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					Options: the basics
                Readings for Options
• The textbook involves intensive mathematics. I therefore suggest
  you use online free sources and the lecture notes as your main
  reference
• Main lecture materials are drawn from (compulsory):
  Chicago Broad of Options Exchange learning center Online tutorials
       Click on “Options Basics” and read:
       [1] Options Overview
       [2] Introduction to Options Strategies
       [3] Expiration, Exercise and Assignment
       [4] Options Pricing 1
• If you want to know more about options, such as the specific
  regulations for trading options, you can go to:
       [1] The Option Industry Council Website
       [2] The Options Clearing Corporations Website
       [3] Check out the various websites for US exchanges.
                   Content
•   What is option?
•   Terminology
•   No arbitrage
•   Pricing Options
    – The Binomial Option Pricing Model
    – The Black-Scholes Model
           Why do we study options?
    Economics: It is all about maximizing happiness:
                                 Happiness = U(CCurrent,Cfuture)
    We are happier with more Cfuture, but because we are risk-averse, we are
    worried about the fluctuation of Cfuture.


    As you will see later, options provide a special payoff structure.
    In words:
    Our happiness is derived not only from current consumption but also from
    future consumptions which inherently involves uncertainty. This ultimately
    constitutes our risk concern over the future payoffs of assets that we own.
    Because of the special payoff structure of options, holding options enables
    us to adjust our risk exposure, and ultimately vary our level of happiness.



Options?           Terminology            Arbitrage          Binomial     Black-Scholes
                     Who trade options?
    A quote from Chicago Board Options Exchange (CBOE):

           “The single greatest population of CBOE users are not huge financial
           institutions, but public investors, just like you. Over 65% of the
           Exchange's business comes from them. However, other participants in
           the financial marketplace also use options to enhance their performance,
           including:

                1.        Mutual Funds
                2.        Pension Plans
                3.        Hedge Funds
                4.        Endowments
                5.        Corporate Treasurers”




Options?              Terminology       Arbitrage         Binomial          Black-Scholes
                                          How big is option trading?
   Figure 6.1: Total number of option contracts traded in a year in ALL US
   Exchanges 1973 – 2003 (Source: CBOE)

                                          1400
           No. of Contracts in Millions




                                          1200
                                          1000
                                          800
                                          600
                                          400
                                          200
                                            0
                                                 1973
                                                        1975
                                                               1977
                                                                      1979
                                                                             1981
                                                                                    1983
                                                                                           1985
                                                                                                  1987
                                                                                                         1989
                                                                                                                1991
                                                                                                                       1993
                                                                                                                              1995
                                                                                                                                     1997
                                                                                                                                            1999
                                                                                                                                                   2001
                                                                                                                                                            2003
                                                                                                    Year

Options?                                            Terminology                      Arbitrage                         Binomial                           Black-Scholes
           How big is option trading?
    Figure 6.2: CBOE average daily trading volume 1973 – 2003 (Source: CBOE)

                                           1.6
            No. of Contracts in Millions




                                           1.4
                                           1.2
                                            1
                                           0.8
                                           0.6
                                           0.4
                                           0.2
                                            0
                                                 1973
                                                        1975
                                                               1977
                                                                      1979
                                                                             1981
                                                                                    1983
                                                                                           1985
                                                                                                  1987
                                                                                                         1989
                                                                                                                1991
                                                                                                                       1993
                                                                                                                                1995
                                                                                                                                         1997
                                                                                                                                                1999
                                                                                                                                                       2001
                                                                                                                                                              2003
                                                                                                    Year




Options?                                         Terminology                         Arbitrage                                Binomial                               Black-Scholes
                                  How big is option trading?
                   Figure 6.3: CBOE year-end options open-interest dollar amount (in
                   thousands)1973 – 2003 (Source: CBOE)

                                  350000000
     Year-End Open Interest (in




                                  300000000
        thousand dollars $)




                                  250000000
                                  200000000
                                  150000000
                                  100000000
                                  50000000
                                         0
                                           73

