 # Asset Price Volatility and Price Extrema

The relationship between price volatilty and a market extremum is examined using a fundamental economics model of supply and demand. By examining randomness through a microeconomic setting, we obtain the implications of randomness in the supply and demand, rather than assuming that price has randomness on an empirical basis. Within a very general setting the volatility has an extremum that precedes the extremum of the price. A key issue is that randomness arises from the supply and demand, and the variance in the stochastic differential equation govening the logarithm of price must reflect this. Analogous results are obtained by further assuming that the supply and demand are dependent on the deviation from fundamental value of the asset.

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## 1. Introduction

### 1.1. Overview

In financial markets two basic entities are the expected relative price change and volatility. The latter is defined as the standard deviation of relative price change in a specified time period. The expected relative price change is, of course, at the heart of finance, while volatility is central to assessing risk in a portfolio. Volatility plays a central role in the pricing of options, which are contracts whereby the owner acquires the right, but not the obligation, to buy or sell at a particular price within a specified time interval.

In classical finance, it is generally assumed that relative price change is random, but volatility is essentially constant for a particular asset .

In this way, price change and volatility are essentially decoupled in their treatment. In particular, the relative price change per unit time

is given by a sum of a deterministic term that expresses the long term estimate for the growth, together with a stochastic term given by Brownian motion.

Hence, the basic starting point for much of classical finance, particularly options pricing (see e.g., [2, 3]), is the stochastic equation for as a function of (the sample space) and given by

 (1.1) dlogP=μdt+σdW.

where is Brownian motion, with so is normal with variance , mean and independent increments (see [4, 5]). While and are often assumed to be constant, one can also stipulate deterministic and time dependent or stochastic and The stochastic differential equation above is short for the integral form (suppressing in notation) for arbitrary

 (1.2) logP(t2)−logP(t1)=∫t2t1μdt+∫t2t1σdW

For constant, and one can write

 (1.3) ΔlogP:=logP(t2)−logP(t1)=μΔt+σΔW.

The classical equation (1.1) can be regarded as partly an empirical model based on observations about volatility of prices. It also expresses the theoretical construct of infinite arbitrage that eliminates significant distortions from the expected return of the asset as a consequence of rational comparison with other assets such as risk free government (i.e., Treasury) bonds. Hence, this equation can be regarded as a limiting case (as supply and demand approach infinity) of other equations involving finite supply and demand  (Appendix A). Thus, it does not lend itself to modification based upon random changes in finite supply and demand. An examination of the relationship between volatility and price trends, tops and bottoms requires analysis of the more fundamental equations involving price change. A suitable framework for analyzing these problems is the asset flow approach based on supply/demand that have been studied in [7, 8, 9, 10], and references therein.

An intriguing question that we address is the following. Suppose there is an event that is highly favorable for the fundamentals of an asset. There is the expectation that there will be a peak and a turning point, but no one knows when that will occur. By observing the volatility of price, can one determine whether, and when, a peak will occur in the future? In general, our goal is to delve deeper into the price change mechanism to understand the relationship between relative price change and volatility.

Our starting point will be the basic supply/demand model of economics (see e.g., [11, 12, 13]). We argue that there is always randomness in supply and demand. However, for a given

supply and demand, one cannot expect nearly the same level of randomness in the resulting price. Indeed, for actively traded equities, there are many market makers whose living consists of exploiting any price deviations from the optimal price determined by the supply/demand curves at that moment. While there will be no shortage of different opinions on the long term prospects of an investment, each particular change in the supply/demand curve will produce a clear, repeatable short term change in the price.

Given the broad validity of the Central Limit Theorem, one can expect that the randomness in supply and demand of an actively traded asset on a given, small time interval will be normally distributed. Thus, supply and demand can be regarded as bivariate normally distributed random variables, with a correlation that will be close to

since the random factors that increase demand tend to decrease supply.

In Sections 2 and 3 we explore the implications of this basic price equation that involves the ratio of demand/supply. By assuming that the supply and demand are normally distributed with a ratio of means that are characterized by a maximum, we prove that an maximum in the expectation of the price is preceded by an extremum in the price volatility. This means that given a situation in which one expects a market bottom based on fundamentals, the variance or volatility can be a forecasting tool for the extremum in the trading price. Furthermore, in pricing options, this approach shows that the assumption of constant volatility can be improved by understanding the relationship between the variance in price and the peaks and nadirs of expected price.

