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

tRt,0t,0 eSF

t

1)

S

Fln( Rate Repo pliedIm

t0,

t0,

Futures-Spot Arbitrage

• Decision:

• If the implied repo > actual repo ratesell 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 0 5.2-St

Buy silver -4.65 St

Total 0 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

tRt,0tt,0 eFF

t

1)

F

Fln(R

t,0

tt,0

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• 2 year futures price of silver is 5.85• 1year repo rate is 8% and the two year repo rate is 9%

Trade today 1year 2 year

Buy the 1y Futures

0 S1-5.2

sell the 2y Futures

0 0 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 0 0.13

Example

• 1 year futures price of silver is 5.2

• 2 year futures price of silver is 5.55

• 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• RCCAinvest 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

Effective Yield

Sept contract (Sep 21=52 days) 10%

52 days Tbill 3%

142 days Tbill 8%

We are on Aug. 1st ; you observe the following spot and futures interest rates. Propose an arbitrage strategy…

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 2Look 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)-1R=(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 strategyToday 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 0 97.6454-St

Total 0 =.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