on the mathematics and economics assumptions of continuous-time models...
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On the Mathematics and Economics Assumptions of Continuous-Time Models
BaoheWangBaoheWang
[email protected]@sina.com
3.1 Introduction
• This chapter attempts to (1) bridge the gap by using only elementary
probability theory and calculus to derive the basic theorems required for continuous time analysis.
(2) make explicit the economics assumptions implicitly embedded in the mathematical assumption.
• The general approach is to keep the assumption as weak as possible.
• But we need make a choice between the losses in generality and the reduction in mathematical complexity.
• The substantive contributions of continuous time analysis to financial economic theory:
(1) trading take place continuously in time (2) the underlying stochastic variables follow diffu
sion type motions with continuous sample paths• The twin assumptions lead to a set of behavioral
equations for intertemporal portfolio selection that are both simpler and rich than those derived from the corresponding discrete trading model.
• The continuous trading is an abstraction from physical reality.
• If the length of time between revisions is very short, the continuous trading solution will be a approximation to the discrete trading solution.
• The application of continuous time analysis in the empirical study of financial economic data is more recent and less developed.
• In early studies, we assume that the logarithm of the ratio of successive prices had a Gaussian distribution.
• But the sample characteristics of the time series were frequently inconsistent with these assumed population properties.
• Attempts to resolve these discrepancies proceeded along two separate paths.
• The first maintains the independent increments and stationarity assumptions but replaces the Gaussian with a more general stable (Parato-Levy) distribution.
• The stable family frequently fit the tails of the empirical distributions better than the Gaussian.
• But there is little empirical evidence to support adoption of the stable Paretian hypothesis over that of any leptokurtotic distribution.
• Moreover, the infinite variance property of the non-Gaussian stable distributions implis :
(1) most of our statistical tools, which are based upon finite-moment assumptions are useless.
(2) the first-moment or expected value of the arithmetic price change does not exist.
• The second, Cootner (1964) consider the alternative path of finite-moment processes whose distributions are nonstationary .
• The general continuous time framework requires that the underlying process be a mixture of diffusion and Poisson-directed processes.
• The general continuous time framework can accommodate a wide range of specific hypotheses including the “ reflecting barrier” model.
• Rosenberg (1972) shows that a Gaussion model with a changing variance rate appear to explain the observed fat-tail characteristics of return.
• Rosenberg (1980) has developed statistical techniques for estimating the parameters of continuous time processes.
• As discussed by Merton, If the parameters are slowly varying functions of time, then it is possible to exploit the different “ time scales” to identify and estimate these parameters.
• The second distribution still required more research before a judgment can be made as to the success of this approach.
• Their finite moment properties make the development of hypothesis tests considerably easier for these processes than for the stable Pareto-Levy processes.
• With this as a background, we development the assumptions of continuous time models.
• If denote the price of a security at time t, the change in the price of the security:
where h denote the trading horizon, , denote
( )X t
1
( ) (0) [ ( ) ( 1)]n
X T X X k X k 0T nh
( ) ( 1)X k X k ( ) [( 1) ]X kh X k h
• The continuous time trading interval assumption implies that the trading interval h is equal to the continuous time infinitesimal dt
• Note: it is unreasonable to assume that the equilibrium distribution of returns on a security over a specified time period will be invariant to the trading interval for that security.
• Because investors’ optimal demand function will depend upon how frequently they can revise their portfolio.
• Define the random variables by:
• is the unanticipated price change in the security between and , conditional on being at time .
• Because for , hence the partial sums form a martingale.
• The theory of martingales is usually associated in the financial economics literature with the “ Efficient Marker Hypothesis”.
( )k1( ) ( ) ( 1) { ( ) ( 1)}kk X k X k E X k X k
( )k1k k
1k { ( )} 0k jE k 1, ,j k
1( )
n
nS k
• Two economics assumptions:• Assumption 1: For each finite time interval [0,T]
there exists a number , independent of the number of trading intervals n, such that where .
• Assumption 2: For each finite time interval [0,T], there exists a number , independent of n, such that .
