Asset Price Volatility and Price Extrema
aa r X i v : . [ q -f i n . M F ] J u l Unspecified JournalVolume 00, Number 0, Pages 000–000S ????-????(XX)0000-0
ASSET PRICE VOLATILITY AND PRICE EXTREMA
CAREY CAGINALP AND GUNDUZ CAGINALP
Abstract.
The relationship between price volatility and a market extremumis examined using a fundamental economics model of supply and demand.By examining randomness through a microeconomic setting, we obtain theimplications of randomness in the supply and demand, rather than assumingthat price has randomness on an empirical basis. Within a very general settingthe volatility has an extremum that precedes the extremum of the price. Akey issue is that randomness arises from the supply and demand, and thevariance in the stochastic differential equation governing the logarithm of pricemust reflect this. Analogous results are obtained by further assuming that thesupply and demand are dependent on the deviation from fundamental valueof the asset. Introduction
Overview.
In financial markets two basic entities are the expected relativeprice change and volatility. The latter is defined as the standard deviation ofrelative price change in a specified time period. The expected relative price changeis, of course, at the heart of finance, while volatility is central to assessing riskin a portfolio. Volatility plays a central role in the pricing of options, which arecontracts whereby the owner acquires the right, but not the obligation, to buy orsell 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 [1].In this way, price change and volatility are essentially decoupled in their treat-ment. In particular, the relative price change per unit time P − dP/dt = d log P/dt is given by a sum of a deterministic term that expresses the long term estimate forthe growth, together with a stochastic term given by Brownian motion.Hence, the basic starting point for much of classical finance, particularly optionspricing (see e.g., [2, 3]), is the stochastic equation for log P as a function of ω ∈ Ω(the sample space) and t given by(1.1) d log P = µdt + σdW. where W is Brownian motion, with ∆ W := W ( t ) − W ( t − ∆ t ) ∼ N (0 , ∆ t ) , so W is normal with variance ∆ t , mean 0 , and independent increments (see [4, 5]). While µ and σ are often assumed to be constant, one can also stipulate deterministic andtime dependent or stochastic µ and σ. The stochastic differential equation above is
Received by the editors July 31, 2018.
Key words and phrases.
Volatility and Price Trend, Modelling Asset Price Dynamics, PriceExtrema, Market Tops and Bottoms. c (cid:13) short for the integral form (suppressing ω in notation) for arbitrary t < t (1.2) log P ( t ) − log P ( t ) = Z t t µdt + Z t t σdW For µ, σ constant, and ∆ t := t − t , one can write(1.3) ∆ log P := log P ( t ) − log P ( t ) = µ ∆ t + σ ∆ W. The classical equation (1.1) can be regarded as partly an empirical model basedon observations about volatility of prices. It also expresses the theoretical constructof infinite arbitrage that eliminates significant distortions from the expected returnof the asset as a consequence of rational comparison with other assets such as riskfree government (i.e., Treasury) bonds. Hence, this equation can be regarded as alimiting case (as supply and demand approach infinity) of other equations involv-ing finite supply and demand [6] (Appendix A). Thus, it does not lend itself tomodification based upon random changes in finite supply and demand. An exam-ination of the relationship between volatility and price trends, tops and bottomsrequires analysis of the more fundamental equations involving price change. A suit-able framework for analyzing these problems is the asset flow approach based onsupply/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 eventthat is highly favorable for the fundamentals of an asset. There is the expectationthat there will be a peak and a turning point, but no one knows when that willoccur. By observing the volatility of price, can one determine whether, and when, apeak will occur in the future? In general, our goal is to delve deeper into the pricechange mechanism to understand the relationship between relative price changeand 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 levelof randomness in the resulting price. Indeed, for actively traded equities, thereare many market makers whose living consists of exploiting any price deviationsfrom 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 ofan investment, each particular change in the supply/demand curve will produce aclear, repeatable short term change in the price.Given the broad validity of the Central Limit Theorem, one can expect that therandomness in supply and demand of an actively traded asset on a given, small timeinterval will be normally distributed. Thus, supply and demand can be regardedas bivariate normally distributed random variables, with a correlation that will beclose to − SSET PRICE VOLATILITY AND PRICE EXTREMA 3 improved by understanding the relationship between the variance in price and thepeaks and nadirs of expected price.Subsequently, in Section 3, we generalize the dependence on demand/supply inthe basic model, and find that under a broad set of conditions one has neverthelessthe result that the extremum in variance precedes the expected price extremum.In Section 4 we introduce the concept of price change that depends on supplyand demand through the fundamental value. The trader motivations are assumedto be classical in that they depend only on fundamental value; however, the priceequation involves the finiteness of assets, which is a non-classical concept. Withoutintroducing non-classical concepts such as the dependence of supply and demand onprice trend, we obtain a similar relationship between the volatility and the expectedprice.In Section 5, we prove that within the assumptions of this model and general-izations, the peak of the expected log price occurs after the peak in volatility.1.2.
General Supply/Demand model and stochastics.
We write the generalprice change model in terms of the price, P, the demand, ˜ D , and supply, ˜ S . Inparticular, the relative price change is equal to a function of the excess demand, (cid:16) ˜ D − ˜ S (cid:17) / ˜ S (see e.g., [11], [12]). That is, we have(1.4) P − dP/dt = G (cid:16) ˜ D/ ˜ S (cid:17) where G : R + → R satisfies ( a ) G (1) = 0 , ( b ) G ′ ( x ) > x ∈ R + . If symmetrybetween ˜ D and ˜ S is assumed, then one can also impose ( c ) G (1 /x ) = − G ( x ) . Aprototype function with properties ( a ) − ( c ) is given by G ( x ) := x − /x. A basic stochastic process based on (1 .
