Theoretical Model of Stock Trading Behavior with Biases

Transcription

1 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. Theoreical Model of Sock Trading Behavior wih Biases Ke Liu, Kin Keung Lai, Jerome Yen Absrac Sock invesors are no fully raional during heir rading, and many behavioral biases affec heir rading behavior, such as represenaive bias and disposiion effec. However, mos of he lieraure on behavioral finance cas effors on explaining empirical phenomena observed in financial markes, bu lile on how individual invesors rading performance is affeced by heir behavioral biases. As agains he common percepion ha behavioral biases are always derimenal o invesmen performance, we conjecure ha hese biases can someimes yield beer rading oucomes for invesors. Focusing on represenaive bias and disposiion effec, we consruc a mahemaical model in which he represenaive invesor follows a Bayesian rading sraegy based on an underlying Markov chain, swiching beween Trending regime and Mean-reversion regime. By his model, we are able o underake scenario analysis o rack invesor behavior and performance along he ime, under differen paerns of marke movemens. Resuls validae our conjecure by showing ha he effec of behavioral biases can someimes be posiive on invesor performance. Index Terms Represenaive Bias, Disposiion Effec, Bayesian Invesor, Trading Behavior I. INTRODUCTION Alhough EMH (Efficien Marke Hypohesis, by Fama [] sounds heoreically beauiful and serves as he foundaion for mos modern financial models, many empirical researches in finance have widely found evidence of anomalies agains EMH [2][3]. This incompeence of modern finance wih explaining many observed phenomena is due o he building blocks of i, i.e. expeced uiliy heory and absolue arbirage assumpion. Thus, raher han reaing agens as raional in modern finance, behavioral finance argues ha people can make sysemaic errors when making decisions because of misaken beliefs and psychological biases. Research in his field is mainly cas in wo direcions. The firs direcion is o discover hose behavioral biases by psychological experimens, or applying cogniive psychology and heories on behavioral biases for he explanaions of empirical phenomena observed in financial markes, especially hose anomalies modern finance fails o expound [4][5]. However, in spie of he grea explanaory Ke Liu is wih he Deparmen of Managemen Sciences, College of Business, Ciy Universiy of Hong Kong, Ta Chee Avenue, Kowloon, Hong Kong. (corresponding auhor, phone: ; msliuke@ciyu.edu.hk K.K. Lai is chair professor in he Deparmen of Managemen Sciences, Ciy Universiy of Hong Kong. ( mskklai@ciyu.edu.hk Prof. Jerome Yen is wih he Deparmen of Finance and Economics, Tung Wah College, Wylie Road, Kowloon, Hong Kong. ( risksoluion@gmail.com ISBN: ISSN: (Prin; ISSN: (Online power his line of research offers, conradicions among various biases ogeher wih he qualiaive and descripive naure of i cause large difficulies, when applying hese findings o pracical problems solving ex ane. Therefore, he furher developmen of behavioral finance is growingly demanding he second direcion of research: modeling of behaviors quaniaively and precisely. Though, owing o he uncerain and equivocal propery of human behaviors, exised lieraure in his line ofen consruc models wih weigh oo simplified assumpions and largely differ from realiy. For example, many simply use he price difference o deermine he rading amoun of shares, in order for modeling momenum raders. Obviously, momenum raders can no be rading in so simple a sraegy. In his ligh, his paper assumes he represenaive invesor following a Bayesian rading sraegy, whereas he invesor believes ha marke sae can swich beween rending and mean-reversion, as modeled by an underlying Markov chain. By his model, we are able o underake scenario analysis o rack he invesor s behavior and performance along he ime, under each ypical paern of marke movemens. The main purpose of his paper is o see wheher he effec