What does the financial market pricing do?
A simulation analysis with a view to systemic volatility, exuberance and vagary
Abstract
Biondi et al. (2012) develop an analytical model to examine the emergent dynamic properties of share market price formation over time, capable to capture important stylized facts. These latter properties prove to be sensitive to regulatory regimes for fundamental information provision, as well as to market confidence conditions among actual and potential investors. Regimes based upon marktomarket (fair value) measurement of traded security, while generating higher linear correlation between market prices and fundamental signals, also involve higher market instability and volatility. These regimes also incur more relevant episodes of market exuberance and vagary in some regions of the market confidence space, where lower market liquidity further occurs.
Keywords: financial regulation, asset pricing, financial bubbles , market exuberance, market microstructure, common knowledge
JEL Codes: C63, D02, D47, D82, E17, E37, G1, G17, M41, M48.
1 Introduction and Related Literature
Share Exchanges (financial markets) are socioeconomic devices that enable interactions and exchanges between potential buyers and potential sellers while dynamically fixing market prices of exchanged securities. Financial economists have developed elegant models that explain the eventual equilibrium results of this process, without going into the details of pricesetting mechanisms. This literature generally identifies two drivers guiding the market pricesetting process: matching of supply and demand, and information, with the latter being emphasized by recent contributions on efficient financial markets (Bouchaud et al. 2009; Shiller 2013; Fama 2013; Stiglitz and Gallegati 2011; Markowitz 2005). Informationally efficient markets fully and correctly integrate any new (i.e. unexpected) information that affects the fundamental value of traded securities into their price. Due to rational expectations, market prices are expected to change only with unexpected news: informational efficiency implies then that current price () is the best predictor of future ones, making the price time series a random walk. Formally:
Accordingly, fundamental value is supposed to exist and to be correctly known by at least some informed investors who can then act as arbitrageurs between ongoing market prices and that reference value, making a temporary (or illusory) profit while driving back the market price towards its fundamental benchmark, through the very impact of their trades. Indeed the arbitrage mechanism stands at the core of the equilibrium approach to efficient financial markets and can be used to justify the unpredictability of market prices in the long run (Malkiel 2003; LeBaron 2006; Iori and Porter 2012; LeRoy 2004). Alternatively, each agent can observe the fundamental value with some noise, which may then be progressively eliminated through emergent information extracted, over time, from evolving market prices and interactions. Following Hayek (1945), markets can also be studied as aggregators of dispersed information. No investor can then observe fundamental value, but the market aggregating process enables its emergent collective discovery.
As equilibrium approaches do not disentangle the specifics of the trading process, the interactional nature of its dynamics and its epistemic preconditions, concerns have been raised that this modelization strategy does not correctly lead our understanding of share markets activity especially in the context choosing regulatory designs and regimes (Shubik 2007; Kirman 1999; Biondi 2011b). In particular, Sunder (1997) chapter 7, noted that information from outside the market pricing process needs to be gathered and interpreted by market players in order to be integrated in the market prices; quoting the author: ‘the hypothesis of instantaneous adjustment of price to new information leads paradoxically to the conclusion that such an adjustment cannot occur due to the absence of private incentives to gather information’. Being subject to investigation and interpretation by market players, fundamental value becomes quite an abstract concept which cannot be computed with arbitrary accuracy at every instant in time, with whatever amount of required information available. At the very least, some irreducible margin of judgment and intrinsic error shall remain. As Pareto (1906) page 30 argued:
In a volume on economics recently published we find that “the price is a concrete manifestation of value.” We are already familiar with the incarnations of Buddha. To them we are now asked to add the incarnations of Value.
In contrast, a reasonable (realistic) modelization strategy is to accept that investors idiosyncratically interpret evolving signals of fundamental performance that deliver noisy information about the shareissuing corporate group. As each investor interprets information idiosyncratically, no universal consensus is expected. Investors interact one with another to form their focal price opinions. Trade is then based on disagreement (Stout 2011; Biondi 2011a) and investors are confronted to the collective dynamic dimension of aggregating supply and demand through the Share Exchange. This dimension introduces an opportunity to profit from the ongoing evolution of aggregate market prices over time. Arbitrage strategies can then be based upon the dynamics of fundamental signals or market prices. A dynamic tension may especially exist between speculative (or chartist) strategies based upon aggregate market prices, and fundamentalist strategies based upon fundamental signals of reference. Furthermore, the process that generates information about the fundamental performance of the shareissuing entity is not a phenomenon neutral to the market dynamics. Fundamental signals are generated through institutional devices external to the market that facilitate its working (Phelps 1987; Frydman 1982; Fama and French 1992). These institutions provide common knowledge that nurtures fundamental financial analysis processes. Let us define the whole of financial reporting and disclosure, financial analysis techniques and related standards as a common knowledge regime. These regimes assume specific forms, including distinctive accounting regimes that define financial performance and position of the shareissuing corporate group over time. Last but not least, investors factually do not take their decisions in isolation but are subject to social influence and interaction. The study of the impact of this social dimension on agents’ behaviour has been developed by a recent strand of economic literature (see Aldashev et al. 2011; Ozsoylev and Walden 2011; Kukacka and Barunik 2013). While this paper does not analyze the dynamic effect of social dimension, it deals with an overall market confidence space that comprises the combination of all the possible states of confidence among and across both sides of the market.
This paper elaborates on the analysis and results by Biondi et al. (2012) which, drawing upon this modelisation strategy, develop an analytical model where investors’ decisionmaking is influenced by four dynamic drivers:

Heterogeneity at individual and group levels, leading to trade on disagreement;

A fundamental information signal generating common knowledge over time;

An emergent social interaction dynamics shaping individual and group beliefs, expectations and forecasts;

A market pricing mechanism that aggregates the decisions of investors (orders to sell, to buy or just wait) which disagree according to possibly diverse opinions and profit opportunities.
