Download 5.4 Fundamental Theorems of Asset Pricing

5.4 Fundamental Theorems of
Asset Pricing
報告者：何俊儒
Introductions
• Extending the discussions of above sections to
the case of multiple stocks driven by multiple
Brownian motions
• Developing and illustrating the two
fundamental theorems of asset pricing
• Giving the precise definitions of the basic
concepts of derivative security in continuoustime models
5.4.1 Girsanov and Martingale
Representation Theorems
• Throughout this section,
W (t )  (W1 (t ), ,Wd (t ))
is a multidimensional Brownian motion on a
probability space (, F , P ) .
• Note that:
– P is the actual probability measure
– Associated with this Brownian motion, we have a
filtration F(t)
Theorem 5.4.1 (Girsanov, multiple dimension)
• Let T be a fixed positive time, and let
(t )  (1 (t ), , d (t )) be a d-dimensional adapted
process. Define t
t
1
2
Z (t )  exp{  (u )  dW (u )   (u ) du},
0
2 0
t
W (t )  W (t )   (u )du ,
0
and assume that
t
E  (u) Z (u)du  
0
2
2
(5.4.1)
Theorem 5.4.1 (Girsanov, multiple dimension)
• Set Z = Z(t). Then E(Z) = 1, and under the
probability measure P given by
P( A)   Z ()  dP() for all A  F ,
A
the process W (t ) is a d-dimensional Brownian
motion
Remark of the Theorem 5.4.1
• The Ito integral in (5.4.1) is
 (u)  dW (u)   
t
t
d
0
0
j 1
 j (u)dW j (u)   j 1   j (u)dW j (u)
d
t
0
• In (5.4.1), (u) denotes the Euclidean norm
(u )  ( j 1  (u))
• (5.4.2) is shorthand notation for
with
d
t
2
j
Wj (t )  W j (t )    j (u)du,
0
1/ 2
j  1,
,d
The conclusions of the multidimensional
Girsanov Theorem
• The component process of W (t ) are
independent under P
– The component process of W (t ) are independent
under P, but each of the  j (u ) processes can
depend in a path-dependent, adapted way on all
of the Brownian motions W1 (t ), ,Wd (t )
– Under P, the components of W (t ) can be far from
independent, however, after the change to the
measure P , these components are independent
• The proof of Theorem 5.4.1 is like that of the
one-dimensional Girsanov Theorem
Theorem 5.4.2
(Martingale representation, multiple dimensions )
• Let T be a fixed positive time, and assume that
F(t), 0  t  T , is the filtration generated by the
d-dimensional Brownian motion W(t), 0  t  T.
Let M(t), 0  t  T , be a martingale with respect
to this filtration under P. Then there is an
adapted, d-dimensional process (u)  ( (u), ,  (u)),
0  t  T , such that
1
t
M (t )  M (0)   (u)dW (u), 0  t  T .
0
d
Theorem 5.4.2
(Martingale representation, multiple dimensions )
• If, in addition, we assume the notation and
assumptions of Theorem 5.4.1 and if M (t ), 0  t  T ,
is a P -martingale, then there is an adapted, ddimensional process (u )  (1 (u ), ,  d (u)) such
that
t
M (t )  M (0)   (u)dW (u), 0  t  T .
0
5.4.2 Multidimensional Market Model
• We assume there are m stocks, each with
stochastic differential
dSi (t )  i (t )Si (t )dt  Si (t ) j 1 ij (t )dW j (t ), i  1,
d
,m
(5.4.6)
• Assume that the mean rate of return and the
volatility matrix are adapted process.
• These stocks are typically correlated
• To see the nature of this correlation, we set  i (t ) 
d
2

 j 1 ij (t ) which we assume is never zero
• We define processes
 ij (u )
Bi (t )   j 1 
dW j (u ), i  1, , m
0  (u )
i
• Being a sum of stochastic integrals, each Bi (t )
d
t
is a continuous martingale2 . Furthermore,
 ij (u )
dBi (t )dBi (t )   j 1 2 dt  dt
 i (u )
• According to Levy’s Theorem, Bi (t ) is a
d
Brownian motion
• Rewriting (5.4.6) in terms of the Brownian
motion Bi (t ) as
dSi (t )  i (t )Si (t )dt  Si (t ) i (t )dBi (t )
• For i  k the Brownian motion
typically not independent
Bi (t )
and Bk (t ) are
• To see this, we first note that
 ij (t ) kj (t )
dBi (t )dBk (t )   j 1
dt  ik (t )dt
 i (t ) k (t )
d
where
1
d
ik (t ) 
 (t ) kj (t )

