A Modified Levy Jump-Diffusion Model Based on Market Sentiment Memory for Online Jump Prediction
AA Modified L´evy jump diffusion model Based on Market Sentiment Memory forOnline Jump Prediction
Zheqing Zhu, Jian-guo Liu, Lei Li
Duke UniversityDurham, North Carolina 27708
Abstract
In this paper, we propose a modified L´evy jump diffusionmodel with market sentiment memory for stock prices, wherethe market sentiment comes from data mining implementa-tion using Tweets on Twitter. We take the market sentimentprocess, which has memory, as the signal of L´evy jumps inthe stock price. An online learning and optimization algo-rithm with the Unscented Kalman filter (UKF) is then pro-posed to learn the memory and to predict possible pricejumps. Experiments show that the algorithm provides a rel-atively good performance in identifying asset return trends.
Introduction
Stock price is considered as one of the most attractive in-dex that people like to predict (Steele 2012; Geman 2002).An important model for the stock price that has been builtin the academia is L´evy process (Papapantoleon 2000;Cont and Tankov 2003; Ornthanalai 2014), which is a classof stochastic processes essentially show three features: a lin-ear drift, a Brownian motion and a compound Poisson pro-cess. This model gains some success but it is pure randomregarding the fluctuation.With the external information of market and macroeco-nomics, a purely random compound Poisson process doesnot accurately reflect the fluctuation of the financial assetin a constantly changing financial environment. The neces-sity of developing a model that incorporates external signalfrom the market is critical in improving accuracy in financialderivative pricing. One of the incentive for accurate pricingis to avoid financial crisis. In 2007-2008, part of the finan-cial crisis was caused by unforeseen drop in option prices.Researchers have tried to develop distributions other than anormal distribution for pricing noise (Borland 2002), but notmany have incorporated external signal for online learningand prediction.Instead of trying to develop a model which takes accuratenoise into consideration, we aim to develop in this paper amodified L´evy jump diffusion model with market sentimentmemory to follow volatility clustering of financial assets,and a UKF algorithm to predict possible price jumps on-line. We intend to address the jump-diffusion effect in the fi-nancial market with big-data and machine learning technol-ogy to exploit market sentiment from Twitter. Different from previous approaches in financial asset pricing, the model in-volves non-Markovian processes with exponentially decay-ing memory, which can then be transformed into Markovianprocesses with higher dimension.The main results of this paper include: • Incorporating market sentiment memory which hasmemory in financial asset pricing and developing a modifiedL´evy jump-diffusion model. We take the market sentimentmemory as the signal of L´evy jumps in the pricing model(see Equations (16) and (17)). • Developing an unscented Kalman filter algorithm thatactively learns market sentiment memory and accuratelypredicts asset trend accordingly (see Section
Market Senti-ment Memory UKF Optimization Algorithm ). • Capturing majority of big asset price movements (assetprice jumps) from market sentiment with relatively high ac-curacy. We found that outbreak of market sentiment indeedcan predict majority of price jumps (see, for example, Fig-ures 2 (a) and (c)).