The present paper assesses the systemic properties of the financial system in terms of market volatility, exuberance, vagary, liquidity and stability, through the analysis of the impact of these drivers on the formation of market prices over time and contexts. These systemic properties prove to be sensitive to common knowledge regimes (which correspond to stylized accounting models of reference for financial reporting and disclosure, namely historical cost and fair value accounting models). Through numerical simulation, we develop a comprehensive economic analysis of the influence of different common knowledge regimes coupled with various combinations of speculative and fundamentalist beliefs across supply and demand sides. Simulations generate distinctive market price series for every regulatory regime across financial system conditions, enabling comparative analysis of financial systemic performances across time and contexts.
From a policy perspective, our numerical analysis contributes to the efforts to imagine and design regulatory regimes which show higher degrees of systemic stability and sustainability under a large set of circumstances.
Furthermore, our numerical results provide relevant theoretical points that may further improve conceptual design of control systems such as accounting and prudential regulation. In particular, our disagreementbased analytical model captures and explains Shiller (2000)’s market exuberance that is endogenously generated by the market pricing process over space and time. It also provides a theoretical explanation of fair value accounting procyclical contribution to market bubbling (Enria et al. 2004; Boyer 2007). Furthermore, these results may be empirically tested against actual share market dynamics, while helping to improve on existing empirical tests for market volatility, exuberance, vagary, liquidity and stability.
The rest of the paper is organized as follows. Section 2 introduces the building blocks of the model and related notation. This is followed by a discussion (Section 3) of the calibration used to test the model. The remaining sections provide numerical results regarding the impact of the drivers discussed above on share price formation, pointing to market pricing (Section 4.1), volatility (Section 4.2), exuberance (Section 4.3), and liquidity (Section 4.4). Furthermore, information quality is assessed in term of linear correlation with and linear forecasting power of market prices over time (Section (4.5)).
2 Model and Notation
Biondi et al. (2012) develop a heterogeneous agents model that generalizes received equilibrium approaches to financial market pricing process. This paper presents results based on an extensive study of a simplified version of that model. This section summarizes it in its main features and in its basic assumptions.
According to Aoki and Yoshikawa (2011) chapter 9, two broad categories of chartism and fundamentalism account for most of possible investment strategies. Following Hirota and Sunder (2007) and Heemeijer et al. (2009), we consider a large population of heterogeneous investors which form their focal price expectations (upon which they base their trading strategies) according to the following generic function:
(1) 
where
(2) 
Each generic investor can belong to one of two groups . Group is formed by those investors that do not hold shares, implying potential demand, while group is formed by investors which hold shares (shareholders), implying potential supply. Equation 1 comprises four elements. The first is the past market clearing price . The second is the signal generated by the market about the aggregate price trend . The importance given to this market signal is weighted by the market confidence . The third element is the individual forecast revision . It consists of the difference between investor’s past price expectation and the last clearing market price that was actually realized (Equation 2), weighted by which captures both group and individual heterogeneities. The forth element denotes the formation of an individual opinion based upon available signal of fundamental performance , which is common knowledge for both market sides and all the individual investors. This opinion is weighted then by the individual parameter , while captures both group and individual heterogeneities.
According to this framework of analysis, the financial system is embedded in a dual institutional structure which drives market pricing process: the enterprise entity side is subsumed by the signal of fundamental performance , while the market side is captured by the market price trend . In particular, the institutional signal that incorporates the fundamental performance (named ) of the corporate group whose security is traded is generated by an exogenous mechanism of the type: with . This signal is assimilated by each agent idiosyncratically: each investor is then individually characterized by a certain degree of fundamentalism (chartism), captured by the weight he attributes to the signal of fundamental performance . ^{1}^{1}1Nevertheless, the actual degree of fundamentalism (chartism) for the whole marketplace is endogenously determined by financial system dynamics, and does not depend only on these exogenous subjective attitudes or beliefs.
The institutional signal’s generating process is inspired by distinctive accounting models of reference, namely Fair Value Accounting (marktomarket); Historical Cost Accounting and one theoretical benchmark derived from equilibrium approaches, involving targetbased signaling provision. The latter regime assumes the very existence and relatively accurate knowledge of fundamental value of reference by informed investors who wish to exploit related signaling. While we may refer to the first two regimes as actual regimes, denoting stylized practical modes of accounting, the latter regime can be only thought in a vacuum.
Formally, the Historical Random Trend Accounting (HRT) regime implies an evolving exogenous signal of fundamental performance that is orthogonal to market price dynamics, composed by a stochastic component resulting from positive and negative flows together with a stochastic trend component (Biondi 2011c; Anthony 2004), i.e.:^{2}^{2}2As robustness check we also simulated our model with a pure historical cost approach without the trend: outcomes are extremely similar and relegated to appendix.
(3) 
with representing a value extracted from a normal distribution with mean 0 and standard deviation 1, bounded between and and representing a value extracted from a uniform distribution between the two indicated values.
A the opposite side of the spectrum, the Fair Value Accounting (FVA) regime implies a pure marktomarket accounting system that replicates market information with one time lag (Kothari 2001; Nissim and Penman 2008):
(4) 
Finally, Stochastic Target Reverting Accounting (TRAS) regime implies a reverting fundamental performance signal that targets a given core value that is bounded within a stochastic band of width due to white noise error of estimation and other random effects:^{3}^{3}3As robustness check we also simulated our model on a signal that reverts toward a precise value: results are very similar and relegated to appendix.
(5) 
with further bounded between . This assumption about implies that, under this regime, investors know deterministically the range of fundamental values that the traded security can reach.
