Remark. Adding to (5.9) the f0(x) from (5.2), we see that asymptotic expression for density fA(x) (for Δt≪1 and x≫1) satisfies bounds obtained from (5.10) so that fA(x)∼O(1/x2) for x≫1.
6. The density with arbitrary correlation between the number of jumps in the supply and demand
In this section we allow for the possibility that the number of jumps, ΔNt in the demand (i.e., numerator) has an arbitrary correlation, ρJ, with the supply (i.e., denominator). For positive reals λ01, λ02, λ12 and k,l∈ℕ, we write the bivariate Poisson probability (see [35]) as equation (6.1) where 𝔼[ΔNt]=Var[ΔNt]=λ01+λ12=:λ1, 𝔼[ΔÑ t]=Var[ΔÑ t]=λ02+λ12=:λ2. Summing over l yields the individual probabilities, equation, and similarly for ℙ(ΔÑ t=l). Also, the covariance is given by Cov[ΔNt,ΔÑ t]=λ12 and, hence, the correlation by equation.
Thus the correlation in the number of jumps can be adjusted through the parameters λ01,λ02, and λ12. The analysis of Sections 2 and 3 can now be adapted for this general case by noting that the joint density for the random variables R̂ i(ki) defined by (2.4) is still given by (2.7) and (2.8). At this point we do not assume independence of ΔNt and ΔÑ t but use the joint probability density above, and (2.9) is now replaced by equation, and similarly, (2.12) is identical except for the p(k1,k2) substitution. The result (2.14) is then equation. (6.3) One needs to show convergence of the sums above. Noting that μR1/σR1∼k1/21  and the exponential is bounded by unity, one can prove the boundedness of the series by noting that equation.
Hence, we can state the conclusion analogous to Theorem 2.3. In addition, the analysis for R1 and R2 conditioned on positivity can be similarly carried out as in Section 3, with the following result.
Theorem 6.1. Let Ra:=R1/R2 where R1=D,R2=S are described by jump-diffusion processes (i.e., Brownian motion plus a compound Poisson process) through (2.3). Assume that the two Poisson processes, ΔNt and ΔÑ t are bivariate Poisson with means λ1=λ01+λ12 and λ2=λ02+λ12 respectively, and covariance λ12. Let ρ∈(−1,1) be the correlation between R1 and R2 conditioned on a fixed pair of jumps (k1,k2). With Gε be defined via (2.20), the density fc of Rc:=Gε(Ra) satisfies the asymptotic relation equation. The density fc(y |R1,R2>0) of Rc:=Gε(Ra) conditioned on positive R1 and R2 also satisfies the asymptotic relation equation.
The total correlation, ρT, as a function of the parameters is calculated in the Appendix C.
Summarizing the results, we see that the issue of correlation between supply and demand is complex. Given k1 jumps or shocks for demand and k2 for supply, we can impose a correlation, ρ∈(−1,1), between supply and demand conditioned on (k1,k2). However, the overall correlation (not conditioned on k1 and k2) for demand and supply will have a smaller magnitude if the number of jumps are independent, and is calculated in Appendix C.
In this section, we have considered the problem when the number of jumps in the supply and demand, ΔNt and ΔÑ t has an arbitrary correlation, ρ(P), given by (6.2) in which case one can obtain the full range (−1,1) of correlation for the supply and demand, as stated in Theorem 6.1.
The limiting case in which the correlation between supply and demand is given by −1, i.e., anti-correlation, has been considered in Theorem 5.1.
7. Conclusions
We have shown that fat tails can be obtained solely by modeling price adjustment through supply and demand, each of which are random variables consisting of Brownian motion and shocks from a Poisson distribution. This consistent with the empirical evidence for many asset markets. The classical theory embodied in (1.1) is based partly on empirical observations near the mean (and not the tails) of price changes. The fact that the normal distribution is a reasonable model near the mean – and is generally convenient mathematically – does not provide any evidence that the tail of density should behave as the normal does, i.e., f(x)∼e−x2/2. Prices evolve through re-balancing supply and demand. In this paper we have assumed a model that embodies the perspective of supply/demand without invoking the fine microstructure, and show that the quotient of two Levy processes of jump-diffusion type has a tail distribution that is a power law.
In essence, exponential decay in the classical finance models is a feature of the assumptions made without compelling theoretical arguments. Prices are not distributed by a random process; rather they are the consequence of changes in supply and demand which have randomness. In fact, it is almost a tautology to say that all price change is a direct consequence of changes in supply and demand, which have randomness. If the supply and demand were deterministic, there would be no randomness in price change. Agammaegate supply and demand involve the sum of a large number of decision makers so that the Central Limit Theorem can be applied. The addition of a compound Poisson process to the randomness incorporates the influx of a smaller number of large orders that are manifested as jumps in supply or demand.
A natural way to model price change is through an excess demand model which involves a quotient of supply and demand. The important mathematical idea is that a quotient of two normals is not normal, though it resembles a normal near the mean for a range of parameters. The tail of the density of a quotient of normals has previously been shown to have a power law decay. In this paper, we showed that this persists even with the introduction of shocks in the form of a Poisson distribution when there is an arbitrary correlation in the number of shocks and an arbitrary total correlation between supply and demand.