4) for log P is defined by(1.5) d log P ( t, ω ) = a ( t, ω ) dt + b ( t, ω ) dW ( t, ω )for some functions a and b in H [0 , T ], the space of stochastic processes with asecond moment integrable on [0 , T ] (see [5]). The terms a ( t, ω ) and b ( t, ω ) canbe identified from G and the nature of randomness that is assumed. In any timeinterval ∆ t, there is a random term in ˜ D and ˜ S. The assumption is that thereare a number of agents who are motivated to place buy orders. The relative frac-tion is subject to randomness so that the deterministic demand, D ( t ) , multipliedby 1 + σ R ( t ; ω ) for some random variable R ( t ; ω ). Likewise, one has the deter-ministic supply, S ( t ) , by 1 − σ R ( t ; ω ). This yields, for sufficiently small σ , theapproximation(1.6) D ( t ; ω ) S ( t ; ω ) − → D ( t ) (cid:8) σ R (cid:9) S ( t ) (cid:8) − σ R (cid:9) − D ( t ) S ( t ) − D ( t ) S ( t ) σR, with σ being either constant, time dependent or stochastic. We can then write G (cid:16) ˜ D/ ˜ S (cid:17) ˜= G ( D/S ) + G ′ ( D/S ) (cid:18) σ DS R (cid:19) and thereby identify a ( t ; ω ) = G ( D/S ) and b ( t ; ω ) = σ DS G ′ ( D/S ) . Note that weview the randomness as arising only from the σR term, so we can assume that D and S are deterministic functions of t at this point. Later on in this paperwe consider additional dependence on D and S. By assuming that the randomvariable R is normal with variance ∆ t and w ( t + ∆ t ) − w ( t ) is independent of CAREY CAGINALP AND GUNDUZ CAGINALP w ( t ) − w ( t + ∆ t ), one obtains the stochastic process below (in which D ( t ) and S ( t ) are deterministic).By differentiating ( c ), we note1 x G ′ (cid:18) x (cid:19) = xG ′ ( x ) , and thereby write the stochastic differential equation d log P ( t, ω ) = G ( D/S ) dt + 12 (cid:26) DS G ′ (cid:18) DS (cid:19) + SD G ′ (cid:18) SD (cid:19)(cid:27) dW ( t, ω ) . In particular, for G ( x ) := x − /x one has d log P = (cid:18) DS − SD (cid:19) dt + σ (cid:26) DS + SD (cid:27) dW. We are interested in the relationship between volatility and market extrema, andfocus on market tops by using the simpler equation for the function G ( x ) := x − c ) holds only approximately near D/S = 1 . The equation is then (seeAppendix)(1.7) d log P ( t, ω ) = (cid:18) D ( t ) S ( t ) − (cid:19) 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 σ deterministic or even constant below. If we were toassume that the supply and demand have randomness that is not necessarily thenegative of one another, then we can write instead,(1.8) D (1 + σ a R a ) S (1 − σ b R b ) ˜= (1 + σ a R a + σ b R b ) DS − . yielding the analogous stochastic process,(1.9) d log P ( t, ω ) = (cid:18) D ( t ) S ( t ) − (cid:19) dt + D ( t ) S ( t ) { σ a dW a + σ b dW b } . Derivation of the stochastic equation.
We make precise the ideas aboveby starting again with (1.4) where D ( t ; ω ) and S ( t ; ω ) are random variables thatare anticorrelated bivariate normals with means µ D ( t ) and µ S ( t ) and both havevariance σ . We can regard the means as the deterministic part of the supply anddemand at any time t , so that with Σ as the covariance matrix [14], we write(1.10)( D ( t ; ω ) , S ( t ; ω )) ∼ N ( ~µ ( t ) , Σ) with ~µ := ( µ D , µ S ) , Σ := (cid:18) σ ( t ) − − σ ( t ) (cid:19) . For any fixed t , one can show that the density of D/S is given by(1.11) f D/S ( x ) = 1 + µ D /µ S √ π σ µ S ( x + 1) e − ( x − µD/µS ) (cid:18) σ µS (cid:19) x +1)2 . Other approximations in different settings have been studied in [15, 16, 17] andreferences therein.
SSET PRICE VOLATILITY AND PRICE EXTREMA 5
For values of x near the mean of D/S , one has(1.12) ( x + 1) ˜= (cid:18) µ D µ S + 1 (cid:19) . We can use this to approximate the density, using σ R q := σ µ S (cid:16) µ D µ S + 1 (cid:17) as theapproximate variance of D/S, as(1.13) f D/S ( x ) ˜= 1 √ πσ R q e − ( x − µD/µS ) σ Rq ; f DS − ( x ) ˜= 1 √ πσ R q e − ( x − µD/µS +1 ) σ Rq . With this expression for the density of R := D/S − , we can write the basicsupply/demand price change equation as(1.14) ∆ log P ∆ t ˜= R ∼ N (cid:18) µ D µ S − , σ R q (cid:19) , where each variable depends on t and ω. Subtracting out the µ D µ S −
1, defining R ∼ N (cid:16) , σ R q (cid:17) , and noting that R depends on t through σ R , we write(1.15) ∆ log P ˜= (cid:18) µ D µ S − (cid:19) ∆ t + σ R R ∆ t. By definition of Brownian motion, we can write(1.16) ∆ log P ˜= (cid:18) µ D µ S − (cid:19) ∆ t + σ R q ∆ W. With σ and µ D held constant, an increase in µ S leads to a decrease in thevariance σ R . We would like to approximate this under the condition that µ D /µ S ≈ . By rescaling the units of µ D , µ S , σ together and assuming that each of µ D and µ S are sufficiently close to 1 that we can consider the leading terms in a Taylorexpansion, and write(1.17) µ D = 1 + δ D , µ S = 1 + δ S . Note that µ D and µ S will be nearly equal unless one is far from equilibirium.Ignoring the terms higher than first order one has σ R q = σ (1 + δ S ) (cid:18) δ D δ S (cid:19) ˜=4 σ (1 − δ S + δ D ) . (1.18)We are considering − δ S = δ D =: δ so that(1.19) σ R q = 4 σ (1 + 4 δ ) . Using Taylor series approximation, one has(1.20) (cid:18) µ D µ S (cid:19) = (cid:18) δ − δ (cid:19) ˜=1 + 4 δ. We can thus write the stochastic equation above as(1.21) ∆ log P ˜= (cid:18) µ D µ S − (cid:19) ∆ t + 2 σ µ D µ S ∆ W, CAREY CAGINALP AND GUNDUZ CAGINALP so that the differential form is given in terms of f := µ D /µ S − d log P ( t ) = f ( t ) dt + σ ( f ( t ) + 1) dW ( t )This is in agreement with the heuristic derivation above, with σ = 2 σ and σ asthe variance of each of S and D. Location of maxima of Supply/Demand versus price
The deterministic model.
We will show that if
D/S − f , then the stochastic equation above will imply that thevariance over a small time interval ∆ t will have an extremum before the price hasits extremum.Once we do this simplest case, it will generalize it to the situation where f := D/S − P − dPdt = DS − f, i.e., ddt log P ( t ) = f ( t )Assume that f is a prescribed function of t that is C ( I ) for I ⊃ ( t , ∞ ) ⊃ ( t a , t b )satisfying:( i ) f ( t ) > t a , t b ) , f ( t ) < I \ ( t a , t b ) and f + 1 > I ;( ii ) f ′ ( t ) > t < t m , f ′ ( t ) < t > t m , f ′ ( t m ) = 0;( iii ) f ′′ ( t ) < t ∈ ( t a , t b ) . Then log P ( t ) is increasing on t ∈ ( t a , t b ) and decreasing on t ∈ ( t b , ∞ ) and hasa maximum at t b . In other words, the peak of f occurs at t m while the peak of log P is attainedat t b > t m . This demonstrates the simple idea that price peaks some time after thepeak in demand/supply. In fact, during pioneering experiments Smith, Suchanekand Williams [18] 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 thatwas predicted in [7] was confirmed in experiments starting with [8].2.2.
The stochastic model.
Recall that µ D and µ S are deterministic functions oftime only. We model the problem as discussed above so the only randomness belowis in the dW variable. The stochastic equation given by (1 .
