Microsoft Word - CUSJ 29.docx ________________________________________________________________________________ CUSJ 2019 __________________________________________________________________________________________ 29 Abstract — This paper presents an agent-based model of a stock market in which investors trade based on heterogeneous and changing beliefs. The model extends that from (Goodman, 2016), in which each agent trades based on the theory of dynamic investment in (Merton, 1969), with his own belief on the return of the stock. The baseline model implies that optimistic investors buy stocks from pessimistic investors. Then, the model is extended such that the beliefs of agents are time-varying: each agent adjusts his belief differently with the arrival of new information. The key findings from the simulation of the extended model are that (1) trade volume increases with higher heterogeneity of agents' response to news arrivals and (2) return volatility decreases with more positive or more negative average level of response to news arrivals. I. INTRODUCTION Stock prices are determined by trading, which arises from differences in opinions among investors. If all investors had the same expectations on stock returns, no trades would occur, because everyone would seek to trade in the same direction. Thus, it makes sense to study the stock price as a representation of heterogeneous beliefs on stock returns materialized through trading. Figure 1. Time Series of Analyst Forecasts of Amazon's EPS from I.B.E.S. It is also reasonable to think that these beliefs on stock returns change over time, with the influx of new information. More specifically, Figure 1 illustrates the analysts’ Amazon's earnings per share (EPS) forecast from 2010. Figure 1 reveals that the opinions are time-varying and heterogeneous across analysts. This may suggest that investors make different conclusions from the same set of information. This motivates modeling the heterogeneity of beliefs as an accumulation of the idiosyncratic response to the identical set of randomly arriving news information. There is a large body of literature regarding empirical evidence on the effect of heterogeneous beliefs on asset price dynamics, many of which including (Diether, Malloy, & Scherbina, 2002) (Chen, Hong, & Stein, 2002) (Park, 2005) (Berkman, Dimitrov, Jain, Koch, & Tice, 2009) (Qu, Starks, & Yan, 2003), document a negative relationship between belief dispersion and the mean returns of stocks. (Goetzmann & Massa, 2005) and (Yu, 2011) find that dispersion in beliefs is negatively correlated with future returns. On the other hand, (Avramov, Chordia, Jostova, & Philipov, 2009) show that such negative relation is limited to worst-rated firms, while (Doukas, Kim, & Pantzalis, 2006) find that the relationship is positive. There are also papers that study the effect of belief dispersion on return volatility and trading volume. For instance, (Banerjee, 2011) finds evidence that stock volatilities increase in belief dispersion and (Goetzmann & Massa, 2005) finds positive relationship between trading volume and belief dispersion. To study how interactions among investors with heterogeneous and changing beliefs affect stock prices, I apply an Agent Based Model (ABM) approach. The ABM approach offers several advantages that representative agent models do not. Firstly, an ABM allows its agents to have heterogeneous beliefs, not just heterogeneous utilities. While representative agent models allow for heterogeneous utilities of agents, which means that agents can have different preferences and thus different objectives, all of its agents must agree on the return of the assets. The Representative Agent Theorem requires the assumption that all agents should have homogeneous beliefs. However, ABM operates without that assumption, a property which I find useful for studying markets in which the beliefs on asset returns are heterogeneous across investors. Secondly, agents in an ABM follow a simple behavioral rule, instead of optimizing with the full knowledge of the state of the world. Thirdly, the key dynamics in an ABM are generated endogenously from the interaction among the agents. In relation to ABM and economics, (Westerhoff & Franke, 2012) illustrates the usefulness of ABM in designing economic policies with their example of technical traders, fundamental traders, and a central authority model to study the impact of simple intervention strategies on asset price dynamics. (Lengnick, 2011) constructs an ABM for business cycles in an economy and illustrates that an ABM can reproduce many stylized facts without the strict assumption of rationality. (Lengnick, 2011) also finds that the aggregate behavior generated by this ABM is not equal to the results of a microeconomic optimization by the representative agent. A more empirical application was conducted by (Baptista, Hinterschweiger, Farmer, Low, & Uluc, 2016), who developed an ABM for the UK housing market to examine the effect of macroprudential policies on key Agent Based Trading Model of Heterogeneous and Changing Beliefs Jaehoon Jung* NYU Courant