Subsequently, in Section 3, we generalize the dependence on demand/supply in the basic model, and find that under a broad set of conditions one has nevertheless the result that the extremum in variance precedes the expected price extremum.

In Section 4 we introduce the concept of price change that depends on supply and demand through the fundamental value. The trader motivations are assumed to be classical in that they depend only on fundamental value; however, the price equation involves the finiteness of assets, which is a non-classical concept. Without introducing non-classical concepts such as the dependence of supply and demand on price trend, we obtain a similar relationship between the volatility and the expected price.

In Section 5, we prove that within the assumptions of this model and generalizations, the peak of the expected log price occurs after the peak in volatility.

### 1.2. General Supply/Demand model and stochastics

We write the general price change model in terms of the price, the demand, , and supply, . In particular, the relative price change is equal to a function of the excess demand, (see e.g., , ). That is, we have

 (1.4) P−1dP/dt=G(~D/~S)

where satisfies for all If symmetry between and is assumed, then one can also impose A prototype function with properties is given by

A basic stochastic process based on for is defined by

 (1.5) dlogP(t,ω)=a(t,ω)dt+b(t,ω)dW(t,ω)

for some functions and in , the space of stochastic processes with a second moment integrable on (see ). The terms  and can be identified from and the nature of randomness that is assumed. In any time interval there is a random term in and The assumption is that there are a number of agents who are motivated to place buy orders. The relative fraction is subject to randomness so that the deterministic demand, multiplied by for some random variable . Likewise, one has the deterministic supply, by . This yields, for sufficiently small , the approximation

 (1.6)

with being either constant, time dependent or stochastic. We can then write

 G(~D/~S)~=G(D/S)+G′(D/S)(σDSR)

and thereby identify and Note that we view the randomness as arising only from the term, so we can assume that and are deterministic functions of at this point. Later on in this paper we consider additional dependence on and By assuming that the random variable is normal with variance and is independent of , one obtains the stochastic process below (in which and are deterministic).

By differentiating , we note

 1xG′(1x)=xG′(x),

and thereby write the stochastic differential equation

 dlogP(t,ω)=G(D/S)dt+12{DSG′(DS)+SDG′(SD)}dW(t,ω).

In particular, for one has

 dlogP=(DS−SD)dt+σ{DS+SD}dW.

We are interested in the relationship between volatility and market extrema, and focus on market tops by using the simpler equation for the function for which holds only approximately near The equation is then (see Appendix)

 (1.7) dlogP(t,ω)=(D(t)S(t)−1)dt+σ(t,ω)D(t)S(t)dW(t,ω).

For market bottoms, one can obtain similar results (see Appendix).

We will specialize to deterministic or even constant below. If we were to assume that the supply and demand have randomness that is not necessarily the negative of one another, then we can write instead,

 (1.8) D(1+σaRa)S(1−σbRb)~=(1+σaRa+σbRb)DS−1 .

yielding the analogous stochastic process,

### 1.3. Derivation of the stochastic equation

We make precise the ideas above by starting again with (1.4) where and are random variables that are anticorrelated bivariate normals with means and and both have variance We can regard the means as the deterministic part of the supply and demand at any time , so that with as the covariance matrix , we write

 (1.10)

For any fixed , one can show that the density of is given by

 (1.11) fD/S(x)=1+μD/μS√2πσ1μS(x+1)2e−12(x−μD/μS)2(σ1μS)2(x+1)2.

Other approximations in different settings have been studied in [15, 16, 17] and references therein.

For values of near the mean of , one has

 (1.12) (x+1)2~=(μDμS+1)2.

We can use this to approximate the density, using as the approximate variance of as

 (1.13) fD/S(x)~=1√2πσRqe−(x−μD/μS)22σ2Rq;   fDS−1(x)~=1√2πσRqe−(x−μD/μS+1)22σ2Rq.

With this expression for the density of we can write the basic supply/demand price change equation as

 (1.14) ΔlogPΔt~=R1∼N(μDμS−1,σ2Rq),

where each variable depends on and Subtracting out the , defining and noting that depends on through we write

 (1.15) ΔlogP~=(μDμS−1)Δt+σRR0Δt.

By definition of Brownian motion, we can write

 (1.16) ΔlogP~=(μDμS−1)Δt+σRqΔW.

With and held constant, an increase in leads to a decrease in the variance We would like to approximate this under the condition that By rescaling the units of , together and assuming that each of and are sufficiently close to that we can consider the leading terms in a Taylor expansion, and write

 (1.17) μD=1+δD, \ μS=1+δS .

