Black-Scholes Model

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					Black-Scholes Model
      Mark Oden
      MAP4413
    March 10th, 2006
               Call Options
 Gives the buyer the
  right to take stocks
  away from the seller
  at the exercise price
 The contract
  increases in value as
  the stock approaches
  or passes the
  exercise price
Put Options
       The buyer of the
        contract has the right
        to sell stocks to the
        seller of the contract
        at the exercise price
       The contract
        increases in value as
        the stock retreats
        from the exercise
        price
            Black-Scholes
 Fischer Black and Myron Scholes
  published the derivation and equation of
  the Black-Scholes Option Pricing Model in
  1973 under several assumptions.
 Bottom line: options of traded stocks are
  fundamentally priced
The Black-Scholes PDE:
V 1   2  V       V          2
   S         rS     rV  0
t 2     S 2
                   S
Call Option Pricing Model:

                   ln( S / K )  (r   2 / 2)T            ln( S / K )  (r   2 / 2)T
C ( S , T )  SN (                                     rT
                                                )  Ke N (                                      T)
                               T                                      T


Put Option Pricing Model:

                                  ln( S / K )  (r   2 / 2)T          ln( S / K )  (r   2 / 2)T
P( S , T )  Ke    rT
                         N(   T                              )  SN (                              )
                                              T                                  T
Example
                   Negative Affects
 Myron Scholes was a partner of Long-Term Capital Management
    (LTCM) established in 1994.
   LTCM would use models and experienced investing knowledge to
    pick differences in short and long securities, profiting from the small
    margin.
   In 1998 the firm had $5B in equity and $125B in liability
   August 17, 1998 Russia devalues the rouble and suspended $13.5B
    of it’s Treasury Bonds, causing LTCM’s equity to decrease
    drastically, ever increasing it’s leverage ratio
   In order to prevent a stock market crash due to the outflux of
    investors and banks from LTCM’s funds the Federal Reserve
    organized a $3.5 B rescue effort to take over LTCM’s management
   Had LTCM lost leverage, all banks and creditors would pull out of
    other firms – including Salomon Brothers, Merrill Lynch.
                             Bibliography
   “ASX – Australian Stock Exchange.” http://www.asx.com.au/. Tuesday, March 7,2006.
   “Black-Scholes.” http://en.wikipedia.org/wiki/Black-Scholes. Tuesday, March 7, 2006.
   “Case Study – LTCM.” http://www.erisk.com/Learning/CaseStudies/ref_case_ltcm.asp. Tuesday,
    March 7, 2006.
   Shah, Ajaj. “Black, Merton and Scholes:Their work and its consequences.”
    http://www.mayin.org/ajayshah/PDFDOCS/Shah1997_bms.pdf. Tuesday, March 7, 2006.
   “YHOO: Options for YAHOO INC.” http://finance.yahoo.com/q/op?s=YHOO&m=2007-01.
    Tuesday, March 7, 2006.

				
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