Review of Current Work
Much academic effort to model the stock market has beendevoted to produce a better mathematical model based onLevy process and geometric Brownian model as founda-tions. Geman in 2002 used Levy process modeled with nor-mal inverse Gaussian model, generalized hyperbolic distri-butions, variance gamma model and CGMY process, whichreduces the complexity of underlying Levy measure, andproduced meaningful statistical estimation of stock prices(Geman 2002). Cheridito in 2001 proposed fractional ge-ometric Browniam motion model in order to gain a betterestimate of stock price (Cheridito 2001).Other academic efforts have been put in the field of senti-ment analysis and prediction based on market data and pub-lic views. Sul et al. in 2014 collected posts about firms inS&P 500 and analyzed their cumulative emotional valenceand compared the return of firms with positive sentimentwith other companies in S&P 500 and found significant cor-relation (Sul, Dennis, and Yuan 2014). Bollen et al. in 2011also produced similar result that Tweet sentiment and stockprice are strongly correlated in short term (Bollen, Mao, andZeng 2011). Zhang and Skiena in 2010 used stock and me-dia data to develop a automatic trading agent that was ableto long and short stock based on sentiment analysis on me- a r X i v : . [ q -f i n . S T ] S e p ia (Twitter and news platforms) data and getting a return aswell as a high Sharpe ratio (Zhang and Skiena 2010). Apartfrom pure sentiment analysis and trading strategies, Vincentand Armstrong in 2010 introduced prediction mechanismsalso based on Twitter data to alert investors of breaking-point event, such as an upcoming recession (Vincent andArmstrong 2010).In this paper, we propose a modified L´evy jump diffu-sion model where we replace the standard compound Pois-son component with a process determined by an exponen-tially decaying market sentiment memory, while the latteris extracted by UKF. UKF is able to take into account thenon-linear transformation between market sentiment and as-set return compared with the linear Kalman filter in (Duanand Simonato 1999), where Duan et al. proposed using lin-ear Kalman filter to model exponential term structures in fi-nancial and economic systems. Background and preliminaries
In this section, we give a brief introduction to the L´evyjump-diffusion model for the price movement of assets andunscented Kalman filter. These are the building blocks forour modified L´evy jump-diffusion model with market senti-ment memory and its algorithm.
L´evy Jump-Diffusion Model
In many theories such as Black-Scholes Model, the pricemovement of financial assets are modeled by the stochas-tic differential equation (SDE) driven by Brownian motion((Steele 2012, Chap. 10)) dP ( t ) = ˜ µP ( t ) dt + σP ( t ) dB ( t ) , where B ( t ) is a standard Brownian motion. The solution ofthis SDE is known to be the geometric Brownian motion P ( t ) = P (0) e µt + σB t , µ = ˜ µ − σ .µ is called the drift factor of the geometric Brownian motionand represents the log annual return of the financial asset. σ represents the volatility of daily return of the asset. Aninteresting fact is that µ does not affect option pricing in theBlack-Scholes model (Steele 2012; Ross 2011). However, if ˜ µ equals the interest rate r , this geometric Brownian motionbecomes risk-neutral and gives the Black-Scholes formulafor the no-arbitrage cost of a call option directly (Ross 2011,Chap. 7).One of the drawbacks of the geometric Brownian motionis that the possibility of discontinuous price jumps is not al-lowed (Cont and Tankov 2003; Ornthanalai 2014). The L´evyJump-Diffusion Model is one that tries to resolve this issue((Ross 2011, Chap. 8), (Cont and Tankov 2003)): P ( t ) = P (0) e µt + σB t + (cid:80) ti =1 ( (cid:80) Nij =1 J ij − λκ ) , (1)where J ij ∼ N ( κ, σ J ) i.i.d. are the jump parameters. N i ∼ P oisson ( λ ) i.i.d. are the Poisson parameters that control theoccurance of the jump. κ, σ J and κ are constant parameters and Z, N i , J ij are assumed to be mutually independent. Thelog return for day t is given by: r ( t ) = ln( P ( t )) − ln( P ( t − µ + Z + ( N t (cid:88) j =1 J tj − λκ ) . (2)The L´evy jump-diffusion model consists