# The Black Scholes Model FINANCE by liaoqinmei

VIEWS: 14 PAGES: 36

• pg 1
```									            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
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）

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
 The exercise price of a warrant sometimes changes
on specified dates.
 If the company is involved in a merger, the
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
 The exercise price of a warrant sometimes changes
on specified dates.
 If the company is involved in a merger, the
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

```
To top