stock price formula for hft

Electronic copy available at httpssrncomabstract=1977561

Trading Rules Over FundamentalsA Stock Price Formula for High Frequency Trading

Bubbles and Crashes

Godfrey Cadogan lowast

Working Paper

Comments welcome

January 2 2012

lowastCorresponding address Information Technology in Finance Institute for Innovation and TechnologyManagement Ted Rogers School of Management Ryerson University 575 Bay Toronto ON M5G 2C5e-mail godfreycadoganryersonca I thank Oliver Martin for his comments on an earlier draft of thepaper which improved its readability Research support from the Institute for Innovation and TechnologyManagement is gratefully acknowledged All errors which may remain are my own


Abstract

In this paper we present a simple closed form stock price formula which captures empirical reg-ularities of high frequency trading (HFT) based on two factors (1) exposure to hedge factor and(2) hedge factor volatility Thus the parsimonious formula is not based on fundamental valuationFor instance it allows us to use exposure to and volatility of E-mini contracts to predict movementsin an underlying index For application we first show that for given exposure to hedge factor andsuitable specification of hedge factor volatility HFT stock price has a closed form double expo-nential representation There in periods of uncertainty if volatility is above historic average arelatively small short selling trade strategy is magnified exponentially and the stock price plum-mets when strategies are phased locked Second we demonstrate how asymmetric response tonews is incorporated in the stock price by and through an endogenous EGARCH type volatilityprocess and find that intraday returns have a U-shaped pattern inherited from HFT strategiesThird we show that the stock price is bounded from below (crash) ie flight to quality but notfrom above (bubble) ie confidence when phased locked trade strategies violate prerequisites ofvan der Corputrsquos Lemma Thus extant regulatory proposals to control price dynamics of selectstocks ie rdquolimit uplimit downrdquo bands over 5-minute rolling windows may mitigate but not stopfuture market crashes or price bubbles from manifesting in underlying indexes that exhibit HFTstock price dynamics

Keywords high frequency trading hedge factor volatility price reversal market crash price bub-bles fundamental valuation van der Corputrsquos Lemma

JEL Classification Codes C02 G11 G12 G13


Contents

1 Introduction 2

2 The Model 521 Trade strategy representation of alpha in single factor CAPM 7

3 High Frequency Trading Stock Price Formula 9

4 Applications 1141 The impact of stochastic volatility on HFT stock prices 1142 Implications for intraday return patterns and volatility 1343 van der Corputrsquos lemma and phase locked stock price 15

5 Conclusion 18

References 19

1

1 Introduction

Recent studies indicate that high frequency trading (HFT) accounts for about 77of trade volume in the UK1 and upwards of 70 of trading volume in US eq-uity markets2 and that rdquo[a]bout 80 of this trading is concentrated in 20 of themost liquid and popular stocks commodities andor currenciesrdquo3 Thus at anygiven time an observed stock price in that universe reflects high frequency tradingrules over price fundamentals4 In fact a recent report by the Joint CFTC-SECAdvisory Committee summarizes emergent issues involving the impact of HFTon market microstructure and stock price dynamics5 This suggests the existenceof stock price formulae for HFT different from the class of Gordon6 dividendbased fundamental valuation7 and present value models popularized in the litera-ture8 This paper provides such a formula by establishing stochastic equivalencebetween recent continuous time trade strategy and alpha representation theoriesintroduced in Jarrow and Protter (2010)Jarrow (2010) and Cadogan (2011) Evi-dently high frequency traders quest for alpha and their concomitant trade strate-gies are the driving forces behind short term stock price dynamics In that caselet (ΩFtge0FP) be a filtered probability space S(tω) be a realized stockprice defined on that space X(tω) be a realized hedge factor βX(tω) be a tradestrategy factor ie exposure to hedge factor σX(tω) be hedge factor stochas-tic volatility Q be absolutely continous with respect to P and BX(uω) be a P-Brownian bridge or Q-Brownian motion that captures background driving pricereversal strategies We claim that the stock price representation for high frequency

1(Sornette and Von der Becke 2011 pg 4)2Brogaard (2010)3Whitten (2011)4(Zhang 2010 pg 5) characterizes algorithmic trading rules thusly rdquo HFT is a subset of algorithmic trading or

the use of computer programs for entering trading orders with the computer algorithm deciding such aspects of theorder as the timing price and order quantity However HFT distinguishes itself from general algorithmic trading interms of holding periods and trading purposesrdquo

5See (Joint Advisory Committee on Emerging Regulatory Issues 2011 pg 2) (rdquoThe Committee believes that theSeptember 30 2010 Report of the CFTC and SEC Staffs to our Committee provides an excellent picture into the newdynamics of the electronic markets that now characterize trading in equity and related exchange traded derivativesrdquo)(emphasis added)

6Gordon (1959)7See eg Tobin (1984)8See eg Shiller (1989) and Shiller and Beltratti (1992) who used annual returns in a variant of Gordonrsquos model of

fundamental valuation for stock prices and reject rational expectations present value models But compare (Madha-van 2011 pp 4-5) who reports that rdquoThe futures market did not exhibit the extreme price movements seen in equitieswhich suggest that the Flash Crash [of May 6 2010] might be related to the specific nature of the equity marketstructurerdquo

2

trading is given by

S(tω) = S(t0)exp(int t

t0σX(u ω)βX(uω)dBX(uω)