of three compo-nents. The first part, µ is the expected logarithmic return ofthe financial asset. The second component, Z is the whitenoise of the price or logarithmic return that is unpredictable.The third component, (cid:80) N i j =1 J ij − λκ , is a compound Pois-son distribution that provides a jump signal and a jump mag-nitude. The standard L´evy jump diffusion model intends toinclude the fat tails that have been generally observed inasset returns in addition to the Gaussian structure that iscommonly assumed. The downside of L´evy jump diffusionmodel is that, although the model works well in a long-termstructure, it fails to recognize market specific informationthat can be extracted from public opinion if used to predictshort-term asset return. Unscented Kalman Filter
In this section, we give a brief introduction to UKF ((Julierand Uhlmann 1997; Wan and Van Der Merwe 2000)), whichis used to give an accurate estimate of the state of nonlineardiscrete dynamic system. Suppose that the state of a systemevolves according to x ( t ) = f t ( x ( t − , u ( t )) + b ( t ) , (3)where x ( t ) represents the state of the system at time t , u ( t ) is the external input and b ( t ) is a noise with mean zero andcovariance Q t . f t is the known model of dynamics for x ( t ) .A measurement z ( t ) is then made to x ( t ) : z ( t ) = h ( x ( t )) + d ( t ) , (4)where d ( t ) is the measurement noise with mean zero and co-variance R t , independent of b ( t ) . h is the known measure-ment function.Suppose that ˆ x ( t − is the belief of the state x at time t − and the covariance matrix of ˆ x ( t − is P t − . Thegeneral process of Kalman filter is given as follows: • Predict: Using the belief ˆ x ( t − , P t − , we obtain a pre-dict (prior belief) of the state ¯ x ( t ) . Denote ¯ P t the covari-ance matrix of x ( t ) − ¯ x ( t ) (current prior belief in processcovariance matrix). • Update: Using the prediction ¯ x ( t ) , we have the predictedmeasurement µ z . When we have the observation z ( t ) , weupdate the belief of the state at t as ˆ x ( t ) = ¯ x ( t ) + K t ( z ( t ) − µ z ) .K t is the Kalman gain matrix computed as follows: firstof all, we compute the approximation of the covariancematrix of the residue z ( t ) − µ z as P z , and the covariancematrix between x ( t ) − ¯ x ( t ) and z ( t ) − µ z as P xz . Then,the Kalman gain matrix is given as K t = P xz P − z . he intuition is that this is the ration between belief instate and belief in measurement. The covariance matrixof ˆ x ( t ) is then computed as P t = ¯ P t − K t P z K (cid:62) t . In the case f t and h are nonlinear, it is usually hard tocompute the mean and covariance of ¯ x ( t ) and ˆ x ( t ) . The un-scented transform computes these statistics as following: (i)generates n + 1 ( n ∈ N + ) (deterministic) sigma points us-ing ˆ x ( t − , P t − and Q t , with certain weights w mi and w ci .(ii) Evolve these sigma points under f t , and obtain n + 1 data Y i . Then, the statistics are approximated using these n + 1 data. The UKF makes use of the unscented trans-form to approximate ¯ x ( t ) , ¯ P t , P z and P xz to second orderaccuracy. Hence, the cycle of UKF can be summarized asfollowing:1. Predict the next state from the posterior belief in thelast step: (UKF.predict()) X = Σ(ˆ x, P ) ,Y i = f t ( X i , u ( t )) , i = 0 , . . . , n, ¯ x = n (cid:88) i =0 w mi Y i ¯ P = n (cid:88) i =0 w ci ( Y i − ¯ x )( Y i − ¯ x ) (cid:62) + Q, (5)Here, Σ is the algorithm to generate sigma points from theposterior belief ˆ x ( t − . The standard algorithm is Vander Merwe’s scaled sigma point algorithm (Van Der Merwe2004), which, with a small number of sampling with corre-sponding weights w mi and w ci , gives a good performance inbelief state representation. See (Van Der Merwe 2004) forthe formulas of w mi and w ci .2. Update from prior belief to posterior belief accordingto current noisy measurement. (UKF.update()) Z i = h ( Y i ) , i = 0 , , . . . , n,µ z = n (cid:88) i =0 w mi Z i P z = n (cid:88) i =0 w ci ( Z − µ z )( Z − µ z ) (cid:62) + R t K = [ n (cid:88) i =0 w ci ( Y − ¯ x )( Z − µ z ) (cid:62) ] P − z ˆ x = ¯ x + K ( z − µ z ) P = ¯ P − KP z K T . (6)Refer to (Julier and Uhlmann 1997) for the details of imple-mentation of a UKF. An important aspect of UKF that canbe utilized in estimating jump-diffusion process is its Gaus-sian belief. We can regard each Gaussian component in thecompound Poisson distribution in jump-diffusion as a statebelief of UKF and apply to UKF to obtain a stable transitionfunction with current sentiment data as input. Market sentiment