                                           76

                                           79

                                           82

                                           85

                                           88

                                           91

                                           94

                                           97

                                           00

                                           03
                                        19

                                        19

                                        19

                                        19

                                        19

                                        19

                                        19

                                        19

                                        19

                                        20

                                        20
                                                                   Year


Options?                                 Terminology   Arbitrage          Binomial   Black-Scholes
           Where do we trade options?
      Trading of standardized options contracts on a national exchange started
      in 1973 when the Chicago Board Options Exchange (CBOE), the world's
      first listed options exchange, began listing call options.
      Options are also traded in US in several smaller exchanges, including:
           •   New York - the American Stock Exchange (AMEX)
                         - the International Securities Exchange (ISE)
           •   Philadelphia - the Philadelphia Stock Exchange (PHLX)
           •   San Francisco - the Pacific Stock Exchange (PCX)




Options?                Terminology       Arbitrage         Binomial      Black-Scholes
           Where do we trade options?
    Trading of non-standardized options contracts occurs on the Over-the-
       counter (OTC) market. And it is an even bigger market than the
       exchange-traded market for option trading.


    •      The OTC market is a secondary market that trades securities (stocks,
           options, and other financial assets) which are not traded on an exchange
           due to various reasons (e.g., an inability to meet listing requirements,
           special terms in contracts incompatible with exchanges’ standardized
           terms).
    •      For such securities, brokers/dealers negotiate directly with one another
           over computer networks or by phone. Their activities are monitored by the
           National Association of Securities Dealers. (Conversations over the
           phone are usually taped.)
    •      One advantage of options traded in OTC is that they can be tailored to
           meet particular needs of a corporate treasurer or fund manager.

Options?              Terminology       Arbitrage         Binomial          Black-Scholes
        Standardized VS Non-Standardized
    Standardized Options
    •      The terms of the option contract is standardized.
    •      Terms include:
            1. The exercise price (also called the strike price)
            2. The maturity date (also called the expiration date)
                •   For stock options, this is the third Saturday of the month in which
                    the contract expires.
            3. Number of shares committed on the underlying stocks
                •   In US, usually 1 option contract – 100 shares of stock
            4. Type: American VS European
                •   Stocks options in exchanges are American
    Non-standardized options also involves these terms, but they can be
      anything. For example, 1 contract underlies 95 shares instead of 100
      shares of stock. Terms being more flexible for non-standardized options
      and are traded in OTC market are the two distinct features.
Options?              Terminology        Arbitrage          Binomial           Black-Scholes
                           A screenshot from CBOE
•    As of Oct 30, 2006 at around 12pm ET, Verizon Commuications (VZ) was selling at $37.28 per share.
•    An american call option that allows the holder to buy 100 shares of VZ on or before the third Saturday of
     Nov 2006 for a price of $37.50 has a market price of $240 (=$2.40x100).




    Options?               Terminology            Arbitrage               Binomial               Black-Scholes
                   What is an option contract?
   •       There are 2 basic types of options: CALLs & PUTs
   •       A CALL option gives the holder the right, but not the obligation
            •   To buy an asset
            •   By a certain date
            •   For a certain price
   •       A PUT option gives the holder the right, but not the obligation
            •   To sell an asset
            •   By a certain date
            •   For a certain price


   •       an asset – underlying asset
   •       Certain date – Maturity date/Expiration date
   •       Certain price – strike price/exercise price

Options?                Terminology        Arbitrage         Binomial         Black-Scholes
                              Bunch of Jargons
   Option is a DERIVATIVES – since the value of an option depends on the
      price of its underlying asset, its value is derived.

   In the Money - exercise of the option would be profitable
       Call: market price      >        exercise price
       Put: exercise price     >        market price

   Out of the Money - exercise of the option would not be profitable
      Call: market price      <        exercise price
      Put: exercise price     <        market price

   At the Money - exercise price and asset price are equal

           •   Long – buy
           •   Short – sell

   e.g.,        Long a put on company x – buy a put contract of company x.
                Short a call on company y – sell/write a call contract of company y

Options?               Terminology       Arbitrage         Binomial           Black-Scholes
                         Bunch of Jargons
    American VS European Options


    An American option – allows its holder to exercise the right to purchase (if a
    call) or sell (if a put) the underlying asset on or before the expiration date.


