## What is it about?

Canonically, stock prices are supposed to be determined by rational expectations that are formed by investors in respect of future cash flows and the riskiness of future cash flows. In stated respect, if a stock experiences excess demand - demand outstripping supply - it ought to be because relative to sizes of cash flows, risk is low. In context, as such of rational expectations, only a stock whose valuation already is estimated to be high experiences excess demand, as such arrives at an even higher valuation, that is, arrives at a 'rational valuation bubble' (Stiglitz 1990). So long as such a firm remains innovative in some timely fashion, there is arrival at maintenance of the rational valuation bubble; there is not as such any risk induced by a rational valuation bubble that has credibility, that is, that is premised on right estimates of future cash flows and risk. For concreteness, as recently is established in Obrimah (2022), if stock prices conform with rational expectations, always excess demand is predicated on outcomes of estimates of the extent to which a stock facilitates risk sharing for investors. Relative then to risk-return trade-offs, which have necessarily a first-order effect on stock valuations, demand, equivalently excess demand has always only a second-order (secondary) effect on stock prices. The evidence from stock markets indicates quite strongly, however, non-conformance of stock prices with the canon, that is, finds stock valuations embed lotteries, as such embed prices that are not premised on risk-return trade-offs, equivalently stock prices that are not premised on rational expectations. Relevant studies are inclusive of Barberis, Huang and Thaler (2006), Barberis and Huang (2008), Kumar (2009), and Chiu (2010). Importantly, Cambell and Shiller (1988) do not find any evidence that stock prices are premised on their risk-return trade-offs, rather find they are premised on consumption aspirations of investors, as such premised on sentiments and subjective priors or objectives of investors. It is straightforward that valuation of stocks with reference to consumption aspirations of investors implies, necessarily stock prices that deviate from implications of stocks' risk-return trade-offs. It further is true that valuation of stocks with reference to consumption aspirations violates the principle of 'Fisherian Separation', namely an economy functions best when decision making in respect of consumption aspirations are separated from decision making in respect of investment activities. The evidence is so bad, Arthur, Holland, LeBaron, Palmer, and Tayler (2018) refer to current formulations of stock markets as, 'Artificial Stock Markets'. This study provides a formal theoretical fundamental rationalization that is non-behavioral for why, regardless of market efficiency, stock prices do not conform with rational expectations, that is, are not representative of stocks' risk-return trade-offs.

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## Why is it important?

If we do not understand why exactly stock prices deviate from the direct valuation implications of their risk-return trade-offs, clearly we are unable to arrive at any capacity for remedying of the deviation, a deviation that is source of high return volatilities that subsist in stock markets. The formal theory in this study shows modeling of stock returns in continuous time implies incapacity of a determination as to whether stock prices conform, or not with rational expectations. In stated respect, it is straightforward that the convention in Finance and Economics is modeling of stock returns in continuous time. For concreteness, in presence of modeling of stock returns in continuous time, prices that emerge in stock markets are not functions of risk-return trade-offs, rather are, in entirety functions of outcomes of trading, that is, the extent to which there is arrival at either of excess demand or excess supply. Interestingly, with the studies providing solely empirical evidence, Stoll (2000) and Kim and Stoll (2014) provide exact corroboration for the formal theoretical predictions in this study. The intuition for how exactly stock returns can evolve (asymptotically) efficiently, yet stock prices are premised solely on demand and supply forces is as follows. If stock prices are modeled in discrete time, there is arrival at two dichotomous conditions which signal that investors ought to form a higher or lower price. Once the conditions are ascertained, regardless of materialization of either of excess demand or supply, the next stock price must conform with the recommendation of either of the two conditions. For concreteness, whereas the two conditions are mathematical, their satisfaction, or otherwise is arrived at by investors. The conditions are objective enough, however, that with publicly available information as the standard, which of course is the standard for market efficiency, there exists feasibility of agreement among all investors as to the necessary directionality of the change to any stock price. While some may wonder at recommendation of modeling of stock returns in continuous time, consider that Merton (1976) asserts quite clearly that each of stock prices or returns, or option prices ought ideally be modeled in discrete time. Given the 1997 Nobel Prize Laureate in Economics, Prof. Robert Merton could not at that time himself figure out how to model stock returns in discrete time (most academics did not, at that time, have exposure to the tools necessary, namely tools of Pure Mathematics, as opposed to tools of Calculus; this study succeeds only because it adopts the tools of Pure Mathematics), he did the next best thing, in Merton (1973), he arrived at a discrete result, namely the Intertemporal Capital Asset Pricing Model (ICAPM), which as you probably can guess mostly is tested in continuous time using sophisticated GARCH models. Further, himself - in Merton (1976b) - and Harrison, Pitbladdo, and Schaefer (1984) maintain that an option pricing model is most robust whenever it embeds some discontinuity, that is, 'jumps to the underlying asset price', as such assert that modeling in continuous time, which in of itself does not allow for discontinuities, is less amenable to the modeling of option prices than modeling in discrete time. Given this study is the first to come up with a rubric for the modeling of stock returns in discrete time, no one can be blamed for the less robust convention, which is modeling of stock returns in continuous time. Now, however, there is opportunity to build better models via explorations of the modeling of any of asset prices, asset returns, or option prices in discrete time. For illustration, suppose arrival at demand for a higher price as outcome of rational expectations. If there is excess demand, forecast increase is accentuated. If there is excess supply, the forecast increase is diminished, but the next price still is demanded to be a higher price than the current price. Suppose, on the contrary, demand for a lower price as outcome of rational expectations. Demand for a lower price is rational whenever investors infer, ex post, a mistake in the formation of the current price. If there is excess supply, forecast decrease is accentuated. If, due to liquidity trading or speculation there is excess demand, forecast decrease is mitigated, but the next price still is demanded by market makers to be a lower stock price. The two conditions that facilitate the constraining of stock prices to conform with rational expectations do not exist in context of modeling of stock returns in continuous time. There is not, as such, any mechanism for ascertaining whether stock prices conform, or not with rational expectations. In absence of the two conditions - which for a fact do not exist in public equity markets as market mechanisms of which investors have awareness - investors premise their sense of fairness of stock prices on their subjective priors or sentiments, hence the sense that stock prices are shaped by behaviors, as opposed to risk-return trade-offs that are conditioned on rational preference parameters of investors.

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This page is a summary of: MODELING OF STOCK RETURNS IN CONTINUOUS VIS-À-VIS DISCRETE TIME IS EQUIVALENT, RESPECTIVELY, TO THE CONDITIONING OF STOCK RETURNS ON A RANDOM WALK PROCESS FOR TRADE IMBALANCES VIS-À-VIS A RANDOM WALK PROCESS FOR EVOLUTION OF INFORMATION, Annals of Financial Economics, May 2022, World Scientific Pub Co Pte Lt,

DOI: 10.1142/s2010495222500105.

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