arXiv subjects

Victor Olkhov

Publications and source records attributed to Victor Olkhov.

At least 19 recordsLinked to original sources

arXiv2025

Market-Based Variance of Market Portfolio and of Entire Market

We present the unified market-based description of returns and variances of the trades with shares of a particular security, of the trades with shares of all securities in the market, and of the trades with the market portfolio. We consider the investor who doesn't trade the shares of his portfolio he collected at time t0 in the past. The investor observes the time series of the current trades with all securities made in the market during the averaging interval. The investor may convert these time series into the time series that model the trades with all securities as the trades with a single security and into the time series that model the trades with the market portfolio as the trades with a single security. That establishes the same description of the returns and variances of the trades with a single security, the trades with all securities in the market, and the market portfolio. We show that the market-based variance, which accounts for the impact of random change of the volumes of consecutive trades with securities, takes the form of Markowitz's (1952) portfolio variance if the volumes of consecutive trades with all market securities are assumed constant. That highlights that Markowitz's (1952) variance ignores the effects of random volumes of consecutive trades. We compare the market-based variances of the market portfolio and of the trades with all market securities, consider the importance of the duration of the averaging interval, and explain the economic obstacles that limit the accuracy of the predictions of the returns and variances at best by Gaussian distributions. The same methods describe the returns and variances of any portfolio and the trades with its securities.

econ.GN↗

arXiv2025

Unwitting Markowitz' Simplification of Portfolio Random Returns

In his famous paper, Markowitz (1952) derived the dependence of portfolio random returns on the random returns of its securities. This result allowed Markowitz to obtain his famous expression for portfolio variance. We show that Markowitz's equation for portfolio random returns and the expression for portfolio variance, which results from it, describe a simplified approximation of the real markets when the volumes of all consecutive trades with the securities are assumed to be constant during the averaging interval. To show this, we consider the investor who doesn't trade shares of securities of his portfolio. The investor only observes the trades made in the market with his securities and derives the time series that model the trades with his portfolio as with a single security. These time series describe the portfolio return and variance in exactly the same way as the time series of trades with securities describe their returns and variances. The portfolio time series reveal the dependence of portfolio random returns on the random returns of securities and on the ratio of the random volumes of trades with the securities to the random volumes of trades with the portfolio. If we assume that all volumes of the consecutive trades with securities are constant, obtain Markowitz's equation for the portfolio's random returns. The market-based variance of the portfolio accounts for the effects of random fluctuations of the volumes of the consecutive trades. The use of Markowitz variance may give significantly higher or lower estimates than market-based portfolio variance.

econ.GN↗

arXiv2025

Markowitz Variance May Vastly Undervalue or Overestimate Portfolio Variance and Risks

We consider the investor who doesn't trade shares of his portfolio. The investor only observes the current trades made in the market with his securities to estimate the current return, variance, and risks of his unchanged portfolio. We show how the time series of consecutive trades made in the market with the securities of the portfolio can determine the time series that model the trades with the portfolio as with a single security. That establishes the equal description of the market-based variance of the securities and of the portfolio composed of these securities that account for the fluctuations of the volumes of the consecutive trades. We show that Markowitz's (1952) variance describes only the approximation when all volumes of the consecutive trades with securities are assumed constant. The market-based variance depends on the coefficient of variation of fluctuations of volumes of trades. To emphasize this dependence and to estimate possible deviation from Markowitz variance, we derive the Taylor series of the market-based variance up to the 2nd term by the coefficient of variation, taking Markowitz variance as a zero approximation. We consider three limiting cases with low and high fluctuations of the portfolio returns, and with a zero covariance of trade values and volumes and show that the impact of the coefficient of variation of trade volume fluctuations can cause Markowitz's assessment to highly undervalue or overestimate the market-based variance of the portfolio. Incorrect assessments of the variances of securities and of the portfolio cause wrong risk estimates, disturb optimal portfolio selection, and result in unexpected losses. The major investors, portfolio managers, and developers of macroeconomic models like BlackRock, JP Morgan, and the U.S. Fed should use market-based variance to adjust their predictions to the randomness of market trades.

