Futures Pricing

1. Futures Pricing

2. 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.

3. 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

4. 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

5. 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%)

6. 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

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

8. 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

9. 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%

10. 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

11. 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…)

12. 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)

13. 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)

14. 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

15. 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

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

17. 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

18. 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.

19. Step 3: design the strategy

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

21. 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

22. 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

23. 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

24. 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