For all accounting regimes, the same stochastic error is added to account for estimation error, measurement error and other random effects:
(6) 
The last building block of our model is the mechanism through which the market price is formed at every trade time t. Investors’ bidding strategy is based on their focal price expectations. Actual shareholders can sell or wait for the next period, while potential shareholders can buy or wait for the next period. Accordingly, every shareholder (i.e. each agent with ) wishes to sell if the clearing market price is higher than his focal price expectations, that is, , while every potential buyer () wishes to buy if the clearing market price is smaller than his focal price expectations, that is, . In particular, extreme investors, which express fully speculative () or fully fundamentalist () strategies, form their respective focal price expectations as follows:
(7) 
At every trading time , the model assumes an aggregate matching process (in line with Di Guilmi et al. 2012; Foley 1994; Anufriev and Panchenko 2009; Chiarella and Iori 2002; Horst 2005). The share exchange protocol receives all orders and ranks them on the base of the focal prices expresses by the extreme investors as defined in Equations 7. This ranking is based upon the assumption of a uniform distribution of focal prices between the extremes of each market side (formally, ). This simplifying assumption enables to create a market matching process which generates a complex market dynamics of the market clearing pricing that is analytically treatable (see Biondi et al. (2012) for further details). Therefore, aggregate demand and supply depend on four focal prices expressed by idealtype investors over time, defined as follows:
(8) 
Accordingly, given the imposed uniform distribution of investors over each market side, the Share Exchange Mechanism by aggregating and matching demand and supply delivers the market clearing price, defined as:
(9) 
with being a small tick value that is activated whether demand and offer cannot be matched. In this case a new price is generated that is slightly different from the one of the previous step.^{4}^{4}4The presence of this random error turns out to be without material impact on simulation results as discussed in the next section. Aggregate market price dynamics enriches the passage between the individual and the collective level, making the latter irreducible to the former. Each price pattern becomes then unique over time and space. Replication of several patterns through simulation enables then to infer regularities on the working of this financial system under its distinctive conditions.
Finally, regarding our model, one could object to the absence of exchanged quantities (and of individual agents’ portfolios). However, individual portfolios are irrelevant here, since we study investors that are not budget constrained and that do form their focal prices on past and next period expectations and posting their orders deterministically by comparing their focal prices with past called price.
3 Simulation calibration
The bulk of our analysis is based on numerical simulations that study the systemic properties of financial market price formation under alternative common knowledge regimes and over the market confidence space. The latter space comprises all the possible combinations of market confidence by investors on both market sides. The analytical model requires calibration to perform this simulation analysis. This calibration does not purport here to obtain realistic assumptions for the parameters, but to improve comparability between various parameter sets and distinctive common knowledge regimes over the overall market confidence space.
In particular, while the measures of market confidence do change dynamically as result of a social interaction, the present paper neglects this interaction and does analyse simulations in which parameters value do not change over time. This choice is in line with our objective of exploring the systemic effects of different degrees of market confidence. For the same reason we simplify the general model by imposing the value of and the value for the variable weighting the individual opinion . Both calibrations purport to obtain a symmetric setup around the median investor identified as . This symmetry choice follows from the hypothesis of uniform distribution between extreme focal prices (on both market sides) and from the fact that all stochastic elements are normally distributed.^{5}^{5}5From our assumption that is distributed uniformly among the investors between 0 and 1, it follows that values of the market confidence imply that the investors tend to be speculative. At the opposite, when investors tend to be fundamentalists. The choice of further implies that we exclude, for the sake of this analysis, the existence of a systematic, evolutive, bias in the interpretation of .
With these simplifications to the model, we run numerical simulations varying and between and in steps of . The market price distribution shows a scalefree property. Therefore, for every number of replications, its mean (median) value and variance remains similar for the same market price series length (number of simulated market periods), while, if the number of replications increases, this synthetic measure increases for the same market price series length. Our results are obtained averaging outcomes from series of 500 market prices replicated over 1000 simulation rounds, allowing satisfying comparative analysis between financial systems under distinctive common knowledge regimes. To be sure, if market price distribution is scalefree (power law), these synthetic measures do not characterize the underlying distribution, but, at the same level of series length and replications, they remain comparable between ‘imagined worlds of accounting’ and finance (Sunder 2011). Given the characteristics of our model, each simulation tells a quite unique ’story’. However, we want to avoid simulation’s results depending only on purely transient factors. For this reason, in order to make simulations comparable across types of common knowledge regimes, for each combination of parameters and simulation round, a given random seed is fixed and kept constant whenever the same combination is repeated for another type of common knowledge regime.
In the context of market price formation, whether, at one time step t, demand and supply do not match at all (making thus impossible to generated an updated market price), the auctioneer updates the past market price according to this formula: . This small error is introduced in order to avoid deadends of the market price dynamics. While this device is rarely called under HRT and TRAS, it activates more often under FVA^{6}^{6}6For the FVA this device is called from a minimum of of the times up to a maximum of of the steps. It is called in average of the steps (median ).Detailed about this variable as complementary online material. especially where the market mood of the two sides is very different. However, where and are not extremely different (that is, the areas where our focus and main findings come from), the auctioneer interventions are very limited.
In order to deal with the possibility that, due to the interaction between the market price dynamics and the fundamental signal dynamics, the market price does fall at (or below) zero, an avoidance mechanism is introduced. Whenever the cumulated fundamental signal (pointing to the underlying fundamental performance) falls at or below zero, this mechanism jumpstarts the signal for that period according to the formula: . This mechanism is never used except under the FVA, where it is called very rarely (it activates a maximum of on the total steps number, with an average activation of ).
As noticed discussing the model, to every fundamental signal, we add an error with . The results of the model are especially sensitive to the magnitude of this error under fair value accounting (FVA), since it is  by assumption  the only exogenous shock that is stochastically added to the financial system under this regime. Figures 1 shows that this error magnifies financial instability already at its first order (mean and median market prices) under this regime. For sake of unbiased comparability of simulation results, we calibrate this error at , in order to reduce its relative impact for all common knowledge regimes.
Finally, in order to ensure that the results reported below are accurate, we are interested in knowing whether our simulations produce homoskedastic time series. For each simulation, we run the Brown and Forsythe (1974)’s test for homogeneity of variances (which we chose for his robustness to nonnormal data distributions). This test is passed about of the times (at a significance level of ), providing satisfying confidence that our results are robust and that can be tested with regression analyses.