)(11)

For instance this formula allows us to use exposure (βX) to and volatility (σX) ofE-mini contracts (X) to predict movements in an underlying index (S) Supposethat rdquoX-factorrdquo volatility has a continuous time GARCH(11) representation givenby

dσ2X(tω) = θ(ηminusσ

2X(tω))dt +ξ σ

2X(tω)dWσX (tω) (12)

where θ is the rate of reversion to mean volatility η ξ is a scale parameter andWσX is a background driving Brownian motion for stochastic volatility σX Afterapplying Girsanovrsquos Theorem to remove the drift in 12 and then substituting theexpression in 11 we get the double exponential local martingale representationof the stock price given by

S(tω) =

S(t0)exp(exp(minus 12 ξWσX (t0)))exp

(int t

t0βX(uω)exp(1

2 ξWσX (uω))dBX(uω))

(13)

Further details on derivation of the formula and description of variables are pre-sented in the sequel

As indicated above our parsimonious formula is closed form and it isbased on two factors (1) exposure to hedge factor and (2) hedge factor volatilityThe formula plainly shows that for given exposure βX(uω) to the rdquoX-factorrdquoif uncertainty in the market is such that volatility is above historic average ieσ2

X(tω) gt η is relatively high a portfolio manager might want to reduce expo-sure in order to stabilize the price of S(tω) However if βX(uω) is reduced tothe point where it is negative ie there is short selling then the price of S(tω)will be in an exponentially downward spiral if volatility continues to increaseThus feedback effects9 between S(tω) the rdquoX-factorrdquo and exposure control ortrade strategy βX(uω) determines the price of S10 In this setup so called market

9See (Oslashksendal 2003 pg 237) for taxonomy of admissible [feedback] control functions See also de Long et al(1990) (positive feedback trading by noise traders increase market volatility)

10For instance (Madhavan 2011 pg 5) distinguishes between rules based algorithmic trading and comparativelyopaque high frequency trading in which traders play a signal jamming game of quote stuffing There orders are postedand immediately canceled ie reversed This price reversal strategy is captured by βX (uω) and embedded in ourstock price

3

fundamentals do not determine the priceOne of the key results of our paper is the introduction of an admissible

lower bound of zero (crash) and unbounded upper limit (bubble) for stock pricesin high frequency trading environments The analytics for that results is basedon violation of van der Corputrsquos Lemma11 Recent proposals by the Joint Ad-visory Committee on Emerging Regulatory Issues include so called rdquoPause rulesrdquoextended to limit up limit down bands for the price of select stocks over rolling 5-minute windows12 Assuming without deciding that such recommendations wouldmitigate order imbalances that may trigger excess volatility its unclear how pauserules would affect the behaviour of an underlying basket of stocks or derivativesthat may not be covered by the recommendation In which case an underlyingstock index (comprised of stocks not covered by the recommendation) priced byour formula remains vulnerable to crashes and bubbles The efficacy of our for-mula is supported by recent research on HFT which we review next

An empirical study by Zhang (2010) found that high frequency trading(HFT) increases stock price volatility In particular rdquothe positive correlation be-tween HFT and volatility is stronger when market uncertainty is high a time whenmarkets are especially vulnerable to aggressive to HFTrdquo13 A result predicted byour formula Further he used earnings surprise and analysts forecast as proxiesfor news and firm fundamentals to examine HFT response to price shocks Hefound that rdquothe incremental price reaction associated with HFT are almost entirelyreversed in the subsequent periodrdquo14 That finding is consistent with the price re-versal strategy predicted by our theory And the response to news is captured byour continuous time GARCH(11) specification15 to hedge factor volatility incor-porated in our closed form formula in an example presented in the sequel

(Jarrow and Protter 2011 pg 2) presented a continuous time signallingmodel in which HFT rdquotrades can create increased volatility and mispricingsrdquo whenhigh frequency traders observe a common signal This is functionally equivalentto the trade strategy factor embedded in our closed form formula However thoseauthors suggests that rdquopredatory aspects of high frequency tradingrdquo stem from thespeed advantage of HFT and that perhaps policy analysts should focus on thataspect By contrast our formula suggests that limit on short sales would mitigate

11See eg (Stein 1993 pg 332)12See eg (Joint Advisory Committee on Emerging Regulatory Issues 2011 Recommendation 3 pg 5)13(Zhang 2010 pg 3)14Ibid15See eg (Engle 2004 pg 408)

4

the problem while still permitting the status quoA tangentially related paper16 by Madhavan (2011) used a market segmen-

tation theory based on application of the Herfindahl Index of market microstruc-ture to explain the Flash Crash of May 6 2010 Of relevance to us is (Madhavan2011 pg 4) observation that the recent joint SEC and CFTC report on the FlashCrash identified a traderrsquos failure to set a price limit on a large E-mini futures con-tract used to hedge an equity position as the catalyst for the crash There stockprice movements were magnified by a feedback loop An event predicted by andembedded in our stock price formula for high frequency trading (Madhavan2011 pg 6) also provides a taxonomy of high frequency trading strategies frompapers he reviewed