econ.GN↗

arXiv2025

Market-Based Portfolio Variance

The variance measures the portfolio risks the investors are taking. The investor, who holds his portfolio and doesn't trade his shares, at the current time can use the time series of the market trades that were made during the averaging interval with the securities of his portfolio and assess the current return, variance, and hence the current risks of his portfolio. We show how the time series of trades with the securities of the portfolio determine the time series of trades with the portfolio as a single market security. The time series of trades with the portfolio determine its return and variance in the same form as the time series of trades with securities determine their returns and variances. The description of any portfolio and any single market security is equal. The time series of the portfolio trades define the decomposition of the portfolio variance by its securities, which is a quadratic form in the variables of relative amounts invested into securities. Its coefficients themselves are quadratic forms in the variables of relative numbers of shares of its securities. If one assumes that the volumes of all consecutive deals with each security are constant, the decomposition of the portfolio variance coincides with Markowitz's (1952) variance, which ignores the effects of random trade volumes. The use of the variance that accounts for the randomness of trade volumes could help majors like BlackRock, JP Morgan, and the U.S. Fed to adjust their models, like Aladdin and Azimov, to the reality of random markets.

econ.GN↗

arXiv2024

Expressions of Market-Based Correlations Between Prices and Returns of Two Assets

This paper derives the expressions of correlations between prices of two assets, returns of two assets, and price-return correlations of two assets that depend on statistical moments and correlations of the current values, past values, and volumes of their market trades. The usual frequency-based expressions of correlations of time series of prices and returns describe a partial case of our model when all trade volumes and past trade values are constant. Such an assumptions are rather far from market reality, and its use results in excess losses and wrong forecasts. Traders, banks, and funds that perform multi-million market transactions or manage billion-valued portfolios should consider the impact of large trade volumes on market prices and returns. The use of the market-based correlations of prices and returns of two assets is mandatory for them. The development of macroeconomic models and market forecasts like those being created by BlackRock's Aladdin, JP Morgan, and the U.S. Fed., is impossible without the use of market-based correlations of prices and returns of two assets.

econ.GN↗

arXiv2024

Lower Bounds of Uncertainty of Observations of Macroeconomic Variables and Upper Limits on the Accuracy of Their Forecasts

This paper defines theoretical lower bounds of uncertainty of observations of macroeconomic variables that depend on statistical moments and correlations of random values and volumes of market trades. Any econometric assessments of macroeconomic variables have greater uncertainty. We consider macroeconomic variables as random that depend on random values and volumes of trades. To predict random macroeconomic variables, one should forecast their probabilities. Upper limits on the accuracy of the forecasts of probabilities of macroeconomic variables, prices, and returns depend on the number of predicted statistical moments. We consider economic obstacles that limit by the first two the number of predicted statistical moments. The accuracy of any forecasts of probabilities of random macroeconomic variables, prices, returns, and market trades doesn't exceed the accuracy of Gaussian approximations. Any forecasts of macroeconomic variables have uncertainty higher than one determined by predictions of coefficients of variation of random values and volumes of trades.

econ.GN↗

arXiv2024

Volatility Depends on Market Trades and Macro Theory

We consider the randomness of market trade as the origin of price and return stochasticity. We look at time series of trade values and volumes as random variables during the averaging interval Δ and describe the dependences of market-based volatilities of price and return on the volatilities and correlations of market trade values and volumes. We describe the market-based origin of the lower boundaries of the accuracy of macroeconomic variables and consider, as an example, the accuracy of macroeconomic investments. We highlight that current macroeconomic models describe relations between the 1st order variables determined by sums of trade values or volumes. To predict market-based volatilities of price, return, and volatilities of macroeconomic variables, one should develop econometric methodologies, collect data, and elaborate macroeconomic theories of the 2nd order that model the mutual dependence of the 1st and 2nd order economic variables. The absence of macroeconomic theories of the 2nd order means no economic basis for predictions of market-based volatilities of price and return, as well as volatilities of any macroeconomic variables. In turn, that limits the accuracy of forecasting probabilities of price, return, and the accuracy of macroeconomic variables in the best case by Gaussian distributions.

q-fin.ST↗

arXiv2024

To VaR, or Not to VaR, That is the Question

We consider economic obstacles that limit the reliability and accuracy of value-at-risk (VaR). Investors who manage large market transactions should take into account the impact of the randomness of large trade volumes on predictions of price probability and VaR assessments. We introduce market-based probabilities of price and return that depend on the randomness of market trade values and volumes. Contrary to them, the conventional frequency-based price probability describes the case of constant trade volumes. We derive the dependence of market-based price volatility on the volatilities and correlation of trade values and volumes. In the coming years, that will limit the accuracy of price probability predictions to Gaussian approximations, and even the forecasts of market-based price volatility will be inaccurate and highly uncertain.

