Chapter 12
The Black-ScholesMerton Model
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The Stock Price
Assumption

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
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
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The Lognormal Property

It follows from this assumption that

 2
~  
  2



T , 


T 



 2
~   ln S 0  
  2




T , 


T 

ln S T  ln S 0
or
ln S T

Since the logarithm of ST is normal, ST is
lognormally distributed
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The Lognormal
Distribution
E (ST )  S0 e
T
var ( S T )  S 0 e
2
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2 T
2
(e
 T
 1)
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Continuously Compounded
Return (Equations 13.6 and 13.7), page 279)
If x is the continuously compounded
return
ST  S 0 e
1
ST
x=
ln
T
S0
xT
2
2


 

x  
  2 , T 


5
The Expected Return


The expected value of the stock price is
S0eT
The expected return on the stock is
 – 2/2
E  ln( ST / S 0 )  （   / 2）T
2
ln  E ( ST / S 0 )   T
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 and −2/2


Suppose we have daily data for a period
of several months
 is the average of the returns in each
day [=E(DS/S)]
−2/2 is the expected return over the
whole period covered by the data
measured with continuous
compounding (or daily compounding,
which is almost the same)
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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%
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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  D t
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?
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Estimating Volatility from
Historical Data
1. Take observations S0, S1, . . . , Sn at
intervals of t years
2. Define:
 Si
u i  ln 
 S i 1




3. Calculate the standard deviation, s , of
the ui ´s
s
4. The historical volatility estimate is:  * 
t
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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




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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
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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.
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The Derivation of the
Black-Scholes Differential
Equation
1 of 3:
DS  S Dt  S Dz
2
 ƒ
ƒ
 ƒ 2 2
ƒ
Dƒ 
S 
½
 S  Dt 
S Dz
2
t
S
S
 S

W e se t u p a p o rtfo lio co n sistin g o f
 1: d e riva tive
+
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ƒ
S
: sh a re s
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The Derivation of the
Black-Scholes Differential
Equation
2 of 3:
T h e va lu e o f th e p o rtfo lio  is give n b y
  ƒ
ƒ
S
S
T h e ch a n ge in its va lu e in tim e D t is give n b y
D   D ƒ 
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ƒ
S
DS
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The Derivation of the
Black-Scholes Differential
Equation
3 of 3:

The return on the portfolio must be the risk-free rate.
Hence
D  rD t
  (t )   (0 ) e
rt
ln  ( t )  ln  ( 0 )  rt
d

 rdt

We substitute forD f andD S in these equations to get
the Black-Scholes differential equation:
2

f
2
2
1
 rS

 S
 rf
2
2
t
S
S
f
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f
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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
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–t)
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Risk-Neutral Valuation




The variable  does not appear in the BlackScholes 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
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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
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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
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The Black-Scholes
Formulas
c  S0 N (d1 )  X e
p X e
w h e re
 rT
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N (d 2 )
N ( d 2 )  S0 N ( d1)
d1 
d2 
 rT
ln( S 0 / X )  ( r  
2
/ 2)T
 T
ln( S 0 / X )  ( r  
2
 T
Financial Engineering
/ 2)T
 d1   T
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）
在风险中性世界中
, 假定 :
dS  rSdt   Sdz
ln S T


 2
~   ln S 0   r 
T , 
2 


根据风险中性定价原理
c  e
 rT

T 

:
 rT
Eˆ [max( S T  X , 0 )]  e


X
(S T  X ) g ( S T ) dS T

 
 r 
T
2 

2
ln S T 的均值 m 等于 : m  Eˆ ln( S T )  ln S 0
令Q 
ln S T  m

T
显然 , Q ~  ( 0 ,1), 其概率密度函数为
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Financial Engineering
: h (Q ) 
1
2
e
Q
2
/2
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：
Eˆ [max( S T  X , 0 )] 


e

(ln X  m ) / 
e

TQm
1

e
2

TQm

(e

(ln X  m ) / 
h ( Q ) dQ  X
[  ( Q 
1
2
2
2
e
TQm
 X ) h ( Q ) dQ
T
T
h (Q ) 

(  Q  2

h ( Q ) dQ

(ln X  m ) / 
T
TQ2m )/ 2
2
T )  T  2 m ] / 2
2

e
m  T / 2
2
e
[  ( Q 
2
T ) ]/ 2
e
2
m  T / 2
h (Q  
T)
 Eˆ [max( S T  X , 0 )]
e
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2
m  T / 2

h (Q  

(ln X  m ) / 
T ) dQ  X
T
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
h ( Q ) dQ

(ln X  m ) / 
T
25
BS公式的推导（3）

h (Q  

(ln X  m ) / 
T ) dQ  1  N [(ln X  m ) / 
T 
T]
T
 N [(  ln X  m ) / 
T 
T ]  N[
ln( S 0 / X )  ( r   / 2 )T
2
ln[ Eˆ ( S T ) / X ]   / 2

]
T
2
 N[
同样 ,



(ln X  m ) / 
T
]  N (d1 )
ln( S 0 / X )  ( r   / 2 )T
2
h ( Q ) dQ  N [
m 
 Eˆ [max( S T  X , 0 )]  e

2
T
T
]  N (d 2 )
N ( d 1 )  XN ( d 2 )  Eˆ ( S T ) N ( d 1 )  XN ( d 2 )
 S 0 e N ( d 1 )  XN ( d 2 )
rT
 c  S 0 N (d1 )  e
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 rT
XN ( d 2 )
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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 ( d 1 )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。
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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
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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
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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
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number
of common shares
outstanding.
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.)
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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:
NS T  MbX
认股权证持有者的回报为:
max[ b(
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NS T  MbX
N  Mb
 X ),0 ] 
Nb
N  Mb
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N  Mb
max( S T  X ,0 )
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
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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
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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 exdividend date
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