The rest of the paper proceeds as follows In section 2 we introduce themodel In section 3 we derive the HFT stock price formula The main result thereis Proposition 33 In section 4 we apply the stock price formula in the contextof stochastic volatility in Lemma 41 characterize intraday return patterns in inLemma 42 and the impact of phase locked high frequency trading strategies onstock prices in Proposition 45 In section 5 we conclude with perspectives on theimplications of the formula for trade cycles in the context of quantum cognition

2 The Model

We begin by stating the alpha representation theorems of Cadogan (2011) Jarrowand Protter (2010) in seriatim on the basis of the assumptions in Jarrow (2010)Then we show that the two theorems are stochastically equivalentndashat least for asingle factorndashand provide some analytics from which the HFT stocp price formulais derived

Assumption 21 Asset markets are competitive and frictionless with continuoustrading of a finite number of assets

Assumption 22 Asset prices are adapted to a filtration of background drivingBrownian motion

Assumption 23 Prices are ex-dividend16See also (Sornette and Von der Becke 2011 pp 4-5) who argue that volume is not the same thing as liquidity

and that HFT reduces welfare of the real economy

5

Theorem 24 (Trading strategy representation Cadogan (2011))Let (ΩFt FP) be a filtered probability space and Z = ZsFs 0 le s lt infin bea hedge factor matrix process on the augmented filtration F Furthermore leta(ik)(Zs) be the (ik)-th element in the expansion of the transformation matrix(ZT

s Zs)minus1ZT

s and B = B(s)Fs s ge 0 be Brownian motion adapted to F suchthat B(0) = x Assuming that B is the background driving Brownian motion forhigh frequency trading the hedge factor sensitivity process ie trading strategyγ = γsFs0le s lt infin generated by portfolio manager market timing for Brow-nian motion starting at the point xge 0 has representation

dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1

a(ik)(Zt)dB(tω) xge 0

for the i-th hedge factor i = 1 p and 0le t le 1

Proof See (Cadogan 2011 Thm 46)

Remark 21 (Cadogan 2011 sect22) presented a data mining algorithm based onmartingale system equations to identify excess returns and describe high frequencytrade strategy

Cadogan (2011) computes an alpha vector ααα(tω) = Zγγγ(tω) where Z is a ma-trix of hedge factors and γγγ(tω) is a vector of hedge factor sensitivity that con-stitutes the active portfolio managerrsquos trading strategy This parametrization isconsistent with hedge factor models17 such as Fama and French (1993)

Theorem 25 ((Jarrow and Protter (2010)))Given no arbitrage there exist K portfolio price processes X1(t) XK(t) suchthat an arbitrary securityrsquos excess return with respect to the default free spot in-terest rate ie risk free rate rt can be written

dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)

minus rtdt)+δi(t)dεi(t) (21)

where εi(t) is a Brownian motion under (PFt) independent of Xk(t) for k =1 K and δi(t) = sum

dk=K+1 σ2

ik(t)17See Noehel et al (2010) for a review

6

Proof See (Jarrow and Protter 2010 Thm 4)

In contrast to Cadogan (2011) endogenous alpha (Jarrow 2010 pg 18) sur-mised that rdquo[i]n active portfolio management an econometrician would add a con-stant αi(t)dt to [the right hand side of] this modelrdquo So Cadogan (2011) alpha isadaptive whereas Jarrow (2010) alpha is exogenous Without loss of generalitywe assume that the filtration in each model is with respect to background drivingBrownian motion

21 Trade strategy representation of alpha in single factor CAPM

If our trade strategy representation theory is well defined then it should shed lighton the behavior of alpha in a single factor model like CAPM where there is nohedge factor In particular for cumulative alpha A(t) let18

A(t) = Zγγγ(t) (22)

where Z is a hedge factor matrix and γγγ(t) is a trade strategy vector Thus forsome vector ααα(t) we have

dA(t)|Z=ααα(t)dt = Zdγγγ(t) Let (23)Z = 1n (24)

So that

(ZT Z)minus1ZT = nminus11Tn and a(1k)(Zs) = nminus1 k = 1 n (25)

Substitution of these values in Theorem 24 gives us by abuse of notation thescalar equation for trade strategy alpha

minusα(t)dt =minusdγ(1)(t) =minus x

1minus tdt +dB(t) (26)

18See (Christopherson et al 1998 pp 121-122) for representation of alpha conditioned on a Z-matrix of economicinformation

7

That is the equation of a Brownian bridge starting at B(0) = x on the interval [01]See (Karlin and Taylor 1981 pg 268) So that trade strategy alpha is given by

dγ(1)(t) =minusdBbr(t) (27)

γ(1)(t) = Bbr(0)minusBbr(t) (28)

However there is more According to Girsanovrsquos formula in (Oslashksendal 2003pg 162) we have an equivalent probability measure Q based on the martingaletransform

M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)

dQ(ω) = M(Tω)dP(ω) 0le t le T le 1 (210)

Furthermore we have the Q-Brownian motion

B(t) =minusint t

0

x1minus s

ds+B(t) and (211)

dγ(1)(t) =minusdB(t) =minusdBbr(t) (212)

In other words γ(1) is a Q-Brownian motionndashin this case a Brownian bridgendashthatreverts to the origin starting at x We note that

dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)

Hence the rdquoresidual(s)rdquo εt associated with rate of change of Jensenrsquos alpha havean approximately skewed U-shape pattern on [01] Additionallly (Karlin andTaylor 1981 pg 270) show that the Brownian bridge can be represented by theGaussian process