    A European option – allows its holder to exercise the option only on the
    expiration date.




Options?           Terminology        Arbitrage          Binomial           Black-Scholes
                           Call Option’s payoff
   •       Assume you hold ONE contract of that Calls, i.e., the call contract allows
           you to buy 100 shares of VZ on the third Saturday of November at an
           exercise price of $37.50/share. Assume it is European instead of
           American.
   What is your payoff on the expiration date if VZ at that date is:
   (a) Selling @ $40
            •   You will be very happy. To cash in, you do two things simultaneously:
                 •   [1] exercise your right, and buy 100 VZ at $37.50. Total amount
                     you use is $3,750.
                 •   [2] sell 100 shares of VZ at the market price (i.e., $40/share).
                     Total amount you get is $4,000.
            •   Your payoff is $4,000 - $3,750 = $250.
   (b) Selling @ $30
            •   You will be very sad. You would not exercise the rights. The contract
                is thus expired without exercising.
            •   Your payoff is $0!!!!
Options?                Terminology        Arbitrage          Binomial           Black-Scholes
                Call Option’s payoff (European)
   •       Exercise Price = $37.50/share.
            •   If at maturity, market Price = $40 > $37.50
                (You exercise and get profit, the option you hold is said to be
                “in-the-money” because exercising it would produce profit)
            •   If at maturity, market price = $20 < $27.50
                (You do not exercise, the option you hold is said to be
                “out-of-the-money” because exercising would be unprofitable)
   •       In general, if you hold a call option contract, you want VZ’s stock price to
           skyrocket. The higher is the share price of VZ, the happier you are.
           Because the value of the call option is higher if the underlying asset’s
           price is higher than the exercise price.
   •       In contrast, the value of a call option is zero if the underlying asset’s price
           is lower than the exercise price. Whether it is $30, $29 or $2.50, it does
           not matter, the call option will still worth zero.
   •       Try to derive the payoff for the seller of call using this example.



Options?                Terminology        Arbitrage           Binomial            Black-Scholes
                            Put Option’s payoff
   •       Assume you hold ONE contract of that Puts, i.e., the put contract allows
           you to sell 100 shares of VZ on the third Saturday of November at an
           exercise price of $35/share. (current price of this option = $0.05) Assume
           it is European instead of American.
   What is your payoff on expiration date if Intel at that date is:
   (a) Selling @ $40
            •   You will be very sad. You would not exercise the rights. The contract
                is thus expired without exercising.
            •   Your payoff is $0!!!
   (b) Selling @ $30
            •   You will be very happy. To cash in, you do two things simultaneously:
                 •   [1] you buy 100 shares of VZ at $30, total purchase = $3,000
                 •   [2] exercise your right, and sell 100 VZ at $35. Total amount you
                     get is $3, 500.
            •   Your payoff is $3,500 - $3,000 = $500.
Options?                Terminology       Arbitrage          Binomial          Black-Scholes
              Put Option’s payoff (European)
   •       Exercise Price = $35/share.
   •       If at maturity, market Price = $40 > $35
           (You do not exercise, and the option you hold is said to be “out-of-the-
           money” because exercising would be unproductive)
   •       If at maturity, market price = $30 < $35
           (You exercise, and the option you hold is said to be “in-the-money”
           because exercising would be profitable)
   •       In general, if you hold a put option contract, you want VZ to go broke. If
           VZ is selling at a penny, you will be very rich.
   •       That means, the value of a put option is higher if the underlying asset’s
           price is lower than the exercise price.
   •       In contrast, the value of a put option is zero if the underlying asset’s price
           is higher than the exercise price. Whether it is $40, $39 or $1000, it does
           not matter, the put option will still worth zero.
   •       Again, try to derive the payoff of the seller.


