Introduction to Financial Derivatives

Transcription

1 Introduction to Financial Derivatives Forward and Futures Contracts Week of September 17, 2012 Assignment For week of Sept 17 (This Week) Read: Hull Chapter 3 (Hedging with Futures) Problems (Due September 24) Chapter 3: 4, 7, 10, 17, 18, 20, 22; 26 Chapter 3 (7e): 4, 7, 10, 17, 18, 20, 22; 26 Problems (Due Today, September 17) Chapter 1: 17, 18, 22, 23; 34, 35 Chapter 1 (7e): 17, 18, 22, 23; 30, 31 Chapter 2: 15,16, 21, 22; 30 Chapter 2 (7e): 15, 16, 21, 22; Assignment Where we are For Week of Sept 24 th (Next Week) Read: Hull Chapter 4 (Interest Rates) Problems (Due September 24) Chapter 3: 4, 7, 10, 17, 18, 20, 22; 29 Chapter 3 (7e): 4, 7, 10, 17, 18, 20, 22; 26 Problems (Due October 1 st ) Chapter 4: 5, 8, 9, 11, 12, 14, 16, 22; 32 Chapter 4 (7e): 5, 8, 9, 11, 12, 14, 16, 22; 27 Exams Final Exam: Dec 20th; 9:00am Noon (Gilman 132) Last week Introduced some general ideas & concepts in Options, Forwards, & Futures (Chapter 1, OFOD) Looked at some basic ideas/mechanics of Futures Markets (Chapter 2, OFOD) This week Hedging Approaches/Techniques using Futures The basis and basis risk (Chapter 3, OFOD) Next week: Interest Rates and the Present Value of Future Cash (Chapter 4, OFOD) Midterm: October 17 th (Tentative)

2 Margins Example: Futures Trade (page 27-28) A margin is cash or marketable securities deposited by an investor with the broker Initial Margin Maintenance Margin 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 A Possible Outcome Table 2.1, Page 28 Other Key Points About Futures They are settled daily Closing out a futures position involves entering into an offsetting trade Most contracts are closed out before maturity

3 Collateralization in OTC Markets It is becoming increasingly common for contracts to be collateralized in OTC markets They are then similar to futures contracts in that they are settled regularly (e.g. every day or every week) Another Detail for Cash and Carry Arbitrage Contract price changes with longer term Higher or Lower To this point we have neglected storage cost Lets re-visit no-arbitrage equation F(t0,T) - S(t0) x [(1+r ) T ] = Storage (T) Storage costs ignored in earlier gold example No storage costs for FX Convenience Yield Oil: An Arbitrage Opportunity? Suppose that: - The spot price of oil is US$95 - The quoted 1-year futures price of oil is US$125 - The 1-year US$ interest rate is 5% per annum - The storage costs of oil are 2% per annum Is there an arbitrage opportunity? 2. Oil: Another Arbitrage Opportunity? Suppose that: - The spot price of oil is US$95 - The quoted 1-year futures price of oil is US$80 - The 1-year US$ interest rate is 5% per annum - The storage costs of oil are 2% per annum Is there an arbitrage opportunity?

4 Futures Prices for Gold on Jan 8, 2007: Prices Increase with Maturity Futures Prices for Orange Juice on Jan 8, 2007: Prices Decrease with Maturity Futures Price ($ per oz) Contract Maturity Month 600 Jan-07 Apr-07 Jul-07 Oct-07 Jan-08 Futures Price (cents per lb) Contract Maturity Month 170 Jan-07 Mar-07 May-07 Jul-07 Sep-07 Nov Delivery If a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract. 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 2.15 Some Terminology Open interest: the total number of contracts outstanding equal to number of long positions or number of short positions Settlement price: the price just before the final bell each day used for the daily settlement process Volume of trading: the number of contracts traded in 1 day

5 Convergence of Futures to Spot Questions Spot Price Futures Price Spot Price Futures Price When a new trade is completed what are the possible effects on the open interest? Can the volume of trading in a day be greater than the open interest? Time Time (a) (b) Regulation of Futures Regulation is designed to protect the public interest CFTC the Feds Regulators try to prevent questionable trading practices by either individuals on the floor of the exchange or outside groups NFA the industry Accounting & Tax Ideally hedging profits (losses) should be recognized at the same time as the losses (profits) on the item being hedged Then there is the exchange (LTCM) 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