1 0A
1var( )nS A2
0 1var( ) {[ ( )] }
n
nS E k
2A
2var( )nS A
• The second assumption rules out the variance become unbounded such as Pareto-Levy stable distribution with infinite variance.
• Assumption 3: There exists a number independent of n, such that for where and
• This assumption rules out the possibility that all the uncertainty in the unanticipated price change over [0,T] is concentrated in a few of the many trading periods, such as lottery ticket.
3 3, 1 0A A 3( ) 1, ,V k V A k n
20( ) { ( )}, 1, ,V k E k k n max ( )kV V k
• Proposition 3.1: If Assumption 1,2,3 hold, then That is, and and is asymptotically
proportional to h where the proportionality factor is positive.
Proof:
Suppose then
( ) , 1, , .V k h k n ( ) ( )V k O h
( ) ( )V k o h ( )V k
0 01 1 1 1var( ) { ( ) ( )} { ( ) ( )}
n n n n
nS E k j E k j ,k j k j
0 0{ ( ) ( )} { ( ) { ( )}} 0jE k j E j E k k j
Therefore From Assumption 3 and 2
where , hence From Assumption 3 and1
where hence
1var( ) ( )
n
nS V k
23 21
3
2
3
( ) 1
( )
n A hnVA V k A VAT
A h V V kVAT
2 30 A A ( ) ( )V k O h
1 33 3 3 1
1 1( ) var( )n
A A hV k AV A S A A Tn n
1 3 0A A ( ) ( )V k o h
• Suppose that can take on any one of m distinct values denoted by where m is finite.
• Suppose that there exist a number , independent of n, such than .
• If information available as of time zero}, then from Proposition 3.1 it follow that: and because m is finite it follows that: for every j.
• Without lost generally, we assume for every j .
( )k( ), 1, ,j k j m
M 2j M
( ) { ( ) |j jp k prob k
2
1( )
m
j jp O h 2 ( )j jp O h
2j jp h
• Assumption 4: For and are sufficiently “well-behaved” functions of that there exist numbers and such that and .
• This assumption is stronger than is necessary.• From Assumption 4, we have
So • This say that “ the larger the magnitude of the
outcome, the smaller the likelihood that the even will occur.
1, , , jj m p j
jq jr jq
jp hjr
j h
22 j jq r
j jh p h 2 1 1, ,j jq r j m
• Because and is bounded , both and must be nonnegative, and therefore, we have:
• We can partition its outcomes into three type: (1) “type I” outcome is one such that . (2) “type II” outcome is one such that (3) “type III” outcome is one such that
1jp 2j jq jr
0 1 0 1 2, 1, , .j jq and r j m
12jr
10 2jr
0jr
• Let J denote the set of events j such that the outcomes are of type I.
• For and therefore .• For , .• So for small trading intervals h, virtually all
observations of will be type I outcomes, and therefore an apt name for might be “ the set of rare events.”.
j, 0,jj J q (1)jp ocj J (1)jp o
( )kcJ
3.2 Continuous-Sample-Path Processes With “No Rare Events”
• In this section, it is assumed that all possible outcomes for are of type I, and therefore is empty.
• Define , denote the conditional expected dollar return per unit time on the security.
( ), 1, ,k k n cJ
1{ ( ) ( 1)}k kE X k X k h
• Assumption 5: For every h, it is assumed that exists, and that there exists a number , independent of h, such that
. • This assumption ensures that for all securities
with a finite price the expected rate of return per unit time over the trading horizon is finite.
• From before formula, we have:
k1, ,k n
| |k
( ) ( 1) ( )kX k X k h k
• Proposition 3.2: If, for all possible outcomes for are type I outcomes, then the continuous-time sample path for the price of the security will be continuous.
• Proof: Let A necessary and sufficient condition for continuity of the sample path for X is that, for every .
Define , so For every define function as the s
olution of .
1, , ,k n ( )k
1( ) {| ( ) ( 1) | | }k kQ prob X k X k I
0, ( ) ( )kQ o h 1 2
{ }max | | /j ju h (1)u O0 ( )h
1 2( )h u h
Because and are , for every . Therefore, for every h, , , and hence as .• The sample for X(t) is continuous, but it is almost
nowhere differentiable.
