Options Black Scholes Pricing Related Models Option Valuation and Scholes

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```					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
 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
 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
 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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