Note that and will be nearly equal unless one is far from equilibirium. Ignoring the terms higher than first order one has

 σ2Rq =σ21(1+δS)2(1+1+δD1+δS)2 (1.18) ~=4σ21(1−3δS+δD).

We are considering so that

 (1.19) σ2Rq=4σ21(1+4δ).

Using Taylor series approximation, one has

 (1.20)

We can thus write the stochastic equation above as

 (1.21) ΔlogP~=(μDμS−1)Δt+2σ1μDμSΔW,

so that the differential form is given in terms of by

 (1.22) dlogP(t)=f(t)dt+σ(f(t)+1)dW(t)

This is in agreement with the heuristic derivation above, with

and  as the variance of each of and

## 2. Location of maxima of Supply/Demand versus price

### 2.1. The deterministic model.

We will show that if is given by a deterministic function , then the stochastic equation above will imply that the variance over a small time interval will have an extremum before the price has its extremum.

Once we do this simplest case, it will generalize it to the situation where is also stochastic, and show that the same result holds.

To this end, first consider the simple, purely deterministic case:

 (2.1) P−1dPdt=DS−1=:f, i.e.,  ddtlogP(t)=f(t)

Assume that is a prescribed function of that is for satisfying:

on on and on

if if

if

Then is increasing on and decreasing on and has a maximum at

In other words, the peak of occurs at while the peak of is attained at This demonstrates the simple idea that price peaks some time after the peak in demand/supply. In fact, during pioneering experiments Smith, Suchanek and Williams  observed that bids tend to dry up shortly before a market peak. Also, the important role of the ratio of cash to asset value in a market bubble that was predicted in  was confirmed in experiments starting with .

### 2.2. The stochastic model.

Recall that and are deterministic functions of time only. We model the problem as discussed above so the only randomness below is in the variable. The stochastic equation given by for a continuous function in the integral form, for any and is

 (2.2) ΔlogP=∫t2t1f(z)dz+∫t2t1σ(z)(f(z)+1)dW(z).

Note that for the time being we are assuming that and may depend on time but are deterministic. We compute the expectation 111We let denote . and variance of this quantity:

 (2.3) E[ΔlogP]=∫t2t1f(z)dz

since is deterministic and

 Var[ΔlogP] =E[∫t2t1f(z)dz+∫t2t1σ(z){f(z)+1}dW(z)]2 (2.4) −(E[∫t2t1f(z)dz+∫t2t1σ(z){f(z)+1}dW(z)])2.

The term is deterministic and vanishes when its expectation is subtracted. The expecation of the and the terms vanishes also. We are left with

 Var[ΔlogP] =E[∫t2t1σ(z){f(z)+1}dW(z)]2 (2.5) =∫t2t1σ2(z){f(z)+1}2dz

using the standard result (, p. 68).

We want to consider a small interval so we set and . We have

 V(t,t+Δt) :=Var[logP(t+Δt)−logP(t)] (2.6) =∫t+Δttσ2(z){f(z)+1}2dz. V(t) :=limΔt→01ΔtV(t,t+Δt)=limΔt→01Δt∫t+Δttσ2(z){f(z)+1}2dz (2.7) =σ2(t){f(t)+1}2.
###### Example 2.1.

For the maximum variance of will be when is at a maximum, which is when has its maximum, i.e., at

 (2.8) ddtV(t)=ddt{f(t)+1}2=2{f(t)+1}ddtf(t)

Since in all cases, we see that the derivative of  is of the same sign as the derivative of so the limiting variance is increasing when is increasing and vice-versa. Recall that increases so long as and decreases when In other words, for the peak case, one has if and only if with a maximum at  When has a peak, the maximum of will be at when has its maximum.

To summarize, if the coefficient of is with constant and has a maximum at then will also have a maximum at so that the maximum in will occur after the maximum in since

###### Remark 2.2.

We have shown that has a maximum, at some time that is preceded by a maximum in . We can use this together with Jensen’s inequality to show that for arbitrary Indeed, since we can write

 (2.9) ElogP(tm)P(t1)≥0.

Let and in Jensen’s inequality, , we have

 (2.10) EY=EelogY≥eElogY≥1.

Hence, the expected ratio of price at to the price at any other point is greater than

###### Remark 2.3.

The conclusion above can be contrasted with the standard model adjusted so that has the same property of a peak at some time . Performing the same calculation of - for this model yields the result so that it provides no information on the expected peak of prices.