G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =

s1(1minus s2) s1 lt s2

s2(1minus s1) s2 lt s1(215)

8

This is the so called Doob transformation see (Doob 1949 pp 394-395) and

B(t) = (1+ t)B(

t1+ t

)(216)

(Karatzas and Shreve 1991a pg 359) also provide further analytics which showthat we can write the trade strategy alpha as

dγ(1)(t) =

1minus γ(1)

1minus tdt +dB(t) 0le t le 1 γ

(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)

T (s) = inft|lt M gttgt s (219)G(t) = BltMgtT (t) (220)

where lt M gtt is the quadratic variation19 of the local martingale M

3 High Frequency Trading Stock Price Formula

We use a single factor representation of Theorem 24 and Theorem 25 with K = 1to derive the stock price formula as follows20 Using (Jarrow 2010 eq(5) pg 18)formulation let

α(t)dt =dS(t)S(t)

minus rtdtminusβ1(t)(

dX1(t)X1(t)

minus rtdt)minusσdB(t) (31)

=dS(t)S(t)

minusβ1(t)dX1(t)X1(t)

minus rt(1minusβ1(t))dtminusσdB(t) (32)

19See (Karatzas and Shreve 1991b pg 31)20(Jarrow and Protter 2010 pg 12 eq (12)) refer to the ensuing as a rsquoregression equationrdquo for which an econo-

metrican tests the null hypothesis H0 α(t)dt = 0 However (Cadogan 2011 eq 21) applied asymptotic theoryto econometric specification of a canonical multifactor linear asset pricing model augmented with portfolio managertrading strategy to identify portfolio alpha

9

Comparison with 26 suggests that the two alphas are equivalent if the followingidentifying restrictions are imposed

dγ(1)(t)equiv α(t)dt (33)

rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x

1minus t= (β1(t)minus1)rt (36)

So that

minusdγ(1) =minus(β1(t)minus1)rtdt +dB(t) (37)

In fact if for some constant drift microX volatility σX and P-Brownian motion BX wespecify the rdquohedge factorrdquo dynamics

dX1(t)X1(t)

= microXdt +σXdBX(t) (38)

then after applying Girsanovrsquos change of measure to 38 and by abuse of notation[and without loss of generality] setting βX(tω) = β1(t) we can rewrite 34 as

dS(t)S(t)

= βX(tω)σXdBX(t) (39)

for some Q-Brownian motion BX These restrictions required for functional equiv-alence between the two models are admissible and fairly mild We summarize theforgoing in the following

Proposition 31 (Stochastic equivalence of alpha in single factor models) As-sume that asset prices are determined by a single factor linear asst pricing modelThen the Jarrow and Protter (2010) return model is stochastically equivalent toCadogan (2011) trading strategy representation model

Corollary 32 (Jarrow (2010) trade strategy drift factor)β1(t)tisin[0T ] in Jarrow (2010) is a trade strategy [drift] factor

10

Equation 39 plainly shows that the closed form expression for the stockprice over some interval [t0 t] is now

dln(S(tω))= βX(tω)σXdBX(tω) (310)

rArrint t

t0dln(S(tω))= σX

int t

t0βX(uω)dBX(uω) (311)

rArr S(tω) = S(t0)exp(int t

t0σX︸︷︷︸

volatility

βX(uω)︸︷︷︸exposure

dBX(uω)︸︷︷︸news

)(312)

This formula allows us to use E-mini futures contracts to predict movements in anunderlying index For constant volatility σX this gives us the following

Proposition 33 (High Frequency Trading Stock Price Formula)Let (ΩF FP) be a probability space with augmented filtration with respect toBrownian motion B = B(tω)0 le t lt infin Let S be a stock price and X be ahedge factor with volatility σX adapted to the filtration and βX be the exposureof S to X Then the stock price over the interval [t0 t] is given by


t0σXβX(uω)dBX(uω)

)

4 Applications

We provide three applications of our stock price representation theory First weconsider the case of stochastic volatility when σX has continuous time GARCH(11)representation21 We use Girsanovrsquos Theorem to extend the formula to the caseof double exponential representation of HFT stock price correlated backgrounddriving processes Second we consider the case of intraday return patterns gen-erated by our formula and show how it has EGARCH features There we showhow news is incorporated in the HFT stock price Third we present bounds ofstock prices arising from phase locked processes There we use van der Cor-putrsquos Lemma22 for phase functions to motivate our result There applications are

21 In practice we would use the implied volatility of the rdquoX-factorrdquo Furthermore there are several different spec-ifications for stochastic volatility popularized in the literature See Shephard (1996) for a review However rdquo[t]heGARCH(11) specification is the workhorse of financial applicationsrdquo (Engle 2004 pg 408)

22See (Stein 1993 Prop 2 pg 332)

11

presented in seriatim in the sequel

41 The impact of stochastic volatility on HFT stock prices

This specification of the model permits extension to representation(s) of stochasticvolatility for the rdquoX-factorrdquo to mitigate potential simultaneity bias For expositionand analytic tractability we specify a GARCH(11) representation for σX


2X(tω))dt +ξ σ


where θ is the rate of reversion to mean volatility η ξ is a scale parameter andWσX is a background driving Brownian motion Girsanov change of measure for-mula see eg (Oslashksendal 2003 Thm II pg 164) posits the existence of a Q-Brownian motion WσX (tω) and local martingale representation

dσ2X(tω) = ξ σ


rArr σ2X(tω) = exp

(ξ

int t

t0dWσX (u)