VaderSentiment
We aim to extract information from Twitter for both idiosyn-cratic sentiment information and market sentiment informa-tion. Before we present the implementation of data min-ing and analytics, we introduce VaderSentiment (Hutto andGilbert 2014). For a given sentence s , VaderSentiment pro-duces, vader ( s ) = [ positive, neutral, negative, compound ] , (7)where positive , neutral and negative are respective sig-nals calculated and compound is an overall evaluation ofthe quantitative sentiment of the sentence. Each of these sig-nals are bounded within [0 , . Since we would like to extractinstability of the sentiment analysis, we take vader’s neutralas the noise and the confidence level is defined by the valueof − neutral . For each input sentence s , we extract: sentiment ( s ) = vader ( s ) .compoundnoise ( s ) = vader ( s ) .neutral (8)For a single day t , we have, S ( t ) = (cid:88) s ∈ Set ( t ) (1 − noise ( s )) ∗ sentiment ( s ) ,E ( t ) = (cid:80) s ∈ Set ( t ) noise ( s ) | Set ( t ) | . (9)See Figure 1. At day t, from Twitter, we extract dailytweets from two pairs of keyword inputs, asset name (e.g.MSFT for Microsoft) and its trading market (e.g. NAS-DAQ) for idiosyncratic comments and trading sector (e.g.tech for Microsoft) and its trading market for market-relatedcomments. For each day, we calculate idiosyncratic senti-ment S I ( t ) , macroeconomic sentiment S M ( t ) , and the cor-responding E I ( t ) and E M ( t ) from day t’s Tweets using (9).We then feed them into UKF sentiment memory optimiza-tion algorithm (which will be introduced in Section ) tolearn the exponentially decaying memory model and thenpredict. Market Sentiment Memory
We now consider adding the memory process m ( t ) of a ran-dom variable ξ ( t ) into the model: m ( t ) = (cid:90) t −∞ γ ( t − s ) ξ ( s ) ds + (cid:90) t −∞ γ ( t − s ) dB ( s ) , (10)where γ and γ are memory kernels and the second term rep-resents the white noise. In this work, we assume throughoutthat the effect of white noise is negligible so that γ = 0 .In general, the kernel could be completely monotone. Bythe Bernstein theorem, any completely monotone functionis the superposition of exponentials (Widder 1941). For ap-proximation, we can consider finite of them: γ ( t ) = N (cid:88) i =1 a i exp( − λ i t ) . (11)igure 1: Flowchart for Return-Sentiment Memory Kernel learning system.The advantage of these exponential kernels is that we candecompose m as m ( t ) = (cid:90) t −∞ ( N (cid:88) i =1 a i exp( − λ i ( t − s ))) ξ ( s ) ds ⇒ m = N (cid:88) i =1 m i , so that each m i satisfies the following SDE driven by ξ : dm i = − λ i m i dt + a i ξ ( t ) dt. (12)In this way, the non-Markovian memory process is embed-ded into Markovian processes with higher dimension.In this work, for simplicity, we just assume N = 1 (i.e. thememory kernel is a single exponential mode) and considerthe memories (denoted as η I , η M ) of two individual sen-timent inputs, idiosyncratic sentiment ( S I ( t ) ) and macroe-conomic sentiment ( S M ( t ) ) so that the two sentiment pro-cesses are given by the discretized SDE (12): η I ( t ) = p I η I ( t −
1) + a I S I ( t ) ,η M ( t ) = p M η M ( t −
1) + a M S M ( t ) , (13)where p ∈ [0 , is called unit decay factor, and a > iscalled the inclusion factor. We define the market sentimentmemory process η ( t ) as a linear combination of two compo-nents η I and η M with η ( t ) = c I η I ( t ) + c M η M ( t ) . (14)For algorithmic development purpose, we impose p + a = 1 , (15)to limit the search space. Note that (15) is not a constraintbecause we later we care κη only. Enforcing p + a = 1 onlyselects a scaling for κ . Model and Algorithm
In this section we present a modified L´evy jump model forthe asset. A UKF is then used to predict jump magnitude onthe next day using computed market sentiment.
