The Quotient of Normal Random Variables And Application to Asset Price Fat Tails

02/13/2018 ∙ by Carey Caginalp, et al. ∙ Brown University University of Pittsburgh 0

The quotient of random variables with normal distributions is examined and proven to have have power law decay, with density f( x) ≃ f_0x^-2, with the coefficient depending on the means and variances of the numerator and denominator and their correlation. We also obtain the conditional probability densities for each of the four quadrants given by the signs of the numerator and denominator for arbitrary correlation ρ∈-1,1). For ρ=-1 we obtain a particularly simple closed form solution for all x∈ R. The results are applied to a basic issue in economics and finance, namely the density of relative price changes. Classical finance stipulates a normal distribution of relative price changes, though empirical studies suggest a power law at the tail end. By considering the supply and demand in a basic price change model, we prove that the relative price change has density that decays with an x^-2 power law. Various parameter limits are established.

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1. Introduction

A long-standing puzzle in economics and finance has been the ”fat tails” phenomenon in relative asset price change and other observations that refer to the empirically observed power-law decay rather than the expected classical exponential. In practical terms, this means unusual events are less rare than expected, resulting in broad implications as discussed below. Given the quite general application of the Central Limit Theorem, it is natural to expect that the frequency of the relative price change as a function of the relative price change would result in a normal, or Gaussian, distribution with exponential decay, i.e., the density has the same tail as

where is the mean and is the variance.

The assumption of a normal distribution for relative price changes dates back to Bachelier’s thesis [2] in 1900, and was further popularized in the Black-Scholes work on options pricing [4]. Champagnat et. al. [9]

note several reasons for the saliency of utilizing log-normal price changes. First, they can be ”simply interpreted and estimated. Second, closed-form expression exists for several options. Third, they could be embedded in a continuous time process, as the geometric Brownian motion, which models the evolution of the stock over the time. Indeed, many theories, for example the Capital Asset Pricing Model (CAPM) for portfolio management, take their roots in the Gaussian world.”

One particular application involves ”Value-at-Risk” which addresses questions such as: Can we be sure that the investment will retain at least, say, of its current value within a five year time period with a confidence? This methodology is often at the heart of risk analysis of an investment portfolio. The Gaussian assumption facilitates calculations; however, there have been numerous studies that indicate the risk is understated as a result [9], [36], [37]. An early study by Fama [13]

provided empirical evidence that there were ten times as many observations of relative price changes than would be expected at four standard deviations from the classical theories stipulating normal distributions. Mandelbrot and Hudson

[25] express their perspective in a section heading: ”Markets are very, very risky – more risky than the standard theories imagine.” In a more generalized context, Taleb [33] has long asserted that unusual events occur far more often than one would expect from the normal distribution. Another aspect of work in this area has involved modeling [23], [24]. The empirical observations of fat tails have also been noted in high frequency trading [10].

Thus, the question of the tail of the distribution of relative price changes is important in several key areas of finance and economics. As a practical matter it would appear that one can investigate empirically, and implement the conclusions without reference to a particular model. While data seems to be abundant, the problem is that obtaining a large amount of it often entails using data from older time periods that may well be irrelevant. Hence, the theoretical examination of the origins of the tail of the distribution becomes crucial. Various explanations have been offered for the observation of fat tails. These differ from our approach in that they stipulate an exogenous influence that alters the natural normal decay. For example, it has been argued that large institutional investors placing trades in less  liquid markets tend to cause large spikes [14]. Theoretical models, e.g., [24], using random walk have been used to explain fat tails. We refer to the book by Kemp [21] for a review of the literature. While these may be important factors that lead to fat tails, we demonstrate below that fat tails also arise from the endogenous price formation process.

In the classical approach to finance the basic starting point is the stochastic differential equation for the relative price change, as a function of time, and (with as the sample space):

(1.1)

Here is the standard Brownian motion, so i.e., is normal with the variance as and mean , and has independent increments. The stochastic differential equation above is shorthand for the integral form (suppressing in notation)

(1.2)

where and can be constant, functions of , or random variables. For constant, and one can write

(1.3)

With the assumption that is nearly constant over time, classical finance clearly stipulates that the relative temporal changes in asset prices should be normal. The basic equation (1.1) is obtained from the idea that all information about the asset is incorporated into the price, and that random world events alter the value on each time interval. Further, the assumption of the existence of a vast arbitrage capital means that the changes in the valuation are immediately reflected in the price, notwithstanding any bias or mistake on the part of the less knowledgeable investors. Consistent with the Central Limit Theorem, it is assumed that the events that alter the asset valuation are normally distributed. But the important and tacit second assumption is that relative price changes inherit this property.