## 3. Additional randomness In Supply and Demand

### 3.1. Stochastic Supply and Demand.

Let be a stochastic function such that and . With and we write the SDE in differential and integral forms as

 (3.1) dX=fdt+σ(1+f)dW
 (3.2) X(t+Δt)−X(t)=∫t+Δttf(s)ds+∫t+Δttσ(s)(1+f(s))dW(s).

where we will assume is a continuous, deterministic function of time, though we can allow it to be stochastic in most of the sequel.

One has since and one obtains again the identities

 (3.3) EΔX=∫t+ΔttEf(s)ds,
 Var[ΔX] =E[∫fds+∫σ(1+f)dW]2 −(E[∫fds+∫σ(1+f)dW])2 (3.4) =Var[∫fds]+2E[∫fds∫σ(1+f)dW]+E[∫σ(1+f)dW]2

where all integrals are taken over the limits and .

###### Lemma 3.1.

Let Then for some depending on this bound, one has

 (3.5) ∣∣∣E∫t+Δttf(s′)ds′∫t+Δttσ(s){1+f(s)}dW(s)∣∣∣≤C(Δt)3/2.
###### Proof.

We apply the Schwarz inequality to obtain

 ∣∣∣E∫t+Δttf(s′)ds′∫t+Δttσ(s){1+f(s)}dW(s)∣∣∣ (3.6) ≤⎧⎨⎩E(∫t+Δttf(s′)ds′)2⎫⎬⎭1/2⎧⎨⎩E(∫t+Δttσ(s){1+f(s)}dW(s))2⎫⎬⎭1/2.

We bound each of these terms. Using the Schwarz inequality on the integral, we obtain using generic throughout,

 (3.7) E(∫t+Δttf(s′)ds′)2≤C(Δt)2.

The second term is bounded using the fact that is deterministic,

 E(∫t+Δttσ(s){1+f(s)}dW(s))2 =∫t+Δttσ2(s)E{1+f(s)}2ds (3.8) ≤CΔt.

Taking the square roots of (3.7) and (3.8), and combining with proves the lemma. ∎

###### Lemma 3.2.

Let be a continuous, deterministic function and assume Then

 (3.9) ∣∣∣Var[ΔX]−∫t+Δttσ2(s)E{1+f(s)}2ds∣∣∣≤C(Δt)3/2
###### Proof.

Basic stochastic analysis yields

 (3.10) E(∫t+Δttσ2(s){1+f(s)}dW)2=∫t+Δttσ2(s)E{1+f(s)}2ds.

Thus, using (3.4) and we have the result (3.9). ∎

Now, we would like to determine the maximum of and show that it precedes the maximum of the expected log price.From the calculations above, one has

###### Lemma 3.3.

In the general case, assuming on but allowing stochastic such that one has

 (3.11)
###### Lemma 3.4.

Suppose and is a deterministic continuous function on then one has

 (3.12) V(t)=σ2{1+Ef}2+σ2Varf.

and the extrema of occur at such that

 (3.13) 2σσ′{[1+Ef]2+Varf}+σ2{2[1+Ef](Ef)′+(Varf)′}=0.
###### Proof.

Using Lemma 3.3, we write

 V(t) =σ2E[1+2f+f2]=σ2{1+2Ef+(Ef)2+Ef2−(Ef)2} (3.14) =σ2(1+Ef)2+σ2Varf.

Differentiation implies the second assertion. ∎

###### Lemma 3.5.

Suppose on , while and are constant in Then the extremum of occur for such that

 (3.15) ddtEf(t)=0.
###### Proof.

From the previous Lemma, we have , yielding

 (3.16) limΔt→01ΔtV(t,t+Δt)=σ2(1+Ef(t))2+Var[f(t)]

Since we are assuming that is constant in time, we obtain

 ∂∂tlimΔt→0V(t,t+Δt) =∂∂t{σ2(1+Ef(t))2} (3.17) =2σ2(1+Ef(t))ddtEf(t).

Thus, the right-hand side vanishes if and only if i.e., at (by definition of ). Note that we have so that

### 3.2. Properties of f

The condition is easily satisfied by introducing randomness in many forms. For the Lemma above, we would also like to satisfy

Another way of attaining this (up to exponential order) is to define as the stochastic process

 (3.18) df(t)=μf(t)dt+σf(t)dW(t)

where and are both time dependent but deterministic.