)(43)

rArr σX(tω) = exp(

12ξ (WσX (u)minusWσX (t0))

)(44)

Substitution in 312 gives us the double exponential representation


t0exp(1

2ξ (WσX (uω)minusWσX (t0)))βX(uω)dBX(uω))

(45)

= S(t0)exp(exp(minus 12 ξWσX (t0)))

exp(int t

t0βX(uω)exp(1

2 ξWσX (uω))dBX(uω)) (46)

The formula plainly shows that for given stock price exposure βX to the rdquoX-factorrdquo if uncertainty in the market is such that volatility is above historic averageie σ2

X(tω) gt η because uncertainty about the news WσX is relatively high aportfolio manager might want to reduce exposure in order to stabilize the priceof S However if βX is reduced to the point where it is negative ie there isshort selling then the price of S will be in an exponentially downward spiral ifvolatility continues to increase In the face of monotone phase locked strategiespresented in subsection 43 this leads to a market crash of the stock price Thus

12

feedback effects between S the rdquoX-factorrdquo and exposure control or trade strategyβX determines the price of S In this setup so called market fundamentals do notdetermine the price We present this representation in the following

Lemma 41 (Double exponential HFT stock price representation)When stochastic volatility follows a continuous time GARCH(11) process thehigh frequency trading stock price has a double exponential representation givenin 46

42 Implications for intraday return patterns and volatility

In this subsection we derive the corresponding formula for intraday returns rHFT (tω)

for HFT stock price formula Assuming that σX is stochastic in Proposition 33we have

ln(S(tω)) = ln(S(t0))+int t

t0σX(uω)βX(uω)dBX(uω) (47)

Consider the dyadic partition prod(n) of [t0 t] such that

t(n)j = t0 + j2minusn(tminus t0) (48)

d ln(S(tω)) =dS(tω)

S(tω)dt =

σX(tω)βX(tω)dBX(tω)

S(t0)exp(int t

t0 σX(uω)βX(uω)dB(uω)

) (49)

The discretized version of that equation where we write dBX(tω)dt asymp εX(tω) sug-

gests that HFT intraday returns is given by

rHFT (tω) =

σX(tω)βX(tω)εX(tω)S(t0)minus1

exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)

Examination of the latter equation shows that it inherits the U-shaped pattern from213 by and through εX(tω) This theoretical U-shape result is supported by Ad-mati and Pfleiderer (1988) in the context of intraday trade volume and optimal

13

decisions of liquidity traders and informed traders Moreover an admissible de-composition of HFT returns in the context of an ARCH specification is

rHFT (t) = σHFT (t)εHFT (tω) where (411)

σHFT (t) = σX(tω)exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)

εHFT (tω) = βX(tω)εX(tω) (413)

Examination of 412 shows that it admits an asymmetric response to news εX(t(n)j ω)

about and exposure to the hedge factor X This is functionally equivalent to Nelson(1991) EGARCH specification And it may help explain why Busse (1999) foundweak evidence of volatility timing in daily returns for active portfolio managementin a sample of mutual funds based on his EGARCH specification Equation 413shows that high frequency traders response εHFT (tω) to news εX(tω) about therdquoX-factorrdquo is controlled by stock price exposure βX(tω) We summarize thisresult in the following

Lemma 42 (Intraday HFT return behaviour)The volatility of HFT returns depends on contemporaneous hedge factor volatilityand asymmetrically on historic news about and exposure to hedge factors Intra-day HFT returns exhibit EGARCH features and inherit U-shaped patterns fromprice reversal strategies of high frequency traders

We next extend the analysis to correlation between the background driving pro-cesses for hedge factor and hedge factor volatility

Let the X-factor be a forward rate The stochastic alpha beta rho (SABR)model23 for X-dynamics posits

dX(tω) = σX(tω)Xβ (tω)dBX(tω) 0le β le 1 (414)dσX(tω) = ασX(tω)dWσX (tω) α ge 0 (415)

BX(tω) = ρWσX (tω) |ρ|lt 1 (416)

where BX and WσX are background driving Brownian motions as indicated As-suming without deciding that the SABR model is the correct one to specify the

23See Zhang (2011) for a recent review

14

dynamics of X the double exponential representation in 46 plainly shows thatit includes the background driving Brownian motions in 416 In which case weextend the representation to

S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1

2 ξWσX (uω))dWσX (uω)) (417)

That representation plainly shows that now the stock price depends on backgrounddriving dynamics of hedge factor volatility and its correlation with underlyinghedge factor dynamics This suggests that for applications the implied volatilityof hedge factor and or some variant of the VIX volatility index should be factoredin HFT stock price formulae Here again the nature of the correlation coefficientρ ie whether its positive or negative for given exposure βX determines how thestock price responds to short selling We close with the following

Lemma 43 (HFT stock price as a function of background driving processes) HFTstock price dynamics depends on the background driving process for hedge factorstochastic volatility and its correlation with the background driving process forhedge factor dynamics

Remark 41 This lemma implies the existence of a Hidden Markov Model to cap-ture the latent dynamics in hight frequency trading In order not to overload thepaper we will not address that issue

