Derivatives and Investment Management The Use of Futures and Options

History of the Chicago Mercantile Exchange The CME started almost a 100 years ago by listing two commodity contracts: on butter and eggs. Consequently, it was known as the Chicago Butter and Egg Board. Now the CME lists about 50 contracts, including futures on equity indexes, interest rates (especially the Eurodollar future) currencies, and, of course, commodities.

Pricing of Futures/Option Price of forward/future = (Price of underlying asset) - (present value of dividend paid over life of forward) + (interest accrued over maturity). If the dividend yield is less than the interest rate, then the future price will be greater than the spot. Price of Option: Black-Scholes formula.

Examples of Futures Contracts S&P 500: Chicago Mercantile Exchange 1. Value of contract: $250 times the index level. Thus, if the index is at 1000, a contract is equivalent to a long position of $250,000 in the index. Example: suppose you want to invest $5,000,000 in the index at an index level of Because 5,000,000/(250x1075)=18.6. You will buy 19 contracts. If the index increases from 1075 to 1085 the next day, you will have made 250x10=$2500 per contract.

Example of Futures Contract 2. Margin requirement: 10% or less. Thus, for a contract worth $250,000 (if the futures is quoted at 1000), you are likely to require less than $25,000 as margin. Many brokers will charge you only about 5%. 3. Marking to Market: The future is marked to market every day, so that your gains/losses are booked daily. 4. Contracts available: March, June, Sep, Dec 5. Minimum tick size: 0.1 index points, or $250x0.1=$25

Other Examples The Mini S&P contract: 50 times the index with minimum tick size of 0.25 (or $12.50). Russell: $500 times the index with minimum tick size of 0.05 or $25. Nasdaq 100: $100 times the index, with a minimum tick size of 0.05 or $5. DJIA is traded on the Chicago Board of Trade (CBOT) and has a multiplier of 10.

Futures on Growth and Value Indices The S&P 500/BARRA Growth and Value Indices: Each S&P 500 Index company is assigned either to the Growth or Value Index so that the two indices "add up" to the S&P 500. The indices are rebalanced twice a year (January 1 and July 1) based on a variety of factors. They're designed with about 50% of the S&P 500 capitalization in the Value Index and 50% in the Growth Index. Companies in the Growth Index have higher market capitalizations on average than those in the Value Index, so there are many more companies in the Value Index.

Why use the Future? The futures market is important because it allows you to trade quickly and easily, with low transaction costs. What can the futures be used for? It can be used to create a position in the index, without actually having to buy the underlying stocks that make up the composition of the index. It can also be used to change the “beta” or the market exposure of the fund.

Futures and Portfolio Management 1. Can be used to quickly change the beta of the fund. Example: you are currently invested in high beta stocks. Given the current market turmoil, you want to decrease the beta of your fund. But you don’t want to incur costs of trading your stock position. Solution: you sell futures. Suppose your current beta is 1.4, and your total stock position is $100,000,000. A 1% change from the current S&P level is likely to affect your portfolio by (1)(1.4)(100)=$1.4 million. You want to decrease your beta to 1.2.

Example: (continued) You could decrease your beta to 1.2 by selling part of your portfolio and moving to cash. The amount you want to move to cash is (1-1.2/1.4= ) 14.29% of the $100 million. Instead of selling $14.29 million of stock, you can go short an equivalent amount of the S&P 500 futures. Thus, at an index level of 1075, you will short about 14,290,000/(250x1075)=53 contracts.

Futures and Portfolio Management 2. The futures contract can be used to quickly invest the cash that flows in every day. Example: you are told at 3:58 pm that $10 million of new cash has just come into the fund. This has to be invested into the fund at the closing price of the fund at the end of the day. You do not have time to increase the holding of the individual stocks you own in the fund; instead, you invest it temporarily (for a day) into the S&P 500. At an index level of 1075 you will buy 10,000,000/(250x1075)=37 contracts.

What is an option? Call: A call gives you the right to buy the underlying asset at a fixed price within a certain period of time. Put: A put gives you the right to sell the underlying asset at a fixed price within a certain period of time. (Revise the payoff diagram: what is the payoff of a position of long 1 share of stock, and long 1 put?)

Using Puts to Hedge your Portfolio You can use a put to hedge your portfolio. To determine your hedging strategy, you have to ask the following questions: 1. At what price level do you want to protect your portfolio value? (Example: if you want to prevent your portfolio from falling more than x% below its current level, what is x?). 2. What percentage of your assets are you willing to spend to protect yourself?

Costs of Hedging (1) Buying puts can be quite expensive. The cost of the put depends on its strike, the volatility, and the maturity of the put. You can use the Black-Scholes formula to figure out the cost. Here’s a simple way of quickly estimating the cost of an at-the-money put (where the strike of the put is equal to the current index level) for a given volatility (sigma) and maturity (T): Put price = 0.4 (Stock Price) (volatility) sqrt(T). Currently, options are traded at 21% vol. Thus, a 3-month at-the-money put at an index level of 911 (as on 4/22) would cost about (0.4)(911)(0.21)sqrt(0.25) = $38 (or 4.2% of the current index level of 911).