4 Simulation Results
4.1 Market Pricing
By construction, all common knowledge regimes of fundamental signal provision are designed to swing around the same mean (the initial price is fixed for all simulations to ). Since, all stochastic elements are symmetric around this central point and do not impose any systematic bias to the aggregate market pricing and to investor’s beliefs related to the fundamental signal, the mean (median) market price series is then expected to conform to this benchmarking level. Every significative deviation from this central point can only depend on the interaction between the investors’ expectations, the market matching protocol and the evolution of the fundamental signal over time (under each distinctive common knowledge regime). According to Table 1, both HRT and TRAS produce results that remain around this central point. The FVA signal shows a different result. Observing its distribution of mean and median market price, it emerges that, in some regions of market confidence space, they diverge and reach extreme values.
Signal Type  Mean Std  Min  Q1  Median  Q3  Max 

HRT (mean)  1000.25 3.6688  996.472  1000.096  1000.237  1000.378  1035.581 
HRT (median)  996.736  1000.098  1000.239  1000.383  1016.071  
FVA (mean)  2.4e+60 2.1e+61  999.557  1000.490  1000.820  2.1e+17  5.7e+63 
FVA (median)  997.11  1000.48  1000.816  4.2e+08  1.2e+32  
TRAS (mean)  1000.16 1.7446  999.589  1000.064  1000.154  1000.246  1000.550 
TRAS (median)  999.56  1000.062  1000.154  1000.248  1000.563 
In order to gain insights into the reasons for this behavior, it is useful to observe the distribution, over the parameter space, of mean and median market prices in Figure 2. Under FVA regime, mean prices significantly diverge from the theoretical central level of 1000 when at least one market sentiment (either the one of supply or the one of demand side) are speculative, implying or , and especially where both market sides are overconfident in the market signal, i.e. when . Indeed, where this overconfidence condition is met, bidding by either potential investors (demand) or actual shareholders (supply) drives up the market pricing as long as the market price trend goes up, because the market signal is overweighed  relative to other components  in their forecasting. This creates an overwhelming selffulfilling and selfreinforcing loop since fundamental signal and market signal reinforce each other, pushing the market price higher and higher. Only very low levels of market confidence by the other side of the market ( or ) can counterweight speculative beliefs that drive the market price up. This happens because, where the market sides strongly differ in market confidence, matching between demand and supply orders is reduced (see Figure 5). Lower bids on one side of the market tend then to break the loop between market and fundamental signals, keeping the market price dynamics in line with the expected central level. Notably, negative bubbling (that is, bubbling dynamics that tends toward zero) is absent in average over our simulations. This happens because our simplified trading strategies do not allow for short selling, thus making it impossible for investors with price decreasing expectations to drive the market price down. Additionally, even when the market trend enters a negative bubble, the downward movement is limited to the minimum bound of zero, while the upward movement is unbounded. Thus, the presence of negative bubbles in the area of overconfidence (found by Biondi et al. 2012 studying single simulations) is hidden by the much bigger and unconstrained values generated by positive bubbles.^{7}^{7}7The average minimum price, over the FVA regime, is 997.43 (median 999.56) and a minimum of 39.06.
The existence of a significant distortion of market pricing under fair value accounting regime is comforted by the median of the medians of market price (Figure 2, right panel). This measure confirms that the market price distribution is severely strayed away from 1000 under this regime whenever both and , that is, in the region of joint overconfidence by both demand and supply sides. In the region where only one of them is overconfident, conservative levels by the opposite side do counteract, driving the central level back to the theoretical amount. This provides the additional finding that the formation of market price bubbles in those areas is an extreme event that does not occur at the median level of the distribution.
Overall, this results show that, under some conditions, the FVA is prone to generate relevant market bubbles that materialize in our simulation through persistent explosions of market prices. To be sure, simulated market price levels reach unrealistic values because both investors and the corporate group are not budgetconstrained, since a purely theoretical dynamics is under investigation here. These bubbles depend from the selffulfilling and selfreinforcing interaction between the market sentiment and the fundamental signal which, in the case of FVA, communicates back to the market a fundamental information autocorrelated to the market price dynamics. This result also shows that, not only fair value regime adds instability to the market, but also distorts its working in a selfreinforcing endogenously generated chaos which strays the market pricing away from benchmarking level of fundamental performance. From the regulatory design perspective, this preliminary analysis seems to indicate that introducing FVA may be a harmful choice for financial market stability, as for actual market moods shift endogenously and remain outside control by regulatory authorities. As showed here, shifts toward overconfidence in the market increase the probability of formation of market price bubbles.
4.2 Market Volatility
One important characteristic upon which dynamic properties of financial market can be assessed is volatility (pointing to the properties of the market price series at the univariate level). A classic measure of volatility can be computed as the logarithm of standard deviation over the mean of the market price, averaged series by series:
(10) 
Table 2 shows the distributional characteristics of average volatility together with the of the maximum (this latter measure has the objective of obtaining values relative to volatility peak while excluding the most extreme events).
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HRT (mean)  0.0036  0.0017  0.0021  0.0026  0.0036  0.3363 
HRT (75% Peak)  0.0045  0.0021  0.003  0.0032  0.0045  0.4790 
FVA (mean)  1.6436  0.0003  0.0006  0.0009  2.9661  8.7466 
FVA (75% Peak)  1.7882  0.0004  0.0007  0.0010  4.3298  8.7466 
TRAS (mean)  0.0017  0.0011  0.0014  0.0016  0.0019  0.0034 
TRAS (75% Peak)  0.0021  0.0013  0.0017  0.0020  0.0023  0.0042 
Even though no actual common knowledge regime can match the performance achieved by theoretical TRAS, the regime HRT still provides good systemic stability by reducing overall volatility over the whole range of the parameter space of market confidence. This better overall performance is obtained by reducing extreme volatility events, since fair value accounting FVA does have better performances for both minimum volatility, and the first and second quartiles (median). Indeed, the average volatility measures are increased due to the occurrence extreme events with large volatility. Such dramatic events are enabled by FVA but prevented by HRT. This analysis over the whole distribution of volatility is further corroborated by taking a volatility measure at a certain peak point of this distribution, in line with valueatrisk approaches. Indeed, taking the volatility measure at its third quartile^{8}^{8}8This peak point denotes the expected maximum value of volatility under that accounting regime at 75% likelihood. (Table 2, second lines for each signal) we notice that the FVA shows clearly inferior performance compared to both the theoretical TRAS and the actual HRT regime. These results are further corroborated studying the volatility width around the median value, summarized in appendix. Volatility can be interpreted as a measure of the mispricing risk in financial system dynamics. Our results show that, coeteris paribus, FVA increases beyond any fundamental change in the nature of the exchanged security this risk, through the interaction between fundamental signal and market price dynamics.