Modified L´evy Jump Diffusion Model
Recall the logarithmic return in L´evy jump diffusion (2). Aswe state in Section , we would like to incorporate externalinformation from social media to extract market sentiment,which contributes to asset return movements. Here we definethe modified L´evy jump diffusion model, r ( t ) = ln( P ( t )) − ln( P ( t − µ + Z + ( M − ν ) , (16)where M is the jump amplitude random variable. ν is a con-stant to take off the drift trend in M ( t ) (equivalent to λκ inthe L´evy jump diffusion model) and we compute ν in ad-vance using history data.We assume that the jump magnitude is determined by thetotal memory effect of market sentiment, or the market sen-timent process η in (14). In particular, we assume: M ( t ) = κ ( t ) η ( t ) . (17)which indicates that a current jump magnitude is determinedby the sentiment memory. An implication of this setting isthat market sentiment value from an individual day is a kindof volatile velocity to the return of an asset. We assume that κ evolves with momentum so that it satisfies the order 1 au-tocorrelation model (AR(1)): κ ( t ) = φκ ( t −
1) + g + (cid:15) t , (18)with g being a constant for the innovation and (cid:15) t being adiscrete white noise. Market Sentiment Memory UKF OptimizationAlgorithm
In this section, we introduce a UKF optimization algorithm.In the algorithm, the drift µ is determined in advance, whichis the daily return in the long history, we set g = 1 in (18),and preset c ’s in Equation (14) for each iteration. The al-gorithm is used to find the optimal p , a and φ defined inEquation (13) and (18) with in-sample data .State of the system is represented as x ( t ) = r ( t ) κ ( t ) η ( t ) η I ( t ) η M ( t ) , nd the input vector is u ( t ) = (cid:20) S I ( t ) S M ( t ) (cid:21) . The dynamics of the system f t (see (3)) is given by: f ( x ( t ) , u ( t + 1); Λ) = µ + Z + κ ( t ) η ( t ) − νφκ ( t ) + gc I η I ( t ) + c M η M ( t ) p I η I ( t ) + a I S I ( t + 1) p M η M ( t ) + a M S M ( t + 1) , (19)with Λ = [ φ, a I , a M , p I , p M ] being the parameters.We define true return on day r ∗ ( t ) and κ ∗ ( t ) = ( r ∗ ( t ) − µ + ν ) /η ( t ) as the measurements of r ( t ) and κ ( t ) : h ( x ) = [ r ∗ ( t ) , κ ∗ ( t )] . (20)Motivated by the fact that S I , S M are random, we assumethe measurement noise variance R is a combination of thatfor S I and S M : R = a I E I ( t ) + a M E M ( t ) , where E I and E M are confidence level of sentiment valuescomputed by the second equation in (9). The randomness in S I and S M provides noise for the evolution f and h .We introduce here Jenson’s alpha and beta market risk toset c I and c M in (14). Beta market risk is defined as (Jensen1968): β ( t ) = cov( r ( t ) , r M ( t ))var( r M ( t )) , (21)where r M is the market log return. Jenson’s alpha is: α ( t ) = r i ( t ) − [ r f ( t ) + β ( r M ( t ) − r f ( t ))] , (22)where r i is individual return and r f is risk free rate ( r M , r i , r f are all computable from current data). Using Jenson’s α risk, we set c I ( t ) = α ( t ) /r ( t ) ,c M ( t ) = 1 − c I , (23)for computing η ( t ) .We define the objective function in the optimization. U ( JI UKF , JI act )= | JI posUKF (cid:84) JI posact | + | JI negUKF (cid:84) JI negact | T − | JI posUKF \ JI posact | + | JI negUKF \ JI negact | T . (24)In the formula, JI = { J t : | r ( t ) − µ | > . σ } represents the set of jumps 1.96 standard deviations awayfrom the process mean, or 1.96 volatility from the drift fac-tor. JI pos indicates positive jumps and JI neg indicates neg-ative jumps. What we aim to achieve here is that jumps iden-tified by UKF overlaps the most with the actual jumps thathappens. The major goal of UKF Optimization is essentiallytrying to identify a trend in the asset return time series.With the settings above, we present the UKF optimizationalgorithm for searching optimal p , a and φ , with restrictionthat p + a = 1 and φ ∈ (0 , . UKF Optimize ( c o e f e r r , i d i o s e n t [ ] ,m a r k s e n t [ ] , r e t [ ] ) :I n i t i a l i z e x , P , Q, Rp I , p M = 0O p t i m a l = [ a , p , 0 ]f o r p I i n 0 . . 