of behavioral biases can someimes be posiive on invesor s performance. The res of his paper is organized as follows: Secion 2 elaboraes in deails he model consrucion and formulaion; In Secion 3 we simulae differen ypical marke scenarios o invesigae he invesor s behavior and performance under each marke paern, and he impac of disposiion effec; Finally, Secion 4 concludes. II. MODELING OF INVESTOR TRADING BEHAVIOR A. Model Raionale Represenaive bias can cause a wo-edged effec, i.e. on one hand i may yield invesors habi of exrapolaive expecaion (assuming he previous price rend paern o coninue in fuure price movemens and hen chasing he rend; while on he oher hand i may foser invesors belief in mean-reversion. In addiion, he ineresing experimen designed by Andreassen and Kraus [6] provides sriking evidence ha, subjecs can swich heir rading behavior o rend chasing from racking an average price level (i.e. sell in price rise and buy in price fall, as significan changes in price have occurred over some periods. In such sense, we direc our effor o a modeling framework in which invesor s rend chasing behavior ogeher wih mean-reversion belief can be reconciled. Forunaely, he basic concep used o model invesor s response o company s earnings, in he WCE 200

2 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. eminen BSV model [7], can also be employed for our aim o model invesor s response o price evoluion. We assume ha here is a caegory of invesors, whose rading sraegy is quie reasonable and sophisicaed, and seems o represen he realiy beer. They are aware of he imporance of recognizing he marke regime firs. Obviously, he marke will no always walk in rends, nor will i always rever o mean swifly once i drifed away. Thus hey se up a Markov chain governing marke regime swiching in heir mind. For he sake of formulaion expedience, hey are modeled as a single represenaive invesor. This invesor follows a Bayesian sraegy o updae his/her belief on marke using new price evidences coninually along he ime. The invesor s belief can swich beween Trend-chasing and Mean-revering. If he/she believes he marke moving currenly in Regime (Trend-chasing, he/she would probably buy afer observing an uprise of price; whereas if in Regime 2 (Mean-reversion, he/she buys afer a marke fall because he/she believes he marke would much likely o rever o normal. Here, one may argue ha, he invesor represened in his model only makes use of marke informaion (price evoluion o deermine his/her rading behaviors, while in pracice invesors may rely on oher fundamenal or economic informaion for rading decisions. However, his paper aims no a modeling all kinds of invesors ogeher, which may become a remendously arduous endeavor. Indeed, he Bayesian invesor in our model can be viewed as represenaive of chariss, echnical raders and all hose who use hisorical marke informaion o predic he marke. This ype of raders occupies a large share of all marke paricipans, and heir performance under psychological biases is of considerable ineres. Addiionally, in many sock markes especially of hose emerging markes, mos individual invesors end o rade jus by waching paerns of marke movemens, irrespecive of oher ypes of informaion. B. Model Deducion Of he wo-sae Markov chain in he invesor s belief, he ransiional probabiliy marix TP is: TP ( , Where, λ is he believed probabiliy of marke remaining in Regime, and λ 22 is he believed probabiliy of marke remaining in Regime 2. Price change a ime Z is defined as below, and X is he marke price. z X X (2 One main concern of he invesor is he sign of Z which indicaes he price movemen direcion. 