4.3 Market Exuberance and Vagary
Market exuberance points to market prices dynamic that strays away from levels that are reflected in fundamentals. In mathematical terms, this refers to the properties of the market price series relative to the fundamental signal series (at the bivariate level). In order to study this dimension of the financial system, we define the cumulated fundamental signal series that provides an evolving benchmark for ongoing market pricing process over time and space as follows:
(11) 
On this basis, we compute two measures of disconnection between the market price dynamics and this cumulated fundamental signal dynamics:

The first measure (Table 3, first line) is labelled and defined as the relative distance between the current price and its lagged signal of reference, weighted by this latter lagged signal. In order to exclude extreme events from our consideration, we take the third quartile of this measure, providing an assessment of the maximum distance  at 75% likelihood  generated by the financial system over time. Formally:
(12) 
The second measure, the mean exuberance (Table 3, second lines), captures an average peak value in market exuberance on the whole simulation sample as follows:
(13)
Results regarding these measures are summarized in Table 3.
Signal Type  Mean Std  Min  Q1  Median  Q3  Max 

HRT ()  0.0023 0.0043  0.0002  0.0015  0.0019  0.0023  0.2922 
HRT()  2.10e05  2.84e06  1.41e05  1.69e05  1.98e05  1.94e03 
FVA ()  0.1018 0.0160  0.0001  0.0003  0.0003  0.1085  0.8660 
FVA()  2.22e04  9.36e07  2.65e06  2.72e06  2.81e04  1.74e03 
TRAS ()  0.0019 0.0030  0.0002  0.0013  0.0019  0.0025  0.0035 
TRAS ()  1.69e05  2.79e06  1.24e05  1.72e05  2.12e05  2.98e05 
Considering that for complete absence of exuberance this measure would be 0, shows that HRT regime remains in line with TRAS and does not generate any significant amount of exuberance. On the contrary, the FVA shows worse performances, allowing high peaks of volatility in the superior half of the distribution and producing in average a measure of exuberance of around the fundamental signal of reference. Mean exuberance (second lines of Table 3, for each regime) under FVA is significantly higher than both historical accounting regime and the theoretical one. However, more significative is the fact that the overall performance of FVA is skewed: it performs better than HRT for the inferior half of the sample distribution, but worsens in the superior half, even though the maximum value remains in line with historical cost accounting regimes. This points to the spatial organization of its behavior over the overall market confidence space. This organization shows that higher exuberance observed under FVA can be attributed to the more likely occurrence of extreme events in which the market dynamics becomes relatively more exuberant.
Market exuberance is especially important because a persistent distance between the fundamental signal and the market prices over time and contexts implies potential distortion in allocation of resources through the market pricing process. Under exuberant conditions, trades occur either above or below the central point of reference, implying inefficient and unfair transfers of resources across investors and periods.
4.3.1 Market Vagary (Dynamic Disconnection)
From a dynamic perspective, market exuberance consists of a overall (comparatively static or crosssectional) disconnection between market prices and fundamental signals of reference. Biondi et al. 2012 find insightful disconnection effects over time between the two dynamics, leading market prices to stray away from their benchmark level of fundamental performance for long durations. It is then interesting to disentangle here these dynamic effects of disconnection conceptually labelled market vagary or errancy hereafter. For this purpose let define a dissociation measure as: . For sake of this analysis, we choose a very stringent definition of dissociation duration: we count one dissociation duration whenever strays away from its overall mean for more than two standard deviations for at least 10 time periods . We then compute the percentage of time^{9}^{9}9Given this strict definition of dissociation we find relatively low percentages of time that fall into this category. This difference, however, preserves significant differences across common knowledge regimes. periods over the overall length of every simulation in which the fundamental signal is disconnected from the market price (Table 4, first lines for each regime), as well as the average length of the disconnection durations (Table 4, second lines for each regime).
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HRT ()  0.3022  0  0.2606  0.3186  0.3602  0.5078 
HRT (Periods)  16  0  15  16  17  21 
FVA()  0.2722  0  0.0456  0.3608  0.39925  0.518 
FVA (Periods)  14  0  16  17  18  36 
TRAS ()  0.2419  0  0.1660  0.2606  0.3246  0.4446 
TRAS (Periods)  16  0  15  16  16  22 
The dissemination of both measures on the parameter space of market confidence is significant (Figure 3). Theoretical common knowledge regime TRAS shows better performances around the parity line where . This spatial organization is crudely replicated by the HRT regime. Under the latter, the percentage of dissociation time remains quite in line with the theoretical benchmark provided by TRAS. Lower levels are concentrated along the diagonal where market side opinions are quite similar or the same (), especially where both opinions are not too conservative (i.e., outside the corner area where ). Notably, a specific region of dissociation occurs in the second quadrant, including along the diagonal. In this region, overconfidence in the market signal by both demand and supply is in perpetual tension with fundamental signal HRT that remains independent from market price trend. Speculative bubbles then continuously attempt to occur but they eventually burst because of this dynamic tension. At the opposite, for the corner area (where ), high overconfidence seems to predominate, reducing then the impact of this tension. This phenomenon does not occur under TRAS, since investors that care about fundamental signals are assumed to treat a common target fundamental value of reference, under this regime. This theoretical regime does not involve stochastic trends in fundamentals that can trigger temporary speculative bubbles over time.