43 van der Corputrsquos lemma and phase locked stock price

Before we examine the impact of phased locked trade strategies on HFT stockprices we need the following

Proposition 44 (van der Corputrsquos Lemma) (Stein 1993 pg 332)Suppose φ is real valued and smooth in (ab) and that |φ (k)(x)| ge 1 for all x isin(ab) Then |

int ba exp(iλφ(x))dx| le ckλ

minus1k holds when

i k ge 2 or

ii k = 1 and φ prime(x) is monotonic

15

The bounds ck is independent of φ and λ

Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)

be a local martingale for the j-th high frequency trader Furthermore consider thecomplex valued function

S(tω) = S(t0)exp(

iσXφj(t))

(419)

define the oscillatory integrals over some interval alt t lt b for the stochastic phasefunction φ j(tω)

I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)

Consider the differential

φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj

X(tω) be a monotone trade strategy over the trading horizon under con-sideration Without loss of generality assume that BX is Q-Brownian motion sothat

|φ jprime(tω)| ge 1rArr |φ jprime(tω)|2rArr βj2(tω)dt ge 1 (423)

These are the restrictions that must be imposed on trade strategy for Proposition44 to hold However the proposition also suggests that if φ j is real valued andsmooth ie differentiable and φ jprime is monotonic over the interval (ab) then

|I j(σX)| le ckσminus1

kX (424)

From the outset we note that when |φ jprime|lt 1 the conditions of Proposition 44 are

16

violated so that

I j(σX)gt ckσminus1

kX (425)

Furthermore the conditions above are violated because φ j is a Brownian func-tional which oscillates rapidly near 0 so it is not differentiable there24 Thus in amarket with a finite number of traders N say we have

N

sumj=1|I j(σX)| ge Nckσ

minus1k

X (426)

From 420 26 and 425 we have

|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N

sumj=1|I j(σX)|gt Nckσ

minus1k

X (428)

rArr IN(σX) =1N

N

sumj=1|I j(σX)|gt ckσ

minus1k

X (429)

Equation 420 represents the impact of the j-th traderrsquos strategy βj

X on stockprices However 428 shows that in the best case when volatility grows no slowerthan O(N) the cumulative effect of high frequency traders strategies β 1

X βNX

is unbounded from above Moreover 429 represents the phased locked or averagetrade strategy effect on stock prices which is also unbounded from above Thuswe proved the following

Proposition 45 (Phased locked stock price)Let β 1

X βNX be the distribution of high frequency traders [monotone] strate-

gies over a given trading horizon in a market with N traders Let I j(σX) in 420be the impact of the j-th trade strategy on the stock price


t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X

24The rdquo0rdquo can be translated to another time t0 say and the rapid oscillation holds by virtue of the strong Markovproperty of Brownian motion See (Karatzas and Shreve 1991b pg 86)

17

be the phased locked stock price attributed to average high frequency trading strat-egy Then the stock price is bounded from below but not from above In whichcase the lower bound constitutes the price at market crash when ck = 0 Whenck gt 0 the stock price is unbounded from above and determined by high frequencytrading strategies with admissible price bubbles

Remark 42 The path characteristics of the Brownian functional φ j(tω) are suchthat ck = 0 corresponds to very large jumps in φ jprime(tω) in 422 at or near t In otherwords volatility in the stock is extremely high and there arenrsquot enough buyers forthe phased locked short sales strategies Thus markets breakdown as there is aflight to quality and or liquidity in the sense of Akerlof (1970) For rdquoregularrdquoφ jprime(tω) ie there is more confidence and markets are bullish ck gt 0 and we getprice bubbles

5 Conclusion

This paper synthesized the continuous time asset pricing models in Cadogan (2011)(trade strategy representation theorem) and Jarrow and Protter (2010) (K-factormodel for portfolio alpha) to produce a simple stock price formula that cap-tures several empirical regularities of stock price dynamics attributable to highfrequency tradingndashaccording to emergent literature The model presented heredoes not address fundamental valuation of stock prices Nor does it explain whatcauses a high frequency trader to decide to sell [or buy] in periods of uncertaintyHowever recent quantum cognition theories show that subjects emit a behaviouralquantum wave in decision making when faced with uncertainty Thus further re-search in that direction may help explain underlying trade cycles Additionally thedouble exponential representation of HFT stock prices suggests that some kind ofexponential heteroskedasticity correction factor ie EGARCH should be used forperformance evaluation of active portfolio management

18

References

Admati A and P Pfleiderer (1988) A Theory of Intraday Patterns Volume andPrice Variability Review of Financial Studies 1(1) 3ndash40

Akerlof G A (1970 Aug) The Market for rdquoLemonsrdquo Quality Uncertainty andthe Market Mechanism Quarterly Journal of Economics 84(3) 488ndash500

Brogaard J A (2010 Nov) High Frequency Trading and Its ImpactOn Market Quality Working Paper Kellog School of Management De-partment of Finance Northwestern University Available at SSRN eLibraryhttpssrncomabstract=1641387

Busse J A (1999) Volatility Timing in Mutual Funds Evidence From DailyReturns Review of Financial Studies 12(5) 1009ndash1041

Cadogan G (2011) Alpha Representation For Active Portfolio Management andHigh Frequency Trading In Seemingly Efficient Markets In Proceedings ofJoint Statistical Meeting (JSM) Volume CD-ROM Alexandria VA pp 673ndash687 Business and Economic Statistics Section American Statistical Associa-tion

