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					            Chapter 12


   The Black-Scholes-
     Merton Model


2011/9/25     Financial Engineering   1
  The Stock Price
  Assumption
     Consider a stock whose price is S
         dS=μSdt+σSdz
        dS/S= μdt+σdz
     ? dlnS= μdt+σdz No.
     From Ito’s lemma,
        dlnS= (μ- σ2/2)dt+σdz


2011/9/25          Financial Engineering   2
The Lognormal Property
     It follows from this assumption that




     Since the logarithm of ST is normal, ST is
      lognormally distributed

2011/9/25            Financial Engineering         3
  The Lognormal
  Distribution




2011/9/25   Financial Engineering   4
Continuously Compounded
Return (Equations 13.6 and 13.7), page 279)
  If x is the continuously compounded
  return




                                              5
The Expected Return

     The expected value of the stock price is
      S0emT
     The expected return on the stock is
       m – s2/2




2011/9/25          Financial Engineering    6
  m and m−s2/2
      Suppose we have daily data for a period
      of several months
     m is the average of the returns in each
      day [=E(DS/S)]
     m−s2/2 is the expected return over the
      whole period covered by the data
      measured with continuous
      compounding (or daily compounding,
      which is almost the same)
2011/9/25         Financial Engineering    7
  Mutual Fund Returns                     (See Business
  Snapshot 13.1 on page 281)

     Suppose that returns in successive
      years are 15%, 20%, 30%, -20% and
      25%
     The arithmetic mean of the returns is
      14%
     The returned that would actually be
      earned over the five years (the
      geometric mean) is 12.4%
2011/9/25         Financial Engineering              8
      The Volatility
    The volatility of an asset is the standard
     deviation of the continuously compounded
     rate of return in 1 year
    The standard deviation of the return in
     time Dt is
    If a stock price is $50 and its volatility is
     25% per year what is the standard
     deviation of the price change in one day?

    2011/9/25        Financial Engineering      9
Estimating Volatility from
Historical Data
 1. Take observations S0, S1, . . . , Sn at
   intervals of t years
 2. Define:


 3. Calculate the standard deviation, s , of
   the ui ´s
 4. The historical volatility estimate is:


2011/9/25           Financial Engineering      10
Nature of Volatility
   Volatility is usually much greater when
    the market is open (i.e. the asset is
    trading) than when it is closed
   For this reason time is usually
    measured in “trading days” not calendar
    days when options are valued



                                        11
  The Concepts Underlying
  Black-Scholes
           The option price & the stock price depend on the
            same underlying source of uncertainty
           We can form a portfolio consisting of the stock
            and the option which eliminates this source of
            uncertainty
           The portfolio is instantaneously riskless and must
            instantaneously earn the risk-free rate
           This leads to the Black-Scholes differential
            equation


2011/9/25                   Financial Engineering            12
        Assumptions of BS Formula
   The short-term interest rate is known and is
    constant through time.
   The stock price follows a random walk in continuous
    time with a variance rate proportional to the square
    of the stock price.Thus the distribution of stock
    prices is lognormal. The variance rate of the return
    on the stock is constant.
   The sock pays no dividends.
   The option is “European”.
   There are no transaction costs.
   It’s possible to borrow money to buy stocks.
   There are no penalties to short selling.
      2011/9/25          Financial Engineering       13
      The Derivation of the
1 of 3:

Black-Scholes Differential
Equation




 2011/9/25   Financial Engineering   14
        The Derivation of the
  2 of 3:

  Black-Scholes Differential
  Equation




2011/9/25    Financial Engineering   15
        The Derivation of the
  3 of 3:

  Black-Scholes Differential
  Equation
     The return on the portfolio must be the risk-free rate.
      Hence




     We substitute for    and     in these equations to get
      the Black-Scholes differential equation:




2011/9/25               Financial Engineering               16
The Differential Equation
   Any security whose price is dependent on the
    stock price satisfies the differential equation
   The particular security being valued is determined
    by the boundary conditions of the differential
    equation
   In a forward contract the boundary condition is
        ƒ = S – K when t =T
   The solution to the equation is
          ƒ = S – K e–r (T   –t)



    2011/9/25                Financial Engineering   17
      Risk-Neutral Valuation
   The variable m does not appear in the Black-
    Scholes equation
   The equation is independent of all variables
    affected by risk preference
   The solution to the differential equation is
    therefore the same in a risk-free world as it
    is in the real world
   This leads to the principle of risk-neutral
    valuation


    2011/9/25          Financial Engineering        18
  Applying Risk-Neutral
  Valuation
            1. Assume that the expected
              return from the stock price is
              the risk-free rate
            2. Calculate the expected
              payoff from the option
            3. Discount at the risk-free
              rate

2011/9/25            Financial Engineering     19
Valuing a Forward Contract with
Risk-Neutral Valuation