econ.GN↗

arXiv2024

Theoretical Economics as Successive Approximations of Statistical Moments

This paper studies the links between the descriptions of macroeconomic variables and statistical moments of market trade, price, and return. The randomness of market trade values and volumes during the averaging interval Δ results in the random properties of price and return. We describe how averages and volatilities of price and return depend on the averages, volatilities, and correlations of market trade values and volumes. The averages, volatilities, and correlations of market trade, price, and return can behave randomly during the long interval Δ2>>Δ. To describe their statistical properties during the long interval Δ2, we introduce the secondary averaging procedure of trade, price, and return. We explain why, in the coming years, predictions of market-based probabilities of price and return will be limited by Gaussian distributions. We discuss the roots of the internal weakness of the commonly used hedging tool, Value-at-Risk, that cannot be solved and remains the source of additional risks and losses. One should consider theoretical economics as a set of successive approximations, each of which describes the next array of the n-th statistical moments of market trades, price, return, and macroeconomic variables, which are repeatedly averaged during the sequence of increasing time intervals.

econ.GN↗

arXiv2024

Economic Complexity Limits Accuracy of Price Probability Predictions by Gaussian Distributions

We discuss the economic reasons why the predictions of price and return statistical moments in the coming decades, in the best case, will be limited by their averages and volatilities. That limits the accuracy of the forecasts of price and return probabilities by Gaussian distributions. The economic origin of these restrictions lies in the fact that the predictions of the market-based n-th statistical moments of price and return for n=1,2,.., require the description of the economic variables of the n-th order that are determined by sums of the n-th degrees of values or volumes of market trades. The lack of existing models that describe the evolution of the economic variables determined by the sums of the 2nd degrees of market trades results in the fact that even predictions of the volatilities of price and return are very uncertain. One can ignore existing economic barriers that we highlight but cannot overcome or resolve them. The accuracy of predictions of price and return probabilities substantially determines the reliability of asset pricing models and portfolio theories. The restrictions on the accuracy of predictions of price and return statistical moments reduce the reliability and veracity of modern asset pricing and portfolio theories.

q-fin.GN↗

arXiv2024

Theoretical Economics and the Second-Order Economic Theory. What is it?

The economic and financial variables of economic agents determine macroeconomic variables. Current models consider agents' variables that are determined by the sums of values and volumes of agents' trades during some time interval Δ. We call them first-order economic variables. We describe how the volatilities and correlations of market trade values and volumes determine price volatility. We argue that such a link requests consideration of agents' economic variables of the second order that are composed of sums of squares of agents' transactions during Δ. Almost any variable of the first order should be complemented by its second-order pair. Respectively, the sums of agents' second-order variables introduce macroeconomic variables of the second order. The description of the first- and second-order macroeconomic variables establishes the subject of second-order economic theory. We highlight that the complexity of second-order economic theory essentially restricts any hopes for precise predictions of price probability and, at best, could provide estimates of price volatility. That limits the predictions of price probability to Gauss's approximations only.

econ.TH↗

arXiv2024

Price and Payoff Autocorrelations in a Multi-Period Consumption-Based Asset Pricing Model

This paper highlights the hidden dependence of the basic pricing equation of a multi-period consumption-based asset pricing model on price and payoff autocorrelations. We obtain the approximations of the basic pricing equation that describe the mean price "to-day," mean payoff "next-day," price and payoff volatilities, and price and payoff autocorrelations. The deep conjunction of the consumption-based model with other versions of asset pricing, such as ICAPM, APM, etc. (Cochrane, 2001), emphasizes that our results are valid for other pricing models.

econ.GN↗

arXiv2024

Market-Based Price Autocorrelation

This paper assumes that the randomness of market trade values and volumes determines the properties of stochastic market prices. We derive the direct dependence of the first two price statistical moments and price volatility on statistical moments, volatilities, and correlations of market trade values and volumes. That helps describe the dependence of market-based price autocorrelation between times t and t-τ on statistical moments and correlations between trade values and volumes. That highlights the impact of the randomness of the size of market deals on price statistical moments and autocorrelation. Statistical moments and correlations of market trade values and volumes are assessed by conventional frequency-based probabilities. The distinctions between market-based price autocorrelation and autocorrelation that are assessed by the frequency-based probability analysis of price time series reveal the different approaches to the definitions of price probabilities. To forecast market-based price autocorrelation, one should predict the statistical moments and correlations of trade values and volumes.

econ.GN↗

arXiv2024

Market-Based "Actual" Returns of Investors

We describe how the market-based average and volatility of the "actual" return, which the investors gain within their market sales, depend on the statistical moments, volatilities, and correlations of the current and past market trade values. We describe three successive approximations. First, we derive the dependence of the market-based average and volatility of a single sale return on market trade statistical moments determined by multiple purchases in the past. Then, we describe the dependence of average and volatility of return that a single investor gains during the "trading day." Finally, we derive the market-based average and volatility of return of different investors during the "trading day" as a function of volatilities and correlations of market trade values. That highlights the distribution of the "actual" return of market trade and can serve as a benchmark for "purchasing" investors.