j 1 ij
 i (t ) k (t )
• Ito’s product rule implies
d ( Bi (t ) Bk (t ))  Bi (t )dBk (t )  Bk (t )dBi (t )  dBi (t )dBk (t )
and integration of this equation yields
t
t
t
0
0
0
Bi (t ) Bk (t )   Bi (u)dBk (u)   Bk (u)dBi (u)   ik (u)du
Covariance formula
• Taking expectations and using the fact that the
expectation of an Ito integral is zero, we
obtain the covariance formula
t
(5.4.12)
Cov[ Bi (t ), Bk (t )]  E  ik (u)du
0
• If the process  ij (t ) and  kj (t ) are constant, then
so are  i (t ),  k (t ) and ik (t ) .Therefore
(5.4.12) reduces to
Cov[ Bi (t ), Bk (t )]  ik t
• When  ij (t ) and  kj (t ) are themselves random
process, we call ik (t ) the instantaneous
correlation between Bi (t ) and Bk (t )
• Finally, we note from (5.4.8) and (5.4.9) that
dSi (t )dSk (t )   i (t ) k (t ) Si (t ) Sk (t )dBi (t )dBk (t )
 ik (t ) i (t ) k (t ) Si (t ) Sk (t )dt
• Rewriting (5.4.13) in terms of “relative
differential,” we obtain
dSi (t ) dSk (t )

 ik (t ) i (t ) k (t )
Si (t ) S k (t )
(5.4.13)
• The volatility process  i (t ) and  k (t ) are the
respective instantaneous standard deviations
of the relative changes in Si and S k at time k
• The process ik (t ) is the instantaneous
correlation between these relative changes
• However, standard deviations and correlations
can be affected by a change of measure when
the instantaneous standard deviations and
correlations are random
Discount process of stocks
• We define a discount process
t
R ( u ) du

0
D(t )  e

• We assume that the interest rate process R(t)
is adapted
• Besides the stock prices, we shall often work
with discounted stock prices and their
differentials are
d ( D(t ) Si (t ))  D(t )[ dSi (t )  R(t ) Si (t ) dt ]
(5.4.15)
 D(t ) Si (t )[( i (t )  R(t )) dt   j 1 ij (t ) dW j (t )]
d
 D(t ) Si (t )[( i (t )  R(t )) dt   i (t ) dBi (t )], i  1,
,m
5.4.3 Existence of the Risk-Neutral
Measure
Definition 5.4.3
A probability measure P is said to be riskneutral if
– (i) P and P are equivalent (i.e., for every A  F
P( A)  0 and if and only if P( A)  0 )
– (ii) under P , the discounted stock price D(t ) Si (t )
is a martingale for every i = 1, …, m
• To make the discounted stock prices be
martingales, we rewrite d ( D(t )Si (t )) as
d ( D(t ) Si (t ))  D(t ) Si (t ) j 1 ij (t )[ j (t )dt  dW j (t )]
d
(5.4.16)
• If we can find the market price of risk
processes  j (t ) that make (5.4.16) hold, with
one such process for each source of
uncertainty W j (t ) we can then use the
multidimensional Girsanov Theorem to
construct an equivalent probability measure P
• This permits us to reduce (5.4.16) to
d
d ( D(t ) Si (t ))  D(t ) Si (t ) j 1 ij (t )dW j (t )
And hence D(t )Si (t ) is a martingale under P
• The problem of finding a risk-neutral measure
is simply one of finding processes  j (t ) that
make (5.4.15) and (5.4.16) agree
• Since these equations have the same
coefficient multiplying each dW j (t ), they agree
if and only if the coefficient multiplying dt is
the same in both case, which means that
d
i (t )  R(t )   j 1 ij (t ) j (t ), i  1, , m
• We call these the market price of risk
equations where are in the d unknown
processes 1 (t ), , d (t )
Example 5.4.4
• Suppose there are two stocks (m = 2) and one
Brownian motion (d = 1), and suppose futher
that all coefficient processes are constant
• Then, the market price of risk equations are
1  r   1 ,
 2  r   2 .
• These equations have a solution  if and only
if
1  r  2  r

1
2
Example 5.4.4 (conti.)
• If this equation doesn’t hold, then one can
arbitrage one stock against the other
• Suppose that
1  r  2  r

1
2
and define
1  r  2  r


0
1
2
Example 5.4.4 (conti.)
• Suppose that at each time an agent holds
1
shares
S1 (t ) 1
1
2  
shares
S2 (t ) 2
1 
– Stock one:
– Stock two:
– Money market account: at the interest rate r
to set and maintain this portfolio
• The initial capital required to take the stock
1
1

positions is   and to set up the whole
portfolio, including the money market
position, is X(0) = 0
1
2
Example 5.4.4 (conti.)
• The differential of the portfolio value X(t) is
dX (t )  1 (t )dS1 (t )   2 (t )dS2 (t )  r ( X (t )  1 (t )dS1 (t )   2 (t )dS 2 (t ))dt
1  r
2  r

dt  dW (t ) 
dt  dW (t )  rX (t )dt
1
2
  dt  rX (t )dt
• The differential of the discounted portfolio
value is
d ( D(t ) X (t ))  D(t )(dX (t )  rX (t )dt )   D(t )dt
where  D (t ) is strictly positive and
nonrandom