• Which diverges as .
u (1)O ( ) 0h 0 0 ( )h h
| ( ) ( 1) |X k X k ( ) 0kQ ( )lim[ ] 0kQ
h 0h
1/ 2( ) ( 1) / / (1 )k jX k X k h h h
0h
• So we need a generalized calculus and corresponding theory of stochastic differential equations.
• Some moment properties for .( ) ( 1)X k X k 2 2
1{ ( )}/ (1) 1, ,k kE k h O k n 2 2
1 1
2
{[ ( ) ( 1)] } {( ( )) }
0( )
k k k
k
E X k X k E h k
h h h
for
Hence, the unconditional Nth central and noncentral absolute moments of
are the same.
0 1
2 / 2
1
{| ( ) | } | |
( ) ( )
NmNj j
Nm N N Nj
E k p
p u h u h o h
2N
1/ 20
/ 2 2
{| ( ) ( 1) | } ( )
( )
N N
N N N
E X k X k h uh
u h o h
( ) ( 1)X k X k
• Because the unconditional Nth central and noncentral do not depend on the probabilities of specific outcomes . Therefore:
• Define , where so and
{ }jp
1{| ( ) | } ( ) 2NkE k o h for N
/ 21 1{| ( ) ( 1) | } {| ( ) | } ( )N N N
k kE X k X k E k o h
2 1/ 2( ) ( ) /( )ku k k h 2 1 2/( ) (1), 1, , ;j j ku h O j m
21 1{ ( )} 0 { ( )} 1;k kE u k E u k
1{| ( ) | } (1), 2NkE u k O N
• We have:• This form makes explicit an important property
frequently observed in security returns: the realized return on a security over a short trading interval will be completely dominated by its unanticipated.
• However , it does not follow that in choosing an optimal portfolio the investor should neglect differences in the expected returns among stocks, because the first and second moments of the returns are of the same order.
1/ 2( ) ( 1) ( )k kX k X k h u k h
• Let , where , is a function with bounded third partial derivatives.
• Denote by X the known value of , then
• We use Taylor’s theorem
( ) ( , )F t f X t ( )X t X f 2C
( 1)X k 1/ 2 1, ,j k k jX X h u h j m
1/ 21
1/ 2 22 11
( , ) ( , 1) ( , 1)( )
1( , 1) ( , 1)( )2
j k k j
k k j j
f X k f X k f X k h u h
f X k h f X k h u h R
• Where• For each and every j,• So we have
2 1/ 222 12
3 2111 112
2 3122 222
1 ( , 1) ( , 1)( )21 1( , )( ) ( , )( )6 21 1( , )( ) ( , )2 6
j k k j
j j j j j j
j j j j j
R f X k h f X k h u h h
f X X f X X h
f X X h f h
( ) ( 1)j j j j jX X X and k v 3/ 2| | ( ) ( ) 1, ,jR O h o h j m
1/ 21
2 22 11
( , ) ( , 1) ( , 1)( )
1( , 1) ( , 1) ( )2
j k k j
k j
f X k f X k f X k h u h
f X k h f X k u h o h
• We can describe the dynamics for
• Applying the conditional expectation to both side
1
2 22 11
1/ 21
( ) ( 1) { [ ( 1), 1]
1[ ( 1), 1] [ ( 1), 1] }2
[ ( 1), 1] ( )
k
k j
k j
F k F k f X k k
f X k k h f X k k u h
f X k k u h o h
( )F k
1 1
22 11
[ ( ) ( 1)] { [ ( 1), 1]
1[ ( 1), 1] [ ( 1), 1] } ( )2
k k
k
E F k F k f X k k
f X k k h f X k k h o h
• Define
• So
1[ ( ) ( 1)] /k kE F k F k h
1 2
211
{ [ ( 1), 1] [ ( 1), 1]
1 [ ( 1), 1] } (1)2
k k
k
f X k k f X k k
f X k k o
211
1/ 21
( ) ( 1)
1 [ ( 1), 1] [ ( ) 1]2
[ ( 1), 1] ( ) ( )
k
k
k
F k F k h
f X k k u k h
f X k k u k h o h
• It is clear that the realized change in F over a very short time interval is completely dominated by the component of the unanticipated change.
