Introduction to Options

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Introduction to Options
B, K M Chapters 20 21
End-of-chapter problems 1-9,17,18,20,24,25
1-10,12-15,17-20
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Types of Options

• American Call Option
• Contract giving its owner the right to purchase
  a given number of shares (100) of a specific
  security at the strike price (X) at any time
  prior to maturity (T)
• A call is not an obligation to buy 100 shares at
  X per share. The owner of the call option only
  exercises if he/she finds it in his/her interest
• Recall however that for each long side of the
  contract there is a short side. If you sell the
  option, you will be required to sell at X per
  share when the option is exercised (which will
  only occur when the price of the underlying
  security is greater than X)
• Zero sum game

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Types of Options

• American Put Option
• Contract giving its owner the right to sell a
  given number of shares (100) of a specific
  security at the strike price (X) at any time
  prior to maturity (T)
• Again, if you sell this option, you must buy
  shares at X if the option is exercised

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Types of Options

• European Call and Put Options
• Like American call and put options except that
  exercise is only possible at expiration
• The geographical labeling is a misnomer. Most
  options traded here and in Europe are American
  options. Foreign currency options and some stock
  index options traded on the CBOE are important
  exceptions however. (The SP500 options contract
  is a European option while the SP100 contract is
  American.)

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Notation

• Co Current price of a call option (per share).
• Po Current price of a put option.
• X Strike or exercise price.
• T Expiration date.
• t any time between issue date and maturity date
• So Current price of the underlying security
  (stock).
• X lt So Call is in-the-money (put is
  out-of-the-money).
• X gt So Call is out-of-the-money (put is
  in-the-money).

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Notation

• Strike Price
• Option contracts on equities are introduced with
  strike prices in increments of 5 above and below
  the current stock price (10 increments may be
  used for stocks selling for more than 100, and
  2.50 for stocks trading at less than 30)
• New contracts are introduced as the stock price
  increases or decreases

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Notation

• Expiration Dates
• Most Puts and calls traded on equity options
  have maturities lt 9 months. Options expire at the
  end of the third Friday of the expiration month.
• Typically, a stock has option contracts with
  three expiration dates (separated by three
  months)
• More heavily traded options have four expiration
  months (nearest two months and the next two
  months in the normal 3-6-9 month cycle)
• For example, IBM in the beginning on January
  will have January, February, April and July
  options traded

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Notation

• Expiration Dates
• The most heavily traded options are the ones
  trading near-the-money with the closest
  expiration date
• LEAPS (Long-term equity anticipation securities)
  are long-term exchange traded options that last
  up to 3 years. They are currently traded on
  indexes on the CBOE and on individual stocks on
  various exchanges 

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Other Types of Options

• Warrants
• Call options issued by a firm
• Exercise requires the firm to issue new shares
  and results in a cash flow to the firm
• Asian Options
• - Payoffs depend on the average price of the
  underlying asset during at least a portion of the
  life of the option

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Other Types of Options

• Barrier Options
• - Depend not only on the asset price at
  expiration, but also on whether or not the asset
  price has crossed through some barrier.
• - A down and out option expires worthless if
  the asset price falls below some level.
• - A down and in option expires worthless unless
  the asset does fall below some barrier al least
  once during the life of the option.

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Other Types of Options

• Lookback Options
• - Payoffs depend in part on the maximum or
  minimum price during the life of the option
• Currency-Translated Options
• - Either the asset price or exercise price is
  denominated in a foreign currency
• Binary Options
• - Provide a fixed payoff if the stock price meets
  some condition (for example, if S gt X, the option
  pays 1000)

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Equity Option Trading

• Puts and Calls on individual stocks and on stock
  indices actively traded on CBOE and the
  International Securities Exchange (an electronic
  market based in NY)
• Before 1973, no trading of standardized options
  in the U.S. There was some trading of options on
  OTC market, with large transaction costs and low
  trading volume
• Options trading dates back to the 17th century
  Holland, where farmers purchased put options on
  tulip bulbs to reduce price uncertainty
• Without a centralized agency to guarantee
  payment in the event of exercise, hard to trade
  option contracts because of solvency issues

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Equity Option Trading

• CBOE started trading calls in 1973
• Completely standardized option contracts
• Few (3 or 4) maturities
• No paper certificates (electronic book entries)
• Higher trading volumes and liquidity, lower
  transaction costs
• All contracts are with the Option Clearing
  Corporation. No risk of default (from perspective
  of buyers)

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Adjustments

• Calls and Puts are not adjusted for cash
  dividends
• X is adjusted for stock splits
• Example Stock has a 2-for-1 split
• St (after split) ½ St-1 (just before split)
• Call (X50) would split into two calls (X25)