6 Foreign Exchange Quotes Futures exchange rates are quoted as the number of USD per unit of the foreign currency Forward exchange rates are quoted in the same way as spot exchange rates. This means that GBP, EUR, AUD, and NZD are quoted as USD per unit of foreign currency. Other currencies (e.g., CAD and JPY) are quoted as units of the foreign currency per USD. Long & Short 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 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 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

7 Convergence of Futures to Spot (Hedge initiated at time t 1 and closed out at time t 2 ) Basis Risk Futures Price Spot Price t 1 t 2 Time Basis is the difference between spot & futures Basis risk arises because of the uncertainty about the basis when the hedge is closed out Basis risk is also about the reality that the asset to be hedged may not be the same as that underlying the futures Short Hedge Basis Risk Long Hedge Basis Risk Suppose that F 1 : Initial Futures Price F 2 : Final Futures Price S 1 : Initial Asset Price S 2 : Final Asset Price You hedge the future sale of an asset (one in which you now might be long) by entering into a short futures contract P/L = (S 1 F 1 ) + (S 2 F 2 ) = (S 2 S 1 ) (F 2 F 1 ) = [S 2 (F 2 F 1 )] S 1 = [F 1 + (S 2 F 2 )] S 1 Price t 2 (w/hedging) in selling long asset = S 2 +(F 1 F 2 )= F 1 + Basis (at 2) 2.27 Suppose that F 1 : Initial Futures Price F 2 : Final Futures Price S 1 : Initial Asset Price S 2 : Final Asset Price You hedge the future purchase of an asset (one you now might be short) by entering into a long futures contract P/L = (S 1 F 1 ) (S 2 F 2 ) = (S 1 S 2 ) (F 1 F 2 ) = S 1 [S 2 (F 2 F 1 )] = S 1 [F 1 + (S 2 F 2 )] Price t 2 (w/hedging) to buy-in short asset = S 2 (F 2 F 1 )= F 1 + Basis (at 2)

8 Choice of Contract to Minimize Basis Risk 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. Optimal Hedge Ratio Hedging using Futures Proportion of the exposure in a cash position that should optimally be hedged is S where F S is the standard deviation of S, the change in the spot price (for the cash instrument) 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 Optimal Hedge Ratio Minimum Variance Result The number of futures, NF, in ratio h (the hedge ratio, or NF = h x NA), required to hedge NA units of an asset between times t 1 and t 2 follows from P/L of the hedged position over that time: P/L = (NAS2 NFF2) (NAS1 NFF1) (long asset, hedge w/short futures) = NA ΔS NF ΔF = NA x (ΔS h ΔF) ( NF = h x NA ) where S 1, F 1 and S 2, F 2 ( ΔS & ΔF ) are the spot price of the asset, S, and the futures, F, at t 1 and t 2, respectively (or their respective changes, Δ) The minimum variance hedge ratio can be found by minimizing the variance of the P/L (w/r to the hedge ratio) 2.31 Optimal Hedge Ratio Minimum Variance Result Minimizing the variance of the P/L P/L = NA x (ΔS h ΔF) Is equivalent to min [Var(ΔS h ΔF)] where the variance of this linear combination of F & S, ν = Var [ΔS h ΔF] = S2 + h 2 F2 2 h S F We minimize ν when h is such that dν/dh = 0 if d 2 ν/dh 2 > 0 or 2 h F2 2 S F = 0 which implies S h F

9 Optimal Hedge Ratio Regressing change in asset price to change in futures Optimal Hedge Ratio minimum variance hedge 2.33 Optimal Hedge Ratio Minimum Variance Result Example 3.3 (Pg 59) Minimum Variance Hedge on Jet Fuel using Heating Oil Futures HR =.78 (see below right) Heating oil contract = 42,000 gal Jet Fuel to be Hedged = 2 mm gal Number of Contracts (#Contr) S.0263 h F ,000,000 #Contr ,000 Month (=i) Change in Futures price per gallon (=xi) Change in Fuel price per gallon (=yi) stddev correl HR Tailing the Hedge Tailing the Hedge Two ways to determine the number of contracts to use for hedging are Compare the exposure to be hedged with the exposure of the assets underlying one futures contract (previous slide) Compare the value to be hedged with the value of one futures contract ( = futures price times the size of one futures contract ) (next slide) The second approach incorporates an adjustment for the daily settlement of futures For the Jet Fuel Example the tail suggests N h V * * A VF Where we need to know the fuel (A) price ( = $1.94) and the futures (F) price ( = $1.99) Now we have V 2,000, ,880,000 So the optimal number of contracts is A F V 42, ,580 h V N * * A 0.783,880,000 VF 83,