• We have:
1/ 21[ ( 1), 1] ( )kf X k k u k h
21
21
{[ ( ) ( 1)] }
{ [ ( 1), 1] } ( )
k
k
E F k F k
f X k k h o h
/ 2
1{[ ( ) ( 1)] } ( ) ( )N NkE F k F k O h o h
• So the conditional moment of and
is same.
• The co-moments between and is also same
( ) ( 1)F k F k
( ) ( 1)X k X k
1
21
{[ ( ) ( 1)][ ( ) ( 1)]}
{ [ ( 1), 1] } ( )
k
k
E F k F k X k X k
f X k k h o h
1
/ 2
{[ ( ) ( 1)] [ ( ) ( 1)] }
( ) ( )
j N jk
N
E F k F k X k X k
O h o h
( ) ( 1)F k F k
( ) ( 1)X k X k
• The conditional correlation coefficient per unit time between contemporaneous changes in F and X is
• Now we study contribution to the change in F over a finite time interval.
• Define
1
1
1 (1) [ ( 1), ( 1)] 0
1 (1) [ ( 1), ( 1)] 0k o if f X k k
o if f X k k
( )O h
1/ 21
( ) ( 1) ( ) ( 1)
[ ( 1), 1] ( )
k
k
G k G k F k F k h
f X k k u k h
• If we define
• We have
• By construction , and therefore . Therefore , the partial
sums form a martingale.• Because , from the Law of
Large Numbers for martingales that
2 211( ) [ ( 1), 1] [ ( ) 1] / 2ky k f X k k u k
( ) ( 1) ( ) ( )G k G k y k o h
1{ ( )} 0kE y k
{ ( )} 0 1, ,k jE y k j k
1( )
ny k
2 20 1{ ( ) / }
nE y k k
1 1
1lim[ ( )] lim[ ( )] 0
n nh y k T y k as n
n
• We have that for fixed T
• Taking the limit we have• So, the cumulative error of the approximation
goes to zero with probability one.• Hence, for , we have
1 1 1( ) (0) ( ) ( ) ( ) (1)
n n nG T G h y k o h h y k o
( ) (0) 0G T G
0T
1
1/ 211 1
( ) (0) [ ( ) ( 1)]
[ ( 1), 1] ( )
n
n n
k k
F T F F k F k
h f X k k u k h
• Hence the stochastic term will have a negligible effect on the change in over a finite time interval.
• By the usual limiting arguments for Riemann integration, we have
F
1 0lim( ) ( )
Tn
kn
h t dt
1/ 2
11
1/ 210
lim{ [ ( 1), 1] ( ) }
[ ( ), ] ( ) ( )( )
n
kn
T
f X k k u k h
f X t t t u t dt
• So we have
• The stochastic differential for F is
• There is no difference because the contribution of this stochastic term to the moments of dF over the infinitesimal interval dt is , and over finite intervals it disappears.
1/ 210 0
( ) (0) ( ) [ ( ), ] ( ) ( )( )T T
F T F t dt f X t t t u t dt
1/ 21( ) ( ) [ ( ), ] ( ) ( )( )dF t t dt f X t t t u t dt
( )O dt
( )o dt
• Let , we get the stochastic differential for X
• Throughout this analysis, the only restrictions on the distribution for where (a)
(b) (c) (d)the distribution for is discrete.
( , )f X t X
1 2
0 0( ) (0) ( ) ( ) ( )( )
T TX T X t dt t u t dt
1/ 2( ) ( ) ( ) ( )( )dX t t dt t u t dt
( )u t {()}0, Eut
2{ ( ) } 1E u t ( ) (1)u t O( )u t
• Assumption 6: The stochastic process for is a Markov process.
• The Assumption 6 can be weakened to say that the conditional probabilities for X depend on only a finite amount of past information.
