Portfolio Management - Chapter 17

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Portfolio Management - Chapter 17 Powered By Docstoc
					       Chapter 17

       Principles of
Options and Option Pricing


                       Prof. Rushen Chahal

         Prof. Rushen Chahal                 1
We sent the first draft of our paper to the Journal of Political
Economy and promptly got back a rejection letter. We then sent
 it to the Review of Economics and Statistics, where it was also
                             rejected.

Merton Miller and Eugene Fama…then took an interest in the
paper and gave us extensive comments on it. They suggested to
   the JPE that perhaps the paper was worth more serious
     consideration. The journal then accepted the paper.


                                              - Fischer Black



                        Prof. Rushen Chahal                        2
                  Outline
• Introduction
• Option principles
• Option pricing




                      Prof. Rushen Chahal   3
               Introduction
• Innovations in stock options have been among
  the most important developments in finance
  in the last 20 years
• The cornerstone of option pricing is the Black-
  Scholes Option Pricing Model (OPM)
  – Delta is the most important OPM progeny to the
    portfolio manager



                     Prof. Rushen Chahal             4
              Option Principles
•   Why options are a good idea
•   What options are
•   Standardized option characteristics
•   Where options come from
•   Where and how options trade
•   The option premium
•   Sources of profits and losses with options

                      Prof. Rushen Chahal        5
   Why Options Are A Good Idea
• Options:
  – Give the marketplace opportunities to adjust risk
    or alter income streams that would otherwise not
    be available

  – Provide financial leverage

  – Can be used to generate additional income from
    investment portfolios


                      Prof. Rushen Chahal               6
            Why Options Are
          A Good Idea (cont’d)
• The investment process is dynamic:
  – The portfolio managers needs to constantly
    reassess and adjust portfolios with the arrival of
    new information


• Options are more convenient and less
  expensive than wholesale purchases or sales
  of stock

                       Prof. Rushen Chahal               7
           What Options Are
• Call options
• Put options




                 Prof. Rushen Chahal   8
                Call Options
• A call option gives you the right to buy within
  a specified time period at a specified price

• The owner of the option pays a cash premium
  to the option seller in exchange for the right
  to buy



                     Prof. Rushen Chahal            9
Practical Example
 of A Call Option




    Prof. Rushen Chahal   10
                Put Options
• A put option gives you the right to sell within
  a specified time period at a specified price

• It is not necessary to own the asset before
  acquiring the right to sell it




                     Prof. Rushen Chahal            11
             Standardized
         Option Characteristics
• All exchange-traded options have standardized
  expiration dates
  – The Saturday following the third Friday of
    designated months for most options

  – Investors typically view the third Friday of the
    month as the expiration date



                       Prof. Rushen Chahal             12
           Standardized
   Option Characteristics (cont’d)
• The striking price of an option is the
  predetermined transaction price
  – In multiples of $2.50 (for stocks priced $25.00 or
    below) or $5.00 (for stocks priced higher than
    $25.00)

  – There is usually at least one striking price above
    and one below the current stock price


                       Prof. Rushen Chahal               13
           Standardized
   Option Characteristics (cont’d)
• Puts and calls are based on 100 shares of the
  underlying security
  – The underlying security is the security that the
    option gives you the right to buy or sell

  – It is not possible to buy or sell odd lots of options




                       Prof. Rushen Chahal                  14
     Where Options Come From
• Introduction
• Opening and closing transactions
• Role of the Options Clearing Corporation




                    Prof. Rushen Chahal      15
                Introduction
• If you buy an option, someone has to sell it to
  you

• No set number of put or call options exists
  – The number of options in existence changes every
    day
  – Option can be created and destroyed



                     Prof. Rushen Chahal            16
              Opening and
          Closing Transactions
• The first trade someone makes in a particular
  option is an opening transaction
  – An opening transaction that is the sale of an
    option is called writing an option