However, it is fair value common knowledge regime FVA that shows the worst performances (higher disconnection durations). This happens everywhere but in the region where volatility is the highest and market prices are distorted at their first order (see Sections 4.1 on market pricing and 4.2 on market volatility). In this region, overconfident opinions by both sides of the market continuously agree on upward market price trends, making the market less vagarious but dysfunctional. The market pricing mechanism fixes a clearing price that, even though it remains in line with the fundamental signal, is distorted (since in this area the market price and the signal are autocorrelated by construction). The fundamental signal is then unable to provide any rebalancing benchmark. It cannot act as an ‘accounting lighthouse’ for investors (Biondi 2011a, c), leaving them alone with their consensual but distorting beliefs. In this sense, the better performance of FVA is achieved at the cost of the possibility of much higher volatility, exuberance and more likely occurrence of extreme events regarding market price dynamics. In sum, FVA shows its better performance where it becomes unable to drive financial system dynamics in line with benchmark level of fundamental performance, while it shows worst performance elsewhere.
This inference is comforted by exploring the mean time length of dissociation (Table 4, second lines, and Figure 4). Again, the performance of FVA in general is obtained by compensating worst results away with those, distorted, from the overconfidence area (second quadrant) of the parameter space of market confidence. On the contrary, all the other common knowledge regimes obtain their best performances (implying shorter dissociation time length) in association with both the parity line and the absolute parity point where .
It is important to study measures of market vagary together with measures of market exuberance since both provide complementary information about the relationship between market price and fundamental signal, clearly showing tendency by FVA to generate market bubbles over that fundamental level of reference. This implies that, not only FVA involves more volatile market price dynamics (as discussed in Section 4.1), but may also precludes the market pricing process to properly and timely incorporate fundamental signals. Financial market becomes then vagarious, evolving in a fundamentally erratic way that goes even beyond market exuberance over those same fundamental levels of performance.
4.4 Market Liquidity
By market liquidity, we mean the ability by the Share Exchange to satisfy trade orders posted by potential investors wishing to buy (potential demand) or sell (potential supply), at any trade time . The degree of market liquidity measures then whether and how much the market matching process enables investors which want to trade to do so.
In particular, our aggregative market matching protocol denotes two areas: one larger area comprises all investors potentially willing to trade (market area); another  smaller  area comprises all investors that may actually contribute to the clearing process given the bids on the opposite market side (clearing area). The study of the ratio between these two areas (by construction comprised between zero and one) provides some understanding of the timely satisfaction of investors’ orders at each market period , pointing to market liquidity as measured by posted orders satisfaction. While the numerical values are quite in line for all accounting regimes, their organization over the parameter space of market confidence (Figure 5) results to be distorted in the case of FVA, especially around the second quadrant denoting market overconfidence shared by both supply and demand.
Under TRAS and HRT, market liquidity is symmetric along the diagonal. Orders satisfaction is then increased as long as market confidence is similar on both sides of the market, and maximal where independently from how conservative or speculative market sentiments are. Orders satisfaction is minimal at the corner where market sentiments on the two sides of the market are opposite, that is, where and viceversa.
In case of conservative weight attributed to the market price trend, the market matching process works under FVA as well, enabling a high proportion of orders satisfaction for investors willing to exchange. However, under FVA, this result does not hold for the region where at least one market side sentiment tends to be speculative, implying and especially in the area where . Here, high market liquidity is maintained only around the diagonal (for market moods by both market sides that are very similar to each other), while the rest of the region shows lower efficiency relative to HRT. This phenomenon results from market instability in that region: both sides are highly confident in the market signal but they do not agree on the precise weight to assign it, thus generating lower levels of market liquidity. This interpretation is confirmed by studying the percentage of satisfied supply in that region (Figure 5). In the part above the diagonal, supply is more overconfident about the market signal than demand, leading to higher levels of unsatisfied supply orders. At the opposite, in the part below the diagonal, demand is more overconfident than supply, leading to lower levels of unsatisfied supply orders. Potential suppliers (buyers) are here overconfident about the market trend relative to potential buyers, willing to selling (buying) at a higher degree than potential buyers (sellers). This generates over supply (demand) that cannot be absorbed by an aggregate matching process based upon disagreement. Orders remain then unsatisfied at a higher degree because of such overdisagreement. In the other regions, however, the market matching process remains satisfyingly around its expected level of one half (implied by the assumption of uniform distribution on both sides of the market).
4.5 Information quality
In the context of accounting and economics literature, the linear correlation between market price and market signal (denoted here by ) is generally employed to assess the quality of accounting information (Lintner 1956; Demsetz 1997; Lev and Zarowin 1999).^{10}^{10}10The relationship between this hypothesis and an equilibrium approach is acknowledged but not investigated here. This section provides results for the distributional characteristics of crosssectional and lagged correlations between these two variables in Tables 5 and 6.
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HRT (mean)  0.8177  0.3988  0.7330  0.8191  0.9059  0.9895 
HRT(median)  0.5030  0.8261  0.8891  0.9444  0.9919  
FVA (mean)  0.9585  0.8157  0.9210  0.9759  0.9980  1 
FVA (median)  0.9380  0.9724  0.9907  1  1  
TRAS (mean)  0.4803  0.0644  0.2735  0.4592  0.6917  0.9698 
TRAS (median)  0.0847  0.3548  0.5667  0.7828  0.9771 
As one can observe, FVA shows high degrees of crosssectional correlation between market price and fundamental signal, both in absolute terms and compared to the HRT and TRAS regimes. Some might be then tempted to conclude that FVA provides better accounting information quality. However, as we have seen in the previous sections, this comes at the cost of higher instability, exuberance and vagary imposed to the financial system dynamics. By taking the theoretical accounting regime as benchmark, only the latter shows some negative crosssectional correlation that is expected when the market price series overshootsandreverts around the fundamental signal series. Contrary to the common interpretation of this variable, higher (average) correlation seems then to imply worse systemic performances, with respect to theoretical common knowledge regime benchmark. On this basis, historical cost accounting regime HRT seems to perform better, since its average (median) correlation is less than fair value accounting regime.