( )X t
1 2, 0 1{ ( ) | , } { ( ) | }t t tE X t X X X E X t X
( , ) ( , ; , )
{ ( ) | ( ) },
p x t p x t X T
prob X T X X t x t T
• Therefore , provide that p is a well behaved function of x and t, it will satisfy all the properties previous derived for F(t)
• This is a “Kolmogorov backward equation”.• Therefore, subject to boundary condition, the for
mula completely specifies the transition probability densities for the two price.
1( ) [ ( ) ( 1)] / 0tt E p t p t h
211 1 2
10 ( , ) ( , ) ( , ) ( , ) ( , )
2x t p x t x t p x t p x t
• The only characteristics of the distribution for the that affect the asymptotic distribution for the security price are the first and second moments.
• Hence, in the limit of continuous trading, nothing of economic content is lost by assuming that the are independent and identical.
{ ( )}u t
{ ( )}u t
• Define is a random variable
• When
• When are independent and identically distributed with a zero mean and unit variance, we have is normal distribution.
( )Z t
1
1/ 2
1 1
( ) (0) [ ( ) ( 1)]
( )
n
n n
k k
Z T Z Z k Z k
h u k h
0, 1k k 1/ 2 1
1/ 2
( )( ) (0)
nu k
Z T Z Tn
( ) (0)Z T Z
{ ( )}u k
• Then the solution to the “Kolmogorov backward equation” with
• We define• When the are independent and distribution
standard normal, the process is called a Wiener or Brownian.
2 1, 0 2
1/ 2
exp[ ( ) / 2( )]( , ; , )
[2 ( )]
X x T tp x t X T
T t
{ ( )}u t
1/ 2( ) ( )( )dZ t u t dt
dZ
• Then the dynamics of can be wrote as:
• And
• The class of continuous-time Markov processes whose dynamics can be written in the two form are call Ito’s processes.
0 0( ) (0) [ ( ), ] [ ( ), ] ( )
T TX T X X t t dt X t t dZ t
[ ( ), ] [ ( ), ] ( )dX X t t dt X t t dZ t
( )X t
• Ito’s lemma: Let be a function define on and take the stochastic integral defined by the two form, then the time-dependent random variable is a stochastic integral and its stochastic differential is:
• Where the product of the differentials is defined by the multiplication rules and
( , )f X t 2C
[0, ]R
( , )F f X t
21 2 11
1( , ) ( , ) ( , )( )
2dF f X t dX f X t dt f X t dX
2( ) , 0dZ dt dZdt 2( ) 0dt
• If the economic structure to be analyzed is such that Assumption 1-6 obtain and can have only type I outcomes.
• Then in continuous-trading model, security-pricing dynamics can always be described by Ito processes with no loss of generality.
• Note:• (1) The normality assumption for the
imposes no further restrictions on the process beyond those of Assumptions 1-6.
• (2) The distribution for the security price change over a finite interval [0,T] may not be normally distribution. Such as
can be shown to have a log-normal distribution
[ ( ), ] [ ( ), ] ( )dX X t t dt X t t dZ t
[ ( ), ] , [ ( ), ]X t t aX X t t bX
( )X t
• The continuous time models with Ito process have substantive benefits.
• Such as, (1) the analysis of corporate liability and option pricing is simplified by Ito process.
(2) in solving the intertemporal portfolio selection problem, the optimal portfolio demand functions will depend only upon the first two moment of the security return distributions.
Continuous-Sample-Path Processes With “ Rare Events”
• In this section, it is assumed that the outcomes for can be either of type I or type II, but not type III.
• The principal conclusion of this analysis will be that, in the limit of continuous trading, the distributional properties of security return are indistinguishable form those of section 3.2.
( ), 1, ,k k n
• Proposition 3.3: If, for all possible outcomes for are either type I or type II outcome, then the continuous-time sample path for the price of the security will be continuous.
• Proof: Let A necessary and sufficient condition for continuity of the sample path for X is that, for every .
