Lecture 21: Options Markets

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Lecture 21 Options Markets
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Options

• With options, one pays money to have a choice in
  the future
• Essence of options is not that I buy the ability
  to vacillate, or to exercise free will. The
  choice one makes actually depends only on the
  underlying asset price
• Options are truncated claims on assets

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Options Exchanges

• Options are as old as civilization. Option to buy
  a piece of land in the city
• Chicago Board Options Exchange, a spinoff from
  the Chicago Board of Trade 1973, traded first
  standardized options
• American Stock Exchange 1974, NYSE 1982

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Terms of Options Contract

• Exercise date
• Exercise price
• Definition of underlying and number of shares

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Two Basic Kinds of Options

• Calls, a right to buy
• Puts, a right to sell

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Two Basic Kinds of Options

• American options can be exercised any time
  until exercise date
• European options can be exercised only on
  exercise date

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Buyers and Writers

• For every option there is both a buyer and a
  writer
• The buyer pays the writer for the ability to
  choose when to exercise, the writer must abide by
  buyers choice
• Buyer puts up no margin, naked writer must post
  margin

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In and Out of the Money

• In-the-money options would be worth something if
  exercised now
• Out-of-the-money options would be worthless if
  exercised now

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Put-Call Parity Relation

• Put option price call option price present
  value of strike price present value of
  dividends price of stock
• For European options, this formula must hold (up
  to small deviations due to transactions costs),
  otherwise there would be arbitrage profit
  opportunities

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Limits on Option Prices

• Call should be worth more than intrinsic value
  when out of the money
• Call should be worth more than intrinsic value
  when in the money
• Call should never be worth more than the stock
  price

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Binomial Option Pricing

• Simple up-down case illustrates fundamental
  issues in option pricing
• Two periods, two possible outcomes only
• Shows how option price can be derived from
  no-arbitrage-profits condition

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Binomial Option Pricing, Cont.

• S current stock price
• u 1fraction of change in stock price if price
  goes up
• d 1fraction of change in stock price if price
  goes down
• r risk-free interest rate

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Binomial Option Pricing, Cont.

• C current price of call option
• Cu value of call next period if price is up
• Cd value of call next period if price is down
• E strike price of option
• H hedge ratio, number of shares purchased per
  call sold

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Hedging by writing calls

• Investor writes one call and buys H shares of
  underlying stock
• If price goes up, will be worth uHS-Cu
• If price goes down, worth dHS-Cd
• For what H are these two the same?

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Binomial Option Pricing Formula

• One invested HS-C to achieve riskless return,
  hence the return must equal (1r)(HS-C)
• (1r)(HS-C)uHS-CudHS-Cd
• Subst for H, then solve for C

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Formula does not use probability

• Option pricing formula was derived without regard
  to the probability that the option is ever in the
  money!
• In effect, the price S of the stock already
  incorporates this probability
• For illiquid assets, such as housing, this
  formula may be subject to large errors

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Black-Scholes Option Pricing

• Fischer Black and Myron Scholes derived
  continuous time analogue of binomial formula,
  continuous trading, for European options only
• Black-Scholes continuous arbitrage is not really
  possible, transactions costs, a theoretical
  exercise
• Call T the time to exercise, s2 the variance of
  one-period price change (as fraction) and N(x)
  the standard cumulative normal distribution
  function (sigmoid curve, integral of normal
  bell-shaped curve) normdist(x,0,1,1) Excel (x,
  mean,standard_dev, 0 for density, 1 for cum.)

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Black-Scholes Formula
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Implied Volatility

• Turning around the Black-Scholes formula, one can
  find out what s would generate current stock
  price.
• s depends on strike price, options smile
• Since 1987 crash, s tends to be higher for puts
  or calls with low strike price, options leer or
  options smirk

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VIX Implied VolatilityWeekly, 1992-2004
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Implied and Actual Volatility Monthly Jan
1992-Jan 2004
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Actual SP500 Volatility Monthly1871-2004
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Using Options to Hedge

• To put a floor on ones holding of stock, one can
  buy a put on same number of shares
• Alternatively, one can just decide to sell
  whenever the price reaches the floor
• Doing the former means I must pay the option
  price. Doing the latter costs nothing
• Why, then, should anyone use options to hedge?

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Behavioral Aspects of Options Demand

• Thalers mental categories theory
• Writing an out-of-the-money call on a stock one
  holds, appears to be a win-win situation
  (Shefrin)
• Buying an option is a way of attaining a more
  leveraged, risky position
• Lottery principle in psychology, people
  inordinately attracted to small probabilities of
  winning big
• Margin requirements are circumvented by options

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Option Delta

• Option delta is derivative of option price with
  respect to stock price
• For calls, if stock price is way below exercise
  price, delta is nearly zero
• For calls, if option is at the money, delta is
  roughly a half, but price of option may be way
  below half the price of the stock.
• For calls, if stock price is way above the
  exercise price, delta is nearly one and one pays
  approximately stock price minus pdv of exercise
  price, like buying stock with credit pdv(E)

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