econ.GN↗

arXiv2024

Three Remarks On Asset Pricing

We consider the consumption-based asset pricing model, derive a new modified basic pricing equation, and present its successive approximations using the Taylor series expansions of the investor's utility during the averaging time interval. For linear and quadratic Taylor approximations, we derive new expressions for the mean price, mean payoff, volatility, skewness, and the asset's amount that define the maximum of the investor's utility. We discuss the market-based origin of price probability. We use volume weighted average price (VWAP) as a market-based average price and introduce market-based price volatility. The use of VWAP results in zero correlations between the price p and trade volume U. We derive a correlation between price p and squares of trade volume {U^2} and between squares of price {p^2} and volume {U^2}. To predict market-based price volatility, one should forecast the 2-d statistical moments of the market trade values and volumes at the same horizon T.

econ.GN↗

arXiv2023

Market-Based Probability of Stock Returns

This paper describes the dependence of market-based statistical moments of returns on statistical moments and correlations of the current and past trade values. We use Markowitz's definition of value weighted return of a portfolio as the definition of market-based average return of trades during the averaging period. Then we derive the dependence of market-based volatility and higher statistical moments of returns on statistical moments, volatilities, and correlations of the current and past trade values. We derive the approximations of the characteristic function and the probability of returns by a finite number q of market-based statistical moments. To forecast market-based average and volatility of returns at horizon T, one should predict the first two statistical moments and correlation of current and past trade values at the same horizon. We discuss the economic reasons that limit the number of predicted statistical moments of returns by the first two. That limits the accuracy of the forecasts of probability of returns by the accuracy of the Gaussian approximations. To improve the reliability of large macroeconomic and market models like BlackRock's Aladdin, JP Morgan, and the U.S. Fed., the developers should use market-based statistical moments of returns.

econ.GN↗

arXiv2022

Why Economic Theories and Policies Fail? Unnoticed Variables and Overlooked Economics

Accuracy of economic theories and efficiency of economic policy strictly depend on the choice of the economic variables and processes mostly liable for description of economic reality. That states the general problem of assessment of any possible economic variables and processes chargeable for economic evolution. We show that economic variables and processes described by current economic theories constitute only a negligible fraction of factors responsible for economic dynamics. We consider numerous unnoted economic variables and overlooked economic processes those determine the states and predictions of the real economics. We regard collective economic variables, collective transactions and expectations, mean risks of economic variables and transactions, collective velocities and flows of economic variables, transactions and expectations as overlooked factors of economic evolution. We introduce market-based probability of the asset price and consider unnoticed influence of market stochasticity on randomness of macroeconomic variables. We introduce economic domain composed by continuous numeric risk grades and outline that the bounds of the economic domain result in unnoticed inherent cyclical motion of collective variables, transactions and expectations those are responsible for observed business cycles. Our treatment of unnoticed and overlooked factors of theoretical economics and policy decisions preserves a wide field of studies for many decades for academic researchers, economic authorities and high-level politicians.

econ.GN↗

arXiv2022

Market-Based Asset Price Probability

The random values and volumes of consecutive trades made at the exchange with shares of security determine its mean, variance, and higher statistical moments. The volume weighted average price (VWAP) is the simplest example of such a dependence. We derive the dependence of the market-based variance and 3rd statistical moment of prices on the means, variances, covariances, and 3rd moments of the values and volumes of market trades. The usual frequency-based assessments of statistical moments of prices are the limited case of market-based statistical moments if we assume that all volumes of consecutive trades with security are constant during the averaging interval. To forecast market-based variance of price, one should predict the first two statistical moments and the correlation of values and volumes of consecutive trades at the same horizon. We explain how that limits the number of predicted statistical moments of prices by the first two and the accuracy of the forecasts of the price probability by the Gaussian distribution. This limitation also reduces the reliability of Value-at-Risk by Gaussian approximation. The accounting for the randomness of trade volumes and the use of VWAP results in zero price-volume correlations. To study the price-volume empirical statistical dependence, one should calculate correlations of prices and squares of trade volumes or correlations of squares of prices and volumes. To improve the accuracy and reliability of large macroeconomic and market models like those developed by BlackRock's Aladdin, JP Morgan, and the U.S. Fed., the developers should explicitly account for the impact of random trade volumes and use market-based statistical moments of asset prices.

econ.GN↗