   Payoff is ST – K
   Expected payoff in a risk-neutral
    world is S0erT – K
   Present value of expected payoff is
         e-rT[S0erT – K]=S0 – Ke-rT



                                          20
  The Black-Scholes
  Formulas




2011/9/25   Financial Engineering   21
The N(x) Function
   N(x) is the probability that a normally
    distributed variable with a mean of zero
    and a standard deviation of 1 is less
    than x
   See tables at the end of the book




                                          22
Properties of Black-Scholes Formula

       As S0 becomes very large c tends to
        S0 – Ke-rT and p tends to zero

    •   As S0 becomes very small c tends to
        zero and p tends to Ke-rT – S0



                                              23
  BS公式的推导(1)




2011/9/25   Financial Engineering   24
  BS公式的推导(2)
将上述对ST的积分转换成对Q的积分,有:




2011/9/25   Financial Engineering   25
  BS公式的推导(3)




2011/9/25   Financial Engineering   26
       BS公式的解释
   S0N(d1)是Asset-or-noting call option的价值,
    -e-rTXN(d2)是X份cash-or-nothing看涨期权空
    头的价值。
   N(d2)是在风险中性世界中期权被执行的概率,
    或者说ST大于X的概率, e-rTXN(d2)是X的风险
    中性期望值的现值。 S0N(d1)是得到ST的风险
    中性期望值的现值。
               是复制交易策略中股票的数量,
    S0N(d1)就是股票的市值, -e-rTXN(d2)则是复
    制交易策略中负债的价值。

     2011/9/25      Financial Engineering     27
  Implied Volatility
     The implied volatility of an option is the
      volatility for which the Black-Scholes
      price equals the market price
     The is a one-to-one correspondence
      between prices and implied volatilities
     Traders and brokers often quote implied
      volatilities rather than dollar prices

2011/9/25          Financial Engineering     28
  Causes of Volatility
     Volatility is usually much greater when
      the market is open (i.e. the asset is
      trading) than when it is closed
     For this reason time is usually measured
      in “trading days” not calendar days
      when options are valued



2011/9/25         Financial Engineering    29
The VIX S&P500 Volatility Index




    Chapter 24 explains how the index is calculated
                                                      30
     Warrant Valuation
The analysis of warrants is much more complicated
  than that of options, because:
 The life of a warrant is typically measured in years,
  rather than in months, so the variance rate may
  change substantially.
 The Exercise price of the warrant is usually not
  adjusted at all for dividends.
 The exercise price of a warrant sometimes changes
  on specified dates.
 If the company is involved in a merger, the
  adjustment that is made in the terms of the warrant
  may change its value.
 The exercise of a large number of warrants may
  sometimes result in a significant increase in the
  number of common shares outstanding.
     2011/9/25           Financial Engineering       31
       Warrants & Dilution
   When a regular call option is exercised the stock
    that is delivered must be purchased in the open
    market
   When a warrant is exercised new stock is issued
    by the company
   If little or no benefits are foreseen by the market
    the stock price will reduce at the time the issue of
    is announced.
   There is no further dilution (See Business
    Snapshot 13.3.)

     2011/9/25           Financial Engineering         32
  Warrant Valuation
     某公司有N股普通股和M份欧式认股权证,
      每份权证可以在T时刻按每股X价格购买b
      股股票.令S表示公司股票价格,则认股
      权证被行使后股票的除权价格为:

  认股权证持有者的回报为:


2011/9/25   Financial Engineering   33
  Dividends
     European options on dividend-paying
      stocks are valued by substituting the
      stock price less the present value of
      dividends into Black-Scholes
     Only dividends with ex-dividend dates
      during life of option should be included
     The “dividend” should be the expected
      reduction in the stock price expected

2011/9/25          Financial Engineering    34
    American Calls

   An American call on a non-dividend-paying
    stock should never be exercised early
   An American call on a dividend-paying stock
    should only ever be exercised immediately
    prior to an ex-dividend date




2011/9/25           Financial Engineering         35
  Black’s Approach to Dealing with
  Dividends in American Call Options


     Set the American price equal to the
    maximum of two European prices:
    1. The 1st European price is for an
    option maturing at the same time as the
    American option
    2. The 2nd European price is for an
    option maturing just before the final ex-
    dividend date

2011/9/25           Financial Engineering       36
                      h(Q )dQ
            (ln X  m ) /     T                                             (ln X  m ) /     T

                                                 1 (  Q 2  2
        e        T Qm
                              h(Q )                e                      T Q2 m ) / 2

                                                 2
               1 [  ( Q              T ) 2  2T  2 m ] / 2
                 e
               2
         e m T / 2 [  ( Q  T )2 ] / 2
                     2

