Download [SC]5.2.2~5.2.5

5.2Risk-Neutral Measure
Part 2
報告者：陳政岳
5.2.2 Stock Under the Risk-Neutral
Measure
• W (t ), 0  t  T is a Brownian motion on a
probability space (, F , ) , and F t  , 0  t  T ,
is a filtration for this Brownian motion. T is a
fixed final time.
• A stock price process whose differential is
dS  t    t  S t  dt   t  S t  dW t  ,0  t  T
where  t  :the mean rate of return and   t 
:the volatility are adapted processes.
Assume that,t 0, T  , t  is almost surely
not zero.
• The stock price is a generalized Brown
motion (see Example 4.4.8), and an
equivalent way of writhing is
t
 t
1 2
 
S  t   S  0  exp     s  dW ( s)      s     s   ds  .
0
2

 
0
• Supposed we have an adapted interest
rate process R(t). We define the discount
process
t
  R  s  ds
D t   e 0
and
dD t   R t  D t  dt.
• Define I  t   0 R  s  ds so that dI t   R t  dt
and dI t  dI t   0.
x
f
x

e
,


• First, we introduction the function
for which f   x    f  x  , f   x   f  x  , and
use the Ito-Doeblin formula to write
dD  t   df  I  t  
t
1
 f   I  t   dI  t   f   I  t   dI  t  dI  t 
2
  f  I  t   R  t  dt
  R  t  D  t  dt
• Although D(t) is random, it has zero
quadratic variation. This is because it is
“smooth.” Namely, D(t )   R(t ) D(t ) one
does not need stochastic calculus to do
this computation.
• The stock price S(t) is random and has
nonzero quadratic variation. If we invest in
the stock, we have no way of knowing
whether the next move of the driving
Brownian motion will be up or down, and
this move directly affects the stock price.
• Considering a money market account with
variable interest rate R(t), where money is
rolled over at this interest rate. If the price
of a share of this money market account at
time zero is 1, then the price of a share of
this money market account at time t is
t
R s  ds
1

0
e

.
D(t )
• If we invest in this account, over short
period of time we know the interest rate at
the time of the investment and have a high
degree of certainty about what the return.
• Over longer periods, we are less certain
because the interest rate is variable, and at
the time of investment, we do not know the
future interest rates that will be applied.
• The randomness in the model affect the
money market account only indirectly by
affecting the interest.
• Changes in the interest rate do not affect
the money market account instantaneously
but only when they act over time.( Warning: .
For a bond, a change in the interest rate
can have an instantaneous effect on price.)
• Unlike the price of the money market
account, the stock price is susceptible to
instantaneous unpredictable changes and is,
in this sense, “more random” than D(t).
Because S(t) has nonzero quadratic
variation, D(t) has zero quadratic variation.
• The discounted stock price process is
t
 t
1 2  
D  t  S  t   S  0  exp    s  dW  s       s   R  s     s   ds 
0
2

 
0
(5.2.19) and its differential is
d  D  t  S  t      t   R  t   D  t  S  t  dt    t  D t  S t  dW t 
(5.2.20)
 t   R t 

  t  D t  S t  
dt  dW  t  
   t 

   t  D  t  S  t    t  dt  dW  t  
where we define the market price of risk to
be   t     t   R  t 
 t 
• (5.2.20) can derive either by applying the
Ito-Doblin formula or by using Ito product
rule.
• The first line of (5.2.20), compare with
dS  t    t  S t  dt   t  S t  dW t  ,
shows that the mean rate of return of the
discounted stock price is  t   R t , which is
the mean rate   t  of the undiscounted
stock price, reduced by the interest rate
R(t).
• The volatility of the discounted stock price
is the same as the volatility of the
undiscounted stock price.
• The probability measure P defined in
Girsanov’s Theorem, Theorem 5.2.3, which
 t   R t 
uses the market price of risk
 t  
 t 
In terms of the Brownian motion of that
theorem, we rewrite (5.2.20) as
d  D  t  S  t      t  D  t  S  t  dW  t  .
(5.2.22)
• We call P , the measure defined in
Girsanov’s Theorem, the risk-neutral
measure because it is equivalent to the
original measure P
.
• According to (5.2.22),
t
D  t  S  t   S  0    u  D u  S u dW u  ,
0
t
and P under the process 0  u  D u  S u dW u 
is an Ito-integral and is a martingale.
• The undiscounted stock price S(t) has
mean rate of return equal to the interest
rate under P , as one can verify by making
the replacement dW t    t  dt  dW t  in
dS  t    t  S t  dt   t  S t  dW t  .
 dS  t   R t  S t  dt   t  S t  dW t  .
• We can solve this equation for S(t) by
simply replacing the Ito integral    s ds
t
t
by its equivalent 0   s dW  s   0   s   R  s ds
in S  t   S  0 exp 0t   s  dW  s   0t    s   1  2  s   ds 
2