22) for a continuousfunction f := µ D /µ S − , in the integral form, for any t < t and ∆ log P :=log P ( t ) − log P ( t ) is(2.2) ∆ log P = Z t t f ( z ) dz + Z t t σ ( z ) ( f ( z ) + 1) dW ( z ) . Note that for the time being we are assuming that σ and f may depend on timebut are deterministic. We compute the expectation and variance of this quantity:(2.3) E [∆ log P ] = Z t t f ( z ) dz since f is deterministic and E [ dW ] = 0; We let E [ Y ] denote E (cid:2)(cid:0) Y (cid:1)(cid:3) . SSET PRICE VOLATILITY AND PRICE EXTREMA 7
V ar [∆ log P ] = E (cid:20)Z t t f ( z ) dz + Z t t σ ( z ) { f ( z ) + 1 } dW ( z ) (cid:21) − (cid:18) E (cid:20)Z t t f ( z ) dz + Z t t σ ( z ) { f ( z ) + 1 } dW ( z ) (cid:21)(cid:19) . (2.4)The R f ( z ) dz term is deterministic and vanishes when its expectation is subtracted.The expecation of the dW and the dzdW terms vanishes also. We are left with V ar [∆ log P ] = E (cid:20)Z t t σ ( z ) { f ( z ) + 1 } dW ( z ) (cid:21) = Z t t σ ( z ) { f ( z ) + 1 } dz (2.5)using the standard result ([5], p. 68).We want to consider a small interval ( t, t + ∆ t ) so we set t → t and t → t + ∆ t .We have V ( t, t + ∆ t ) := V ar [log P ( t + ∆ t ) − log P ( t )]= Z t +∆ tt σ ( z ) { f ( z ) + 1 } dz. (2.6) V ( t ) := lim ∆ t → t V ( t, t + ∆ t ) = lim ∆ t → t Z t +∆ tt σ ( z ) { f ( z ) + 1 } dz = σ ( t ) { f ( t ) + 1 } . (2.7) Example 2.1.
For σ := 1 , the maximum variance of ∆ log P will be when { f ( z ) + 1 } is at a maximum, which is when f has its maximum, i.e., at t m . (2.8) ddt V ( t ) = ddt { f ( t ) + 1 } = 2 { f ( t ) + 1 } ddt f ( t )Since 1 + f ( t ) > V ( t ) is of the same signas the derivative of f, so the limiting variance V ( t ) is increasing when f is increasingand vice-versa. Recall that log P increases so long as f > , and decreases when f < . In other words, for the peak case, one has f ( t ) > t ∈ ( t a , t b )with a maximum at t m . When f has a peak, the maximum of V ( t ) will be at t m when f ( t ) has its maximum.To summarize, if the coefficient of dW is σ { f ( t ) } with σ constant and f hasa maximum at t m then V ( t ) will also have a maximum at t m so that the maximumin E log P will occur after the maximum in V ( t ) since ∂ t E log P ( t ) = f ( t ) . Remark . We have shown that E log P ( t ) has a maximum, at some time t m thatis preceded by a maximum in V ( t ). We can use this together with Jensen’s inequal-ity to show that E [ P ( t m ) /P ( t )] ≥ t. Indeed, since E log P ( t m ) ≥ E log P ( t ) we can write(2.9) E log P ( t m ) P ( t ) ≥ . CAREY CAGINALP AND GUNDUZ CAGINALP
Let Y := P ( t m ) /P ( t ) and g ( x ) := e x in Jensen’s inequality, Eg ( Y ) ≥ g ( E [ Y ]),we have(2.10) EY = Ee log Y ≥ e E log Y ≥ . Hence, the expected ratio of price at t m to the price at any other point t is greaterthan 1 . Remark . The conclusion above can be contrasted with the standard model(1 .
1) adjusted so that µ ( t ) := µ D ( t ) µ s ( t ) has the same property of a peak at some time t m . Performing the same calculation of (2 . .
8) for this model yields the result V ( t ) = σ so that it provides no information on the expected peak of prices.3. Additional randomness In Supply and Demand
Stochastic Supply and Demand.
Let f := D/S − − ≤ Ef and E | f | ≤ C . With X ( t ) := log P ( t ) and ∆ X := X ( t + ∆ t ) − X ( t ) , we write the SDE in differential and integral forms as(3.1) dX = f dt + σ (1 + f ) dW (3.2) X ( t + ∆ t ) − X ( t ) = Z t +∆ tt f ( s ) ds + Z t +∆ tt σ ( s ) (1 + f ( s )) dW ( s ) . where we will assume σ is a continuous, deterministic function of time, though wecan allow it to be stochastic in most of the sequel.One has since EdW = 0 and E [ dsdW ] = 0 one obtains again the identities(3.3) E ∆ X = Z t +∆ tt Ef ( s ) ds,V ar [∆ X ] = E (cid:20)Z f ds + Z σ (1 + f ) dW (cid:21) − (cid:18) E (cid:20)Z f ds + Z σ (1 + f ) dW (cid:21)(cid:19) = V ar (cid:20)Z f ds (cid:21) + 2 E (cid:20)Z f ds Z σ (1 + f ) dW (cid:21) + E (cid:20)Z σ (1 + f ) dW (cid:21) (3.4)where all integrals are taken over the limits t and t + ∆ t . Lemma 3.1.
Let sup [0 ,T ] E | f | ≤ C . Then for some C depending on this bound,one has (3.5) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E Z t +∆ tt f ( s ′ ) ds ′ Z t +∆ tt σ ( s ) { f ( s ) } dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (∆ t ) / . Proof.
We apply the Schwarz inequality to obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) E Z t +∆ tt f ( s ′ ) ds ′ Z t +∆ tt σ ( s ) { f ( s ) } dW ( s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E Z t +∆ tt f ( s ′ ) ds ′ ! / E Z t +∆ tt σ ( s ) { f ( s ) } dW ( s ) ! / . (3.6) SSET PRICE VOLATILITY AND PRICE EXTREMA 9
We bound each of these terms. Using the Schwarz inequality on the R ds integral,we obtain using generic C throughout,(3.7) E Z t +∆ tt f ( s ′ ) ds ′ ! ≤ C (∆ t ) . The second term is bounded using the fact that σ is deterministic, E Z t +∆ tt σ ( s ) { f ( s ) } dW ( s ) ! = Z t +∆ tt σ ( s ) E { f ( s ) } ds ≤ C ∆ t. (3.8)Taking the square roots of (3.7) and (3.8), and combining with (3 .
6) proves thelemma. (cid:3)
Lemma 3.2.
Let σ be a continuous, deterministic function and assume sup [0 ,T ] E | f | ≤ C . Then (3.9) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)
V ar [∆ X ] − Z t +∆ tt σ ( s ) E { f ( s ) } ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (∆ t ) / Proof.
Basic stochastic analysis yields(3.10) E Z t +∆ tt σ ( s ) { f ( s ) } dW ! = Z t +∆ tt σ ( s ) E { f ( s ) } ds. Thus, using (3.4) and f ∈ H [0 , T ] we have the result (3.9). (cid:3) Now, we would like to determine the maximum of V ( t ) and show that it precedesthe maximum of the expected log price. From the calculations above, one has Lemma 3.3.
In the general case, assuming E | f ( t ) | < C on t ∈ [0 , T ] but allow-ing stochastic σ such that Eσ < C one has (3.11) V ( t ) := lim ∆ t → t V ( t, t + ∆ t ) = E h σ ( t ) (1 + f ( t )) i . Lemma 3.4.
Suppose sup [0 ,T ] E | f ( t ) | < C and σ is a deterministic continuousfunction on t ∈ [0 , T ] then one has (3.12) V ( t ) = σ { Ef } + σ V arf. and the extrema of V ( t ) occur at t such that (3.13) 2 σσ ′ n [1 + Ef ] + V arf o + σ (cid:8) Ef ] ( Ef ) ′ + ( V arf ) ′ (cid:9) = 0 . Proof.