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Basic Transactions in Calls

• Buyer purchases American call at time 0
• At any time t (0lttltT) the buyer can do
• Exercise the call by paying 100(X) dollars. He
  then receives 100 shares of stock worth 100 (St)
  dollars in exchange
• Cancel the position by selling the call for 100
  (Ct) dollars
• Hold on to the call
• The maximum loss is limited to the initial
  investment 100 (C0)
• Writer or seller of an American call option is
  obligated to deliver 100 shares of the underlying
  stock in exchange for 100 (X) dollars

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Basic Transactions in Calls

• At any time t (0lttltT) the writer can
• Close out the position by buying a call for 100
  (Ct) dollars
• Do nothing
• Maximum loss is unlimited, so margins must be
  posted to guarantee payment in the event of an
  exercise
• Naked position in options refer to long or short
  positions in an option that are not combined with
  a position in the underlying security
• Here, buying a call is bullish strategy, while
  buying a put is a bearish strategy
• Although buying a call gives you unlimited
  upside gain, it is risky (You stand to lose your
  initial investment)
• Options can be used by speculators as leveraged
  stock positions, but they can also be used
  creatively to manage risk exposures as the
  following example makes clear

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Basic Transactions in Calls

• EXAMPLE Assume IBM currently sells for 70 and
  your analysis indicates a significant increase in
  price. Of course, you cannot forecast
  firm-specific shocks, and IBM could fall in
  price.
• Assume as well
• The price of a 6 month call option with X 70
  currently sells for 7
• The interest rate for the period is 3
• What are the profits (returns) to the following
  three strategies for investing 7,000 as a
  function of IBMs stock price in 6 months?
• Purchase 100 shares of IBM stock
• Purchase 1000 call options w/ X 70 (10
  contracts)
• Purchase 100 calls for 700. Invest remaining
  6300 in T-Bills (6300 11.03 6489)

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19
Option Strategies Protective Put at Option
Expiration
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Covered Call Position at Expiration
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Other Option Strategies

• Straddle A long straddle is established by
  buying both a call and a put on a stock with the
  same expiration date and strike price. This is a
  bet on higher volatility than expected by the
  market

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Some Option Strategies (cont.)

• In Class Exercise
• What are the payoffs and profits to shorting a
  call and a put?

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Other Strategies
Bull Spread
Bear Spread
Butterfly Spread
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Put-Call Parity

• Until now, we have simply assumed the prices for
  calls and puts
• As an introduction to the valuation of options,
  we consider the relationship between the price of
  a European call and a European put on a
  non-dividend paying stock
• Basic setup
• No transaction costs
• No taxes
• Ability to borrow and lend at the same
  (risk-free) interest rate

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Put-Call Parity (cont.)

• Put-Call parity gives the following relation
  between the price of a put and a call
• Recall that S0, P0 and C0 are the prices today
  of the stock, put and call respectively
• You can think of the third term as the present
  value of the strike price or, alternatively, an
  investment in a zero-coupon risk-free bond that
  will grow in value to the strike price at the
  expiration of the call and put options

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Put-Call Parity (cont.)
This relation can be demonstrated as follows. At
expiration (at time T), there are two
possibilities ST lt X or ST gt X, We can value the
portfolios from either side of the equality above
as follows
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Put-Call Parity (cont.)

• Note that in either case, the two portfolios
  have equal value. Therefore their prices today
  must be equal. If they are not it will be
  possible to earn an arbitrage profit!
• This is accomplished by buying the portfolio
  that is undervalued and financing this purchase
  by selling short the portfolio that is
  overvalued. This will result in a positive cash
  flow today with no future risk because the short
  and long positions will cancel each other out at
  time T
• In class exercise Suppose European put and
  calls exist on the same stock, each with X 75
  and the same expiration date. The current stock
  price is 68. The current price of the put is
  6.50 higher than that of the call, and a
  risk-free investment over the life of the options
  will yield 3. Devise a strategy that will earn
  risk-free arbitrage profits.

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Put-Call Parity on Dividend Paying Stocks

• The more general form in the case where the
  stock pays a known dividend over the life of the
  options is given as follows
• where the last term is the present value of
  dividends to be paid during the life of the
  option
• Note that we have reduced the stock price by the
  present value of the dividend

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The No-Arbitrage Principle and Boundaries on
Option Prices

• Consider a call option that pays no dividends
• The value of the call cannot be negative -- This
  is straightforward since the holder of the option
  will only exercise it if it is in-the-money.
• C0 ? S0 -- No one would pay more than 60 for
  the right to buy a stock currently worth 60!
• C0 ? S0 - X/(1 rf) -- Consider two
  portfolios
• A) A call option on one share of stock
• B) One share of stock and borrow X/(1 rf) (the
  PV of the strike price)

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The No-Arbitrage Principle and Boundaries on
Option Prices
At expiration
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The No-Arbitrage Principle and Boundaries on
Option Prices
Although we have placed no-arbitrage bounds on
the price of a call, we obviously need more
precision. On to option pricing!