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On the variance of stock price distributions
Economics Letters North-Holland
171
33 (1990) 171-173
ON THE VARIANCE
OF STOCK
PRICE
DISTRIBUTIONS
Marcello BRAGLIA Polrtecnico
Received Accepted
di Milano,
Milan, Italy
25 October 1988 6 October 1989
A stochastic approach is used to describe the temporal behaviour of stock prices. It is shown that standard unable to give the correct variance of the stock price distribution if the trend of variation is price dependent.
techniques
are
Since the first claim by Shiller (1981) that the movements of stock prices cannot be fully explained by (subsequent) changes in dividends, a number of papers have been published in an attempt to assess the validity of Shiller’s indication and to find a possible explanation of this phenomenon. At present, small sample bias, speculative bubbles and time-dependent expected returns, seem to have lost the original support as possible reasons for the observed excess of volatility. This has suggested a role for non-standard models [e.g., West (1987) and references therein]. In this note we intend to call attention to a possible volatility contribution (say ‘fads’ effect) which can be lost by conoentional techniques. To this end, consider a (t-dependent) random variable X, and let P(x, t) be the relevant probability density function at time t. Below, X, is supposed to be the stock price pr, but it could be also an appropriate function of pt. The behavior of P(x, t) can be described by an appropriate autoregressive process, e.g. x,=+x,-i+n,,
(1)
where + is a parameter, possibly dependent on x and t, and u, an appropriate distribution. Alternatively and much more efficiently, the behaviour of P(x, t) can be described by a diffusion equation, e.g. a Fokker-Planck (henceforth F.P.) equation of the form atp(x,
t) =
a,{-b,(x,
t)F’(x,
t) + ;%[b,(x,
t>P(x,
t>l >v
(2)
where the drift term b,(x, t) and the diffusion term b,(x, t) are given functions of x and t. To give an example, suppose that b, = b,(t) and b, = b*(t) are functions of time only. The resulting F.P. equation is a,P(x,
t) = -bi(t)a,P(x,
t) + :b,(t)8,2P(x,
For P(x, 0) = S(x -x0) eq. (3) has a Gaussian variance CJ* = o*(t) given by the equations yi?, = b,(t) 0165-1765/90/$3.50
and
t).
(fundamental)
d* = b,(t)
0 1990 - Elsevier Science Publishers
solution
with mean m, = ml(t)
and
(4) B.V. (North-Holland)
172
M. Braglia / On the variance of stock price distributions
with the conditions m,(O) =x0 and a’(O) = 0. If P(x, 0) = N[m,(O), a*(O)] then P(x, t) = N[m,, a*], with m, and a* given by eqs. (4) and the appropriate initial conditions. Consider now a process of form (1) with u, = u = N[O, u,“] and constant +. This is a usual assumption when studying stock price volatility. A similar process is easily shown to lead to P(x, t)‘s which satisfy eqs. (3) and (4). In fact, if P(x, 0) = 6(x-x0) or P(x, 0) = N[m,(O), a*(O)], eq. (1) yields P(x, t) = N[m,, a*] with m, = ml(t) and u* = u*(t) which satisfy equations of the general forms
(ml(t) - d4)/(48
-m&Q>) = cp”’
and (u’( t)-u2(co))/(u2(0)-a2(co))=e~2p', where p= if
(5)
- In 191 and u*(cc) = a,*/(1 - $*). In other words, eqs. (1) and (3) have the same solution
b,(t)
= -+1(t)
It is worth noting
-m&o>]
and
that it can also be assumed
b, =b,(x)
= -p[x
- m,(cc)]
b2(t)
= -2p[u*(t)
- u2(oo)].