The correlation between the market price at time t () and the lagged fundamental signal () at time (Table 6) points to the forecast power of the common knowledge information over the market price series onestepahead. It is notable that this correlation is unstable over time and contexts, confirming that no trading strategies can reasonably exploit it systematically across situations. Furthermore, the theoretical regime TRAS shows inferior linear forecast power, surely because of market price overshootandrevert effect around its benchmark level. From this perspective, the superior forecast power showed by fair value accounting FVA may be assessed negatively, since it depends on its autocorrelation more than its capacity to drive investment behaviors to fit with benchmark levels of fundamental performance.
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HRT (mean)  0.8265  0.40404  0.74102  0.82773  0.91566  1 
HRT(median)  0.50969  0.83423  0.89812  0.95387  1  
FVA (mean)  0.95928  0.8167  0.92969  0.97717  0.99853  1 
FVA (median)  0.9387  0.97309  0.99184  1  1  
TRAS (mean)  0.51089  0.006575  0.30085  0.48942  0.72541  1 
TRAS (median)  0.016519  0.3866  0.60198  0.81851  1 
5 Conclusive Remarks
Through numerical simulations and visualizations, this paper develops a comprehensive economic analysis of the theoretical model developed by Biondi et al. (2012). It assesses market price formation and behavior under alternative common knowledge regimes coupled with various combinations of speculative and fundamentalist beliefs on supply and demand sides, with a view to financial market stability, volatility, exuberance, vagary and liquidity. These systemic properties prove to be sensitive to regulatory regimes for fundamental information provision that correspond to stylized accounting models of reference for financial reporting and prudential regulation.
Through mere observation of linear correlation between market prices and fundamental signals, some might conclude that common knowledge provision regimes based upon marktomarket measurement of traded security (FVA) represent a better source of public information about the fundamental performance of securityissuing corporate group. However, we show that this high linear correlation comes together with relevant reduction in systemic properties of the financial system as a whole. In particular, FVAbased regimes prove to involve market misplacing and higher volatility. They also imply higher degree of exuberance and errancy (pointing to the disconnection of the market price series from the underlying fundamental signal series of traded security) coupled with lower levels of orders’ satisfaction for both sides of the financial trading (which point to possible liquidity issues). Under FVAbased regimes, therefore, there exists higher likelihood and materiality of financial market instability and bubbling, related to market price erratic behavior over time and contexts.
Our results are particularly significant since the systemic properties of common knowledge regimes under investigation do radically depend on investors’ relative confidence on the market price dynamics. As the latter factor is not generally under control by regulatory bodies which can shape the common knowledge provision regimes, this dependence implies that adoption and implementation of marktomarket regimes may increase likelihood of market bubbling and inefficiency when market conditions become overconfident. Our study thus recommends to test the economic consequences of regulatory designs and control systems under a large set of situations through simulation, experiment and exploratory field studies.
Appendix 1  Summary of common knowledge regimes
Together with the HRT, FVA and TRAS common knowledge regimes (introduced in Equations 3, 4 and 5) we analyze here two further regimes as robustness check of the results for HRT and TRAS respectively.
The Historical Cost Accounting (HCA) regime implies an evolving exogenous signal of fundamental performance that is orthogonal to market price dynamics and results from stochastic positive and negative flows (Biondi 2011c):
(14) 
The Fixed Target Reverting Accounting (TRAF) regime implies instead a reverting fundamental performance signal that targets a fixed core value of reference:
(15) 
Again, for these two accounting regimes the same stochastic error (introduced in Equation 6) is added to account for estimation error, measurement error and other random effects.
Overall, we propose in this paper results for five different mechanisms (summarized in Table 7) generating common knowledge information inspired from distinctive accounting regimes: two belonging to the historical cost accounting model family; one belonging to the fair value (current value, marktomarket) accounting model family and two providing a theoretical benchmark derived from equilibrium approaches, involving targetbased signaling provision. We refer to the first three regimes as actual regimes as they denote stylized practical modes of accounting, while referring to the last two regimes as theoretical regimes that are and can be only thought in a vacuum.
Actual accounting regimes 



Theoretical accounting regimes  Fixed target reverting accounting (TRAF)  
Stochastic target reverting accounting (TRAS) 
Appendix 2  Data tables (comparisons across common knowledge regimes)
In this appendix we reproduce the tables displayed in the main text including also the omitted results for the theoretical Fixed Target Reverting Accounting (TRAF) regime and for the actual Historical Cost Accounting (HCA) regime. Tables 10 and 13 are instead only reported here to confirm results observed through other variables in the main text.
Market Pricing
Signal Type  Mean Std  Min  Q1  Median  Q3  Max 

HCA(mean)  1000.251 3.5212  996.986  1000.098  1000.238  1000.377  1022.292 
HCA (median)  996.7165  1000.098  1000.239  1000.379  1003.872  
HRT (mean)  1000.253 3.6688  996.472  1000.096  1000.237  1000.378  1035.581 
HRT (median)  996.736  1000.098  1000.239  1000.383  1016.071  
FVA (mean)  2.4e+60 2.1e+61  999.557  1000.490  1000.820  2.1e+17  5.7e+63 
FVA (median)  997.108  1000.483  1000.816  4.2e+08  1.2e+32  
TRAF (mean)  1000.154 0.1983  999.963  1000.071  1000.144  1000.232  1000.394 
TRAF (median)  999.962  1000.072  1000.144  1000.232  1000.398  
TRAS (mean)  1000.155 1.7446  999.589  1000.064  1000.154  1000.246  1000.550 
TRAS (median)  999.568  1000.062  1000.154  1000.248  1000.563 
Market Volatility
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HCA(mean)  0.0035  0.0017  0.0021  0.0025  0.0036  0.3205 
HCA ( Peak)  0.0043  0.0020  0.0025  0.0031  0.0043  0.4823 
HRT(mean)  0.0037  0.0018  0.0022  0.0027  0.0037  0.3363 
HRT ( Peak)  0.0045  0.0021  0.0026  0.0032  0.0045  0.4791 
FVA (mean)  1.6436  0.0003  0.0006  0.0010  2.9661  8.7466 
FVA ( Peak)  1.7882  0.0004  0.0007  0.0010  4.3298  8.7466 
TRAF(mean)  0.0002  0.0001  0.0002  0.0002  0.0002  0.0003 
TRAF ( Peak)  0.0002  0.0001  0.0002  0.0002  0.0003  0.0004 
TRAS (mean)  0.0017  0.0012  0.0014  0.0016  0.0019  0.0035 
TRAS ( Peak)  0.0021  0.0014  0.0017  0.0020  0.0024  0.0042 
The volatility width around its median () further comforts the results that regard the Market volatility. Lets define:
(16) 
where Q1, Q2 and Q3 respectively represents the first quartile, the median value and the third quartile of the market volatility distribution. We can observe in Table 10 that historical cost regimes remain in line with performance by theoretical regimes while the fair value regime shows anomalously high values of volatility width.
