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 continuous-time models

5.4.1 Girsanov and Martingale Representation Theorems

• Throughout this section,

is a multidimensional Brownian motion on a probability space .

• Note that:– P is the actual probability measure– Associated with this Brownian motion, we have a

filtration F(t)

1( ) ( ( ), , ( ))dW t W t W t

( , , )F P

Theorem 5.4.1 (Girsanov, multiple dimension)

• Let T be a fixed positive time, and let be a d-dimensional adapted

process. Define

and assume that

1( ) ( ( ), , ( ))dt t t

2

0 0

0

1( ) exp{ ( ) ( ) ( ) },

2

( ) ( ) ( ) ,

t t

t

Z t u dW u u du

W t W t u du

2 2

0( ) ( )

tE u Z u du

(5.4.1)

Theorem 5.4.1 (Girsanov, multiple dimension)

• Set Z = Z(t). Then E(Z) = 1, and under the probability measure given by

the process is a d-dimensional Brownian motion

( ) ( ) ( ) ,A

P A Z dP for all A F

P

( )W t

Remark of the Theorem 5.4.1

• The Ito integral in (5.4.1) is

• In (5.4.1), denotes the Euclidean norm

• (5.4.2) is shorthand notation for with

( )u

1 10 0 0( ) ( ) ( ) ( ) ( ) ( )

t t td d

j j j jj ju dW u u dW u u dW u

2 1/ 2

1( ) ( ( ))

d

jju u

0( ) ( ) ( ) , 1, ,

t

j j jW t W t u du j d

The conclusions of the multidimensional Girsanov Theorem

• The component process of are independent under– The component process of are independent under

P, but each of the processes can depend in a path-dependent, adapted way on all of the Brownian motions

– Under P, the components of can be far from independent, however, after the change to the measure , these components are independent

• The proof of Theorem 5.4.1 is like that of the one-dimensional Girsanov Theorem

( )W t

P

( )W t

( )j u

1( ), , ( )dW t W t( )W t

P

Theorem 5.4.2(Martingale representation, multiple dimensions )

• Let T be a fixed positive time, and assume that F(t), is the filtration generated by the d-dimensional Brownian motion W(t), Let M(t), be a martingale with respect to this filtration under P. Then there is an adapted, d-dimensional process

such that

0 ,t T

0 .t T 0 ,t T

1( ) ( ( ), , ( )),du u u

0 ,t T

0( ) (0) ( ) ( ), 0 .

tM t M u dW u t T

Theorem 5.4.2(Martingale representation, multiple dimensions )

• If, in addition, we assume the notation and assumptions of Theorem 5.4.1 and if is a -martingale, then there is an adapted, d-dimensional process such that

( ), 0 ,M t t T

P

1( ) ( ( ), , ( ))du u u

0( ) (0) ( ) ( ), 0 .

tM t M u dW u t T

5.4.2 Multidimensional Market Model

• We assume there are m stocks, each with stochastic differential

• 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 which we assume is never zero

1( ) ( ) ( ) ( ) ( ) ( ), 1, ,

d

i i i i ij jjdS t t S t dt S t t dW t i m

( )i t

2

1( )

d

ijjt

(5.4.6)

• We define processes

• Being a sum of stochastic integrals, each is a continuous martingale . Furthermore,

• According to Levy’s Theorem, is a Brownian motion

1 0

( )( ) ( ), 1, ,

( )

td iji jj

i

uB t dW u i m

u

( )iB t

2

21

( )( ) ( )

( )

d iji i j

i

udB t dB t dt dt

u

( )iB t

• Rewriting (5.4.6) in terms of the Brownian motion as

• For the Brownian motion and are typically not independent

( )iB t

( ) ( ) ( ) ( ) ( ) ( )i i i i i idS t t S t dt S t t dB t

i k ( )kB t( )iB t

• To see this, we first note that

where

• Ito’s product rule implies

and integration of this equation yields

1

( ) ( )( ) ( ) ( )

( ) ( )

d ij kji k ikj

i k

t tdB t dB t dt t dt

t t

1

1( ) ( ) ( )

( ) ( )

d

ik ij kjji k

t t tt t

( ( ) ( )) ( ) ( ) ( ) ( ) ( ) ( )i k i k k i i kd B t B t B t dB t B t dB t dB t dB t