[Braglia (1988) and references
and
b, = b*(x)
= 2pu2(oo).
therein]
(6) that: (7)
In fact, this is the necessary and sufficient condition for the preservation of the Gaussian form of the fundamental solution if the coefficients of the F.P. equation are x-dependent. But in eq. (7) the coefficients depend one” on x, and eq. (2) can also be given the steady-state form
[x -m,(co)]P(x)
+ +o)P’(x)
(8)
= 0,
whose solution P(x) = N[m,(co), u*(co)] is just the same as one obtains from process (I) with the mentioned restrictions on + and u,. This simple example shows how the same stochastic process can be described by an autoregression or a diffusion equation. However, the second approach is more general and not so dangerous. In fact, suppose that the stochastic process is time-dependent, i.e., that P(x, t) has a trend and a (bounded) variance, possibly both dependent on time. Such a process is certainly much more appropriate to represent stock prices than that of our previous example. In this case b, and b, will be functions of both x and t and, under general conditions, we will have
riz,=
j
b,(x,
t)P(x,
h2 = /[2xb,(x,
t) dx = (b,),
t) + b,(x,
t)]P(x,
(9)
t) dx = (2x&
+ b,);
(10)
that is, d2= (2(x - m,)b,)
+ (b2).
This important equation shows that, if b, is x-dependent, neglected. In other words, one is not allowed to neglect
(11) there is a contribution to u* that cannot be the x-dependence of the coefficients in the
173
M. Braglia / On the variance of stock price distributions
F.P. equation [e.g. to substitute (6,) and (b2) to b,(x, t) and b,(x, t)] without changing the variance law. The significance of this conclusion is understood when considering that x-dependence of b, means x-dependence of drift speed. Thus, for example, if b, is an increasing function of x [say (6, - m,) > 0 for x > m, and (b, - m,) < 0 for x < m,], the first term of the right-hand side of eq. (11) can give a very important positive contribution to the variance. This is easily seen if conditions for canonical invariance and steady-state (7) are satisfied. But, in the general case, the Gaussian form will not be preserved if b, is x-dependent, as a non-constant drift speed will soon destroy an initial Gaussian distribution. Even the symmetric form can be lost and such a process is difficult to be exactly represented by eq. (1). In any case, the process cannot be correctly represented with standard autoregressive processes with constant parameters. From a mathematical point of view, the result is explained when considering that F.P. eq. (2) is not equivalent to a F.P. equation of the same form (3) of our previous example, with proper coefficients depending only on t. In fact, eq. (2) is equivalent to an infinite differential equation of the form &P(x,
t)=
-p,(t)aXP+(p,(t)/2!)a,2P-(&(t)/3!)a,3P+
. ..)
(12)
where PI(t) = yit, and &(t) = 6’ with Cz, and ti2 given by eqs. (9) and (lo), respectively. The exact temporal behaviour of P(x, t) requires also to consider terms of higher order than the second. Thus, a distribution P(x, t) which would be found to preserve the Gaussian form when supposing & = 0 for n > 2, will generally be turned into a skewed distribution [Braglia (1988) and references therein]. As one can see, the x-dependence of b, and b, has important consequences on both the form of P( x, t) and the variance law. The Gaussian form can be exponentially preserved. But, if non-Gaussian forms are observed for P(x, t) and/or conditional distributions P( y, t + 6 1x, t), one is allowed to consider the experimental result to be an indication of a correlation between x and drift speed which is sufficiently pronounced to produce observable effects. This seems to be the case for stock prices [e.g., Fama (1965)] where a contribution to a’(t) = j&(t)dt from the x-dependence of b, could be important to explain part of stock price volatility. This x-dependence is reasonably a ‘fads’ effect, i.e., the result of (trading) forces possibly due to naive investors. In fact, a correct way to introduce fads into equations for P(x, t) (say an autoregressive process) cannot neglect that the coefficient b,(x, t) of the corresponding F.P. equation must have the correct dependence on the value of the random variable. In the contrary case, theory gives variances which cannot be directly compared with empirical values.
References Braglia, M., 1988, Stochastic theory of stock market, Unpublished manuscript (Politecnico di Milano, Milan, Italy). Fama, E.F., 1965, The behaviour of stock market prices, Journal of Business 38, 34-105. Stiller, R.J., 1981, Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71, 421-436. West, K.D., 1987, Bubbles, fads and stock price volatility tests: A partial evaluation, Manuscript (Princeton University, Princeton, NJ).