0 price up z 0 price unchanged 0 price down ISBN: ISSN: (Prin; ISSN: (Online The poserior probabiliy of Regime given curren price informaion a ime is defined as: q Pr( S z, q (3 Then, following he probabiliy ransiion along he wo-regime Markov chain, he prior probabiliy of Regime esimaed a previous ime period is: Pr( S z, q q ( 22 ( q (4 Obviously, he prior probabiliy of Regime 2 esimaed a previous ime is one minus prior probabiliy of Regime 2. According o he Bayes Theorem, we can updae he invesor s belief on he curren sae (a ime + being Regime, using he new price evidence Z +, as formulaed below: q Pr( z S, z Pr( S z, q / [Pr( z S, z Pr( S z, q (5 Pr( z S 2, z Pr( S 2 z, q ] The wo marke regimes can be seen as hidden saes, whereas wha he invesor can observe is price movemens along he ime and his measurable evidence is used o updae belief on he hidden saes. If he marke currenly dwells in Regime, hen he probabiliy of price momenum (coninuaion denoed as λ T can be defined as below: Pr( z 0 S, z 0 T (6 Pr( z 0 S, z 0 T Similarly, he probabiliy of price mean-reversion given marke dwelling in Regime 2 is defined as below: Pr( z 0 S 2, z 0 M (7 Pr( z 0 S 2, z T, 0.5 M Then, based on he esimaed prior probabiliy of Regime in nex period Pr(S + = z, q, he Bayesian invesor forms in his/her belief he probabiliy for winessing a rise or fall in nex period, given curren price informaion. The four siuaions are all considered and corresponded in (8-(. Pr( z 0 z 0, q Pr( z 0 S, z 0, q Pr( S z, q Pr( z 0 S 2, z 0, q Pr( S 2 z, q (8 T ( q ( 22 ( q ( M ( q ( 22 ( q Pr( z 0 z 0, q Pr( z 0 z 0, q (9 Pr( z 0 z 0, q Pr( z 0 S, z 0, q Pr( S z, q Pr( z 0 S 2, z 0, q Pr( S 2 z, q (0 ( T ( q ( 22 ( q M ( q ( 22 ( q Pr( z 0 z 0, q Pr( z 0 z 0, q ( Under he condiion ha no shor-sell is allowed, he Bayesian invesor s rading behavior is assumed as follows: Based on expeced possibiliy of experiencing a price rise a M WCE 200

3 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. nex period (i.e. p +, currenly he invesor would make a decision of his/her posiion (denoed as H depiced in (3. p Pr( z 0 z, q (2 H A max{ p ( p, 0} (3 Amax{2 p, 0} Where, A is a consan used o link posiion holding decision and expeced probabiliy of price rise. Then, rading behavior in each period can be deermined from he difference beween posiions held by he invesor in wo adjoining periods. Equaion (4 & (5 depic he formulas calculaing buy value and sell value in each period, respecively. B V max{ H H, 0} (4 S V max{ H H, 0} (5 The formulaion derived above for he Bayesian invesor s rading behavior doesn accoun for he disposiion effec (abbr. DE ye. Therefore, in wha follows, sell behavior influenced by he behavior bias is modeled. We assume ha he sell value, should any, is impaced by a coefficien B, which is furher deermined by he DE coefficien denoed as D herein. Then, he invesor s posiion a ime can be adjused as (6 shows. From (7, he effec on sell value produced by he invesor s loss aversion can be quanified by he DE coefficien D. DE DE B S H max{ H V B V, 0} (6 ( D while earning B breakeven (7 D while losing A. Scenario Seing III. SCENARIO ANALYSIS In his secion, how he Bayesian invesor would behave and perform under four differen possible marke scenarios, in erms of he process of evoluion of price, is discussed. Fig. (a illusraes hree basic paerns of he price evoluion process, i.e. linear uprend (Scenario, downrend (Scenario 2, and single-cycle oscillaion (Scenario 3, where price firs rises and hen falls back. Subsequenly, in case of price oscillaion, one may wonder wheher he frequency of oscillaion maers in deermining he invesor s rading behavior and performance. For his, four sub-scenarios (Scenario 4 are designed o invesigae possible invesor behaviors under differen raes of oscillaion of he marke, as shown in Fig. (b. The purpose is o analyze he Bayesian invesor s beliefs and behaviors under each scenario from hree angles, i.e. believed probabiliy of a price rise, posiion held by he invesor, and cumulaive profi. Also, he disposiion effec is inensively sudied. Then, we are ineresed in how he invesor would perform, given differen ses of characerisic parameers of he invesor. B. Summary of Resuls To summarize he resuls obained in he scenario analysis, Table liss hree oucomes concerning he invesor s performance under each price scenario. The invesor realizes