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Options - NYU Stern

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									Session 2: Options I

C15.0008 Corporate Finance
          Topics
      Summer 2006
                   Outline
•   Call and put options
•   The law of one price
•   Put-call parity
•   Binomial valuation
 Options, Options Everywhere!
• Compensation—employee stock options
• Investment/hedging—exchange traded and OTC
  options on stocks, indexes, bonds, currencies,
  commodities, etc., exotics
• Embedded options—callable bonds, convertible
  bonds, convertible preferred stock, mortgage-
  backed securities
• Equity and debt as options on the firm
• Real options—projects as options
Example..
                     Options
The right, but not the obligation to buy (call) or sell
(put) an asset at a fixed price on or before a given
date.
Terminology:
      Strike/Exercise Price
      Expiration Date
      American/European
      In-/At-/Out-of-the-Money
        An Equity Call Option
• Notation: C(S,E,t)
• Definition: the right to purchase one share
  of stock (S), at the exercise price (E), at or
  before expiration (t periods to expiration).
Where Do Options Come From?
• Publicly-traded equity options are not
  issued by the corresponding companies
• An options transaction is simply a
  transaction between 2 individuals (the
  buyer, who is long the option, and the
  writer, who is short the option)
• Exercising the option has no effect on the
  company (on shares outstanding or cash
  flow), only on the counterparty
         Numerical example
• Call option
• Put option
    Option Values at Expiration
• At expiration date T, the underlying (stock) has market
  price ST
• A call option with exercise price E has intrinsic value
  (“payoff to holder”)




• A put option with exercise price E has intrinsic value
  (“payoff to holder”)
          Call Option Payoffs
         Long Call                 Short Call
Payoff                    Payoff




             E       ST                 E       ST
           Put Option Payoffs
         Long Put                 Short Put
Payoff                   Payoff


E



             E      ST                E       ST


                         E
         Other Relevant Payoffs
                      Risk-Free Zero Coupon Bond
         Stock        Maturity T, Face Amount E
Payoff                  Payoff

                        E




                 ST                   ST
        The Law of One Price
• If 2 securities/portfolios have the same payoff
  then they must have the same price
• Why? Otherwise it would be possible to make an
  arbitrage profit
   – Sell the expensive portfolio, buy the cheap
      portfolio
   – The payoffs in the future cancel, but the
      strategy generates a positive cash flow today
      (a money machine)
                  Put-Call Parity
         Stock + Put
Payoff                              Payoff
                            =       E


             E         ST                    E   ST


         Call +Bond
Payoff                              Payoff
                                =
                                    E


             E         ST                    E   ST
             Put-Call Parity
Payoffs:
           Stock + Put = Call + Bond

Prices:
           Stock + Put = Call + Bond
           Stock = Call – Put + Bond
              S = C – P + PV(E)
Introduction to binomial trees
     What is an Option Worth?
Binomial Valuation
Consider a world in which the stock can take on
only 2 possible values at the expiration date of the
option. In this world, the option payoff will also
have 2 possible values. This payoff can be
replicated by a portfolio of stock and risk-free
bonds. Consequently, the value of the option must
be the value of the replicating portfolio.
                     Payoffs

      Stock            Bond (rF=2%)             Call (E=105)
              137                 102                    32
100                 100                     C
              73                  102                    0


 1-year call option, S=100, E=105, rF=2% (annual)
 1 step per year
 Can the call option payoffs be replicated?
            Replicating Strategy
Buy ½ share of stock, borrow $35.78 (at the risk-free rate).



                                             Payoff
                                   (½)137 - (1.02) 35.78 = 32
          Cost
(1/2)100 - 35.78 = 14.22
                                                Payoff
                                       (½)73 - (1.02) 35.78 = 0


  The value of the option is $14.22!
Solving for the Replicating Strategy
The call option is equivalent to a levered position in the
stock (i.e., a position in the stock financed by borrowing).
                       137 H - 1.02 B = 32
                        73 H - 1.02 B = 0
Þ H (delta) = ½ = (C+ - C-)/(S+ - S-)
    B = (S+ H - C+ )/(1+ rF) = 35.78

Note: the value is (apparently) independent of probabilities
and preferences!
       Multi-Period Replication

      Stock          156.25                         51.25
                                   Call (E=105)
              125
                                        C+
100                  100                             0

              80                        C-
                     64                             0
 1-year call option, S=100, E=105, rF=1% (semi-annual)
 2 steps per year
            Solving Backwards
• Start at the end of the tree with each 1-step binomial
  model and solve for the call value 1 period before the
  end
             156.25             51.25        rF = 1%
 125                  C+
              100                0

• Solution: H = 0.911, B = 90.21 Þ C+ = 23.68
• C- = 0 (obviously?!)
                   The Answer
• Use these call values to solve the first 1-step binomial
  model

                125             23.68
     100                                rF = 1%
                 80             0
• Solution: H = 0.526, B = 41.68 Þ C = 10.94
• The multi-period replicating strategy has no intermediate
  cash flows
            Building The Tree
                S++   S+ = uS
    S+
                      S- = dS
S               S+-   S++ = uuS

        -             S-- = ddS
    S
                S--   S+- = S-+ = duS = S
               The Tree!
u =1.25, d = 0.8

                         156.25
                   125

     100                 100

                   80
                         64
        Binomial Replication
• The idea of binomial valuation via
  replication is incredibly general.
• If you can write down a binomial asset
  value tree, then any (derivative) asset
  whose payoffs can be written on this tree
  can be valued by replicating the payoffs
  using the original asset and a risk-free,
  zero-coupon bond.
      An American Put Option
What is the value of a 1-year put option with
exercise price 105 on a stock with current price
100?

The option can only be exercised now, in 6 months
time, or at expiration.

s = 31.5573%      rF = 1% (per 6-month period)
       Multi-Period Replication

      Stock         156.25                 0
                             Put (E=105)
              125
                                  P+
100                 100                    5

              80                  P-
                    64                     41
         Solving Backwards
            156.25               0       rF = 1%
125                  P+
            100                 5
      H = -0.089, B = -13.75 Þ P+ = 2.64

            100                  5
80                   P-                    rF = 1%
            64                   41
       H = -1, B = -103.96 Þ P- = 23.96 25!!
                                   -------
The put is worth more dead (exercised) than alive!
                The Answer


             125             2.64
100                                  rF = 1%
              80             25.00

      H = -0.497, B = -64.11 Þ P = 14.42
              Assignments
• Reading
  – RWJ: Chapters 8.1, 8.4, 22.12, 23.2, 23.4
  – Problems: 22.11, 22.20, 22.23, 23.3, 23.4,
    23.5
• Problem sets
  – Problem Set 1 due in 1 week

								
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