Costs of Hedging (2) Thus, it would cost about 4% of your portfolio every quarter to hedge yourself completely from any loss. –This may be worth doing if you have done extremely well, so that even after paying 4% you have still beaten your benchmark for the quarter. If you want to hedge for the year, its cheaper to buy a 1-year option than four 3-month options. A 1-year at-the-money put will cost you twice as much as the 3-month put. At a volatility of 21%, a 1-year put will cost 8.4% of the asset value. For all practical purposes, hedging at the current index level is expensive, and it is unlikely that funds would do this except under unique conditions.

Costs of Hedging (3) Because of the high cost of buying an at-the- money put, most portfolio managers will typically buy out-of-the-money puts, with strikes lower than the current index level. The lower strike also implies that the portfolio manager will have to be willing to lose a certain portion of his portfolio. Because of the large hedging demand for out-of- the-money puts, we typically find that these puts are priced at a higher volatility than at-the-money puts.

Costs of Hedging (4) Here are some examples of current option prices (as of close on 4/22/2003): The S&P 500 index closed at The July option expires on 19 July (expires on the third Friday of the month). The 900 put traded at bid=$30.00, ask=$ The ask price corresponds to a Black-Scholes implied volatility of about 21.7%. The open interest is 833 contracts. The number of trades were 116 contracts. The 800 put was traded at bid=6.70 ask=7.70. The number of contracts traded were 427 with open interest of 504.The ask price corresponds to an implied volatility of about 27.50%. –At an implied volatility of 21.7%, the put would have been priced at $4.53. In other words, the put is much more expensive than the at-the-money option – in fact, it is almost 70% more expensive than what it would be worth if priced at the implied vol of 21.70%. –If you buy the 800 put, you can never lose more than ( )/911=12%. But to buy this protection, it will cost you 7.70/911=0.8% of your portfolio value.

Portfolio Insurance

Portfolio Insurance (1) Recall that managers can use puts to keep a floor on their portfolio. However, puts can be very expensive (especially for out-of-the-money puts). In particular, we observed that the volatility that is used to price out-of-the-money puts appears very high relative to actual volatility. Also, using S&P 500 index puts to trade is always an approximation, as your portfolio may imperfectly correlated to the S&P 500. An alternative to using puts to hedge is to use a trading strategy called “portfolio insurance”. Portfolio insurance attempts to replicate a put by trading the underlying portfolio.

Main Idea Behind Portfolio Insurance (2) Suppose your current NAV is 10. You want to ensure that your NAV does not fall below 9. One way to achieve this goal is to sell your portfolio as the market falls, so that you are completely into cash by the time your NAV has reduced to 9. Your trading strategy is to increase your cash position as the market declines (with a goal of 100% cash at a portfolio value of 9), and decrease your cash position as the market increases. This trading strategy is called “portfolio insurance”. Qt: how much do you buy/sell with market fluctuations?

Portfolio Insurance (3) To determine how much to buy or sell, you use the put’s delta (or hedge ratio) as a guide. Suppose the current NAV is 10.00, and you want to put a floor at $8.50. Instead of buying a put with strike, 8.50, you want to implement a portfolio insurance strategy. The delta or the hedge ratio of the put (from the black-Scholes spreadsheet) of a strike of 8.50 is about Thus, your initial position at a NAV of 10 will be 18% in cash, and 82% in your portfolio. You will change the weights as the price moves.

Portfolio Insurance (4) For different levels of the price you can figure out the level of cash from the black-Scholes formula. For example, at a price of 10,9,8,7,6,5,4 your cash position will be 18%, 31%, 50%, 70%, 87%,97%, and 100% respectively. To figure out (approximately) the average price you receive on your portfolio, we can add up how much we sold at each price level. Let us assume that you sold your position at the average price between each price level. Thus, between the price of 9 and 8, we will assume that you sold 19% ( ) of your portfolio at an average price of 8.50.

Portfolio Insurance (5) We will assume that your initial position in cash at 0.18 can be achieved by selling part of your portfolio at its current price of $10.0, and that you liquidate completely at a price of Average price = 10.00x x( )+ 8.50x( ) x( ) x( ) x( ) + 4.5x(1-0.97) = $7.94. Thus, your “portfolio insurance” strategy has resulted in a floor that is close to $8.50 (that you wanted to achieve).

Portfolio Insurance (6) What are the advantages/disadvantages of this strategy? A. Advantages: 1. You do not have to worry about finding an option on an index that is correlated with your portfolio.

Portfolio Insurance (6) B. Disadvantages: 1. It may fail precisely when you require it the most - when there is a big crash - as you may not be able to quickly sell at your average price. 2. Also, you are selling when the price is falling - if enough portfolio managers are selling, they will make the price fall even more, which will in turn trigger even greater selling! In fact, many regulators and the press blames portfolio insurance for causing/exacerbating the 1987 crash. Portfolio insurance went out with a bang after the ’87 crash.