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There exists a well-developed statistical theory predicting extreme price values for financial markets known as extreme value theory (EVT). This approach relies on the seemingly obvious, but rarely analyzed, assumption that price displacement extremes actually exist for various markets. This paper attempts to describe the behavior of financial markets as a set of functions in terms of the dynamic variables price and time based on the net difference between ask and bid volumes over a unit period, thereby offering evidence to support the assumption that price extremes exist. Yet, it’s not meaningful to show merely that extremes exist. If the extreme negative price displacement simply represents a complete market collapse then the assumption becomes trivial. Accordingly, the paper also introduces a method to determine whether price displacements are constrained by non-trivial extremes. This description might have implications for EVT and market risk management in approximating the magnitude of “Black Swan” events. The paper also shows that if one can closely approximate the magnitude of such a rare event, one cannot also predict when the event will occur with any meaningful degree of certainty.

A “Black Swan” refers to a highly improbable event that lies significantly outside of normal expectations.^{1} For financial markets, such events typically manifest themselves as extreme price variations.

There exists a well-developed theory predicting these extreme price variations called Extreme Value Theory (EVT) [

Yet, merely demonstrating that extreme values exist is not enough. In the context of financial markets, the extreme measure that investors and market risk managers care about most is usually a negative price displacement. If the extreme negative price displacement―measured as a percent decrease in prices over a defined period―is simply unity (i.e., a complete market collapse to zero) then the assumption that price displacement extremes exist becomes trivial. This paper introduces a heuristic method to determine whether negative price displacements over a defined period are constrained by extremes that are non-trivial.

There are two reasons to focus on price reductions instead of price increases. The first is the reason just mentioned; it is the focus of most market risk managers. The second is that price increases are theoretically infinite while price decreases are limited to the displacement between the current price and zero. Although we use an extreme price increase variable as part of our mathematical formalism, this paper is careful to only claim a method for approximating price reductions.

Further, this method is only potentially useful for markets that are robust and actively traded, and where the mean value of the price displacement ratio is close to zero. This last requirement implies a form of mean reversion, which remains controversial in the literature [

Lastly, this heuristic is based on an uniformitarian-like assumption that, to some extent, historical data can be used to predict extreme price displacements. There is significant disagreement in the literature over whether this postulate holds [

Although EVT deals with extreme deviations from a probability distribution median, this paper does not use a statistical approach in arguing that extreme prices exist. The paper does employ some basic probability methods in §3, but these are transitional in arriving at a conclusion of non-triviality. Even though these are not the standard probability methods of EVT, the paper’s conclusions offer an approximation of the magnitude of a price displacement extreme, which is a consistent element in traditional EVT analysis [

The paper is divided into five sections. The first is this brief introduction. The second defines a financial market as a system dependent entirely upon the interactions of generalized buyers and sellers and then offers an inductive argument that suggests the existence of price extremes. The third section offers a way to calculate these price extremes to determine if they are non-trivial. The fourth provides predictive limitations of the heuristic as to the timing of extreme events in the form of an uncertainty principle and other limitations generally. The fifth section offers a brief summary.

We begin by defining the time and price variables of a financial market and then examine the different perspectives from which one can view these variables as coordinates. We will then derive a general equation of market dynamics based on these perspectives and show that the solution suggests the existence of negative price extremes.

Define a unit period as ^{2}

An asset also has a set of possible price displacement configurations as functions of elapsed time

Define

Because we are primarily concerned with the net price displacement for a unit period, we can imagine a normalized price-time coordinate system K such that

In an active market, traders with the desire to buy an asset place bid orders and traders wishing to sell place ask orders. Yet, desire is not enough to directly affect a change in the price of an asset. There must also exist an interaction with another trader to turn a bid or ask order into a transaction. We’ll call transactions associated with bids “bid volume.” This becomes the scalar quantity B representing the number of bid transactions for a unit period. “Ask volume” is similarly the scalar quantity A representing the number of ask transactions for a unit period.

Analysis of historical data from active markets suggests B has the potential to reduce prices while A has the potential to increase prices. The bid price is defined as the highest current price at which a trader is willing to buy. The ask price is defined as the lowest current price at which a trader is willing to sell. Therefore, as bid transactions transpire, the once highest bid price vanishes and the next highest (but lower than before) price becomes the highest current price. As ask transactions transpire, the once lowest ask price vanishes and the next lowest (but higher than before) price becomes the lowest current price. Thus, B is selling volume and A is buying volume [

When a financial market experiences more buying volume than selling volume, there are more traders buying at the ask price. This “pushes” the asset up in price. When a market is experiencing more selling volume than buying volume, there are more traders selling at the bid price, which “pushes” the asset down in price. Thus, the effect of bid volume on price displacement in

From this we see that the net effect of all bid volume for a unit period changes the state of the asset along the stationary price dimension in the negative direction from