1 s t e p c o e f e r r :f o r p M i n 0 . . 1 s t e p c o e f e r r :f o r p h i i n 0 . . 1 s t e p c o e f e r r :f o r t i n l e n ( r e t [ ] ) :UKF . p r e d i c t ( x , P ,f ( p I , p M , p h i ) )UKF . R = [(1 − p I ) ˆ 2 ∗ e r r o r I ˆ 2 ( S I ( t ) )+ (1 − p M ) ˆ 2 ∗ e r r o r M ˆ 2 ( S M ( t ) ) ]UKF . u p d a t e (R( t + 1 ) )u = U( JI UKF , J I a c t )U p d a t e O p t i m a l ( a , p , p h i , u )r e t u r n O p t i m a l . a , O p t i m a l . p , O p t i m a l . p h iwhere U is the objective function of UKF-Optimization al-gorithm defined in (24). UpdateOptimal(a , p , φ, u) meansif u is bigger than the old u , we update (a , p , φ ) to the newparameter and keep the old values otherwise.We first use UKF Optimize with in-sample data to findthe optimal p I , p M and φ for the maximum coverage on theactual jumps. After the optimal parameters are obtained, weuse UKF predict the Stoke price using Model (16) online.The UKF is generally used for state transition learningwhere the transition rules and noises are relatively stable.One reason is that during a near-stationary process, statebelief is generally strengthened such that state transitionconverges. The Kalman gain factor, due to a strong beliefin state, with very small covariance, quickly approaches 0.Consequently, UKF has learned pattern of state transitionand is only mildly adjusted by input. In our case, the eco-nomic process has different trends in different time windowswhile the UKF is hardly used to model a non-stationary pro-cess. A critical idea of learning the non-stationary economicmodel using UKF in our model is that, we would not want tomodel observations from the market as a sensor with fixedvolatility. The volatility clustering effect of asset returns cangreatly impact the training result. Here we model the volatil-ity clustering effect with sentiment error term. The signif-icance of this algorithm is that with a small modification,UKF can be used to learn multiple exponentially decayingsentiment memory with guaranteed process covariance con-vergence performance even given a chaotic non-linear sys-tem(Feng, Fan, and Chi 2007) with a quadratic time com-plexity over the standard UKF by searching the coefficientspace with some acceptable coefficient error. Note that theoutput of UKF is not a strictly exponentially decaying mem-ory due to its non-pre-deterministic Kalman gain parameter. Experiment
We now present the experimental results for Facebook (FB),Microsoft (MSFT) and Twitter (TWTR).Figure 2 (a) and 2 (b) represent the actual return and theUKF return prediction based the modified L´evy jump diffu- a ) ( b ) ( c ) Figure 2: Experimental results for FB ( a ) ( b ) ( c ) Figure 3: Enlarged plots for FB ( a ) ( b ) ( c ) Figure 4: Experimental results for MSFTsion model for FB during the time period from 2016-02-03to 2017-02-02 (note that there are only trading days ina year). The parameters p, a, φ are trained by the UKF opti-mization algorithm using date from 2013-02-02 to 2016-02-02. The jump prediction precision is . . The in-sampleprediction precision for time period 2013-02-02 to 2016-02-02 is . . Figure 2 (c) shows η ( t − (we have offset inthe memory plot because we use η ( t − to do the predictionfor day t ) .The spikes in η indicate outbreaks of market sentiment. Tosee how these spikes affect the jump prediction, we zoom inthe plots for FB from Day to Day in Figure 3. Thereare evident spikes in η ( t − for t = 30 , , , . For t = 30 and t = 47 , the real stock price curve has abnor-mal jumps, and our prediction of