Using Lemma 3.3, we write V ( t ) = σ E (cid:2) f + f (cid:3) = σ n Ef + ( Ef ) + Ef − ( Ef ) o = σ (1 + Ef ) + σ V arf. (3.14)Differentiation implies the second assertion. (cid:3)
Lemma 3.5.
Suppose E | f ( t ) | < C on t ∈ [0 , T ] , while σ and V ar [ f ( t )] areconstant in t. Then the extremum of V ( t ) occur for t such that (3.15) ddt Ef ( t ) = 0 . Proof.
From the previous Lemma, we have V ( t, t + ∆ t ) := R t +∆ tt σ ( s ) E [1 + f ( s )] ds ,yielding(3.16) lim ∆ t → t V ( t, t + ∆ t ) = σ (1 + Ef ( t )) + V ar [ f ( t )]Since we are assuming that V ar [ f ( t )] is constant in time, we obtain ∂∂t lim ∆ t → V ( t, t + ∆ t ) = ∂∂t n σ (1 + Ef ( t )) o = 2 σ (1 + Ef ( t )) ddt Ef ( t ) . (3.17)Thus, the right-hand side vanishes if and only if ddt Ef ( t ) = 0 , i.e., at t m (bydefinition of t m ). Note that we have 1 + f > Ef > . (cid:3) Properties of f . The condition E | f | < C is easily satisfied by introducingrandomness in many forms. For the Lemma above, we would also like to satisfy V ar [ f ( t )] = const. Another way of attaining this (up to exponential order) is to define f as thestochastic process(3.18) df ( t ) = µ f ( t ) dt + σ f ( t ) dW ( t )where µ f and σ are both time dependent but deterministic.We can assume that f ( t ) is a given, fixed value, and obtain (see e.g., [4], [5])(3.19) V ar [ f ( t )] = E (cid:20)Z tt σ f ( s ) dW ( s ) (cid:21) = Z tt σ f ( s ) ds since σ f ( s ) is deterministic . In particular, if one has σ f ( s ) := e − s/ , then V ar [ f ( t )] ≤ e − t while R t σ ( s ) ds =1 − e − t so one has approximately constant variance for t ≥ t for large t . In par-ticular, one has(3.20) ddt V ar [ f ( t )] = ddt Z tt σ f ( s ) ds = σ f ( t ) = e − t . General coefficient of dW . The stochastic differential equation (3.1) en-tails a coefficient of dW that is proportional to D/S.
One can also consider theimplications of a coefficient that is proportional to the excess demand
D/S − h ( t ) := g ( f ( t )) for an arbitrarycontinuous function g leading to the stochastic differential equation(3.21) d log P = f dt + σhdW, where σ can also be stochastic or deterministic function of time.From this stochastic equation one has immediately(3.22) dE [log P ] dt = Ef similar to the completely deterministic model, except that f is replaced by Ef.
SSET PRICE VOLATILITY AND PRICE EXTREMA 11
From the integral version of the stochastic model, we can write the expectationand variance as(3.23) E [∆ log P ] = Z t +∆ tt Ef ( s ) dsV ( t, t + ∆ t ) := V ar [∆ log P ] = V ar "Z t +∆ tt f ( s ) ds + 2 E "Z t +∆ tt σ ( s ) h ( s ) dW ( s ) + Z t +∆ tt E [ σ ( s ) h ( s )] ds. (3.24)The middle term on the right-hand side vanishes while the first term is of order(∆ t ) , yielding the following relation for V ( t ). Lemma 3.6.
Let h ( t ) := g ( f ( t )) and σ satisfy Eh < C, Eσ < C . Then onehas (3.25) V ( t ) := lim ∆ t → t V ( t, t + ∆ t ) = E [ σ ( t ) h ( t )] . Next, we examine whether V ( t ) occurs prior to the maximum of log P ( t ) inseveral examples. Example 3.7.
Consider the function g ( z ) = z q where q ∈ N . Let σ := 1 and f ∈ L [0 , t ] be deterministic. From the Lemma above, we obtain(3.26) V ( t ) = h ( t ) = f ( t ) q , ddt V ( t ) = 2 qf ( t ) q − ddt f ( t ) . When f := D/S − t a , t b ) itis positive (as demand exceeds supply) and f has its maximum for some value t m ∈ ( t a , t b ) . The identity above implies that V ( t ) has a maximum when f hasa maximum. Also, the defining stochastic equation above implies E log P has itsmaximum at t b > t m . Example 3.8. (Symmetry between D and S and more general coefficients) If wehypothesize that the level of noise is proportional essentially to the magnitude (orits square) of the difference between D and S divided by the sum (which is a proxyfor trading volume), then we can write that coefficient as(3.27) σ ( D − S ) ( D + S ) . We can consider a more general case in which we write, for example, for σ = const, (3.28) d log P ( t ) = (cid:18) DS − (cid:19) dt + σ (cid:18) D − SD + S (cid:19) p dW where p ∈ N can be either even or odd. Note that we can write all terms as functionsof f := D/S − , so f + 2 = D/S + 1 > D and S are positive, and we have(3.29) d log P ( t ) = f dt + σ (cid:18) ff + 2 (cid:19) p dW. We write(3.30) V ( t ) := lim ∆ t → V ( t, t + ∆ t )∆ t = E (cid:20) σ ( t ) (cid:18) f ( t ) f ( t ) + 2 (cid:19) p (cid:21)
22 CAREY CAGINALP AND GUNDUZ CAGINALP If f is deterministic and σ is constant, we have upon differentiation,(3.31) ddt V ( t ) = 4 pσ f p − [ f + 2] p +1 dfdt Recalling f + 2 > ddt V depends only on f p − df /dt. Notice that itmakes no difference whether p is even or odd.If f has a single maximum at t m ∈ ( t a , t b ) such that f ( t ) > t ∈ ( t a , t b ), and f < t [ t a , t b ] then we have a relative maximum in V at t m .Hence, we see that if the coefficient of dW is a deterministic term of the form(( D − S ) / ( D + S )) p and f has a maximum, whether p is even or odd (i.e., thecoefficient increases or decreases with excess demand), then the limiting volatility V also has a maximum. Example 3.9.