While (1.1) is the basis for a large majority of papers on asset prices and related issues in finance, it is difficult to generalize in some directions. Indeed, with the assumption of infinite arbitrage already built into the model, what can one subtract from infinity in order to obtain the randomness that arises from the finiteness of trader assets and order flow? What is needed then is an approach that takes into account more of the microstructure of trading, i.e., the supply and demand of the asset submitted to the market clearinghouse (see e.g., [5], [27],[29] and references therein).

In this paper we examine the temporal evolution in percentage price changes by modeling price change that utilizes a fundamental approach of supply/demand economics analysis. We show that if one assumes supply and demand are normally distributed, the mathematics of the quotient of normals suffices to yield fat tails. In particular, in the limit of large deviations from the mean, one obtains the result for the density for large where the constant depends on the means, variances and correlations of supply and demand.

Thus modeling of the relative price change in terms of finite supply and demand lead naturally to the basic statistical problem of the distribution of the quotient of two normal random variables. While there is a long history of the problem, surveyed below, most results concern the mid-range of the distribution, rather than the tail. We obtain a number of exact representations and rigorous bounds on the density conditioned upon the signs of the numerator and denominator, as well as the overall density for all correlations such that For we obtain a particularly simple exact expression for the density for all

2. The Model and the Quotient of Normals

We focus on modeling price change in economics; the issues are similar in other disciplines in which quotients arise. Classical economics stipulates that prices remain constant in time when supply and demand are equal [35], [20], thereby defining equilibrium. When demand exceeds supply, prices rise to restore equilibrium, and analogously in the other direction. There is theoretical [5], experimental [7] and empirical [6] evidence that asset price change can be modeled as basic goods, with the supply and demand depending on a number of factors such as the cash/asset ratios.

Let and be the supply and demand, respectively. The most general expression for relative price change (see [5] and references, and, [17] p. 165 for motivation for the linear equation below) can be written as

(2.1)

where is the time scale, is the unit price of the asset, and is a differentiable function such that reflecting the assumption that prices do not change when demand equals supply, and for some positive constant which stipulates that prices rise when demand exceeds supply. If trading is very active, prices will react to small changes in supply/demand imbalances, and the function can be assumed to be linear. Also, the constants and can be incorporated into a dimensionless time, yielding our basic starting point:

(2.2)

Rather than considering the randomness directly in terms of prices, as in the classical approach, we assume that supply and demand are random variables. In a liquid market (i.e., frequently traded), the randomly flowing orders for supply and demand can be approximated by normal distributions (as discussed below). However, supply and demand are not likely to be independent, as a random event that increases buying is likely to diminish selling. In general, we can assume that and are bivariate normal random variables. This leads to the question of estimating the tail of a distribution of (2.2) above, i.e., the quotient of bivariate normal random variables. Our particular interest is for negatively correlated and but the analysis below will be for the full range of correlations, as similar issues arise in other problems, e.g., physical, biological, in which there is a quotient or random variables.

The supply and demand on a given interval consists of buy and sell orders submitted to the market with some random distribution. The orders will be influenced by news which will arrive from many independent sources, so that the Central Limit Theorem will apply. That is, under a broad set of conditions, the arrival of random orders, and thus, supply and demand, will be approximated well by the normal distribution. However, price formation evolves through a process that is almost deterministic. In other words, if one had a large sample of the same supply/demand graphs, the resulting relative price change would exhibit only a small variance as market makers and short term traders can readily see the very short term market direction. This is a basic consequence of economic game theory (see

[35], [29] and references therein). In more practical terms, if we consider the stock of a major company that trades with high volume, there will be a large number of market makers whose only business is to profit from any deviations from the ”correct” price, given the order flow. Given a particular supply/demand graph, a significant deviation can only arise from, not one, but many of these market makers erring in the same direction. In summary, the randomness is inherently in the supply and demand curves. For fixed supply/demand functions, the price evolution is nearly deterministic.

The empirical assumption that relative price changes are normal (often stated as price change is log-normal) has been tested, with results that indicate significant deviations from normality (see e.g., [13], [9], [36]). While there are numerous studies on the distribution of trading prices, empirical studies testing the normality of buy/sell orders for major stocks and commodities would be instrumental in understanding the problem of fat-tails in relative price change.

We assume conditions on the orders that are compatible with the Central Limit Theorem, so that, for large, the supply and demand are governed by a bivariate normal distribution. Let and

(2.3)

We assume that constitutes a bivariate normal distribution having density, for

(2.4)

where and is the covariance matrix,

(2.5)

with and for One has the basic bound . Upon defining one has the result that exists if and only if . Note that is said to have a singular bivariate normal distribution if there exists real numbers such that and are identically distributed, with This is equivalent to . We will consider the case later and assume for now that (2.5) is nonsingular. An excellent source for relations on multivariate normal distributions is Tong [31].