Define , so For every define function as the
solution of
1, , ,k n
( )k
1( ) {| ( ) ( 1) | | }k kQ prob X k X k I
0, ( ) ( )kQ o h
{ }max | | rj ju h (1)u O
0 ( )h
( )rh u h
Because and are , for every . Therefore, for every h, , , and hence as .• From Assumption 5: is
asymptotically proportional to h, and therefore so is .
• From Proposition 3.1 the unconditional variance of is asymptotically proportional to h.
u (1)O ( ) 0h 0 0 ( )h h
| ( ) ( 1) |X k X k ( ) 0kQ ( )lim[ ] 0kQ
h 0h
1{ ( ) ( 1)}kE X k X k
0{ ( ) ( 1)}E X k X k
( ) ( 1)X k X k
• The Nth unconditional absolute moment of
• The Nth unconditional noncentral moments:
• The second conditional moments is
( )k( 2) 1
0 1 1
( 2) 1
{| ( ) |} | | ( )
( ) ( ) 2
j
j
Nm m N rNj j
N r
E k p O h
O h o h for N
0{[ ( ) ( 1)] } ( )
( )
N r N
N Nr
E X k X k h uh
u h o h
2 21 1
2
{[ ( ) ( 1)] } {( ( )) }
0( )
k k k
k
E X k X k E h k
h h h
• The Nth conditional moments is
• Hence, the moment relations for are identical with those derived in section 3.2 where only type I outcomes were allowed.
• Define is a function with a bounded order derivative, then from Taylor’s theorem, we have:
1 1{[ ( ) ( 1)] } {( ( )) } ( )N Nk k kE X k X k E h k o h
( 2)N
( ) ( 1)X k X k
( ) ( ) ( )F t f X if X t X 2C( 1)K th
(1)1
(2) 2
[ ( ) ( 1)] { [ ( 1)]
1 [ ( 1)] } ( )2
k k
k
E F k F k f X k
f X k h o h
• Hence, the order relation for the conditional moments of is the same as for the conditional of .
• Over short intervals of time, the unanticipated part of the change will be dominated by
21
21
{[ ( ) ( 1)] }
{ [ ( 1), 1] } ( )
k
k
E F k F k
f X k k h o h
1{[ ( ) ( 1)] } ( ) ( )N NrkE F k F k O h o h
( ) ( 1)F k F k ( ) ( 1)X k X k
(1)[ ( 1)] ( )f X k k
• We now examine stochastic properties for the change in F over a finite time interval.
• Define random variables
• Which by Taylor’s theorem can be rewritten as
( ) 1[ ( ) ( 1)] {[ ( ) ( 1)] }( ) [ ( 1)]
!
j jj k
j
X k X k E X k X ky k f X k
j
(1)1
( ) ( 1) ( ) ( 1)
{ ( ) ( 1)} [ ( 1)] ( )k
G k G k F k F k
E F k F k f X k k
11( ) ( 1) ( )
K
j KG k G k y k R
• For some . But is bounded and .
• So
( 1)1
1 11
[ ( 1) (1 ) ( )]
[ ( ) ( 1)] {[ ( ) ( 1)] }
( 1)!
KK
K Kk
R f X k X k
X k X k E X k X k
K
, 0 1 ( 1)kf
1 ( 1)[ ( )] ( ) ( )K r Kkh k O h o h
1 ( )KR o h
1( ) ( 1) ( ) ( )
K
jG k G k y k o h
• The unconditional variance of is
• Where is the least upper bound on
( ) ( 1)G k G k
0 2 2
02 2
1
1
2 1
var[ ( ) ( 1)] { ( ) ( )} ( )
{{[ ( ) ( 1)]
[ ( ) ( 1)] }
{[ ( ) ( 1)]
[ ( ) ( 1)] }}/ ! ! ( )
( ) ( ) ( )
K K
i j
K K ii j
ik
j
jk
r
G k G k E y k y k o h
M M E X k X k
E X k X k
X k X k
E X k X k i j o h
O h o h o h
iM ( )| |if
• For the finite tiem interval [0,T], we have
• forms a martingale, so
• As , the variance of goes to zero.