Generalizing this concept further, we define a function H ( z ) suchthat H ( z ) > z ∈ R and(3.32) sgnH ′ ( z ) = sgn ( z ) . We consider the stochastic equation, with f deterministic(3.33) d log P = f dt + σ (cid:26) H (cid:18) ff + 2 (cid:19)(cid:27) / dW so that V ( t ) = σ H (cid:16) f ( t ) f ( t )+2 (cid:17) with σ = const. While in principle, f ( t ) := D ( t ) /S ( t ) − ∈ ( − , ∞ ), except under conditionsthat are very far from equilibrium, one can assume f ( t ) ∈ ( − a/ , a/
2) for somesmall a, at least a ∈ (0 , σ − ddt V = ddt H (cid:18) ff + 2 (cid:19) = H ′ (cid:18) ff + 2 (cid:19) f + 2) dfdt . (3.34)Based on this calculation, one concludes if f has a maximum, recalling that f := D/S − d V /dt has the same sign as df /dt. Soa maximum in V corresponds to a maximum in f , while log ( P ) has its maximumat t b > t m . 4. Supply and Demand as a function of valuation
We consider the basic model (1.4) now with the excess demand, i.e.,
D/S − , depending on the valuation, P a ( t ) , which can be regarded either as a stochasticor deterministic function. It is now commonly accepted in economics and financethat the trading price will often stray from the fundamental valuation [18, 19]. Wewrite the price equation for the time evolution as(4.1) ddt log P ( t ) = DS − P a ( t ) P ( t ) . The right hand side of equation (4.1) is a linearization (as discussed in Section1.3) and the right hand side of has the same linearization as ( P a − P ) /P . Theequation 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 SSET PRICE VOLATILITY AND PRICE EXTREMA 13 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) d log P ( t, ω ) = log P a ( t, ω ) P ( t, ω ) dt + σ ( t, ω ) (cid:18) P a ( t, ω ) P ( t, ω ) (cid:19) dW ( t, ω ) . At this point we allow both P a and σ to be stochastic, with EP a < C and Eσ < C but will specialize to given and deterministic P a and σ after the first result. Wealso assume 1 + log ( P a /P ) > , i.e., P a /P > e − , i.e., the fundamental value, P a , and trading price, P, do not differ drastically. Notation 1.
Let X := log P, X a := log P a , y := E log P, y a := E log P a , z := E (log P ) . When log P a and log P are deterministic, we write lower case x a and x, respectively. The equation (4 .
2) is short for the integral form (using the notation above) forany t > t > t , (4.3) X ( t ) − X ( t ) = Z t t X a − Xds + Z t t σ ( s, ω ) (1 + X a − X ) dW ( s ) . Noting that E R f ( t ) dW = 0 , we find the expectation of (4.3) as(4.4) y ( t ) − y ( t ) = Z t t y a ( s ) ds − Z t t y ( s ) ds i.e., one has the ODE, with y := y ( t ) := E [log P ( t )] , (4.5) ddt y ( t ) = y a ( t ) − y ( t ) , y ( t ) := y This has the unique solution, for known y a ( t ) , (4.6) y ( t ) = e t − t y ( t ) + e − t Z tt y a ( s ) e s ds. Note that if we eliminate randomness altogether, i.e., σ := 0 and deterministic P a ( t ),(4.7) ddt log P ( t ) = log P a ( t ) P ( t ) , with solution(4.8) x ( t ) = e t − t x ( t ) + e − t Z tt e s x a ( s ) ds. where x ( t ) := log P ( t ) and x a ( t ) := log P a ( t ). We note that the solution of y ( t ) = E log P ( t ) of log P in terms of y a ( t ) = E log P a ( t ) is the same as log P ( t )in terms of log P a ( t ), i.e. both expected value and deterministic log P satisfy thesame equation.4.1. The stochastic problem.
We write the SDE (4 .
2) as(4.9) dX = ( X a − X ) dt + σ (1 + X a − X ) dW. We say X is a solution to a SDE if X ∈ H [0 , T ] and solves the integral versionof the SDE for almost every ω ∈ Ω. The solution to the stochastic equation (4.2), X ( t, ω ) is unique, belongs to H [0 , T ] and is continuous in t ∈ [0 , T ] for almost every ω ∈ Ω ([5] p. 94). We denote the remaining set Γ , so that X ( t, ω ) is continuous in t for all ω ∈ Ω \ Γ . One has by basic measure theory (e.g., [20]), that for anymeasurable function such as X or X one has E Z t +∆ tt X ( s, ω ) ds = Z Ω Z t +∆ tt X ( s, ω ) dsdP ( ω )= Z Ω \ Γ Z t +∆ tt X ( s, ω ) dsdP ( ω ) + Z Γ Z t +∆ tt X ( s, ω ) dsdP ( ω ) . (4.10)Thus from here on we can ignore the set Γ and assume that X ( t, ω ) is continuouswhen the expectation value is computed .Next, using (4.4) we compute the variance, of ∆ X := X ( t + ∆ t, ω ) − X ( t, ω )and later we will determine the terms that vanish upon dividing by ∆ t,V ( t, t + ∆ t ) := E [ X ( t + ∆ t ) − EX ( t )] − ( E [ X ( t + ∆ t ) − X ( t )]) = E "Z t +∆ tt X a − Xds + Z t +∆ tt σ (1 + X a − X ) dW ( s ) − E "Z t +∆ tt X a − Xds + Z t +∆ tt σ (1 + X a − X ) dW ( s ) . (4.11)Note that with ∆ X := X ( t + ∆ t ) − X ( t ) we have V ( t, t + ∆ t ) = V ar [ X ( t + ∆ t ) − X ( t )] = V ar (cid:20) log P ( t + ∆ t ) P ( t ) (cid:21) = V ar (cid:20) log (cid:18) ∆ PP + 1 (cid:19)(cid:21) ≃ V ar (cid:20) ∆ PP (cid:21) . (4.12)so that V ( t, t + ∆ t ) is essentially a measure of the variance of relative price change.Since E R t +∆ tt σ (1 + X a − X ) dW ( s ) = 0 one has V ( t, t + ∆ t ) = E "Z t +∆ tt X a − Xds + Z t +∆ tt σ (1 + X a − X ) dW ( s ) − E Z t +∆ tt X a − Xds ! = V ( t, t + ∆ t ) + V ( t, t + ∆ t ) + V ( t, t + ∆ t )(4.13)where V ( t, t + ∆ t ) := E Z t +∆ tt X a − Xds ! − E Z t +∆ tt X a − Xds ! ,V ( t, t + ∆ t ) := 2 E "Z t +∆ tt X a − Xds Z t +∆ tt σ (1 + X a − X ) dW ( s ) V ( t, t + ∆ t ) := E Z t +∆ tt σ (1 + X a − X ) dW ( s ) ! = Z t +∆ tt E [ σ (1 + X a − X )] ds (4.14) SSET PRICE VOLATILITY AND PRICE EXTREMA 15
Lemma 4.1.