                      Prof. Rushen Chahal           17
            Opening and
    Closing Transactions (cont’d)
• The trade that terminates a position by closing
  it out is a closing transaction
  – Options have fungibility
     • Market participants can reverse their positions by
       making offsetting trades

     • E.g., the writer of an option can close out the position
       by buying a similar one



                         Prof. Rushen Chahal                      18
            Opening and
    Closing Transactions (cont’d)
• The owner of an option will ultimately:
  – Sell it to someone else

  – Let it expire or

  – Exercise it




                       Prof. Rushen Chahal   19
            Role of the
   Options Clearing Corporation
• The Options Clearing Corporation (OCC):
  – Positions itself between every buyer and seller
  – Acts as a guarantor of all option trades
  – Regulates the trading activity of members of the
    various options exchanges
  – Sets minimum capital requirements
  – Provides for the efficient transfer of funds among
    members as gains or losses occur


                      Prof. Rushen Chahal                20
     OCC-Related
Information on the Web




      Prof. Rushen Chahal   21
  Where and How Options Trade
• Options trade on four principal exchanges:
  – Chicago Board Options Exchange (CBOE)

  – American Stock Exchange (AMEX)

  – Philadelphia Stock Exchange

  – Pacific Stock Exchange


                      Prof. Rushen Chahal      22
           Where and How
         Options Trade (cont’d)
• AMEX and Philadelphia Stock Exchange
  options trade via the specialist system
  – All orders to buy or sell a particular security pass
    through a single individual (the specialist)
  – The specialist:
     • Keeps an order book with standing orders from
       investors and maintains the market in a fair and orderly
       fashion
     • Executes trades close to the current market price if no
       buyer or seller is available


                         Prof. Rushen Chahal                  23
          Where and How
        Options Trade (cont’d)
• CBOE and Pacific Stock Exchange options trade
  via the marketmaker system
  – Competing marketmakers trade in a specific
    location on the exchange floor near the order
    book official

  – Marketmakers compete against one another for
    the public’s business


                      Prof. Rushen Chahal           24
           Where and How
         Options Trade (cont’d)
• Any given option has two prices at any given
  time:
  – The bid price is the highest price anyone is willing
    to pay for a particular option

  – The asked price is the lowest price at which
    anyone is willing to sell a particular option



                       Prof. Rushen Chahal                 25
         The Option Premium
• Intrinsic value and time value
• The financial page listing




                    Prof. Rushen Chahal   26
   Intrinsic Value and Time Value
• The price of an option has two components:
  – Intrinsic value:
     • For a call option equals the stock price minus the
       striking price
     • For a put option equals the striking price minus the
       stock price
  – Time value equals the option premium minus the
    intrinsic value



                         Prof. Rushen Chahal                  27
           Intrinsic Value and
           Time Value (cont’d)
• An option with no intrinsic value is out of the
  money

• An option with intrinsic value is in the money

• If an option’s striking price equals the stock
  price, the option is at the money


                     Prof. Rushen Chahal            28
      The Financial Page Listing
• The following slide shows an example from
  the online edition of the Wall Street Journal:
  – The current price for a share of Disney stock is
    $21.95
  – Striking prices from $20 to $25 are available
  – The expiration month is in the second column
  – The option premiums are provided in the “Last”
    column



                      Prof. Rushen Chahal              29
The Financial Page Listing




          Prof. Rushen Chahal   30
                The Financial
             Page Listing (cont’d)
• Investors identify an option by company,
  expiration, striking price, and type of option:

               Disney JUN 22.50 Call



   Company
                 Expiration      Striking    Type
                                  Price

                       Prof. Rushen Chahal          31
                The Financial
             Page Listing (cont’d)
• The Disney JUN 22.50 Call is out of the money
   – The striking price is greater than the stock price
   – The time value is $0.25


• The Disney JUN 22.50 Put is in the money
   – The striking price is greater than the stock price
   – The intrinsic value is $22.50 - $21.95 = $0.55
   – The time value is $1.05 - $0.55 = $0.50