 

to obtain the formula
t
0
t
 t
1 2
 
S  t   S  0  exp     s  dW  s     R  s     s   ds 
0
2

 
0
• In discrete time: the change of measure
does not change the binomial tree, only
the probabilities on the branches of the
tree.
• In continuous time, the change from the
actual measure P to the risk-neutral
measure P change the mean rate of return
of the stock but not the volatility.
• After the change of measure , we are still
considering the same set of stock price
paths, but we shifted the probability on
them.
• If  t   R t  , the change of measure puts
more probability on the paths with lower
return so return so that the overall mean
rate of return is reduced from   t  to R(t)
5.2.3 Value of Portfolio Process
Under the Risk-Neutral Measure
• Initial capital X(0) and at each time t, 0  t  T ,
holds   t  shares of stock, investing or
borrowing at the R(t).
• The differential of this portfolio value is
given by the analogue of (4.5.2)
dX  t   rX t  dt   t   r  S t  dt   t  S t  dW t 
5.2.3 Value of Portfolio Process
Under the Risk-Neutral Measure
• dX  t     t  dS  t   R t   X t    t  S t   dt
   t    t  S  t  dt    t  S  t  dW  t    R  t   X  t     t  S  t   dt
 R  t  X  t  dt    t    t   R  t   S  t  dt    t    t  S  t  dW  t 
 R  t  X  t  dt    t    t  S  t     t  dt  dW  t  
• By Ito product rule, dD t   R t  D t  dt
(5.2.18) and d  D  t  S  t      t  D  t  S t   t  dt  dW t 
(5.2.20) imply
5.2.3 Value of Portfolio Process
Under the Risk-Neutral Measure
• d  D  t  X t    X t  dD t   D t  dX t   dD t  dX t 
 X  t    D  t  R  t  dt  
D  t   R  t  X  t  dt    t    t  S  t     t  dt  dW  t    
  D t  R t  dt   R t  X t  dt   t  t  S t    t  dt  dW t 
   t    t  D  t  S  t    t  dt  dW  t 
  t  d  D t  S t 
5.2.3 Value of Portfolio Process
Under the Risk-Neutral Measure
• Changes in the discounted value of an
agent’s portfolio are entirely due to
fluctuations in the discounted stock price.
We may use
d  D  t  S  t      t  D  t  S  t  dW  t 
(5.2.22)to rewrite as
d  D  t  X  t      t    t  D  t  S  t  dW  t  .
5.2.3 Value of Portfolio Process
Under the Risk-Neutral Measure
• We has two investment options:
(1) a money market account with rate of
return R(t),
(2) a stock with mean rate of return R(t)
under P.
• Regardless of how the agent invests, the
mean rate of return for his portfolio will be
R(t) under P, and the discounted value of
his portfolio, D(t)X(t), will be a martingale.
5.2.4 Pricing Under the RiskNeutral Measure
• In Section 4.5, Black-Scholes-Merton
equation for the value the European call
have initial capital X(0) and portfolio   t 
process an agent would need in order
hedge a short position in the call.
• In this section, we generalize the question.
5.2.4 Pricing Under the RiskNeutral Measure
• Let V(T) be an F(T)-measurable random
variable. This payoff is path-dependent
which is F(T)-measurability.
• Initial capital X(0) and portfolio process
  t  , 0  t  T , we wish to know that an agent
would need in order to hedge a short
position, i.e., in order to have
X(T) = V(T) almost surely.
• We shall see in the next section that this
can be done.
5.2.4 Pricing Under the RiskNeutral Measure
• In section 4.5, the mean rate of return,
volatility, and interest rate were constant.
• In this section, we do not assume a
constant mean rate of return, volatility, and
interest rate.
• D(T)X(T) is a martingale under implies
D  t  X  t   E  D T  X T  F  t  
 E  D T V T  F  t   .
5.2.4 Pricing Under the RiskNeutral Measure
• The value X(t) of the hedging portfolio is
the capital needed at time t in order to
complete the hedge of the short position in
the derivative security with payoff V(T).
• We call the price V(t) of the derivative
security at time t, and the continuous-time
of the risk-neutral pricing formula is
D  t  X  t   E  D T V T  F  t  , 0  t  T .
(5.2.30)
5.2.4 Pricing Under the RiskNeutral Measure
• Dividing (5.2.30) by D(t), which is F(t)measurable. We may write (5.2.30) as
  t
V  t   E e

T

V  t  F  t  ,0  t  T .