Let X be a solution to the SDE (4 . with σ ( t, ω ) and X a ( t, ω ) continuous for all t ∈ [0 , T ] and all ω ∈ Ω , with bounded second moments. Then ( i ) | V ( t, t + ∆ t ) | ≤ C (∆ t ) so lim ∆ t → V ( t, t + ∆ t ) / ∆ t = 0 , and,( ii ) | V ( t, t + ∆ t ) | ≤ C (∆ t ) / so lim ∆ t → V ( t, t + ∆ t ) / ∆ t = 0 . Proof. ( i ) ( a ) We consider the first term in V , namely, E Z t +∆ tt X a − Xds ! = Z Ω Z t +∆ tt X a − Xds ! dP ( ω )= Z Ω \ Γ Z t +∆ tt X a − Xds ! dP ( ω )(4.15)where we have omitted the set of measure zero, Γ , outside of which X is continuousin t on a closed bounded interval. Hence, one can bound the integrand by C (∆ t ) . Thus we have(4.16) E Z t +∆ tt X a − Xds ! ≤ C (∆ t ) . ( i ) ( b ) Similarly the second term can be bounded as E Z t +∆ tt X a − Xds ! = Z Ω \ Γ Z t +∆ tt X a − Xds ! dP ( ω ) ! ≤ C (∆ t ) . (4.17)Hence, part ( i ) of the lemma has been proven.( ii ) Using the Schwarz inequality on the second term we have12 V ( t, t + ∆ t ) = E ( Z t +∆ tt X a − Xds ! Z t +∆ tt σ (1 + X a − X ) dW ( s ) !) ≤ E Z t +∆ tt X a − Xds ! / E Z t +∆ tt σ (1 + X a − X ) dW ( s ) ! / . (4.18)Using continuity properties, we have the following bound on the first term,(4.19) E Z t +∆ tt X a − Xds ! / ≤ C (∆ t ) . For the second we use the basic property used above, E Z t +∆ tt σ (1 + X a − X ) dW ( s ) ! / = (Z t +∆ tt E [ σ (1 + X a − X )] ds ) / = (Z Ω \ Γ Z t +∆ tt E [ σ (1 + X a − X )] ds ) / ≤ C (∆ t ) / . (4.20) Hence, the proof of the second part of the lemma follows from the followingbound: V ( t, t + ∆ t ) ≤ E Z t +∆ tt X a − Xds ! / E Z t +∆ tt σ (1 + X a − X ) dW ( s ) ! / ≤ C (∆ t ) / (4.21)This proves the second part of the Lemma. (cid:3) Thus, Lemma 4.1 indicates that in analyzing V ( t, t + ∆ t ) / ∆ t in the limit of∆ t → V ( t, t + ∆ t ) / ∆ t. At this point we assume that both P a and σ are deterministic but need not beconstant in time, and we now use lower case, x a := log P a . Lemma 4.2.
Let σ and P a be deterministic, and X ( t ) as solution to the SDE (4 . . Then (4.22) V ( t, t + ∆ t ) = Z t +∆ tt σ [1 + x a ( s ) − EX ( s )] ds + Z t +∆ tt σ V ar [ X ( s )] ds. Proof.
Using the expression (4 .
14) above, the identity follows upon adding andsubtracting EX ( s ) in the integrand. (cid:3) Lemma 4.3.
Let σ and P a be deterministic and continuous. Then V ( t ) := lim ∆ t → V ( t, t + ∆ t )∆ t = lim ∆ t → V ( t, t + ∆ t )∆ t = lim ∆ t → t (Z t +∆ tt σ [1 + x a − y ] + V ar [ X ] ds ) = σ [1 + x a − y ] + V ar [ X ] . (4.23)Next, we will compute V ar [ X ] starting with E (cid:2) X (cid:3) and assuming that P a and σ are deterministic. Lemma 4.4.
Let σ and x a be deterministic and continuous. Then z ( t ) := EX ( t ) satisfies the ODE dzdt = (cid:0) σ − (cid:1) z + (cid:0) − σ (cid:1) x a y − σ y + σ (1 + x a ) z ( t ) = y ( t ) =: y . (4.24) SSET PRICE VOLATILITY AND PRICE EXTREMA 17
Proof.
The stochastic process for X ( t ), i.e., (4 .
9) can be written A ( t, ω ) := ( x a − X ) , B ( t, ω ) := σ (1 + x a − X )(4.25) dX ( t, ω ) = A ( t, ω ) dt + B ( t, ω ) dW ( t )Ito’s formula provides the differential for a smooth function of X as df ( X ( t ) , t ) = (cid:20) ∂f ( X ( t ) , t ) ∂t + A ( t ) ∂f ( X ( t ) , t ) ∂x + B ( t )2 ∂ f ( X ( t ) , t ) ∂x (cid:21) dt + B ( t ) ∂f ( X ( t ) , t ) ∂x dW ( t ) . (4.26)For f ( x ) := x we have then from Ito’s formula, dX = h(cid:0) σ − (cid:1) X + (cid:0) − σ (cid:1) x a X − σ X + σ (1 + x a ) i dt + σ (1 + x a − X ) (2 X ) dW (4.27)Hence, we can write in the usual way, as EdW vanishes:(4.28) E (cid:2) X ( t ) − X ( t ) (cid:3) = Z tt (cid:0) σ − (cid:1) EX + (cid:0) − σ (cid:1) x a EX − σ EX + σ (1 + x a ) ds Using the notation y ( t ) := E (log P ) and z ( t ) := E (log P ) we have(4.29) z ( t ) − z ( t ) = Z tt (cid:0) σ − (cid:1) z ( t ) + (cid:0) − σ (cid:1) x a y ( t ) − σ y ( t ) + σ (1 + x a ) ds. Differentiation with respect to t yields the result and proves the lemma. (cid:3) In the sequel, we assume for simplicity that σ is constant in time, and x a ( t ) is de-terministic and smooth. We can solve for z directly but it will be more illuminatingif we write the solution in the following form. Lemma 4.5.
Let x a be a continuous function. The unique solution to dz dt = − z + 2 x a y (4.30) z ( t ) := y ( t ) (4.31) is given by z ( t ) = y ( t ) . Proof.
Note that x a = y a = EX a since X a is deterministic under our currentassumption. We know that y ( t ) is a solution to the equation(4.32) ddt y ( t ) = y a ( t ) − y ( t ) , y ( t ) := y so we can substitute x a = y ′ + y into (22) and obtain(4.33) z ′ + 2 z = 2 yy ′ + 2 y = 2 y ( y ′ + y ) = 2 x a y. Hence, z ( t ) := y ( t ) solves (4 . , (4 .
31) and from basic ODE theory, the solutionis unique so long as x a is continuous. (cid:3) Lemma 4.6.
The unique solution to (4 . is given by (4.34) z ( t ) := z ( t ) + σ z ( t ) = y ( t ) + σ z ( t ) with z ( t ) defined by (4.35) z ( t ) = Z tt e ( − σ ) ( s − t ) [ y ( s ) − (1 + x a ( s ))] ds. Proof.
Let z be defined by z ( t ) = z ( t ) + σ z ( t ) = y ( t ) + σ z ( t ) . Substitutinginto (4 .
24) yields z ′ + σ z ′ = (cid:0) σ − (cid:1) (cid:0) z + σ z (cid:1) + (cid:0) − σ (cid:1) x a y − σ y + σ (1 + x a ) = σ z − z + (cid:0) σ − (cid:1) σ z + (cid:0) − σ (cid:1) x a y − σ y + σ (1 + x a ) (4.36)so that the terms z ′ and − z + 2 x a y vanish from both sides.. Using z = y wehave left, upon dividing by σ , the equation for z (4.37) z ′ + (cid:0) − σ (cid:1) z = [ y − (1 + x a )] , and elementary methods yield the solution (4.34 - 4.35). (cid:3) Note that although σ ∈ R was used in this proof, comparable result can beobtained in the general case in which σ is a continuous and deterministic function.Thus, Lemmas 4.5 and 4.6 yield the following identity for V ar [ X ( t )] . Theorem 4.7.
Let σ ∈ R and x a ( t ) be deterministic and continuous. Let c := (cid:0) − σ (cid:1) and (4.38) σ I ( t, t + ∆ t ) := V ar [ X ( t + ∆ t )] − V ar [ X ( t )] . (4.39) w ( s ) := [1 + x a ( s ) − y ( s )] . Then one has the following identities:
V ar [ X ( t )] = σ Z tt e c ( s − t ) [ y ( s ) − (1 + x a ( s ))] ds (4.40) I ( t, t + ∆ t ) = Z t +∆ tt e c ( s − t ) w ( s ) ds. (4.41) Proof.