Options?                Terminology         Arbitrage          Binomial           Black-Scholes
                        Some more Jargons

    If the underlying asset of an option is:


    (a) A stock – then the option is a stock option
    (b) An index – the option is an index option
    (c) A future contract – the option is a futures option
    (d) Foreign currency – the option is a foreign currency option
    (e) Interest rate – the option is an interest rate option


    •      ECMC49 will only focus on stock option. But you should know that there
           are other options trading in the market. You should definitely know them
           when you are interviewed by a firm or an i-bank for financial position. You
           will fail your CFA exam if you don’t know them. You also need to know
           the differences between options, futures/forwards and warrants.

Options?              Terminology        Arbitrage           Binomial         Black-Scholes
                     Stock options VS stocks
    Let’s say you hold a option contract for VZ. How does that differ from directly
        holding Verizon Communications’ stocks?
    Similarities:
    •      VZ’s options are securities, so does VZ’s stocks.
    •      Trading VZ’s options is just like trading stocks, with buyers making bids and
           sellers making offers.
    •      Can easily trade them, say in an exchange.
    Differences:
    •      VZ’s options are derivatives, but VZ’s stocks aren’t
    •      VZ’s options will expire, while stocks do not.
    •      There is no fixed number of options. But there is fixed number of shares of VZ’s
           stocks available at any point in time.
    •      Holding VZ’s common stocks entitles voting rights, but holding VZ’s option does
           not
    •      VZ has control over its number of stocks. But it has no control over its number of
           options.

Options?                Terminology           Arbitrage           Binomial            Black-Scholes
                                   Notations

  Strike price = X
  Stock price at present = S0
  Stock price at expiration = ST
  Price of a call option = C
  Price of a put option = P
  Risk-free interest rate = Rf
  Expiration time = T
  Present time = 0
  Time to maturity = T – 0 = T




Options?             Terminology     Arbitrage   Binomial   Black-Scholes
                         Payoff of Long Call
           If you buy (long) a call option, what is your payoff at expiration?
                                     Payoff to Call Holder at expiration
  Strike price = X                              (ST - X) if ST >X
  Stock price at present = S0                       0    if ST < X
                                     Profit to Call Holder at expiration
  Stock price at expiration = ST
                                               Payoff – Purchase Price (time adjusted)
  Price of a call option = C
                                       $
  Price of a put option = P                                               Payoff

  Risk-free interest rate = Rf
  Expiration time = T                                                     Profit

  Present time = 0
                                                                                            ST
  Time to maturity = T – 0 = T          Purchase price     x
                                        adjusted by time




Options?             Terminology       Arbitrage               Binomial          Black-Scholes
                        Payoff of Short Call
           If you sell (short) a call option, what is your payoff at expiration?
                                     Payoff to Call seller at expiration
  Strike price = X                              -(ST - X) if ST >X
  Stock price at present = S0                       0     if ST < X
                                     Profit to Call seller at expiration
  Stock price at expiration = ST
                                               Payoff + Selling Price (time adjusted)
  Price of a call option = C
                                        $
  Price of a put option = P
  Risk-free interest rate = Rf
  Expiration time = T                       Selling price
                                            adjusted by time
  Present time = 0
                                                                                              ST
  Time to maturity = T – 0 = T                                 x
                                                                              Profit

                                                                              Payoff

Options?             Terminology        Arbitrage                  Binomial        Black-Scholes
                          Payoff of Long Put
           If you buy (long) a put option, what is your payoff at expiration?
                                   Payoff to Put Holder at expiration
  Strike price = X                                0    if ST >X
  Stock price at present = S0                 (X – ST) if ST < X
                                   Profit to Put Holder at expiration
  Stock price at expiration = ST
                                             Payoff - Purchasing Price (time adjusted)
  Price of a call option = C
                                       $
  Price of a put option = P
  Risk-free interest rate = Rf
  Expiration time = T
                                                                             Payoff
  Present time = 0
                                                                                             ST
  Time to maturity = T – 0 = T                        x   Purchasing price
                                                          adjusted by time
                                                                             Profit




