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Options Black Scholes Pricing Related Models Option Valuation and Scholes

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Options Black Scholes Pricing Related Models Option Valuation and Scholes Powered By Docstoc
					Black-Scholes Pricing
 & Related Models
          Option Valuation
      and Scholes
 Black
  Call Pricing
  Put-Call Parity
  Variations
Option Pricing: Calls
 Black-Scholes        Model:
       C  S N ( d )  X e rT N ( d )    C = Call
                  1                 2
                                          S = Stock Price
                                          N = Cumulative Normal
                S   
                            2
             ln     r 
                      
                              T                  Distrib. Operator
                X        2
       d                                 X = Exercise Price
        1           T
                                          e = 2.71.....
                                          r = risk-free rate
       d  d  T
        2 1                               T = time to expiry
                                           = Volatility
       N ( d )  Probabilty of Exercise
 Call Option Pricing Example
    IBM is trading for $75. Historically, the volatility is 20% (). A call
     is available with an exercise of $70, an expiry of 6 months, and
     the risk free rate is 4%.

        ln(75/70) + (.04 + (.2)2/2)(6/12)
d1 = -------------------------------------------- = .70, N(d1) = .7580
                 .2 * (6/12)1/2

d2 = .70 - [ .2 * (6/12)1/2 ] = .56, N(d2) = .7123

C = $75 (.7580) - 70 e -.04(6/12) (.7123) = $7.98
Intrinsic Value = $5, Time Value = $2.98
Put Option Pricing
   Put priced through
    Put-Call Parity:
    Put Price = Call Price + X e-rT - S
    (or : P = Xe - rT N ( - d ) - SN ( - d ) )
                             2            1
     From Last Example of IBM Call:

     Put     =      $7.98 + 70 e -.04(6/12) - 75
             =      $1.59
     Intrinsic Value = $0, Time Value = $1.59
     Black-Scholes Variants
 Options on Stocks with Dividends
 Futures Options
  (Option that delivers a maturing futures)
   Black’s Call Model (Black (1976))
   Put/Call Parity
 Options on Foreign Currency
   In text (Pg. 375-376, but not req’d)
   Delivers spot exchange, not forward!
   The Stock Pays no Dividends
     During the Option’s Life
 Ifyou apply the BSOPM to two
  securities, one with no dividends and
  the other with a dividend yield, the
  model will predict the same call
  premium
 Robert Merton developed a simple
  extension to the BSOPM to account
  for the payment of dividends
  The Stock Pays Dividends During
      the Option’s Life (cont’d)
Adjust the BSOPM by following (=continuous dividend yield):
                         T                    RT
          C   *
                   e          SN ( d )  Xe
                                       1
                                        *                  *
                                                      N (d 2 )
          where
                        S            2              
                    ln     R   
                                                       T
                                                        
                        X            2               
          d 1*    
                               T
          and
          d   *
              2
                   d 1*         T
Futures Option Pricing Model
 Black’sfutures option pricing model
 for European call options:

       C  e  RT FN ( a )  KN (b)
                       F     2
                    ln      T
       where a        K   2
                          T
       and     b  a  T
Futures Option Pricing Model
          (cont’d)
 Black’sfutures option pricing model
 for European put options:

       P  e  RT KN (b)  FN (a)
               value the put option
 Alternatively,
 using put/call parity:
                        RT
            P  C e          (F  K )
    Assumptions of the Black-
        Scholes Model
 European   exercise style
 Markets are efficient
 No transaction costs
 The stock pays no dividends during
  the option’s life (Merton model)
 Interest rates and volatility remain
  constant, but are unknown
Interest Rates Remain Constant
 There   is no real “riskfree” interest
  rate
 Often use the closest T-bill rate to
  expiry
               Calculating
           Volatility Estimates
 from   Historical Data:
 S, R, T that just was, and  as standard deviation of
 historical returns from some arbitrary past period

 from   Actual Data:
 S, R, T that just was, and  implied from pricing of
 nearest “at-the-money” option (termed “implied
 volatility).
  Intro to Implied Volatility
 Instead  of solving for the call
  premium, assume the market-
  determined call premium is correct
 Then solve for the volatility that
  makes the equation hold
 This value is called the implied
  volatility
 Calculating Implied Volatility
 Setup  spreadsheet for pricing “at-the-
  money” call option.
 Input actual price.
 Run SOLVER to equate actual and
  calculated price by varying .
         Volatility Smiles
           smiles are in contradiction
 Volatility
  to the BSOPM, which assumes
  constant volatility across all strike
  prices
 When you plot implied volatility
  against striking prices, the resulting
  graph often looks like a smile
                              Volatility Smiles (cont’d)
                                                 Volatility Smile
                                              Microsoft August 2000

                         60
                                                       Current Stock
                                                       Price
                         50
Implied Volatility (%)




                         40


                         30


                         20


                         10


                         0
                              40   45   50   55   60   65   70   75    80   85   90   95   100   105
                                                        Striking Price
       Problems Using the Black-
             Scholes Model
   Does not work well with options that are deep-in-
    the-money or substantially out-of-the-money

   Produces biased values for very low or very high
    volatility stocks
     Increases as the time until expiration increases

   May yield unreasonable values when an option
    has only a few days of life remaining

				
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