The identities follow immediately from Lemma 4.6 and the definition ofvariance in terms of z and y. I.e.,
V ar [ X ( t )] = E [ X ( t )] − [ EX ( t )] = z ( t ) − y ( t ) = σ z ( t )= σ Z tt e ( − σ ) ( s − t ) [1 + x a ( s ) − y ( s )] ds. (4.42) (cid:3) Remark . The maximum value of
V ar [ X ( t + ∆ t )] − V ar [ X ( t )] occurs for t suchthat the average weighted value of w ( s ) with exponential weighting of (cid:0) − σ (cid:1) ismaximal on ( t, t + ∆ t ) . SSET PRICE VOLATILITY AND PRICE EXTREMA 19
Using the lemmas above, we obtain directly the following result.
Theorem 4.9.
Let x a be continuous. Then we have the identities, lim ∆ t → σ − (∆ t ) − V ( t, t + ∆ t ) = lim ∆ t → σ − (∆ t ) − V ( t, t + ∆ t )= w ( t ) + V ar [ X ( t )] i.e., σ − V ( t ) = w ( t ) + σ Z tt e (2 − σ ) ( s − t ) w ( s ) ds (4.43) Q ( t ) := ddt lim ∆ t → σ − V ( t, t + ∆ t )∆ t = σ − ddt V ( t )= w ′ ( t ) + σ w ( t ) − σ (cid:0) − σ (cid:1) Z tt e (2 − σ ) ( s − t ) w ( s ) ds. (4.44) 5. Market extrema
The main objective of this section is to apply the results above understand thetemporal relationship between the extrema of the (log) fundamental value, x a ( t ),and the expected (log) trading price, y ( t ) . Price Maxima.Notation 2.
Let t be the initial time, and t m be defined by x ′ a ( t m ) = 0 , i.e.,the time at which the fundamental value, x a , attains its maximum. The time t ∗ isdefined as the first time at which y ′ ( t ∗ ) = x a ( t ∗ ) − y ( t ∗ ) vanishes, and the curves x a ( t ) and y ( t ) first intersect. Notation 3.
Let ˆ x a ( t ) := e t x a ( t ) , ˆ y ( t ) := e t y ( t ) , ˆ y := e t ˆ y ( t ) . Condition σ . Let σ ∈ (0 ,
1) be a constant, so c := 2 − σ ∈ (1 , . We will assume this condition throughout, though some results are valid withoutit.
Condition C. ( i ) The function x a : [ t , ∞ ) → (0 , ∞ ) has the property that forsome t m ∈ (0 , ∞ ) one has( i ) x ′ a ( t ) > if t < t m ; x ′ a ( t m ) = 0; x ′ a ( t ) < if t > t m . ( ii ) Let y ( t ) =: y ∈ (0 , ∞ ) one has(5.1) x a ( t ) − x ′ a ( t ) < y < x a ( t ) . ( iii ) For some δ, m ∈ (0 , ∞ ) one has(5.2) − x ′ a ( t ) > m > if t > t m + δ. Remarks.
We set y =: y ( t ) , so the two inequalities in Condition C ( ii ) statethat initially (i.e., at t ) the price is below the fundamental value, i.e., underval-uation ( y ( t ) = y < x a ( t )). Using the ODE y ′ = x a − y one has that the firstinequality in Condition C ( ii ) is equivalent to x ′ a ( t ) > y ′ ( t ) > C ( iii ) canbe relaxed to some extent although the condition then appears more technical. Condition E. With t ∗ be defined as above, assume 2 x ′ a ( t ∗ ) + σ e c ( t − t ∗ ) < . Remarks.
Note that this condition is satisfied automatically if t → −∞ . Solong as there is an interval ( t m , t ∗ ) on which x ′ a ( t ∗ ) < − σ e c ( t − t ∗ ) (the latter isexponentially small if t ∗ − t >>
1) the Condition E will be satisfied.Recalling that y ( t ) is given by (4 . y ( t ) = ˆ y ( t ) + Z tt ˆ x a ( s ) ds. since y a = x a as the latter is deterministic.Initially, we have from C ( ii ) that x a ( t ) > y ( t ) . We want to first prove that y intersects with x a at some value t ∗ and that this value t ∗ occurs after t m (i.e., thetime at which x a has its peak). Theorem 5.1.
Assume that C holds. Then there exists a least value t ∗ ∈ ( t m , ∞ ) such that for t < t ∗ one has y ( t ) < x a ( t ) , and, y ( t ∗ ) < x a ( t ∗ ) , i.e., (5.4) ˆ y ( t ∗ ) = ˆ y + Z t ∗ t ˆ x a ( s ) ds = ˆ x a ( t ∗ ) . Since y ′ = x a − y, the maximum of y is attained at t ∗ .Proof. Let I ( t ) := ˆ x a ( t ) − ˆ y − R tt ˆ x a ( s ) ds, so I ( t ) > C ( ii ) . Computing the derivative and using Condition C ( i ) yields(5.5) I ′ ( t ) = ˆ x ′ a ( t ) − ˆ x a ( t ) = e t x ′ a ( t ) > if t < t m . Hence, one has I ( t ) < t < t m . On the other hand, by Condition C ( iii ), when t > t m + δ one has(5.6) I ′ ( t ) = e t x ′ a ( t ) ≤ e t m ( − m )so that I ( t ∗ ) = 0 for some finite t ∗ > t m . (cid:3) Lemma 5.2.
Under C ( i ) , ( ii ) there exists a t ∈ ( t , t m ) such that w ′ ( t ) = 0 , w ′ ( t ) > if t ∈ [ t , t ) , and w ′ ( t ) < if t m < t < t ∗ . Consequently, we have (5.7) t < t < t m < t ∗ . Proof.
Recall (4 .
43) and note w ′ = 2 [1 + x a − y ] ( x ′ a − y ′ ) , whose sign is deter-mined by(5.8) S ( t ) := x ′ a ( t ) − y ′ ( t ) = x ′ a ( t ) − x a ( t ) + y ( t )when t < t ∗ [i.e., when x a ( t ) > y ( t )]. For t we have from C ( ii ) that S ( t ) > . For t m < t < t ∗ we have from C ( i ) that x ′ a ( t ) < y ′ ( t ) = x a ( t ) − y ( t ) > S ( t ) = x ′ a ( t ) − x a ( t ) + y ( t ) < . By continuity, there exists a t ∈ ( t , t m ) such that S ( t ) = 0 and S ( t ) > t < t . I.e., t is the first crossing for S ( t ) and hence for w ( t ) . The ordering (5 . (cid:3) Lemma 5.3.
Assuming Condition C, one has Q ( t ) > . SSET PRICE VOLATILITY AND PRICE EXTREMA 21
Proof.
Since t < t m < t ∗ one has x a ( t ) > y ( t ) and consequently w ( t ) exceeds1 and is thus positive. Hence, we can replace w ( s ) by w ( t ) in the integral, andfactor, in order to obtain the inequality Q ( t ) ≥ σ w ( t ) − σ c Z t t e c ( s − t ) w ( t ) ds = σ w ( t ) n − (cid:16) − e c ( t − t ) (cid:17)o = σ w ( t ) e c ( t − t ) > . (5.10) (cid:3) Lemma 5.4.