                           Prof. Rushen Chahal            32
             The Financial
          Page Listing (cont’d)
• As an option moves closer to expiration, its
  time value decreases
  – Time value decay


• An option is a wasting asset
  – Everything else being equal, the value of an option
    declines over time



                       Prof. Rushen Chahal            33
        Sources of Profits and
         Losses With Options
• Option exercise
• Exercise procedures




                   Prof. Rushen Chahal   34
              Option Exercise
• An American option can be exercised at any
  time prior to option expiration
  – It is typically not advantageous to exercise prior
    to expiration since this amount to foregoing time
    value


• European options can be exercised only at
  expiration

                      Prof. Rushen Chahal                35
           Exercise Procedures
• The owner of an option who decides to
  exercise the option:
  – Calls her broker
  – Must put up the full contract amount for the
    option
     • The premium is not a downpayment on the option
       terms




                       Prof. Rushen Chahal              36
    Exercise Procedures (cont’d)
• The option writer:
  – Must be prepared to sell the necessary shares to
    the call option owner

  – Must be prepared to buy shares of stock from the
    put option owner




                       Prof. Rushen Chahal             37
    Exercise Procedures (cont’d)
• In general, you should not buy an option with
  the intent of exercising it:
  – Requires two commissions

  – Selling the option captures the full value
    contained in an option




                       Prof. Rushen Chahal        38
                Profit and Loss Diagrams
• For the Disney JUN 22.50 Call buyer:

                                                 Maximum profit
                   Breakeven Point = $22.75      is unlimited




          $0
      -$0.25


 Maximum loss
                                $22.50

                           Prof. Rushen Chahal                    39
                 Profit and Loss Diagrams
• For the Disney JUN 22.50 Call writer:


Maximum profit                    Breakeven Point = $22.75


         $0.25
           $0

                                                 Maximum loss
                                                 is unlimited

                             $22.50

                        Prof. Rushen Chahal                     40
               Profit and Loss Diagrams
• For the Disney JUN 22.50 Put buyer:

                  Maximum profit = $21.45


                                        Breakeven Point = $21.45



          $0
      -$1.05


Maximum loss                  $22.50

                         Prof. Rushen Chahal                       41
                 Profit and Loss Diagrams
• For the Disney JUN 22.50 Put writer:

Maximum profit
                   Breakeven Point = $21.45

       $1.05
          $0
                                      Maximum loss = $21.45



                              $22.50

                              Prof. Rushen Chahal             42
                Option Pricing
•   Determinants of the option premium
•   Black-Scholes Option Pricing Model
•   Development and Assumptions of the model
•   Insights into the Black-Scholes Model
•   Delta
•   Theory of put/call parity
•   Stock index options



                      Prof. Rushen Chahal      43
         Determinants of the
          Option Premium
• Market factors
• Accounting factors




                   Prof. Rushen Chahal   44
               Market Factors
• Striking price
  – For a call option, the lower the striking price, the
    higher the option premium


• Time to expiration
  – For both calls and puts, the longer the time to
    expiration, the higher the option premium



                       Prof. Rushen Chahal                 45
        Market Factors (cont’d)
• Current stock price
  – The higher the stock price, the higher the call
    option premium and the lower the put option
    premium


• Volatility of the underlying stock
  – The great the volatility, the higher the call and put
    option premium


                       Prof. Rushen Chahal              46
        Market Factors (cont’d)
• Dividend yield on the underlying stock
  – Companies with high dividend yields have a
    smaller call option premium than companies with
    low dividend yields


• Risk-free interest rate
  – The higher the risk-free rate, the higher the call
    option premium


                       Prof. Rushen Chahal               47
             Accounting Factors
• Stock splits:
   – The OCC will make the following adjustments:
      • The striking price is reduced by the split ratio
      • The number of options is increased by the split ratio

   – For odd-lot generating splits:
      • The striking price is reduced by the split ratio
      • The number of shares covered by your options is
        increased by the split ratio