Options?             Terminology       Arbitrage            Binomial              Black-Scholes
                         Payoff of Short Put
           If you sell (short) a put option, what is your payoff at expiration?
                                    Payoff to Put seller at expiration
  Strike price = X                                  0    if ST >X
  Stock price at present = S0                  -(X – ST) if ST < X
                                    Profit to Put seller at expiration
  Stock price at expiration = ST
                                              Payoff + Selling Price (time adjusted)
  Price of a call option = C
                                        $
  Price of a put option = P
  Risk-free interest rate = Rf                                                Profit
  Expiration time = T                                      Selling price
                                                           adjusted by time   Payoff
  Present time = 0
                                                                                              ST
  Time to maturity = T – 0 = T                         x




Options?             Terminology        Arbitrage             Binomial             Black-Scholes
                  Payoff of Long Put & Short Call
           If you buy (long) a put option and sell (short) a call, assuming their
           exercise prices are the same, what is your payoff at expiration?
    Payoff to Call seller at expiration Payoff to Put Holder at expiration
              -(ST - X) if ST >X                      0    if ST >X
                                       +
                   0 if ST < X                    (X – ST) if ST < X


$                                                      $




                                                                                    Payoff
                                                  ST                                                ST
                  x                                             x



                                        Payoff

       Options?           Terminology            Arbitrage       Binomial           Black-Scholes
              Payoff of Long Put & Short Call
       If you buy (long) a put option and sell (short) a call, assuming their
       exercise prices are the same, what is your payoff at expiration?
Payoff to Call seller at expiration Payoff to Put Holder at expiration
          -(ST - X) if ST >X                      0    if ST >X        = -(ST – X)
                                   +
               0 if ST < X                    (X – ST) if ST < X       = (X - ST)


                      $




                                                                          ST
                                    x
                                                      Payoff




   Options?           Terminology         Arbitrage            Binomial         Black-Scholes
       Long Put & Short Call & Long stock
    If you buy (long) a put option and sell (short) a call, as well as holding 1
    stock. If the two options have the same exercise price and the same
    expiration date, what is your payoff at expiration?

       -(ST – X) if ST >X                                                         X    if ST >X
                                     +   Stock price (ST) at time T =             X    if ST < X
         (X - ST) if ST < X
                                                           Payoff (long the stock)
                       $

                                                                Total Payoff (risk-free)
                   x


                                                                                  ST
                                          x
                                                           Payoff (short call & long put)




Options?               Terminology             Arbitrage               Binomial             Black-Scholes
                                         Put-Call Parity
            What we just do introduces a very important concept for pricing options.
            Holding a portfolio with        (a) 1 stock                     (which costs S0)
                                            (b) selling one call            (which earns C)
                                            (c) buying one put              (which costs P)
            Total value of constructing portfolio = S0 + P - C

                                 Payoff (long the stock)         The payoff at maturity/expiration
    $                                                            is always X !!!


                                         Total Payoff (risk-free)
x


                                                        ST
                    x
                                 Payoff (short call & long put)


        Options?           Terminology            Arbitrage            Binomial           Black-Scholes
                                 Put-Call Parity
Total value of constructing portfolio = S0 + P – C
Get back X at maturity for sure.
Thus X discounted at the risk-free rate should equal to the portfolio value now.
           Thus,                 S0 + P – C = X/(1+Rf)T
In words: “Current stock price plus price of a corresponding put option at exercise
price X minus the price of a corresponding call option with exercise price X is equal
to the present value of X at maturity discounted at risk-free rate.
                                                            Payoff (long the stock)
                                 $

                                                                     Total Payoff (risk-free)
                                    x


                                                                                   ST
                                                      x
                                                             Payoff (short call & long put)


Options?           Terminology            Arbitrage       Binomial              Black-Scholes

				
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