Institute of Mathematical Sciences, 251 Mercer St #801, New York, NY 10012 CUSJ 2019________________________________________________________________________________ __________________________________________________________________________________________ 30 housing market indicators. Their results imply that a larger buy-to-let sector might amplify house price cycles and lead to higher price volatility. These papers show that the ABM can be economically useful in reproducing stylized facts without strong assumptions on agents or equilibriums; the ABM also generates non-standard aggregate behavior that is markedly different from that of a representative agent. In this paper, I extend the baseline model from (Goodman, 2016) to construct a model in which agents trade based on heterogeneous and changing beliefs. In the baseline model from (Goodman, 2016), each agent trades based on his beliefs and the theory of dynamic investment in (Merton, 1969), in which an agent maximizes his power utility by holding a fixed proportion of his wealth in risky assets. This baseline model implies that optimistic investors buy stocks from pessimistic investors. I then extend the model by having the agents adjust their beliefs in a manner that differs from randomly arriving news information. By simulating the extended model, I find that the trade volume increases with the higher heterogeneity of the agents' response to the news arrivals. I also find that the return volatility decreases with a more positive or negative average response level to news arrivals. II. MODEL The baseline model from (Goodman, 2016) uses the result from (Merton, 1969) to define the trading behaviors of the agents. An agent maximizes power utility V(T) by allocating his wealth W(t) between a riskless asset and a risky asset, the value of which evolves with mean ! and volatility σ. !"#$%[' ( ) ] subject to dW = rW t dt + µ − r X t dt + σX t d56 The solution to this problem is to keep a fixed proportion of his wealth in risky assets. This proportion # is a function of the risky asset's excess return µ − r, return volatility σ, and the agent's risk aversion parameter γ. 8 9 = :((9), where : = =-? (@AB)CD This proportion # determines the agents' trading behavior: each agent will buy or sell stocks to maintain that proportion of wealth in stocks. For instance, if the price of the risky asset goes up while # does not change, the agent will sell the risky asset to maintain the proportion of his wealth in the risky asset. In the baseline model, each agent trades based on the Merton proportion and there are n such agents in the market. An important assumption of this model is that each agent has his own belief: agent k has beliefs EF and GF and risk aversion parameter HF. It is further assumed that the total number of stocks in the market does not change and that there is only one risky asset (stock) and one riskless asset (cash). From the assumptions of the model, the following identities hold: First, the wealth of each agent is the sum of her cash and stock value: (F 9 = IF 9 + JF 9 K(9) Then, the wealth of the economy can be expressed as: (F 9 L FM@ = (IF 9 + JF 9 K 9 ) L FM@ Second, in equilibrium, each agent has the optimal ratio of risky assets. (F 9 :F 9 = 8F 9 = JF 9 K(9) The wealth of the economy at equilibrium can thus be expressed as: (F 9 L FM@ = 1 :F 9 JF 9 K(9) L FM@ Third, total number of stocks remains invariant: JF(9@) L FM@ = JF(9O) L FM@ Let N denote the total number of stocks in the market. J = JF(9) L FM@ Fourth, total cash in the economy remains invariant: IF(9@) L FM@ = IF(9O) L FM@ Fifth, from 3 and 4, I find that the wealth of the economy is increased only by the rise in stock price: (F(9O) L FM@ - (F 9@ L FM@ = J(K 9O -K 9@ ) Consider a simple case in which only one of the agents, agent 1, increases his E@ and thus his :@. When only agent one increased its E@ and thus �@, he would not be able to transact at current price if all the other agents are content with their portfolios. Therefore, agent 1 should keep bidding higher prices to buy stocks until he achieves his new proportion :@ of his wealth in stocks. When agent 1 pushes the price up from S to K + PK, wealth of agent k, (F, will increase by d(F = JFPK. As :F has not changed for Q > 1, the change in the optimal stock value for agent k would be P8F = :FJFPS. If agent k buys ________________________________________________________________________________ CUSJ 2019 __________________________________________________________________________________________ 31 !"# number of stocks to rebalance to the new optimal allocation, the value of his stock holdings would change from "#$ to ("# + !"#)($ + !$). Therefore, the following equation holds for each agent other than agent 1: !(# = *#!+# = *#"#!$ = "# + !"# $ + !$ − "#$ = "#!$ + $!"# Rearranging the equation above, the number of stocks that agent k should buy is: !"# = − "# 1 − *# $!$ This suggests that in response to agent 1's optimism, all the other agent will sell stocks (assuming that *# ≤ 1)to maintain their Merton proportion of their wealth in stocks. By separating the variables, I get: !"