Another important issue is that the price adjustment equation (1.5) is not necessarily linear. Hence, we consider appropriate nonlinear functions, Gε, (2.20)\ that are characterized by an exponent q−1 for q>0. The salient feature of this function is that supply and demand enter on equal footing.
For this family of functions we have calculated the power of the decay exponent for the density of the relative price change P−1ΔP, as f(x)∼x−q−1. The exponent is a function of the shape of the price function, Gε, so that there is a direct link between the form of the price function and the decay exponent. Other possibilities for the function Gε for supply and demand yields similar results. Roughly speaking if the dependence of G on the quotient of demand to supply is characterized by the power q−1, then the decay will be f(x)∼|x|−α with α=1+q.
Empirical studies of the tail of distributions for relative stock price changes (denoted by r below) have mainly shown (see [4,23,24,47,51,53] and references therein) that large capitalization stocks have a decay rate given by ζ=̃ 3 or at least within a range ζ∈[3,5] where P({|r|>x})∼x−ζ, so that f(x)∼x−ζ−1. An early paper studied 40 million data points and hypothesized a ζ=3 "universal cubic law." Thus, we have α=q+1, i.e., ζ=q. This would predict that the Gε defined above should have q−1=1/3 corresponding to α=4ζ=3. This means that if the nonlinear price function Gε has a behavior that is somewhat less responsive than linear, the implied decay will be within the empirical range. The analysis also suggests that it would be useful to do empirical research on determining the form of the function Gε, as well as examining the correlation between the changes in supply and demand to determine how close it is to anti-correlation. Some studies (e.g., [40]) have indicated that the tail exponent varies to some extent as one examines different countries and markets such as emerging markets. While stock price movements have been studied extensively, there is much less empirical work on the nature of the supply and demand curves, particularly under extreme conditions. This would be a useful empirical work that could open up a new avenue to understand rare events as discussed below. This approach can be utilized not only for the tail but for the entire distribution so volatility can be modeled in its entirety.
Our model is applicable to any price formation, and is not restricted to financial instruments. In some markets for goods, the exponential tail may hold [1,37]. Our analysis may provide a link between the tails of financial assets and goods/services. In our model, this would be attained by a function G that is very slowly responding for large imbalance e.g., involving a logarithmic function. This would mean that as the supply/demand balance becomes extreme, the relative price does not continue to change nearly as much. It seems intuitively reasonable that the supply and demand curves for goods/services have this feature, since there is usually no possibility of resale. However, further research is needed on this issue.
In general, G is a function to be calibrated. The essential point, however, is that G is a deterministic function, while D and S are Levy processes of jump-diffusion type. To simplify the discussion, let us suppose that they are simply Gaussian. The central thesis is that price change is essentially deterministic given a specific supply and demand. Suppose that one is interested in estimating the probability of a 5% drop in an asset. In order to establish this probability directly from empirical data, one would need a very large amount of data since it occurs infrequently. However, using the formalism above, one would need many fewer observations that would determine the value of the price drop as a function of the imbalance in supply and demand. Once this connection is established, it is then a matter of understanding the probability of that imbalance. The supply, S, and demand, D, are each Gaussian, with means and variances that are easily estimated with a moderate amount of data (i.e., without necessarily observing rare events) so we know the density of the normalized excess demand, namely, (D−S)/S.
In summary, we can establish the deterministic connection between the imbalance of supply/demand and the change in price through the function G. Once this is done, we can estimate the means and variances of the supply and demand individually. The mathematical results then yield the density of (D−S)/S. With the approximation of G we then obtain the probability of a particular size price drop. In particlar, we can estimate the distribution tail exponent, ζ=q.
Note that the use of a smaller data set is not just a matter of convenience. The issue is that as we use much larger sets, we are utilizing data that may be irrelevant, for example, because it is an artifact of a different era. Data sets of 252 trading days and 2520 correspond to one year and ten years, respectively. In the year 2011, for example, the ten year period would include the post-Internet bubble of 2001-2003, as well as the 2007-2008 housing debacle and subsequent recovery. So data from only the period 2010-2011 would be more pertinent in terms of estimating the probabilities of possible rare events in 2012. Then using the data from 252 or 504 relevent trading days, one can find the most extreme price changes and match them up with the (D−S)/S values, so that the values of G for those arguments are set. This is a consequence of G being deterministic. From this data set of 252 or 504 we can reliably determine the mean and variance of each of supply and demand, noting that these are Gaussian by virtue of the Central Limit Theorem for independent agents. Using these parameters, and the analysis of the quotient of Levy jump-diffusion processes, we can calculate a more accurate probability for the extreme events that were observed than would be possible by using just the raw frequency.
The problem of fat tails is part of the general issue of volatility in markets that has been of great interest in recent years (see, e.g., [22]). The methods discussed can be useful within the general setting of volatility problems.
The analysis of this paper has involved normal distributions through the Brownian motion and the distribution of each jump in the Poisson process. An open problem is to extend these results to a broad set of distributions. Using results such as [56], one can consider the quotient other distributions for the supply and demand and analyze the tail behavior.