jumps based on the seti-ment memory has accurately predicted them. There is bigoutbreak of sentiment for t = 39 , and we can see that thereal stock price goes down on Day and . This indicatesthat the jumps in stock price curve are strongly correlated tomarket sentiment memory process and our model is able topredict a significant amount of abnormal jumps.Figure 4 (a) and 4 (b) represent the actual return andthe UKF return prediction for MSFT during the time pe-riod from 2016-02-02 to 2017-02-02, with training data from2010-02-02 to 2016-02-02. The jump prediction precision is . . The in-sample prediction precision for time period2010-02-02 to 2016-02-02 is . . Figure 4 (c) shows themarket sentiment memory process η ( t − (Eq. (14)) .Figure 5 (a) and 5 (b) represent the actual return and the a ) ( b ) ( c ) Figure 5: Experimental results for TWTRUKF return prediction for TWTR during the time periodfrom 2016-02-03 to 2017-02-02, with the training data from2014-02-02 to 2016-02-02. The jump prediction precision is . . The in-sample prediction precision for time period2014-02-02 to 2016-02-02 is . . Figure 5 (c) shows thememory process η ( t − .There are a few significant observations we can draw fromthe results.1. Using UKF generally captures the movement trend ofthe underlying asset, with little guidance with daily returns.More specifically, during periods where sentiment memorykernel value peaks, the stock asset’s return has very strongcorrespondence. However, when there is very minor value inmarket sentiment kernel, the asset return prediction followsthe previous trading day’s return, triggering some inaccu-racy.2. Movements in UKF return prediction are in generalgreater in magnitude than actual returns. This could becaused by high volatility of sentiment values.3. From the sentiment memory graphs (Figures 4- 5(c)), we can observe a strong indication of clustering,which is an evidence of a decaying memory, analogous toGARCH model (Bollerslev 1986) which measures volatilityclustering. This can also be confirmed by trained parametersfrom UKF-optimization algorithm:MSFT: p I = 0 . , p M = 0 . , φ = 0 . .FB: p I = 0 . , p M = 0 . , φ = 0 . .TWTR: p I = 0 . , p M = 0 . , φ = 0 . . Discussion
In this paper, we propose a modified L´evy jump diffusionmodel with market sentiment memory for stock prices. Anonline learning and optimization algorithm with UKF isused to predict possible price jumps. The result from the ex-periments instantiate our theory in market sentiment mem-ory and its impact on asset returns. Our work has signifi-cance in both economics and computer science.Regarding economics, our experiments have shown theexistence of predictability in return by sentiment, which in-dicates market inefficiency in digesting public sentiment.The impact of market sentiment memory on asset returns candramatically change the pricing models for options and fi-nancial derivatives because currently most of these productsrely on the Markovian assumption about financial assets. Toincorporate market sentiment memory into the pricing mod-els, one possible way is to multiplying previous jumps oc-curring in asset’s return history with decaying factors andthen add the models, since jumps are strong indicators ofmarket sentiment outbreak. Another possible way is to in-clude a time series of market sentiment with explicit valuesinto asset pricing models. Clearly, our model adopts the sec-ond strategy.Regarding computer science, our work indicates thatKalman filter techniques (especially UKF) allow onlinelearning for non-observable variables. The market sentimentmemory can not be measured directly and it is an indirectvariable, however, unlike other machine learning techniques,UKF allows online learning of such indirect variables in aniterative manner.
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