10 Hedging Using Index Futures To hedge the risk in a portfolio the number of contracts that should be shorted is P A where P is the value of the portfolio, is its beta, and A is the value of the assets underlying one futures contract 2.37 Example Value of S&P 500 Index is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio? (1 S&P futures contract has value $250 x index) 2.38 Example Changing Beta Value of S&P 500 futures is 1,000 x 250 dollars Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio? P 5,000,000 N* A Short 30 contracts What position is necessary to reduce the beta of the portfolio to 0.75? (15) What position is necessary to increase the beta of the portfolio to 2.0? (long 10)

11 Changing Beta What position is necessary to reduce the beta of the portfolio to 0.75? What position is necessary to increase the beta of the portfolio to 2.0? * If we change the beta of the PF from P *; then short( *) A Hedging Price of an Individual Stock Similar to hedging a portfolio Does not work as well because only the systematic risk is hedged The un-systematic (idiosyncratic) risk that is unique to the stock is not hedged P *; then long( * ) A With 1.5 its short 15 or long 10, respectively Why Hedge Equity Returns May want to be out of the market for a while. Hedging avoids the costs of selling and repurchasing the portfolio but not for free Suppose stocks in your portfolio have an average beta of 1.0, but you feel they have been chosen well and will outperform the market in both good and bad times. Hedging ensures that the return you earn is the risk-free return plus the excess return of your portfolio over the market. Rolling The Hedge Forward We can use a series of futures contracts to increase the life of a hedge Each time we switch from 1 futures contract to another we incur a type of basis risk

12 Hedging with Futures Hedging with Futures In October 2010 a company anticipates it will buy 1mm pounds of copper in each of the following months: Feb 2011, Aug 2011, Feb 2012, & Aug 2012 Assume mkt. price in cents per pound today and at futures dates are as follows: CME contract is for 25,000 pounds Initial margin $2,000 per and maintenance margin is $1,500 per contract Liquidity demands contracts of 13 months or shorter Hedge to 80% of exposure What is a possible hedging strategy? 2.45 What is the impact of the hedging strategy on the price the company pays for copper? What is the initial margin requirements? Is the company subject to any margin calls? 2.46 The End for Today Questions? Schedule Lecture Encounters Monday & Wednesday, 3-4:15pm, Gilman 132 Section: Friday 3-3:50pm, Hodson 213 Section: Thursday 3-3:50, Gilman 77 Final Exam Thursday, December 20 th ; 9:00am Noon Gilman

13 Principals David R Audley, Ph.D.; Sr. Lecturer in AMS david.audley@jhu.edu Office: WH 212A; Office Hours: 4:30 5:30 Monday Teaching Assistant(s) Yu Du, (ydu10@jhu.edu) Tao Wang (twang55@jhu.edu) Tian Xia (summerzju@gmail.com) Office Hours: TBA 1.49 Resources Textbook John C Hull: Options, Futures, and Other Derivatives, Prentice-Hall 2012 (8e) Recommended: Student Solutions Manual On Reserve in Library Text Resources Resources Supplemental Material As directed AMS Website Additional Subject Material Class Resources & Lecture Slides Industry & Street Research (Optional) Consult at your leisure/risk Interest can generate Special Topics sessions Measures of Performance Mid Term Exam (~1/3 of grade) Final Exam (~1/3 of grade) Home work as assigned and designated and Quizzes (~1/3 of grade) Blackboard

14 Protocol Attendance Lecture Mandatory (default) for MSE Fin Math majors Quizzes Section Strongly Advised/Recommended Assignments Due as Scheduled (for full credit) Must be handed in to avoid incomplete Exceptions must be requested in advance