# "# = −(1 − *#) !$ $ By solving this equation and plugging in the initial condition, I derive the number of stocks that agent k will hold in equilibrium: "# 0 = "#(1 2 1 3 ) 4567(3) for 8 ≠ 1 As total number of stocks remains invariant, the number of stocks owned by the agent 1 who changed his views can be expressed as: "4 0 = "- "#(0)( $ 0 $ 0 ) 4567(3) < #=> From the identities and the stock holdings derived above, I now derive the new equilibrium price given that only agent 1 revised its µ4 and thus *4. At the new equilibrium, both identities 1 and 2 should hold: +#(0) < #=4 = (@# 0 + "# 0 $(0) < #=4 = 1 *# 0 "# 0 $(0) < #=4 As total cash in the economy is invariant, (@# 0 + "# 0 $ 0 ) < #=4 = @# 0 < #=4 + "#(0)$(0) < #=4 = 1 *# 0 "# 0 $(0) < #=4 As total number of stocks is invariant, @#(0) < #=4 + "$ 0 = 1 *# 0 "# 0 $(0) < #=4 Now, I separate agent 1 from others: @# 0 < #=4 + "$ 0 = 1 *4 0 "4 0 $ 0 + 1 *# 0 "# 0 $(0) < #=> Plugging in the stock holdings of agents in new equilibrium, @# 0 < #=4 + "$ 0 = 1 *4 0 "- "# 0 $ 0 $ 0 4567 3< #=> $ 0 + 1 *# 0 < #=> "# 0 $ 0 $ 0 4567 3 $(0) By dividing up the sum, I get: @# 0 < #=4 + "$ 0 = 1 *4 0 "$ 0 − 1 *4 0 "# 0 �0 $ 0 4567 3< #=> $ 0 + 1 *# 0 < #=> "# 0 $ 0 $ 0 4567 3 $(0) By rearranging the terms, I get the following equation for S(t): 1 − 1 *4 0 "$ 0 + 1 *# 0 - 1 *4 0 < #=> "# 0 $ 0 4567 3 $(0)67 3 − @# 0 < #=4 = 0 I confirm that the equation above is consistent with the simulation. III. MODEL EXTENSION Now, consider a more realistic case in which agents adjust their beliefs based on the arrival of news information. I assume that news information arrives through Poisson process and the time between each of them is exponentially distributed. Each agent's reaction to news information can be decomposed into two parts: the common response A3 and agent-specific response B#,3. D# 0 + E0 -D# 0 = A3 + B#,3 CUSJ 2019________________________________________________________________________________ __________________________________________________________________________________________ 32 The common response can be considered as the original news information shared by all agents; I will denote it by !". I assume that !" is normally distributed with mean µ$ and variance %$&. In this setting, µ$ would be the average level of response of agents to the news information and %$& would be the variability among the news. !"~((µ$, %$&) Then, for a given original new information, each agent reacts differently as would actual investors in the market. I assume that this idiosyncratic response, denoted as ,-,", is distributed normally with mean µ. and variance %.&. As I want this term to capture only the effect specific to a certain agent, I have the mean µ. equal to 0 and set %.& as the parameter that determines the level of heterogeneity among agents' reactions to news information. !"~((0, %.&) IV. RESULTS I simulated the extended model with news information. Starting from an identical distribution of wealth, stock holdings, and beliefs for all agents, I had the news information arrive in exponentially distributed intervals. Figure 2. Sample Stock Price Path generated from Simulation. Figure 2 shows one of the stock price paths generated by the simulation. The simulated stock price seems to become more volatile as heterogeneity across agent beliefs increases with the dispersion in reaction to news information accumulating over time. Figure 3. Trade Volume Increases in %.& (dispersion in agent response to new information). Figure 3 shows that the trade volume increases with dispersion in reaction to news information, which is the variance in agent-specific response (%.&). This result is in line with my expectation, because this means that agents trade more when they respond to news information more differently. Thus, in this model, heterogeneity in beliefs gives rise to trading in the market. Figure 4. Return Volatility Decreases in µ$ (Average level of response to new information). Figure 4 shows that return volatility decreases with more positive µ$, which means that the average level of response by agents to new information is more positive. I also find that return volatility decreases with more negative µ$, showing no asymmetry. This result is also within my expectation as my model implies that the stock price would be less volatile if all the investors are becoming more optimistic about the stock. ________________________________________________________________________________ CUSJ 2019 __________________________________________________________________________________________ 33 Figure 5. Trade Volume Decreases in News Arrival Frequency. Figure 5 shows that trade volume decreases with more frequent news arrivals. This result is counterintuitive as I expect the heterogeneity in beliefs to accumulate faster with more frequent news arrivals. As of now, I do not have a clear explanation as to why agent trade less if news information arrives more frequently. V. FUTURE WORK As the second extension to the baseline model, I aim to add an extrapolative component to agent beliefs. !" # + %# = 1 − )" !" # + *+ + ,",+ + )"!