Historical data analysis indicates that not all markets given identical ask and bid volumes have the same ask and bid volume effects with respect to positive and negative price displacement ratios. As a result, we can assume that each market additionally has some specific inertial property―let’s call it n―that constrains the asset in its positive or negative movement away from

Of course, this description is incomplete. Ask and bid volumes do not exist in isolation. In an active market, there is always a superposition of specific ask and specific bid volumes over any unit period where the asset is pushed in the positive direction when the ask volume dominates and pushed in the negative direction when the bid volume dominates. This specific net volume gives us

The net effect of the specific net volume―which we’ll call

Again, this is the equation for an asset moving through the stationary dimension of price resulting from the net effect of the specific ask and specific bid volumes for a unit period. It is, in fact, market mechanics as described from the perspective of the vector field

Now let’s look at this behavior from a second frame of reference,

After imagining this behavior from the

where the prime denotes a price derivative just as the dot denotes a time derivative. Combining Equations (1) and (2) gives us the dynamics of a market in terms of both changes in time and price:

But what does

(N.B.: For markets where the mean price displacement for a unit period is not relatively close to zero in K, this assumption cannot hold.)

The first two terms in Equation (3) give us no new information about

Yet, as we just discussed, the only meaningful mean constant for any given unit period is the net (ask and bid) volume for that period since we know from Equations (1) and (2) that

where H is the specific net volume

From Equation (4), we see that

This is consistent with Equation (2).

The general solution to this harmonic approximation is well known:^{3}

Here, z is a complex number,

is the phase or principal argument of z. The change in the phase for each unit period is the radial frequency^{3} Thus, for a unit period,

In solving Equation (5) as an initial value problem, we see that

Additionally,

Of course, this does not necessarily mean that a market has or ever will reach −R, only that the negative extreme theoretically exists for any unit period t. In this way, −R is very similar to a “Black Swan” event from a market risk management perspective [

Now that we’ve provided evidence that negative price extremes exist, the next step is to decide under what conditions these extremes approach triviality, if any. Define triviality as an extreme price displacement ratio

Here we introduce the Lagrangian function^{4} From the time integral of the Lagrangian of Equation (5) we get the following action functional [

We see that

For the probability that either the positive or negative square root of the action functional is on the interval between the extreme values of the square root of the action functional, we can write

where

Because R is an extreme value, this probability must approach certainty since by definition all measures for the positive or negative square root of the action functional must fall between the positive or negative square root of the action functionals containing the positive or negative extreme price displacement

values. Of course, this means that

We can define the phase

This is only an approximation, but we assume the approximation close enough to express the relationship as an equality for practical purposes.

Another way to think about this is to return to the complex plane from §2. If the real price displacement ratio

displacement ratio on K. For example, at

complex vector space, but there is also a phase in the complex conjugate (dual)

vector space where

for every real price displacement ratio as well as a number

If we take the inverse complementary error function of both sides of Equation (9) we get

if we define the constant

Equation (10) still requires knowledge of n for each market to find the value of R. Since n is not a number regularly measured, we should try to approximate R without relying on a known minimized value of n. If we look at the relationship between the absolute value of a specific price displacement ratio

where

1) The measure of centrality (expectation value) of

2) The measure of centrality of the probability that any

3) These measures of centrality coincide.

From these three assumptions, we can define

since the geometric mean of all real numbers on the interval

We can next substitute R for

From this and Equation (9), we see that

If we define the constant

Thus, we find that the magnitude of a market’s extreme price displacement ratio is linearly dependent on the expectation value of its absolute price displacement ratio.

Because both

The method outlined here is not without limitations. Although §3 allows us to closely approximate

It is well known that the time and frequency domains are Fourier transform pairs [

Let x be a function in Hilbert space

Define the uncertainty (standard deviations) of t and

From these definitions, Weyl [

generally, and that

From Equation (6), we see that if we know the measure of

This means

Because knowing the price displacement ratio with arbitrary precision means that we know the time derivative of the phase with the same precision, the more certain we are about the value of

Another significant limitation of this method is that it relies heavily on the sample with which one chooses to calculate

Still, this method might improve our understanding of the rules that undergird market mechanics and, therefore, serve as an additional tool for managing market risk when faced with price displacements that deviate significantly from historic expectations. For example, based on historical data prior to the market close of October 16, 1987, Equation (14) would have predicted that Coca-Cola Company stock (KO) would have had a daily price displacement ratio extreme of

By regarding market mechanics in terms of time and price from two separate perspectives, we have presented theoretical evidence that financial markets have extreme price displacements. These extremes appear non-trivial except in the limits as

I wish to express my gratitude to Christopher L. Culp, Nuno Garoupa, Joseph R. Hanley, James McGrath, Ann M. Manhire, Andrew P. Morriss, Lisa A. Rich, and Saurabh Vishnubhakat for their valuable suggestions. All errors are mine alone and I promise to do better next time.

Manhire, J.T. (2018) Measuring Black Swans in Financial Markets. Journal of Mathematical Finance, 8, 227-239. https://doi.org/10.4236/jmf.2018.81016