171
33 (1990) 171-173
ON THE VARIANCE
OF STOCK
PRICE
DISTRIBUTIONS
Marcello BRAGLIA Polrtecnico
Received Accepted
di Milano,
Milan, Italy
25 October 1988 6 October 1989
A stochastic approach is used to describe the temporal behaviour of stock prices. It is shown that standard unable to give the correct variance of the stock price distribution if the trend of variation is price dependent.
techniques
are
Since the first claim by Shiller (1981) that the movements of stock prices cannot be fully explained by (subsequent) changes in dividends, a number of papers have been published in an attempt to assess the validity of Shiller’s indication and to find a possible explanation of this phenomenon. At present, small sample bias, speculative bubbles and time-dependent expected returns, seem to have lost the original support as possible reasons for the observed excess of volatility. This has suggested a role for non-standard models [e.g., West (1987) and references therein]. In this note we intend to call attention to a possible volatility contribution (say ‘fads’ effect) which can be lost by conoentional techniques. To this end, consider a (t-dependent) random variable X, and let P(x, t) be the relevant probability density function at time t. Below, X, is supposed to be the stock price pr, but it could be also an appropriate function of pt. The behavior of P(x, t) can be described by an appropriate autoregressive process, e.g. x,=+x,-i+n,,
(1)
where + is a parameter, possibly dependent on x and t, and u, an appropriate distribution. Alternatively and much more efficiently, the behaviour of P(x, t) can be described by a diffusion equation, e.g. a Fokker-Planck (henceforth F.P.) equation of the form atp(x,
t) =
a,{-b,(x,
t)F’(x,
t) + ;%[b,(x,
t>P(x,
t>l >v
(2)
where the drift term b,(x, t) and the diffusion term b,(x, t) are given functions of x and t. To give an example, suppose that b, = b,(t) and b, = b*(t) are functions of time only. The resulting F.P. equation is a,P(x,
t) = -bi(t)a,P(x,
t) + :b,(t)8,2P(x,
For P(x, 0) = S(x -x0) eq. (3) has a Gaussian variance CJ* = o*(t) given by the equations yi?, = b,(t) 0165-1765/90/$3.50
and
t).
(fundamental)
d* = b,(t)
0 1990 - Elsevier Science Publishers
solution
with mean m, = ml(t)
and
(4) B.V. (North-Holland)
172
M. Braglia / On the variance of stock price distributions
with the conditions m,(O) =x0 and a’(O) = 0. If P(x, 0) = N[m,(O), a*(O)] then P(x, t) = N[m,, a*], with m, and a* given by eqs. (4) and the appropriate initial conditions. Consider now a process of form (1) with u, = u = N[O, u,“] and constant +. This is a usual assumption when studying stock price volatility. A similar process is easily shown to lead to P(x, t)‘s which satisfy eqs. (3) and (4). In fact, if P(x, 0) = 6(x-x0) or P(x, 0) = N[m,(O), a*(O)], eq. (1) yields P(x, t) = N[m,, a*] with m, = ml(t) and u* = u*(t) which satisfy equations of the general forms
(ml(t) - d4)/(48
-m&Q>) = cp”’
and (u’( t)-u2(co))/(u2(0)-a2(co))=e~2p', where p= if
(5)
- In 191 and u*(cc) = a,*/(1 - $*). In other words, eqs. (1) and (3) have the same solution
b,(t)
= -+1(t)
It is worth noting
-m&o>]
and
that it can also be assumed
b, =b,(x)
= -p[x
- m,(cc)]
b2(t)
= -2p[u*(t)
- u2(oo)].