R u  du
(5.2.31)
• We shall refer to both (5.2.30) and (5.2.31)
as the risk-neutral pricing formula for the
continuous-time model.
5.2.5 Deriving the Black-ScholesMerton Formula
• To obtain the Black-Scholes-Merton price
of a European call, we assume constant
volatility  , constant interest rate r, and
take the derivative security payoff to be
V T    S T   K  .

  t
side of V  t   E e

  Ru du
T
• The right-hand
T
t
V
t

E
e
becomes   

 S T   K 

V  t  F t 


R u  du
F  t  .

5.2.5 Deriving the Black-ScholesMerton Formula
• Because geometric Brownian motion is a
Markov process, this expression depends
on the stock price S(t) and on the time t at
which the conditional expectation is
computed, but not on the stock price prior
to time t.
• There is a function c(t,x) such that

 r T t 

c  t , S  t    E e
S T   K  F  t   .



5.2.5 Deriving the Black-ScholesMerton Formula
• Computing c(t,x) using the Independence
Lemma, Lemma 2.3.4.
• Lemma 2.3.4 (Independence)
Let  , F , P  be a probability space, and let G
be a sub-  -algebra of F. Suppose the
random variables X1 , , X K are G-measurable
and the random variables Y1 , , YL are
independent of G. Let f  x1 , , xK , y1, , yL  be
a function of the dummy variables x1 , , xK
and y1 , , yL , and define
g  x1 , , xK   Ef  x1 , , xK , Y1 , , YL  .
Then
E  f  X 1 ,
, X K , Y1 ,
, YL  G   g  X 1 ,
, X K .
5.2.5 Deriving the Black-ScholesMerton Formula
• With constant  and r, equation
t
 t
1 2
 
S  t   S  0  exp     s  dW  s     R  s     s   ds 
0
2

 
0

1 2 

S  t   S  0  exp  W  t    r    t  ,
2  


becomes
and then rewrite

1 2 

S T   S  t  exp  W T   W  t    r    
2  



1 2 

 S  t  exp   Y   r     ,
2  




• where Y is the standard normal random
W T   W  t 
variableY  
, and  is the “time
T t
to expiration” T-t.
• S(T) is the product of the F(t)-measurable
random variable S(t) and the random
variablewhich is independent of F(t).
• Therefore,

 r T t 

c  t , S  t    E e
S T   K  F  t  




 
 

 1 2 
 r
 E e  x exp   Y   r      K  
 2  
 

 
1

2

1 2

y




1

 r
2
2
 e  x exp   y   r  2     K  e dy

5.2.5 Deriving the Black-ScholesMerton Formula




1


2
• The integrand  x exp   Y   r      K 
2  




is positive if and only if
1 
x 
1 2 
y  d   , x  
log K   r  2    .
  

 

5.2.5 Deriving the Black-ScholesMerton Formula
• Therefore,
1
c t, S t  
2
1

2

d   , x 


d   , x 

1
 y2




1


e r  x exp  y   r   2    K e 2 dy
 2  



 1 2
 2 
x exp  y    y 
 dy
2 
 2
1 d  , x   r  12 y 2

e Ke dy


2
x d  , x 
 1

exp  y   


2
 2


2

 r
dy

e
KN  d   , x  


 xN  d   , x    e  r KN  d   , x  
5.2.5 Deriving the Black-ScholesMerton Formula
• where d   , x   d   , x    
1 
x 
1 2 

log K   r  2    .
  

 
• So, the notation
BSM  , x; K , r ,   

E e  r




1 2 

 x exp   Y   r      K 
2  




• where Y is a standard normal random
variable under P.


,

5.2.5 Deriving the Black-ScholesMerton Formula
• We have shown that
BSM ( , x; K , r ,  ) 
xN  d  , x    e r KN  d   , x   .
• Here we have derived the solution by
device of switching to the risk-neutral
measure.