                          Prof. Rushen Chahal                   48
           Black-Scholes
        Option Pricing Model
• The Black-Scholes OPM:


        C  S  N (d1 )  Ke            rt
                                                N (d 2 ) 

               ln( S / K )   R  ( 2 / 2)  t
                                            
        d1 
                               t
       d 2  d1   t
                        Prof. Rushen Chahal                   49
          Black-Scholes
   Option Pricing Model (cont’d)
• Variable definitions:
  – C = theoretical call premium
  – S = current stock price
  – t = time in years until option expiration
  – K = option striking price
  – R = risk-free interest rate




                       Prof. Rushen Chahal      50
          Black-Scholes
   Option Pricing Model (cont’d)
• Variable definitions (cont’d):
  – = standard deviation of stock returns
      
  – N(x) = probability that a value less than “x” will
    occur in a standard normal distribution
  – ln = natural logarithm
  – e = base of natural logarithm (2.7183)




                       Prof. Rushen Chahal               51
           Black-Scholes
    Option Pricing Model (cont’d)
                           Example

Stock ABC currently trades for $30. A call option on ABC stock
has a striking price of $25 and expires in three months. The
current risk-free rate is 5%, and ABC stock has a standard
deviation of 0.45.

According to the Black-Scholes OPM, but should be the call
option premium for this option?



                           Prof. Rushen Chahal                   52
           Black-Scholes
    Option Pricing Model (cont’d)
                         Example (cont’d)

Solution: We must first determine d1 and d2:

                 ln( S / K )   R  ( 2 / 2)  t
                                              
          d1 
                               t
                 ln(30 / 25)  0.05  (0.452 / 2)  0.25
                                                  
             
                           0.45 0.25
               0.1823  0.0378
                               0.978
                    0.225
                                Prof. Rushen Chahal         53
           Black-Scholes
    Option Pricing Model (cont’d)
                      Example (cont’d)

Solution (cont’d):


                 d 2  d1   t
                      0.978  (0.45) 0.25
                      0.978  0.225
                      0.753
                         Prof. Rushen Chahal   54
           Black-Scholes
    Option Pricing Model (cont’d)
                      Example (cont’d)

Solution (cont’d): The next step is to find the normal probability
values for d1 and d2. Using Excel’s NORMSDIST function yields:



                   N (d1 )  0.836
                   N (d2 )  0.774

                           Prof. Rushen Chahal                  55
            Black-Scholes
     Option Pricing Model (cont’d)
                       Example (cont’d)

Solution (cont’d): The final step is to calculate the option
premium:


         C  S  N (d1 )  Ke rt  N (d 2 )
             $30  0.836  $25e (0.05)(0.25) 0.774
             $25.08  $19.11
             $5.97
                            Prof. Rushen Chahal                56
              Using Excel’s
           NORMSDIST Function
• The Excel portion below shows the input and
  the result of the function:




                   Prof. Rushen Chahal          57
            Development and
        Assumptions of the Model
•   Introduction
•   The stock pays no dividends during the option’s life
•   European exercise terms
•   Markets are efficient
•   No commissions
•   Constant interest rates
•   Lognormal returns



                         Prof. Rushen Chahal               58
               Introduction
• Many of the steps used in building the Black-
  Scholes OPM come from:
  – Physics
  – Mathematical shortcuts
  – Arbitrage arguments

• The actual development of the OPM is
  complicated


                    Prof. Rushen Chahal           59
    The Stock Pays no Dividends
      During the Option’s Life
• The OPM assumes that the underlying
  security pays no dividends

• Valuing securities with different dividend
  yields using the OPM will result in the same
  price



                    Prof. Rushen Chahal          60
    The Stock Pays no Dividends
      During the Option’s Life
• The OPM can be adjusted for dividends:
  – Discount the future dividend assuming continuous
    compounding