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HCA  0.5658  0.4821  0.5495  0.5647  0.5809  1.2735 
HRT  0.5668  0.4795  0.5502  0.5655  0.5818  1.0926 
FVA  28.1821  0  0.5529  0.6397  0.6976  3809.6 
TRAF  0.6081  0.4867  0.5667  0.6019  0.6498  0.7624 
TRAS  0.5513  0.4870  0.5362  0.5507  0.5658  0.6376 
Market Exuberance and Vagary
Signal Type  Mean Std  Min  Q1  Median  Q3  Max 

HCA (75% Peak)  0.0022 0.0041  0.0002  0.0014  0.0018  0.0022  0.2647 
HCA ()  2.00e05  2.52e06  1.35e05  1.62e05  1.89e05  0.0018 
HRT (75% Peak)  0.0023 0.0043  0.0002  0.0015  0.0019  0.0023  0.2922 
HRT ()  2.10e05  2.84e06  1.41e05  1.69e05  1.98e05  0.0019 
FVA (75% Peak)  0.1018 0.0160  9.68e05  0.0003  0.0003  0.1085  0.8660 
FVA ()  2.22e04  9.36e07  2.65e06  2.72e06  2.81e04  1.73e03 
TRAF(75% Peak)  0.0005 0.0003  2.49e05  0.0003  0.0005  0.0007  0.0012 
TRAF()  2.09e06  2.83e07  1.41e06  2.12e06  2.78e06  3.67e06 
TRAS (75% Peak)  0.0019 0.0033  0.0002  0.0013  0.0019  0.0025  0.0035 
TRAS()  1.69e05  2.79e06  1.24e05  1.72e05  2.12e05  2.98e05 
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HCA ()  0.2986  0  0.2562  0.3154  0.3575  0.5108 
HCA (length)  16  0  15  16  17  22 
HRT()  0.3022  0  0.2605  0.3186  0.3602  0.5078 
HRT(length)  16  0  16  16  17  21 
FVA()  0.2722  0  0.0456  0.3608  0.3992  0.5180 
FVA(length)  14  0  16  17  18  36 
TRAF()  0.2618  0  0.1888  0.2892  0.3436  0.4822 
TRAF(length)  16  0  16  17  17  20 
TRAS()  0.24186  0  0.166  0.2606  0.3246  0.4446 
TRAS(length)  16  0  15  16  16  22 
Market Liquidity
Signal Type  Mean Std  Min  Q1  Median  Q3  Max 

HCA (mean)  0.7234 0.2173  0.4916  0.6261  0.7121  0.8153  0.9944 
HCA (median)  0.4985  0.6728  0.7767  0.8822  1  
HRT (mean)  0.7232 0.2166  0.4899  0.6263  0.7124  0.8149  0.9944 
HRT (median)  0.4965  0.6728  0.7767  0.8823  1  
FVA (mean)  0.6925 0.1932  0.4205  0.5918  0.6648  0.7854  0.9923 
FVA (median)  0.3727  0.6174  0.7052  0.8372  1  
TRAF (mean)  0.7205 0.2202  0.4855  0.6188  0.7101  0.8153  0.9921 
TRAF (median)  0.4849  0.6627  0.7751  0.8828  1  
TRAS (mean)  0.7208 0.2179  0.4792  0.6184  0.7111  0.8169  0.9944 
TRAS(median)  0.4831  0.6638  0.7761  0.8838  1 
Information Quality
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HCA (mean)  0.8182  0.4124  0.7335  0.8186  0.9067  0.9897 
HCA (median)  0.5306  0.8269  0.8881  0.9453  0.9922  
HRT (mean)  0.8177  0.3988  0.7330  0.8191  0.9059  0.9895 
HRT(median)  0.5029  0.8261  0.8891  0.9444  0.9919  
FVA (mean)  0.9585  0.8157  0.9290  0.9759  0.9981  1 
FVA (median)  0.9380  0.9724  0.9907  1  1  
TRAF (mean)  0.2845  0.3577  0.0258  0.2391  0.5856  0.9695 
TRAF (median)  0.5335  0.0492  0.3230  0.7066  0.9756  
TRAS (mean)  0.4803  0.0645  0.2735  0.4592  0.6917  0.9697 
TRAS (median)  0.0847  0.3548  0.5667  0.7828  0.9771 
Signal Type  Mean  Min  Q1  Median  Q3  Max 

HCA (mean)  0.8270  0.4177  0.7414  0.8277  0.9165  1 
HCA (median)  0.5339  0.83508  0.8972  0.9542  1  
HRT (mean)  0.8265  0.4040  0.7410  0.8277  0.9156  1 
HRT(median)  0.5097  0.8342  0.8981  0.9539  1  
FVA (mean)  0.9593  0.8167  0.9297  0.9772  0.9985  1 
FVA (median)  0.9387  0.9731  0.9918  1  1  
TRAF (mean)  0.3117  0.3464  0.0002  0.2680  0.6183  1 
TRAF (median)  0.5225  0.0193  0.3598  0.7451  1  
TRAS (mean)  0.5109  0.0066  0.3008  0.4894  0.7254  1 
TRAS (median)  0.0165  0.3866  0.6020  0.8185  1 
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