We let be the density,

the (cumulative) distribution function for the variable

. While the random variables and can take on any values in , the primary interest is in positive values which we consider below, with similar results for the remaining quadrants of presented in Appendix B.


The issue of the quotient of normal variables has been studied in a number of contexts, as it arises in a number of biological and physical problems including constructing the genome mapping of plants, imaging ventilation with inert flourinated gases, and various neurological applications [28]. A classical problem is to estimate parameters (e.g., mean and variance) of the ratio of two populations. An early result by Geary [16] concerned the ratio of two independent normals with zero means. Hinkley [18, 19] obtained a result in the limit as the variance of the denominator approached zero. Kuethe [22] and Marsaglia [26] developed complex expressions for the density of the ratio of two independent normals with strictly positive means and variances.

Diaz-Frances and Rubio [11]

obtained an important result by proving a theorem that establishes bounds on the difference between the true distribution and the proposed approximation. They summarize a number of results and earlier works, noting that the ratio of independent normals with positive means has no finite moments, and that the shape of the distribution of the quotient ”can be bimodal, asymmetric, symmetric, and even close to a normal distribution, depending largely on the values of the coefficient of variation of the denominator.”

Much of the recent focus has been on approximating the ratio of the means by a normal distribution; see Diaz-Frances and Rubio [11] and Diaz-Frances and Sprott [12] for a discussion and references, and also Watson [32], Palomino et al. [28], Schneeweiss [30], and Chamberlin and Sprott [8]. For the most part the results involve the mid-range, where a normal approximation is possible, rather than the tail.

The main focus of our paper will be on the tail of the distribution. Although the terminology is not yet standard, one can broadly define ”heavy tails” or leptokurtosis as distributions with falloff less rapid than the normal, while ”fat tails” consist of power-laws.

While there has been some evidence that the ratio of independent normals, under some conditions on the parameters, will be fat-tailed, there there has not been a comprehensive understanding and analysis of the behavior of the tail of the distribution. In Section 4 we prove that the ratio of two normals, with arbitrary means, variances and correlations has the fat-tail property, with the density, approaching zero as We also prove bounds on the coefficient for large , providing a rigorous description of the tail of the quotient of normals under very general conditions.

3. Calculation of the probability density of , conditioned on positive  and  .

We assume and are all strictly positive. Define

(3.1)

We note that the following probabilities are identical for (and zero for )

(3.2)

Various expressions for the density of bivariate and multivariate normals, such as the one above, can be found in [31].

In order to obtain the conditional density we use the identity

(3.3)

together with an implication of the Dominated Convergence Theorem to write

(3.4)

Defining the quantities

(3.5)
(3.6)
(3.7)

we can express this equation as

(3.8)

Let so and to transform the integral and obtain

(3.9)

We have then the identity

(3.10)

Hence, the density of conditioned on is given by the basic relation (see [3])

(3.11)

where is the set The calculation of is carried out in Appendix A.

We can use to write the following exact expression for the conditional density:

(3.12)

4.  Analysis of the logarithm of the conditional density

In order to extract the behavior of the conditional density for large we analyze its logarithm.

Theorem 4.1.

For one has the bounds

(4.1)

where  .

Proof.

Re-grouping the constants, and taking the logarithm of , we write

(4.2)

The decay exponent will be determined by the large limit of this quantity divided by We first analyze the last term in the expression above:

(4.3)

Taking the logarithm and dividing by we have

(4.4)

We examine the last two terms for

(4.5)

The last logarithm in (4.4) can be bounded (for ) as

(4.6)

Thus one has

(4.7)

concluding the proof. ∎

In order to extract the relevant part of we write

(4.8)

and use this to extract the dependent part of the conditional density.


Theorem 4.2.

(a) For one has the bounds

(4.9)

where satisfies   and more specifically

(4.10)

where    and

(4.11)

where and are defined below.

(b) In the limit one has

(4.12)

and the density can be expressed in the form

(4.13)
Proof.

The main issue is to examine the two terms, and and extract the part that is significant after division by .

One has

(4.14)

Upon dividing by the second of these terms is

(4.15)

and the latter term is bounded by

(4.16)

The last of the terms in (4.14) can be bounded as

(4.17)

Hence, one has the bounds

(4.18)
(4.19)

Next, we examine as we focus on The term has dependence. Note that can be written as

(4.20)

and with the definitions

(4.21)

Using and we write

Basic Taylor series estimates then imply

(4.22)

Upon using so that , we have the bounds

(4.23)

Thus, for large we can thus write, for

(4.24)

where depends on and

This yields the bound

(4.25)

where . Hence, we have

(4.26)

where

(4.27)

That is, one has the bound, with depending on and ,

(4.28)

The proof is concluded by observing that the term can be bounded using the following classical estimates for the error function:

(4.29)