[Braglia (1988) and references
and
b, = b*(x)
= 2pu2(oo).
therein]
(6) that: (7)
In fact, this is the necessary and sufficient condition for the preservation of the Gaussian form of the fundamental solution if the coefficients of the F.P. equation are x-dependent. But in eq. (7) the coefficients depend one” on x, and eq. (2) can also be given the steady-state form
[x -m,(co)]P(x)
+ +o)P’(x)
(8)
= 0,
whose solution P(x) = N[m,(co), u*(co)] is just the same as one obtains from process (I) with the mentioned restrictions on + and u,. This simple example shows how the same stochastic process can be described by an autoregression or a diffusion equation. However, the second approach is more general and not so dangerous. In fact, suppose that the stochastic process is time-dependent, i.e., that P(x, t) has a trend and a (bounded) variance, possibly both dependent on time. Such a process is certainly much more appropriate to represent stock prices than that of our previous example. In this case b, and b, will be functions of both x and t and, under general conditions, we will have
riz,=
j
b,(x,
t)P(x,
h2 = /[2xb,(x,
t) dx = (b,),
t) + b,(x,
t)]P(x,
(9)
t) dx = (2x&
+ b,);
(10)
that is, d2= (2(x - m,)b,)
+ (b2).
This important equation shows that, if b, is x-dependent, neglected. In other words, one is not allowed to neglect
(11) there is a contribution to u* that cannot be the x-dependence of the coefficients in the
173
M. Braglia / On the variance of stock price distributions
F.P. equation [e.g. to substitute (6,) and (b2) to b,(x, t) and b,(x, t)] without changing the variance law. The significance of this conclusion is understood when considering that x-dependence of b, means x-dependence of drift speed. Thus, for example, if b, is an increasing function of x [say (6, - m,) > 0 for x > m, and (b, - m,) < 0 for x < m,], the first term of the right-hand side of eq. (11) can give a very important positive contribution to the variance. This is easily seen if conditions for canonical invariance and steady-state (7) are satisfied. But, in the general case, the Gaussian form will not be preserved if b, is x-dependent, as a non-constant drift speed will soon destroy an initial Gaussian distribution. Even the symmetric form can be lost and such a process is difficult to be exactly represented by eq. (1). In any case, the process cannot be correctly represented with standard autoregressive processes with constant parameters. From a mathematical point of view, the result is explained when considering that F.P. eq. (2) is not equivalent to a F.P. equation of the same form (3) of our previous example, with proper coefficients depending only on t. In fact, eq. (2) is equivalent to an infinite differential equation of the form &P(x,
t)=
-p,(t)aXP+(p,(t)/2!)a,2P-(&(t)/3!)a,3P+
. ..)
(12)
where PI(t) = yit, and &(t) = 6’ with Cz, and ti2 given by eqs. (9) and (lo), respectively. The exact temporal behaviour of P(x, t) requires also to consider terms of higher order than the second. Thus, a distribution P(x, t) which would be found to preserve the Gaussian form when supposing & = 0 for n > 2, will generally be turned into a skewed distribution [Braglia (1988) and references therein]. As one can see, the x-dependence of b, and b, has important consequences on both the form of P( x, t) and the variance law. The Gaussian form can be exponentially preserved. But, if non-Gaussian forms are observed for P(x, t) and/or conditional distributions P( y, t + 6 1x, t), one is allowed to consider the experimental result to be an indication of a correlation between x and drift speed which is sufficiently pronounced to produce observable effects. This seems to be the case for stock prices [e.g., Fama (1965)] where a contribution to a’(t) = j&(t)dt from the x-dependence of b, could be important to explain part of stock price volatility. This x-dependence is reasonably a ‘fads’ effect, i.e., the result of (trading) forces possibly due to naive investors. In fact, a correct way to introduce fads into equations for P(x, t) (say an autoregressive process) cannot neglect that the coefficient b,(x, t) of the corresponding F.P. equation must have the correct dependence on the value of the random variable. In the contrary case, theory gives variances which cannot be directly compared with empirical values.
References Braglia, M., 1988, Stochastic theory of stock market, Unpublished manuscript (Politecnico di Milano, Milan, Italy). Fama, E.F., 1965, The behaviour of stock market prices, Journal of Business 38, 34-105. Stiller, R.J., 1981, Do stock prices move too much to be justified by subsequent changes in dividends?, American Economic Review 71, 421-436. West, K.D., 1987, Bubbles, fads and stock price volatility tests: A partial evaluation, Manuscript (Princeton University, Princeton, NJ).