                                            e m T / 2 h(Q   T )
                                                  2
                   e
             2
          ˆ
        E[max( ST  X ,0)]
                                                                                                      
        e    m  2T / 2
                                                   h(Q   T )dQ  X                                               h(Q )dQ
                              (ln X  m ) /    T                                              (ln X  m ) /    T


2011/9/23                                               Financial Engineering                                                  25
  BS公式的推导(3)
        
                    h(Q   T )dQ  1  N [(ln X  m) /  T   T ]
 (ln X  m ) /  T

                                                  ˆ
                                              ln[ E ( ST ) / X ]   2 / 2
  N [( ln X  m) /  T   T ]  N [                                     ]
                                                         T
      ln( S0 / X )  (r   2 / 2)T
  N[                               ]  N (d1 )
                        T
                                ln( S0 / X )  (r   2 / 2)T
 同样,  h(Q)dQ  N [                                              ]  N (d 2 )
       (ln X  m ) /                         T
                                                             ˆ
   ˆ [max( ST  X ,0)]  e m T N (d1 )  XN (d 2 )  E ( ST ) N (d1 )  XN (d 2 )
                                 2
 E
  S0e rT N (d1 )  XN (d 2 )
  c  S0 N (d1 )  e rT XN (d 2 )


2011/9/23                              Financial Engineering                          26
       BS公式的解释
   S0N(d1)是Asset-or-noting call option的价值,
    -e-rTXN(d2)是X份cash-or-nothing看涨期权空
    头的价值。
   N(d2)是在风险中性世界中期权被执行的概率,
    或者说ST大于X的概率, e-rTXN(d2)是X的风险
    中性期望值的现值。 S0N(d1)是得到ST的风险
    中性期望值的现值。
    D  N (d1 )是复制交易策略中股票的数量,
    S0N(d1)就是股票的市值, -e-rTXN(d2)则是复
    制交易策略中负债的价值。

     2011/9/23      Financial Engineering     27
  Implied Volatility
     The implied volatility of an option is the
      volatility for which the Black-Scholes
      price equals the market price
     The is a one-to-one correspondence
      between prices and implied volatilities
     Traders and brokers often quote implied
      volatilities rather than dollar prices

2011/9/23          Financial Engineering     28
  Causes of Volatility
     Volatility is usually much greater when
      the market is open (i.e. the asset is
      trading) than when it is closed
     For this reason time is usually measured
      in “trading days” not calendar days
      when options are valued



2011/9/23         Financial Engineering    29
The VIX S&P500 Volatility Index




    Chapter 24 explains how the index is calculated
                                                      30
     Warrant Valuation
The analysis of warrants is much more complicated
  than that of options, because:
 The life of a warrant is typically measured in years,
  rather than in months, so the variance rate may
  change substantially.
 The Exercise price of the warrant is usually not
  adjusted at all for dividends.
 The exercise price of a warrant sometimes changes
  on specified dates.
 If the company is involved in a merger, the
  adjustment that is made in the terms of the warrant
  may change its value.
 The exercise of a large number of warrants may
  sometimes result in a significant increase in the
  number of common shares outstanding.
     2011/9/23           Financial Engineering       31
       Warrants & Dilution
   When a regular call option is exercised the stock
    that is delivered must be purchased in the open
    market
   When a warrant is exercised new stock is issued
    by the company
   If little or no benefits are foreseen by the market
    the stock price will reduce at the time the issue of
    is announced.
   There is no further dilution (See Business
    Snapshot 13.3.)

     2011/9/23           Financial Engineering         32
  Warrant Valuation
     某公司有N股普通股和M份欧式认股权证,
      每份权证可以在T时刻按每股X价格购买b
      股股票.令S表示公司股票价格,则认股
      权证被行使后股票的除权价格为:
                                                NST  MbX
                                                 N  Mb
  认股权证持有者的回报为:
             NST  MbX              Nb
      max[b(            X ), ] 
                            0            max(ST  X , )
                                                    0
              N  Mb              N  Mb

2011/9/23               Financial Engineering               33
  Dividends
     European options on dividend-paying
      stocks are valued by substituting the
      stock price less the present value of
      dividends into Black-Scholes
     Only dividends with ex-dividend dates
      during life of option should be included
     The “dividend” should be the expected
      reduction in the stock price expected

2011/9/23          Financial Engineering    34
    American Calls

   An American call on a non-dividend-paying
    stock should never be exercised early
   An American call on a dividend-paying stock
    should only ever be exercised immediately
    prior to an ex-dividend date




2011/9/23           Financial Engineering         35
  Black’s Approach to Dealing with
  Dividends in American Call Options


     Set the American price equal to the
    maximum of two European prices:
    1. The 1st European price is for an
    option maturing at the same time as the
    American option
    2. The 2nd European price is for an
    option maturing just before the final ex-
    dividend date

2011/9/23           Financial Engineering       36

				
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