We can assume that is a given, fixed value, and obtain (see e.g., , )

 (3.19)

since is deterministic

In particular, if one has , then while so one has approximately constant variance for for large . In particular, one has

 (3.20) ddtVar[f(t)]=ddt∫tt0σ2f(s)ds=σ2f(t)=e−t.

### 3.3. General coefficient of dW

The stochastic differential equation (3.1) entails a coefficient of that is proportional to One can also consider the implications of a coefficient that is proportional to the excess demand or a monomial of it. More generally, we can write for an arbitrary continuous function leading to the stochastic differential equation

 (3.21) dlogP=fdt+σhdW,

where can also be stochastic or deterministic function of time.

From this stochastic equation one has immediately

 (3.22) dE[logP]dt=Ef

similar to the completely deterministic model, except that is replaced by

From the integral version of the stochastic model, we can write the expectation and variance as

 (3.23) E[ΔlogP]=∫t+ΔttEf(s)ds
 V(t,t+Δt) :=Var[ΔlogP]=Var[∫t+Δttf(s)ds]+2E[∫t+Δttσ(s)h(s)dW(s)] (3.24) +∫t+ΔttE[σ(s)h(s)]2ds.

The middle term on the right-hand side vanishes while the first term is of order , yielding the following relation for .

###### Lemma 3.6.

Let satisfy . Then one has

 (3.25)

Next, we examine whether occurs prior to the maximum of in several examples.

###### Example 3.7.

Consider the function where . Let and be deterministic. From the Lemma above, we obtain

 (3.26) V(t)=h(t)2=f(t)2q,  ddtV(t)=2qf(t)2q−1ddtf(t).

When has a maximum, note that on some interval it is positive (as demand exceeds supply) and has its maximum for some value The identity above implies that has a maximum when has a maximum. Also, the defining stochastic equation above implies has its maximum at

###### Example 3.8.

(Symmetry between and and more general coefficients) If we hypothesize that the level of noise is proportional essentially to the magnitude (or its square) of the difference between and divided by the sum (which is a proxy for trading volume), then we can write that coefficient as

 (3.27) σ(D−S)2(D+S)2.

We can consider a more general case in which we write, for example, for

 (3.28) dlogP(t)=(DS−1)dt+σ(D−SD+S)pdW

where

can be either even or odd. Note that we can write all terms as functions of

so since and are positive, and we have

 (3.29) dlogP(t)=fdt+σ(ff+2)pdW.

We write

 (3.30) V(t):=limΔt→0V(t,t+Δt)Δt=E[σ(t)(f(t)f(t)+2)p]2

If is deterministic and is constant, we have upon differentiation,

 (3.31) ddtV(t)=4pσ2f2p−1[f+2]2p+1dfdt

Recalling the sign of depends only on Notice that it makes no difference whether is even or odd.

If has a single maximum at such that iff , and iff then we have a relative maximum in at .

Hence, we see that if the coefficient of is a deterministic term of the form and has a maximum, whether is even or odd (i.e., the coefficient increases or decreases with excess demand), then the limiting volatility also has a maximum.

###### Example 3.9.

Generalizing this concept further, we define a function such that for all  and

 (3.32) sgnH′(z)=sgn(z).

We consider the stochastic equation, with deterministic

 (3.33) dlogP=fdt+σ{H(ff+2)}1/2dW

so that with

While in principle, , except under conditions that are very far from equilibrium, one can assume for some small at least .

We compute

 σ−2ddtV =ddtH(ff+2) (3.34) =H′(ff+2)2(f+2)2dfdt.

Based on this calculation, one concludes if has a maximum, recalling that is positive near the maximum, then has the same sign as So a maximum in corresponds to a maximum in , while has its maximum at .

## 4. Supply and Demand as a function of valuation

We consider the basic model (1.4) now with the excess demand, i.e., depending on the valuation, which can be regarded either as a stochastic or deterministic function. It is now commonly accepted in economics and finance that the trading price will often stray from the fundamental valuation [18, 19]. We write the price equation for the time evolution as

 (4.1) ddtlogP(t)=DS−1=logPa(t)P(t).

The right hand side of equation (4.1) is a linearization (as discussed in Section 1.3) and the right hand side of has the same linearization as . The equation simply expresses the idea that undervaluation is a motivation to buy, while overvaluation is a motivation to sell, as one assumes in classical finance. The non-classical feature is the absence of infinite arbitrage. Analogous to Section 1.3, we write the stochastic version of (4.1) as

 (4.2) dlogP(t,ω)=logPa(t,ω)P(t,ω)dt+σ(t,ω)(1+log