ISBN: ISSN: (Prin; ISSN: (Online neiher profi nor loss in he marke of downrend, because he/she never engages in any rading behavior during he course. However, he/she would profi from marke siuaion of uprend and rise-fall oscillaion, since he/she can gradually recognize he upward rend or he downrend in price oscillaion. When rading in flucuaing price process wih differen oscillaion frequency, eiher gain or loss can occur o he invesor. Wih a sufficienly swif oscillaion of sock price, he invesor would assume he marke as evolving in mean-reversion regime, hus will ake advanage of shor price movemens by perfecly buying low and selling high. Neverheless, as he price flucuaes repeaedly in much slower manner, he invesor will be involved in a dilemma in which he/she has grea difficuly in making choice beween rend following sraegy and mean-reversion arbirage. By his way, he/she may be ricked by he marke iself and enailed large loss. As he price oscillaion frequency coninues o decrease o a sufficienly low level, however, he invesor can again realize gains wih his/her Bayesian rading sraegy. Acually, he single-cycle oscillaion in Scenario 3 can be seen as a special case of muli-cycle oscillaion in Scenario 4 when price flucuaion frequency is as low enough as 2 periods per cycle. Therefore, eiher sufficienly low frequency or high frequency of marke flucuaion can render he invesor beer performance, while he invesor would suffer from loss in mids of a range of marke cycling speed. Sock Price / Dollars Sock Price / Dollars Figure (a. Basic Scenarios (-3 of Sock Price Scenario : Uprend Figure (a. Basic Scenarios (-3 of Sock Price Scenario 2: Downrend WCE 200

4 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. Sock Price / Dollars Figure (a. Basic Scenarios (-3 of Sock Price Scenario 3: Single-Cycle Oscillaion fully realizes i while making profis. In his siuaion, DE should conribue o nice rading by making he invesor more conservaive, as he/she liquidaes his/her posiion more decisively when he/she performs sell-high. However, he oucome for he invesor is no improved by DE herein because wih a zero DE he invesor would also empy his/her posiion a higher price. In a marke which oscillaes in he way ha ricks he Bayesian invesor around, as in Scenario 4.2, he loss leads o he invesor s relucance o sell via he impac of DE, hus he invesor sill keeps a posiion ha will parly capure shor uprend, alhough he/she fails o envision i. However, when he shor downrend lass more periods, as in Scenario 4.3, he relucance o liquidae posiion as caused by DE would incur more loss because he invesor fails o imely escape from he downward marke. As he periods for each uprend and downrend increase o sufficien level (as in Scenario 4.4, given a rading profi, he invesor s conservaism caused by DE would impel his/her o more promply wihdraw his/her money from he downward marke. Sock Price / Dollars Figure (b. Scenario 4 Scenario 4.: Two Periods per Cycle Sock Price / Dollars Figure (b. Scenario 4 Scenario 4.3: Six Periods per Cycle Sock Price / Dollars Figure (b. Scenario 4 Scenario 4.2: Four Periods per Cycle Sock Price / Dollars Disposiion effec exers a direc influence upon he invesor s selling behavior when he/she wihou DE shall sell. As a resul, DE has no impac on he invesor s rading when here is no selling a all, as is he case in Scenario and Scenario 2. Wheher DE urns ou o be posiive or negaive upon rading is deermined by wo key facors: gain or loss he invesor is enailed, and he marke volailiy which is represened by he flucuaion frequency. When marke volailiy is high enough (as in Scenario 4., he marke can be seen as moving in mean-reversion regime and he invesor ISBN: ISSN: (Prin; ISSN: (Online Figure (b. Scenario 4 Scenario 4.4: Eigh Periods per Cycle Furher, he Bayesian invesor s sensiiviy (as represened by characerisic parameers o marke movemens can largely affec his/her rading performance. Obviously, he agiliy of acing in he marke direcion is essenial for marke followers performance. Especially in Scenario 4.3, if he sensiiviy is sufficienly high, he invesor would jump ino a WCE 200

