Derivatives and Investment Management

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Derivatives and Investment Management

• The Use of Futures and Options

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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.

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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.

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Examples of Futures Contracts

• SP 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 1075. 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 250x102500 per contract.

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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
  250x0.125

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Other Examples

• The Mini SP 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.

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Futures on Growth and Value Indices

• The SP 500/BARRA Growth and Value Indices Each
  SP 500 Index company is assigned either to the
  Growth or Value Index so that the two indices
  "add up" to the SP 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 SP 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.

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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.

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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
  dont 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 SP level is likely to affect your
  portfolio by (1)(1.4)(100)1.4 million. You want
  to decrease your beta to 1.2.

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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
  (1-1.2/1.41-0.8571) 14.29 of the 100 million.
  Instead of selling 14.29 million of stock, you
  can go short an equivalent amount of the SP 500
  futures. Thus, at an index level of 1075, you
  will short about 14,290,000/(250x1075)53
  contracts.

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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 358 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
  SP 500. At an index level of 1075 you will buy
  10,000,000/(250x1075)37 contracts.

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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?)

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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?

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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.
• Heres 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).

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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.

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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.

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Costs of Hedging (4)

• Here are some examples of current option prices
  (as of close on 4/22/2003)
• The SP 500 index closed at 911.36. The July
  option expires on 19 July (expires on the third
  Friday of the month).
• The 900 put traded at bid30.00, ask31.80. 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 bid6.70 ask7.70. 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 (911-800)/91112. But to buy this
  protection, it will cost you 7.70/9110.8 of
  your portfolio value.

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• Portfolio Insurance

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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 SP 500 index puts to trade is always
  an approximation, as your portfolio may
  imperfectly correlated to the SP 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.

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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?

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Portfolio Insurance (3)

• To determine how much to buy or sell, you use the
  puts 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 0.18.
• 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.

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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
  (0.5-0.31) of your portfolio at an average price
  of 8.50.

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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 4.50.
• Average price 10.00x0.18 9.50x(0.31-0.18)
  8.50x(0.5-0.31) 7.50x(0.7-0.5)
  6.50x(0.87-0.7) 5.50x(0.97-0.87) 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).

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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.

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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.