With Interest Rate Futures TED spread The NOB spread Other spreads with financial futures

TED spread Involves the T-bill futures contract and the eurodollar futures contract Used by traders who are anticipating changes in relative riskiness of eurodollar deposits The TED spread is the difference between the price of the U.S. T-bill futures contract and the eurodollar futures contract, where both futures contracts have the same delivery month If you think the spread will widen, buy the spread

The NOB Spread The NOB spread is “notes over bonds” Traders who use NOB spreads are speculating on shifts in the yield curve If you feel the gap between long-term rates and short-term rates is going to narrow, you could buy T-note futures contracts and sell T-bond futures

LED Spread LED spread is the LIBOR-eurodollar spread LIBOR is the London Inter-Bank Offered Rate Traders adopt this strategy because of a belief about a change in the slope of the yield curve or because of apparent arbitrage in the forward rates associated with the implied yields

MOB Spread The MOB spread is “municipals over bonds” It is a play on the taxable bond market (Treasury bonds) versus the tax-exempt bond market (municipal bonds) Trader buys the futures contract that is expected to outperform the other and sells the weaker contract