Christopherson J A W A Ferson and D A Glassman (1998 Spring) Condi-tional Manager Alphas On Economic Information Another Look At The Per-sistence Of Performance Review of Financial Studies 11(1) 111ndash142

de Long B A Shleifer L H Summers and R J Waldman (1990 June) PositiveFeedback Investment Strategies and Destabilizing Rational Speculation Journalof Finance 45(2) 379ndash395

Doob J L (1949) Heuristic Appproach to The Kolmogorov-Smirnov TheoremsAnn Math Statist 20(3) 393ndash403

Engle R (2004 June) Risk and Volatility Econometric Models and FinancialPractice American Economic Review 94(3) 405ndash420

Fama E and K French (1993) Common Risk Factors in the Return on Bondsand Stocks Journal of Financial Economics 33(1) 3ndash53

19

Gordon M (1959 May) Dividends Earnings and Stock Prices Review ofEconomics and Statistics 41(2) 99ndash105

Jarrow R and P Protter (2010 April) Positive Alphas Abnormal Performanceand Illusionary Arbitrage Johnson School Research Paper 19-2010 Dept Fi-nance Cornell Univ Available at httpssrncomabstract=1593051 Forthcom-ing Mathematical Finance

Jarrow R A (2010 Summer) Active Portfolio Manageement and Positiive Al-phas Fact or fantasy Journal of Portfolio Management 36(4) 17ndash22

Jarrow R A and P Protter (2011) A Dysfunctional Role of High Fre-quency Trading in Electronic Markets SSRN eLibrary Available athttpssrncompaper=1781124

Joint Advisory Committee on Emerging Regulatory Issues (2011 Feb) Recom-mendations Regarding Regulatory Responses to the Market Events of May 62010 Summary Report of the Joint CFTC-SEC Advisory Committee on Emerg-ing Regulatory Issues (Feb 18 2011) Joint CFTC-SEC Advisory Committee onEmerging Regulatory Issues FINRA Conference May 23 2011 Available athttpwwwsecgovspotlightsec-cftcjointcommittee021811-reportpdf

Karatzas I and S E Shreve (1991a) Brownian Motion and Stochastic Calculus(2nd ed) Graduate Text in Mathematics New York N Y Springer-Verlag

Karatzas I and S E Shreve (1991b) Brownian Motion and Stochastic CalculusGraduate Texts in Mathematics New York NY Springer-Verlag

Karlin S and H M Taylor (1981) A Second Course in Stochastic ProcessesNew York NY Academic Press Inc

Madhavan A (2011) Exchange-Traded Funds Market Structure and the FlashCrash SSRN eLibrary Available at httpssrncompaper=1932925

Nelson D (1991 March) Conditional Heteroskedasticity in Asset Pricing ANew Approach Econommetrica 59(2) 347ndash370

Noehel T Z J Wang and J Zheng (2010) Side-by-Side Management of HedgeFunds and Mutual Funds Review of Financial Studies 23(6) 2342ndash2373

Oslashksendal B (2003) Stochastic Differential Equations An IntroductionWith Ap-plications (6th ed) Universitext New York Springer-Verlag

20

Shephard N (1996) Time Series Models in Econometrics Finance and OtherFields Chapter Statistical Aspects of ARCH and Stochastic Volatility pp 1ndash67 London Chapman amp Hall

Shiller R (1989) Comovement in Stock Prices and Comovement in DividendsJournal of Finance 44(2) 719ndash729

Shiller R J and A E Beltratti (1992) Stock prices and bond yields Can their co-movements be explained in terms of present value models Journal of MonetaryEconomics 30(1) 25 ndash 46

Sornette D and S Von der Becke (2011) Crashes and High Frequency TradingSSRN eLibrary Available at httpssrncompaper=1976249

Stein E M (1993) Harmonic Analysis Real Variable Methods Orthogonalityand Oscillatory Integrals Princeton NJ Princeton Univ Press

Tobin J (1984 Fall) A mean variaance approach to fundamental valuation Jour-nal of Portfolio Management 26ndash32

Whitten D (2011 August 17) Has High Frequency TradingFutures Derivatives etc Rigged The Game Against Individ-ual Investors Not Really iStockAnalystcom Available athttpwwwistockanalystcomfinancestory5364511has-high-frequency-trading-futures-derivatives-etc-rigged-the-game-against-individual-investors-not-really

Zhang F (2010) High-Frequency Trading Stock Volatility and Price DiscoverySSRN eLibrary Available at httpssrncompaper=1691679

Zhang N (2011 June) Properties of the SABR Model Working Pa-per Deprsquot of Mathematics Uppasala Uviv Available at http uudiva-portalorgsmashgetdiva2430537FULLTEXT01

