1. November 23, 2010MATH 2510: Fin. Math. 2 1 Forward and futures contracts (Long position) Boths are agreements to buy an underlying asset at future (delivery) date for a specific price. Futures contracts have some added institutional features – most noticably daily settlement. Material: Hull Chapters 2,3 and 5. We’ll skip and jump. (Thus: no need to read Hull like CT1.)

2. November 23, 2010MATH 2510: Fin. Math. 2 2 Futures contracts Available on a wide range of underlyings Exchange traded Specifications need to be defined: – What can be delivered, – Where it can be delivered – When it can be delivered (often a period) Settled daily: On long position you receive the day-to-day change in the futures price.

3. November 23, 2010MATH 2510: Fin. Math. 2 3 Convergence of Futures to Spot If the futures price is above the spot price during the delivery period, arbitrage is possible – Short futures contract – Buy underlying asset – Make delivery And vice versa if the futures price is below the spot price.

4. November 23, 2010MATH 2510: Fin. Math. 2 4 Time (a)(b) Futures Price Futures Price Spot Price

5. November 23, 2010MATH 2510: Fin. Math. 2 5 Futures vs. forward prices If you think that the definition of the futures contract/price seems circular --- I’m inclined to agree. Let’s look a Appendix A in Hull’s chapter 5. This shows that if interest rates are constant, then futures and forward prices are equal.

6. November 23, 2010MATH 2510: Fin. Math. 2 6 If interest rates are stochastic, then forward and futures prices are not (necessarily) equal. If the price of the underlying is positively correlated w/ interest rates, then we receive futures payments in high interest rate scenarios -> futures above forward. But differences are typically small.

7. November 23, 2010MATH 2510: Fin. Math. 2 7 Margins A margin is cash or marketable securities deposited by an investor with his or her broker; serves as collateral. The balance in the margin account is adjusted to reflect daily settlement. Margins minimize the possibility of a loss through a default on a contract.

8. November 23, 2010MATH 2510: Fin. Math. 2 8 Collateral It is becoming increasingly common (post Subprime Crisis/Credit Crunch/Financial Meltdown in particular) for contracts to be collateralized/have margins. They are then similar to futures contracts in that they are settled regularly (e.g. every day or every week)

9. November 23, 2010MATH 2510: Fin. Math. 2 9 Delivery Closing out a futures position involves entering into an offsetting trade; most contracts are closed out before maturity. Though sometimes spectacularly not: Business Snapshot 2.1: Live cattle Amajaro summer 2010: Cocoa

10. November 23, 2010MATH 2510: Fin. Math. 2 10 When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices and Eurodollars) are settled in cash.

11. November 23, 2010MATH 2510: Fin. Math. 2 11 Accounting and Taxation This is really tricky stuff. Ideally hedging profits (losses) should be recognized at the same time as the losses (profits) on the item being hedged Ideally profits and losses from speculation should be recognized on a mark-to-market basis Roughly speaking, this is what the accounting and tax treatment of futures in the U.S.and many other countries attempts to achieve.

12. November 23, 2010MATH 2510: Fin. Math. 2 12 Long & Short Futures Hedges A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price A short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

13. November 23, 2010MATH 2510: Fin. Math. 2 13 Arguments in Favor of Hedging Companies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables.

14. November 23, 2010MATH 2510: Fin. Math. 2 14 Arguments Against Hedging Shareholders are usually well diversified and can make their own hedging decisions It may increase risk to hedge when competitors do not Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

15. November 23, 2010MATH 2510: Fin. Math. 2 15 Basis Risk Basis is the difference between spot and futures prices. Basis risk arises because of the uncertainty about the basis when the hedge is closed out. (Say you can’t match w/ exact delivery date and/or underlying asset for futures.)

16. November 23, 2010MATH 2510: Fin. Math. 2 16 Long Hedge Suppose that F 1 : Initial Futures Price F 2 : Final Futures Price S 2 : Final Asset Price You hedge the future purchase of an asset by entering into a long futures contract Cost of Asset= S 2 – ( F 2 – F 1 ) = F 1 + Basis

17. November 23, 2010MATH 2510: Fin. Math. 2 17 Choice of Contract Choose a delivery month that is as close as possible to, but later than, the end of the life of the hedge When there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging.

18. November 23, 2010MATH 2510: Fin. Math. 2 18 Optimal Hedge Ratio Proportion of the exposure that should optimally be hedged is where  S is the standard deviation of  S, the change in the spot price during the hedging period,  F is the standard deviation of  F, the change in the futures price during the hedging period  is the coefficient of correlation between  S and  F.