If Conditions C and E hold, then Q ( t ∗ ) < . Proof.
We write(5.11) Q ( t ∗ ) = w ′ ( t ∗ ) + σ w ( t ∗ ) − σ c Z t ∗ t e c ( s − t ∗ ) w ( s ) ds, and note that for any t ≤ t ∗ one has x a ( t ) > y ( t ) by Thm 5.1. Consequently, wehave the inequality(5.12) w ( t ) = [1 + x a ( t ) − y ( t )] ≥ w ( t ∗ ) . By using this mimimum value of w that is subtracted, we have(5.13) Q ( t ∗ ) ≤ w ′ ( t ∗ ) + σ w ( t ∗ ) − σ c Z t ∗ t e c ( s − t ∗ ) ds. Also, from Thm 5.1, we have y ′ ( t ∗ ) = x a ( t ∗ ) − y ( t ∗ ) = 0 , so a computation yields(5.14) w ′ ( t ∗ ) = 2 [1 + x a ( t ∗ ) − y ( t ∗ )] ( x ′ a ( t ∗ ) −
0) = 2 x ′ a ( t ∗ ) . Using w ( t ∗ ) = 1, and evaluating the integral, one obtains(5.15) Q ( t ∗ ) ≤ x ′ a ( t ∗ ) + σ e c ( t − t ∗ ) < . The last inequality follows from Condition E . (cid:3) Hence, recalling that t < t < t m < t ∗ , we obtain the result that the maximumof Q, the limiting volatility precedes the peak of y ( t ), which occurs at t ∗ . Theorem 5.5.
There exists a t v ∈ ( t , t ∗ ) such that Q ′ ( t v ) = 0 . In summary, the derivative of y catches up to x a at t . Recalling (4 . Q ( t v ) = σ − d V ( t v ) /dt = 0 corresponds to a maximum in V , and this occursafter t and before t m where x a has its peak. The peak of x a precedes the peak of y at t ∗ . Thus, V has a maximum prior to the maxima of x a and y .In conclusion, we have shown that the limiting volatility V ( t ) attains its maxi-mum prior to that of the expected logarithm of the price, y ( t ). References [1] Bodie, Z., Kane, A., Marcus, A. 2010 Investments, 10 Ed. McGraw Hill, New York.[2] Black, F., Scholes, M. 1973 The pricing of options and corporate liabilities.
J. PoliticalEconomy , 637-654.[3] Wilmott, P., Howison, S. and Dewynne, J. 2008 The Mathematics of Financial Derivatives.Cambridge Univ Press, Cambridge.[4] Billingsley, P. 2012 Probability and Measure, Anniversary Ed. Wiley, Hoboken, NJ. [5] Schuss Z. 2010 Theory and Applications of Stochastic Processes, An Analytical Approach.Springer, New York.[6] Caginalp, G., Desantis, M. 2011 Multi-Group Asset Flow Equations and Stability. Disc. andCont. Dynam. Systems B , 109-150.[7] Caginalp, G., Balevonich, D. 1999 Asset flow and momentum: Deterministic and stochasticequations. Phil. Trans. Royal Soc., Math, Phys., Engr , 2119-2113.[8] Caginalp, G., Porter, D., and Smith, V. 1998 Initial cash/asset ratio and asset prices: anexperimental study.
Proc. Nat. Acad. Sciences , 756-761.[9] H. Merdan, H., Alisen, M. 2011 A mathematical model for asset pricing. Applied Mathematicsand Computation , 1449-1456.[10] Merdan, H. and Cakmak, H. 2012 Liquidity Effect on the Asset Price Forecasting,
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Journal of Economic Behaviorand Organization , 353-383.[14] Tong, Y. 1990 The Multivariate Normal Distribution. Springer-Verlag, New York.[15] D´ıaz-Franc´es, E., Rubio, F. J. 2013 On the existence of a normal approximation to thedistribution of the ratio of two independent normal random variables. Stat. Papers , 1-15.[16] Hinkley, D. 1969 On the ratio of two correlated normal random variables Biometrika ,635–639.[17] Hinkley D. 1970 Correction: On the Ratio of Two Correlated Normal Random Variables Biometrika , , 683.[18] Smith, V.L., Suchanek, G.L Williams, A.W., 1988 Bubbles, crashes, and endogenous expec-tations in experimental spot asset markets. Econometrica , 1119-1151.[19] Shefrin, H. 2005 A Behavioral Approach to Asset Pricing. Elsevier Academic Press, Burling-ton, MA.[20] Rudin, W. 1986 Real and Complex Analysis, 3rd Ed. McGraw-Hill. Appendix
We start with d log P ( t, ω ) = G ( D/S ) dt + 12 (cid:26) DS G ′ (cid:18) DS (cid:19) + SD G ′ (cid:18) SD (cid:19)(cid:27) σdW ( t, ω ) . and set G ( x ) := x − /x, so the model is(5.16) d log P = (cid:18) DS − SD (cid:19) dt + (cid:18) DS + SD (cid:19) σdW in which the supply and demand are on a symmetric footing. In other words, whensupply exceeds demand, the price moves down in the same way as it moves up whendemand exceeds supply. The coefficient for dW is symmetric in S and D. In order to study market tops and bottoms, we would like to simplify this ex-pression. We consider the regimes: ( i ) D and S deviate by a small amount, and( ii ) D and S deviate by a large amount.( i ) Suppose that D = q + δ ′ and S = q − ε ′ where q > δ ′ and ε ′ are smallin magnitude, i.e., one is not far from equilbrium. Then we have with δ := δ ′ /q SSET PRICE VOLATILITY AND PRICE EXTREMA 23 and ε := ε ′ /q DS − δ − ε − δ + ε, − SD = 1 − − ε δ ˜= δ + ε, (cid:18) DS − SD (cid:19) = 12 (cid:18) δ − ε − − ε δ (cid:19) ˜= δ + ε So all three terms are equal up through O ( δ, ε ) . Thus, when one is not too far fromequilibrium, these terms are approximately equal and one can use any of them inthe deterministic part of the price equation.Similarly, under these near equilibrium conditions, the terms
D/S, S/D and(
D/S + S/D ) / O (1) . ( ii ) Next suppose that we are far from equilibrium, and note that DS ˜= DS + SD and DS − DS ˜= DS − SD if D/S >> . Similarly, one has SD ˜= DS + SD and − SD ˜= − SD ˜= DS − SD if S/D >> . Applying these approximations to (5 .
16) we see that for market tops (when D ≥ S ) we can use the model d log P = (cid:18) DS − (cid:19) dt + DS σdW, and analogously, for market bottoms, (when S ≥ D ) we use d log P = (cid:18) − SD (cid:19) dt + SD σdW.
Note that for the coefficient of σdW , we are essentially approximating G ( x ) := x + 1 /x with x when x ≥ /x when x ≤
1. Near x = 1 , of course, thisintroduces a factor of 2 that can be incorporated into σ. Economic Science Institute, Chapman Unviersity, Orange, CA 92866 and MathematicsDepartment, University of Pittsburgh, Pittsburgh, PA 15260 and Mathematics Depart-ment, University of Pittsburgh, Pittsburgh, PA 15260
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