1
1 1
( ) (0) [ ( ) ( 1)]
( ) (1)
n
n K
i
G T G G k G k
y k o
1 1( )
n K
iy k 2var[ ( ) (0)] ( ) (1) (1)rG T G O h o o
0h ( ) (0)G T G
• Hence, we have
• With probability one where
1
(1)
1 1
( ) (0) [ ( ) ( 1)]
[ ( 1)] ( )
n
n n
k
F T F F k F k
h f X k k
(1) (2) 21[ ( 1)] [ ( 1)]2k k kf X k f X k
• So
• Hence, the limit of continuous trading, processes with type I and type II outcomes are indistinguishable from processes with type I outcomes only.
(1)
0
(1)
( ) (0) ( ) [ ( )] ( )
( ) ( ) [ ( )] ( )
T T
oF T F t dt f X t t
dF t t dt f X t t
Discontinuous-Sample-Path Processes With “Rare Events’
• In the concluding section, the general case is analyzed where the outcomes for can be type I, type II or type III.
• Type III resulting sample path for X will be discontinuous.
( )k
• Proposition 3.4: If, for at least one possible outcome for is a type III outcome, then the continuous time sample path for the price of the security will not be continuous.
• Proof:Let A necessary and sufficient condition for continuity of the sample path for X is that, for every . If suppose event j denote a type III outcome for where and is . If is the probability that event j occurs. Then where .
1, , ,k n ( )k
1( ) {| ( ) ( 1) | | }k kQ prob X k X k I
0, ( ) ( )kQ o h ( )k ( ) jk
j (1)O jp
j jp h (1)j O
Define
Note independent of h. For all h such that , if then
. Hence for , any such that if , and therefore . So the path is not continuous.
0
( 1) / 0k j
j k k j
ifh
if
| |j 0, 0h
0 h h ( ) jk | ( ) ( 1) |X k X k
0 h h 0
| ( ) ( 1) |X k X k ( ) jk
( ) ( )k jQ h O h
• These fundamental discontinuities in the sample path manifest themselves in the moment properties of
• The first and second unconditional moments of are asymptotically proportional to• The Nth unconditional absolute moments is
• So none of the moments can be neglected.
( ) ( 1)X k X k
( ) ( 1)X k X k h
0 1
( 2) 1
1
{| ( ) | } | ( ) |
( ) ( )j
mN Nj
m N r
E k p k
O h O h
( )O h
• Define the conditional random variable conditional on having type I outcome.
• Define the conditional random variable conditional on having type I outcome.
• Then
• Where and are all
1/ 2( ) ( ) /u k k h
( ) ( )y k k( )k
( )k
1/ 2 1 ( )( )( )
( )( )
with probability k hu k hk
with probability k hy k
( ), ( ),u k y k ( )k (1)O
• Define
• Because
1
1
1/ 2 1/ 21
1
( ) { ( )}
{ ( ) | }
( ) { ( )}
{ ( ) | }
yk
k
uk
k
y k E y k
E k type III outcome
u k h h E u k
E k type III outcome
1{ ( )} 0kE k 1/ 2
1/ 2( ) ( )( ) ( )
1 ( )
k y k hu k yh o h
k h
• Define
• Then we have:
• Where are all
2 21{ ( ) | }y kh E y k type III outcome
2 21{ ( ) | }u kh E u k type I outcome
2 21{ ( )}k kh E k
2 22 2 2 ( )
1k y
u k y O hh
,,u y k (1)O
• Further for
• Thus, both the type I and type III contribute significantly to the mean and variance of
2N
1 1{ ( )} ( ) { ( )} ( )N y Nk kE k k E y k h o h
( )k
• Let , , where is a function with bounded third partial derivatives.