  – Subtract the present value of the dividend from
    the stock price in the OPM

  – Compute the premium using the OPM with the
    adjusted stock price


                      Prof. Rushen Chahal             61
       European Exercise Terms
• The OPM assumes that the option is European

• Not a major consideration since very few calls
  are ever exercised prior to expiration




                    Prof. Rushen Chahal            62
         Markets Are Efficient
• The OPM assumes markets are informationally
  efficient
  – People cannot predict the direction of the market
    or of an individual stock




                      Prof. Rushen Chahal               63
              No Commissions
• The OPM assumes market participants do not
  have to pay any commissions to buy or sell
• Commissions paid by individual can
  significantly affect the true cost of an option
  – Trading fee differentials cause slightly different
    effective option prices for different market
    participants



                       Prof. Rushen Chahal               64
        Constant Interest Rates
• The OPM assumes that the interest rate R in
  the model is known and constant

• It is common use to use the discount rate on a
  U.S. Treasury bill that has a maturity
  approximately equal to the remaining life of
  the option
  – This interest rate can change


                      Prof. Rushen Chahal       65
          Lognormal Returns
• The OPM assumes that the logarithms of
  returns of the underlying security are
  normally distributed

• A reasonable assumption for most assets on
  which options are available



                   Prof. Rushen Chahal         66
           Insights Into the
         Black-Scholes Model
• Divide the OPM into two parts:

         C  S  N (d1 )  Kert  N (d2 )

                 Part A                 Part B




                      Prof. Rushen Chahal        67
          Insights Into the
    Black-Scholes Model (cont’d)
• Part A is the expected benefit from acquiring
  the stock:
  – S is the current stock price and the discounted
    value of the expected stock price at any future
    point
  – N(d1) is a pseudo-probability
     • It is the probability of the option being in the money at
       expiration, adjusted for the depth the option is in the
       money


                         Prof. Rushen Chahal                   68
          Insights Into the
    Black-Scholes Model (cont’d)
• Part B is the present value of the exercise
  price on the expiration day:
  – N(d2) is the actual probability the option will be in
    the money on expiration day




                       Prof. Rushen Chahal              69
         Insights Into the
   Black-Scholes Model (cont’d)
• The value of a call option is the difference
  between the expected benefit from acquiring
  the stock and paying the exercise price on
  expiration day




                   Prof. Rushen Chahal           70
                           Delta
• Delta is the change in option premium
  expected from a small change in the stock
  price, all other things being equal:

                C
           
                S
         C
 where       the first partial derivative of the call premium
         S
              with respect to the stock price

                           Prof. Rushen Chahal                   71
                Delta (cont’d)
• Delta allows us to determine how many
  options are needed to mimic the returns of
  the underlying stock

• Delta is exactly equal to N(d1)
  – E.g., if N(d1) is 0.836, a $1 change in the price of
    the underlying stock price leads to a change in the
    option premium of 84 cents

                       Prof. Rushen Chahal             72
       Theory of Put/Call Parity
• The following variables form an interrelated
  securities complex:
  – Price of a put
  – Price of a call
  – The value of the underlying stock
  – The riskless rate of interest
• If put/call parity does not hold, arbitrage is
  possible


                      Prof. Rushen Chahal          73
              Theory of
        Put/Call Parity (cont’d)
• The put/call parity relationship:

                       K
        CPS
                   (1  R )T


      where C  price of a call
            P  price of a put
            K  option striking price
            R  risk-free interest rate
            T  time until expiration in years
                      Prof. Rushen Chahal        74
          Stock Index Options
• A stock index option is the option exchanges
  most successful innovation
  – E.g., the S&P 100 index option


• Index options have no delivery mechanism
  – All settlements are in cash




                      Prof. Rushen Chahal        75
    Stock Index Options (cont’d)
• The owner of an in-the-money index call
  receives the difference between the closing
  index level and the striking price

• The owner of an in-the-money index put
  receives the difference between the striking
  price and the index level


                    Prof. Rushen Chahal          76

				
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