+ where )" denotes the degree to which belief of agent k reflects extrapolated market return !+. This would make the model more realistic, because it would be more reasonable that investors also adjust their beliefs based on the realized return, not just on news information. It would also make the belief dynamics more endogenous, since each agent's beliefs will be influenced by the realized return, which is determined by other agents' beliefs. The addition will make this ABM more valuable as the beliefs of agents will be affected by interaction (trading) with other agents with different beliefs, instead of relying solely on the arrival of news information, which is exogenous in this model. VI. CONCLUSION In this paper, I construct a model in which the stock price is determined by investors who trade based on their heterogeneous and changing beliefs. By simulating this model, I find that trade volume is positively correlated with dispersion in investors' response to news arrivals, suggesting that investors trade more when they have more different opinions. This result is consistent with the empirical evidence documented in (Goetzmann & Massa, 2005). However, I do not find a clear positive relationship between dispersion in reaction to news and return volatility, which was found in (Banerjee, 2011). Moreover, I also find that the return volatility is negatively correlated with more positive (or more negative) average level of response by investors to news arrivals, which means that the stock price is less volatile if all investors are becoming more optimistic (pessimistic). Finally, I find that trade volume is negatively correlated to news arrival frequency, contrary to my expectation that more frequent news arrival would accelerate the divergence in agent beliefs and thus increase trade volume. The results from simulations of this model suggests that the model is capable of replicating a number of intuitive results, such investors trading more when they disagree and returns being less volatile when investors are generally more optimistic or more pessimistic, with only simple, reasonable assumptions. I believe that this model can be used as a framework for studying the effect of heterogeneous beliefs on market dynamics under more specific or complex conditions. For example, I could extend the model such that investors adjust their beliefs according to the actual trading price. Then, I would be able to identify the effect of the extrapolative component by comparing the result with that from simulations without the component. In summary, this paper contributes to the literature of heterogeneous beliefs by showing that ABM could be an alternative theoretical approach and providing a simple framework. AUTHOR INFORMATION Corresponding Author *email: jj1419@stern.nyu.edu Funding Sources Received funding from the Summer Undergraduate Research Experience program by the Math Department at Courant Institute of Mathematical Sciences. ACKNOWLEDGMENTS I am grateful to my advisor Jonathan Goodman for his guidance and mentorship. All errors are my own. REFERENCES [1] J. Goodman, "Stock Prices in Metronia," Unpublished Paper, 2016. [2] R. C. Merton, "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, 1969. [3] K. B. Diether, C. J. Malloy and A. Scherbina, "Differences of Opinion and the Cross Section of Stock Returns," Journal of Finance, 2002. [4] J. Chen, H. Hong and J. C. Stein, "Breadth of ownership and stock returns," Journal of Financial Economics, 2002. [5] C. Park, "Stock Return Predictability and the Dispersion in Earnings Forecasts," Journal of Business, 2005. [6] H. Berkman, V. Dimitrov, P. C. Jain, P. D. Koch and S. Tice, "Sell on the news: Differences in opinion, and trading activity," Journal of Financial Economics, 2009. CUSJ 2019________________________________________________________________________________ __________________________________________________________________________________________ 34 [7] S. Qu, L. Starks and H. Yan, "Risk, dispersion of analyst forecasts and stock returns," Working Paper, 2003. [8] W. N. Goetzmann and M. Massa, "Dispersion of opinion and stock returns," Journal of Financial Markets, 2005. [9] J. Yu, "Disagreement and return predictability of stock portfolio," Journal of Financial Economics, 2011. [10] D. Avramov, T. Chordia, G. Jostova and A. Philipov, "Dispersion in analysts' earnings forecasts and credit rating," Journal of Financial Economics, 2009. [11] J. A. Doukas, C. Kim and C. Pantzalis, "Divergence of opinion and equity return," Journal of Financial and Quantitative Analysis, 2006. [12] S. Banerjee, "Learning from prices and the dispersion in beliefs," Review of Financial Studies, 2011. [13] F. Westerhoff and R. Franke, "Agent-based models for economic policy design: two illustrative examples," Working Paper, 2012. [14] M. Lengnick, "Agent-Based Macroeconomics - A Baseline Model -," Working Paper, 2011. [15] R. Baptista, M. Hinterschweiger, J. D. Farmer, K. Low and A. Uluc, "Macroprudential policy in an agent- based model of the UK housing market," Staff Working Paper No. 619, 2016.