• By Taylor’s theorem
( ) ( , )F t f X t ( )X t X f 2C
1
2
[ ( 1) , ] [ ( 1) , 1]
[ ( 1) , 1]
[ ( 1) , 1] ( )
k
k
f X k h y k f X k y k
f X k y k h
f X k y k h o h
1/ 2
1/ 21 2
211
[ ( 1) , ] [ ( 1), 1]
[ ( 1), 1]( ) [ ( 1), 1]
1[ ( 1), 1] ( )
2
k
k
f X k h uh k f X k k
f X k k h uh f X k k h
f X k k u h o h
• The properties of conditional expectation:
1
1
1
211
1 2
1
{ ( ) ( 1)}
( ) { ( ) ( 1)}
[1 ( ) ] { ( ) ( 1)}
1( [ ( 1), 1]2[ ( 1), 1]( ) [ ( 1), 1]
{ [ ( 1) ( ), 1]
[ ( 1, 1)]}) ( )
k
yk
uk
u
k
yk
E F k F k
k hE F k F k
k h E F k F k
f X k k
f X k k y f X k k
E f X k y k k
f X k k h o h
• Define • As , we can write the instantaneous
condition expect change in F per unit time as
• When and there are no type III outcomes, it became into the form of Section 3.2
1{ ( ) ( 1)}/k kE F k F k h
0h
211 1
2
1( ) [ ( ), ] ( ) [ ( ), ][ ( ) ( ) ( )]
2
[ ( ), ] ( ) { [ ( ) ( ), ] [ ( ), ]}
u
yt
t f X t t t f X t t t t y t
f X t t t E f X t y t t f X t t
( ) 0t
• The higher conditional moments
21
1
2
2 21
{[ ( ) ( 1)] }
( { [ ( 1) ( ), 1]
[ ( 1, 1)]}
[ ( 1, 1)] ) ( )
k
yk
u
E F k F k
E f X k y k k
f X k k
f X k k h o h
• For
• Only the type III outcomes contribute significantly to the moment higher than the second.
• The moment of and are same.
2N
1
1
{[ ( ) ( 1)] }
{ [ ( 1) ( ), 1]
[ ( 1, 1)]} ( )
Nk
yk
N
E F k F k
E f X k y k k
f X k k h o h
( ) ( 1)X k X k ( ) ( 1)F k F k
• Let denote the conditional probability density for X(T)=X, at time T, which is a function of security price at time t, so we have p satisfy:
• Hence, knowledge of the functions is sufficient to determine the probability distribution for the change in X between two data.
211 1 2
10 ( , ) ( ) ( , ) ( , )
2
[ ( , ) ( , )] ( ; , )
u p x t y p x t p x t
p x y t p x t g y x t dt
2u g
• Moreover, this process is identical with that of a stochastic process driven by a linear superposition of a continuous sample path diffusion process and a “Poisson directed” process.
• Let be a Poisson distribution random variable.
( ) ( )Q t h Q t
0 1 [ ( ), ] ( )
( ) 1 [ ( ), ] ( )
( ), 2
with probability X t t dt o dt
dQ t with probability X t t dt o dt
N with probability o dt N
• Define , then is an example of a “Poisson direction” process.
• Define • Then the limiting process for the change in X will
be identical with the process described by• The stochastic differential equation of X(t) is
1( ) ( ) ( )dX t y t dQ t 1dX
2dX dt dZ
1 2dX dX
( ) { [ ( ), ] [ ( ), ] ( )}
[ ( ), ] ( ) ( ) ( )
dX t X t t X t t y t dt
X t t dZ t y t dQ t
• Where is the probability per unit time that the change in X is a type III outcome.
• When it is mean only type I outcomes can occur.
• The stochastic differential representation for F
0
211 1
2 1
1( ) { [ ( ), ] ( ) [ ( ), ]
2[ ( ), ]} [ ( ), ] ( )
{ [ ( ) ( ), ] [ ( ), ]} ( )
dF t f X t t y f X t t
f X t t dt f X t t dZ t
f X t y t t f X t t dQ t
• In summary, if the economic structure to be analyzed is such that Assumption 1-5 obtain, then in continuous trading models security price dynamics can always be described by a mixture of continuous sample path diffusion processes and Poisson directed processes with no loss in generality.
• The diffusion process describes the frequent local changes in prices. The Poisson directed process is used to capture those rare events
• The transition probabilities are completely specified by only four functions
• This make the testing of these model structures empirically feasible.
g
The End
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