5 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. posiion of gain from loss. Generally speaking, more agile he invesor is, more successful he/she would be in rading. However, his is in ruh excep a special case (as in Scenario 4.2, in which he marke compleely fools he invesor around by engaging his/her ino a disasrous buy-high-sell-low, while his underperformance is made even worse by higher sensiiviy o marke movemens. Finally, as depiced in Fig. 2, we vary he characerisic parameers over he full range o find he maximum as well as minimum of final cumulaive profi, under each price scenario of oscillaion. As he flucuaion frequency is miigaed, he span beween maximum and minimum of profi is also reduced, which implicaes ha he invesor s marke sensiiviy plays a less imporan role in slow marke oscillaion. Reasonably, highly volaile marke demands higher agiliy o ac, whereas low volailiy marke renders he invesor more ime o recognize he rends and hen follow. We can also find ha he bes performance is aained in Scenario 4. wih srong mean-reversion feaure, while he wors one is in Scenario 4.2 wih he ricky flucuaion fooling he invesor o always buy high and sell low. Given a DE coefficien of 0.4, he picure of paern urns ou o be similar and he discussion above is applicable as well. Table. Summary of Scenario Analysis Gain or Loss? Impac of DE? Impac of Invesor s Higher Sensiiviy? Scenario : Uprend Gain Nil Posiive Scenario 2: Downrend Nil Nil Posiive Scenario 3: Single Cycle Oscillaion (Rise-Fall Gain Posiive Posiive Scenario 4: Muli-Cycle Oscillaion 2 periods per cycle Gain Posiive Posiive 4 periods per cycle Loss Posiive Negaive 6 periods per cycle Loss Negaive Posiive 8 periods per cycle Gain Posiive Posiive 4 x 04 Maximum / Minimum of Cummulaive Profi Max Min Max (DE=0.4 Min (DE= Scenario No. Figure 2. Max. / Min. of Cumulaive Profi a he End Under Differen Oscillaion Scenarios ISBN: ISSN: (Prin; ISSN: (Online WCE 200

6 Proceedings of he World Congress on Engineering 200 Vol I WCE 200, June 30 - July 2, 200, London, U.K. IV. CONCLUSION This paper aemps o answer a seldom asked quesion, i.e. are behavioral biases always harmful o invesor rading performance? This is ofen aken for graned bu sill subjec o proof. For his aim, a heoreical model is formulaed in order o describe precisely a Bayesian invesor's rading behavior and he process of how he/she forms and updaes her belief on marke movemens. Furher, scenario analysis under ypical marke paerns (uprend, downrend, and oscillaion wih differen frequencies presens some ineresing findings, and behavioral bias can be favorable under cerain circumsances. For fuure exension work, oher psychological biases may also be incorporaed ino our model, e.g. confirmaion bias, conservaism, overconfidence. In addiion, his paper serves as a preparaion essenial for some poenial opics, including: how differen ypes of invesors inerac wih each oher? Wha sraegies can arbirageurs possibly adop o ake advanage of invesor s biases o make profi? REFERENCES [] Fama E. (965. The behavior of sock marke prices. Journal of Business, 38 (, [2] Barber, B., Odean, T. (200. Boys will be boys: gender, overconfidence, and common sock invesmen. Quarerly Journal of Economics, 6, [3] Hirshleifer, D. (200. Invesor psychology and asse pricing. Journal of Finance, 56, [4] Garvey, R. and Murphy, A. (2004. Are professional raders oo slow o realize heir losses?. Financial Analyss Journal, 60(4, [5] Shu, P.-G., Yeh, Y.-H., Chiu, S.-B. and Chen, H.-C. (2005. Are aiwanese individual invesors relucan o realize heir losses?. Pacific-Basin Finance Journal, 3, [6] Andreassen, P., Kraus, S. (990. Judgmenal exrapolaion and he salience of change. Journal of Forecasing, 9, [7] Barberis, N., Shleifer, A. and Vishny, R. (998. A model of invesor senimen. Journal of Financial Economics, 49, ISBN: ISSN: (Prin; ISSN: (Online WCE 200