0 0 0( ) ( ) ( ) ( ) ( ) ( ) ( )

t t t

i k i k k i ikB t B t B u dB u B u dB u u du

Covariance formula

• Taking expectations and using the fact that the expectation of an Ito integral is zero, we obtain the covariance formula

• If the process and are constant, then so are and .Therefore (5.4.12) reduces to

0Cov[ ( ), ( )] ( )

t

i k ikB t B t E u du ( )ij t ( )kj t

( ), ( )i kt t ( )ik t

(5.4.12)

Cov[ ( ), ( )]i k ikB t B t t

• When and are themselves random process, we call the instantaneous correlation between and

• Finally, we note from (5.4.8) and (5.4.9) that

• Rewriting (5.4.13) in terms of “relative differential,” we obtain

( )ij t ( )kj t( )ik t

( )iB t ( )kB t

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )i k i k i k i k

ik i k i k

dS t dS t t t S t S t dB t dB t

t t t S t S t dt

( ) ( )( ) ( ) ( )

( ) ( )i k

ik i ki k

dS t dS tt t t

S t S t

(5.4.13)

• The volatility process and are the respective instantaneous standard deviations of the relative changes in and at time k

• The process 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

( )i t ( )k t

iS kS

( )ik t

Discount process of stocks

• We define a discount process

• 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

0( )

( )tR u du

D t e

1

( ( ) ( )) ( )[ ( ) ( ) ( ) ]

( ) ( )[( ( ) ( )) ( ) ( )]

( ) ( )[( ( ) ( )) ( ) ( )], 1, ,

i i i

d

i i ij jj

i i i i

d D t S t D t dS t R t S t dt

D t S t t R t dt t dW t

D t S t t R t dt t dB t i m

(5.4.15)

5.4.3 Existence of the Risk-Neutral Measure

Definition 5.4.3 A probability measure is said to be risk-neutral if– (i) and are equivalent (i.e., for every

and if and only if )

– (ii) under , the discounted stock price is a martingale for every i = 1, …, m

P

P P( ) 0P A

A F( ) 0P A

P ( ) ( )iD t S t

• To make the discounted stock prices be martingales, we rewrite as

• If we can find the market price of risk processes that make (5.4.16) hold, with one such process for each source of uncertainty we can then use the multidimensional Girsanov Theorem to construct an equivalent probability measure

1( ( ) ( )) ( ) ( ) ( )[ ( ) ( )]

d

i i ij j jjd D t S t D t S t t t dt dW t

(5.4.16)

( )j t

( ( ) ( ))id D t S t

( )jW t

P

• This permits us to reduce (5.4.16) to

And hence is a martingale under

• The problem of finding a risk-neutral measure is simply one of finding processes that make (5.4.15) and (5.4.16) agree

( ) ( )iD t S t P1

( ( ) ( )) ( ) ( ) ( ) ( )d

i i ij jjd D t S t D t S t t dW t

( )j t

• Since these equations have the same coefficient multiplying each , they agree if and only if the coefficient multiplying dt is the same in both case, which means that

• We call these the market price of risk equations where are in the d unknown processes

( )jdW t

1( ) ( ) ( ) ( ), 1, ,

d

i ij jjt R t t t i m

1( ), , ( )dt 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

• These equations have a solution if and only if

1 1

2 2

,

.

r

r

1 2

1 2

r r

Example 5.4.4 (conti.)

• If this equation doesn’t hold, then one can arbitrage one stock against the other

• Suppose that

and define

1 2

1 2

r r

1 2

1 2

0r r

Example 5.4.4 (conti.)

• Suppose that at each time an agent holds– 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

positions is and to set up the whole portfolio, including the money market position, is X(0) = 0

11 1

1shares

( )S t

22 2

1shares

( )S t

1 2

1 1

Example 5.4.4 (conti.)

• The differential of the portfolio value X(t) is

• The differential of the discounted portfolio value is

where is strictly positive and nonrandom

1 1 2 2 1 1 2 2

1 2

1 2

( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ( ))

( ) ( ) ( )

( )

dX t t dS t t dS t r X t t dS t t dS t dt

r rdt dW t dt dW t rX t dt

dt rX t dt

( ( ) ( )) ( )( ( ) ( ) ) ( )d D t X t D t dX t rX t dt D t dt

( )D t