21

Introduction

The Model

Trade strategy representation of alpha in single factor CAPM

High Frequency Trading Stock Price Formula

Applications

The impact of stochastic volatility on HFT stock prices

Implications for intraday return patterns and volatility

van der Corputs lemma and phase locked stock price

Conclusion

References


Abstract

In this paper we present a simple closed form stock price formula which captures empirical reg-ularities of high frequency trading (HFT) based on two factors (1) exposure to hedge factor and(2) hedge factor volatility Thus the parsimonious formula is not based on fundamental valuationFor instance it allows us to use exposure to and volatility of E-mini contracts to predict movementsin an underlying index For application we first show that for given exposure to hedge factor andsuitable specification of hedge factor volatility HFT stock price has a closed form double expo-nential representation There in periods of uncertainty if volatility is above historic average arelatively small short selling trade strategy is magnified exponentially and the stock price plum-mets when strategies are phased locked Second we demonstrate how asymmetric response tonews is incorporated in the stock price by and through an endogenous EGARCH type volatilityprocess and find that intraday returns have a U-shaped pattern inherited from HFT strategiesThird we show that the stock price is bounded from below (crash) ie flight to quality but notfrom above (bubble) ie confidence when phased locked trade strategies violate prerequisites ofvan der Corputrsquos Lemma Thus extant regulatory proposals to control price dynamics of selectstocks ie rdquolimit uplimit downrdquo bands over 5-minute rolling windows may mitigate but not stopfuture market crashes or price bubbles from manifesting in underlying indexes that exhibit HFTstock price dynamics

Keywords high frequency trading hedge factor volatility price reversal market crash price bub-bles fundamental valuation van der Corputrsquos Lemma

JEL Classification Codes C02 G11 G12 G13


Contents

1 Introduction 2




5 Conclusion 18

References 19

1

1 Introduction







2

trading is given by



)(11)



2X(tω))dt +ξ σ



S(tω) =


(int t

t0βX(uω)exp(1


(13)






3






4




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References


Contents

1 Introduction 2




5 Conclusion 18

References 19

1

1 Introduction







2

trading is given by



)(11)



2X(tω))dt +ξ σ



S(tω) =


(int t

t0βX(uω)exp(1


(13)






3






4




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References

1 Introduction







2

trading is given by



)(11)



2X(tω))dt +ξ σ



S(tω) =


(int t

t0βX(uω)exp(1


(13)






3






4




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References

trading is given by



)(11)



2X(tω))dt +ξ σ



S(tω) =


(int t

t0βX(uω)exp(1


(13)






3






4




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References






4




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References




2 The Model






5


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References


s Zs)minus1ZT


dγ(i)(tω) =

j

sumk=1

a(ik)(Zt)

[x

1minus t

]dtminus

j

sumk=1







dSi(t)Si(t)

minus rtdt =K

sumk=1

βik(t)(

dXi(t)Xi(t)



dk=K+1 σ2


6





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References





A(t) = Zγγγ(t) (22)



So that






7





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References





M(tω) = exp(int t

0

x1minus s

dB(sω)minusint t

0

(x

1minus s

)2

ds)

(29)



B(t) =minusint t

0

x1minus s

ds+B(t) and (211)



dγ(1)(t)dt

=minusdBbr(t)dt

= εt (213)


G(t) = (1minus t)B(

t1minus t

)(214)

EQ[G(s1)G(s2)] =



8


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References


B(t) = (1+ t)B(

t1+ t

)(216)


dγ(1)(t) =

1minus γ(1)


(1) = 0 (217)

M(t) =int t

0

dB(s)1minus s

(218)





α(t)dt =dS(t)S(t)


dX1(t)X1(t)


=dS(t)S(t)





9



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References



rArr dS(t)S(t)


= 0 (34)

σ = 1 (35)x


So that



dX1(t)X1(t)



dS(t)S(t)





10



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References



rArrint t

t0dln(S(tω))= σX

int t



t0σX︸︷︷︸

volatility



)(312)





)

4 Applications




11





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References





2X(tω))dt +ξ σ



dσ2X(tω) = ξ σ



(ξ

int t

t0dWσX (u)

)(43)



)(44)



t0exp(1


(45)


exp(int t

t0βX(uω)exp(1




12











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References











S(tω)dt =


S(t0)exp(int t


) (49)



rHFT (tω) =


exp(minus

2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)︸︷︷︸

past observations

)(410)


13




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References




2nminus1

sumj=0

σX(t(n)j ω)βX(t

(n)j ω)εX(t

(n)j ω)

)(412)











14


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References


S(tω) =


exp(

ρ

int t

t0βX(uω)exp(1









minus1k holds when

i k ge 2 or


15


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References


Let

φj(tω) =

int t

t0β

jX(uω)dBX(u) (418)


S(tω) = S(t0)exp(

iσXφj(t))

(419)


I j(σX) =int b

aexp(

σXφj(tω)

)dt (420)

I j(σX) =int b

aexp(

iσXφj(tω)

)dt (421)


φjprime(tω) = β

jX(tω)dBX(t) (422)

Let βj





kX (424)


16

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References

violated so that


kX (425)


N


minus1k

X (426)


|I j(σX)| ge I j(σX) (427)

rArrN

sumj=1|I j(σX)| ge

N


minus1k

X (428)

rArr IN(σX) =1N

N


minus1k

X (429)



X βNX






t0σXβ

jX(uω)dBX(uω)

)Let

IN(σX) =1N

N


minus1k

X


17



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References



5 Conclusion


18

References











19













20










21

Introduction

The Model



Applications




Conclusion

References

References











19













20










21

Introduction

The Model



Applications




Conclusion

References













20










21

Introduction

The Model



Applications




Conclusion

References










21

Introduction

The Model



Applications




Conclusion

References

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