Discrete Price Changes Research Articles

Efficient Importance Variational Approximations for State Space Models

Variational methods are a potentially scalable estimation approach for state space models. However, existing methods are inaccurate or computationally infeasible for many state space models. This article proposes a variational approximation that is accurate and fast for any model with a closed-form measurement density function and a state transition distribution within the exponential family of distributions. Our approach constructs a variational approximation to the states that is close to the exact conditional posterior distribution of the states using insights from the efficient importance sampling literature. We show that our method can accurately and quickly estimate a multivariate Skellam stochastic volatility model with high-frequency tick-by-tick discrete price changes of four stocks.

An intraday GARCH model for discrete price changes and irregularly spaced observations

An intraday GARCH model for discrete price changes and irregularly spaced observations

Equivalence between substitutability and M^natural -concavity for set functions under discrete transfers

Fujishige and Yang (2003) prove the equivalence between two fundamental conditions of a valuation function in a market model with indivisibilities, i.e., the gross substitutes (GS) condition and M^natural -concavity. We introduce a weaker variant of the GS condition that concerns discrete price changes rather than continuous price changes. We show that this weaker variant is equivalent to M^natural -concavity if the valuation function takes integer values and has an M^natural -convex effective domain containing the empty set. Our result indicates that assuming the weaker GS condition is sufficient for M^natural -concavity in existing auction models.

Dynamic Discrete Mixtures for High-Frequency Prices

The tick structure of the financial markets entails discreteness of stock price changes. Based on this empirical evidence, we develop a multivariate model for discrete price changes featuring a mechanism to account for the large share of zero returns at high frequency. We assume that the observed price changes are independent conditional on the realization of two hidden Markov chains determining the dynamics and the distribution of the multivariate time series at hand. We study the properties of the model, which is a dynamic mixture of zero-inflated Skellam distributions. We develop an expectation-maximization algorithm with closed-form M-step that allows us to estimate the model by maximum likelihood. In the empirical application, we study the joint distribution of the price changes of a number of assets traded on NYSE. Particular focus is dedicated to the assessment of the quality of univariate and multivariate density forecasts, and of the precision of the predictions of moments like volatility and correlations. Finally, we look at the predictability of price staleness and its determinants in relation to the trading activity on the financial markets.

Intraday Stochastic Volatility in Discrete Price Changes: The Dynamic Skellam Model

ABSTRACTWe study intraday stochastic volatility for four liquid stocks traded on the New York Stock Exchange using a new dynamic Skellam model for high-frequency tick-by-tick discrete price changes. Since the likelihood function is analytically intractable, we rely on numerical methods for its evaluation. Given the high number of observations per series per day (1000 to 10,000), we adopt computationally efficient methods including Monte Carlo integration. The intraday dynamics of volatility and the high number of trades without price impact require nontrivial adjustments to the basic dynamic Skellam model. In-sample residual diagnostics and goodness-of-fit statistics show that the final model provides a good fit to the data. An extensive day-to-day forecasting study of intraday volatility shows that the dynamic modified Skellam model provides accurate forecasts compared to alternative modeling approaches. Supplementary materials for this article are available online.

Healthcare Demand in the Presence of Discrete Price Changes.

Deductibles in health insurance generate nonlinear budget sets and dynamic incentives. Using detailed individual health expenditure data from a Swiss health insurer, we estimate the response in healthcare demand to the discrete price increase generated by resetting the deductible at the start of each calendar year. We find that for individuals with high deductibles, healthcare demand drops by 27%. The decrease is most pronounced for inpatient care and prescription drugs. By contrast, for individuals with low deductibles, there is no significant change in healthcare demand (except for prescription drugs). Overall our results suggest that healthy individuals respond much stronger to the price change.

Estimating the determinants of stock price changes

I develop an approach for estimating the determinants of stock price changes that uses all eligible trade data and other observable parameters of market activity. This approach backs out the unobserved continuous price change distribution from the observable discrete price changes, and does not constrain the determinants to be proportions of the traded bid-ask spread. I show that theoretically impermissible results and skewed estimates of cost components are obtained when the model used for estimating the determinants of stock price changes does not attempt to uncover the mapping between the observed price changes and the underlying unobserved continuous price change process, and does not effectively use all eligible trade data.

Heterogeneous demand responses to discrete price changes: an application to the purchase of lottery tickets

During the survey period of any household expenditure survey price variations may occur. Such variation can be used to identify heterogeneous demand responses to price changes. This is feasible because expenditure surveys usually contain a large number of observations. The principal difficulty for estimation arises because of the sampling process which generates the data. An estimable model of individual purchase is presented and is estimated in the case of the demand for lottery tickets. This model allows identification of heterogeneous responses to changes in the rollover state (i.e. whether last week's jackpot has been added to the current jackpot). The distribution of heterogeneous responses, in the form of a bivariate mixing distribution, is shown to be identified from the available data. The EM algorithm is used to estimate the parameters of the mixing distribution. The estimates imply that there is substantial heterogeneity in the population both in the normal expenditure levels and in the reaction to a jackpot rolled over.

Estimating the Determinants of Stock Price Changes

I develop an approach for estimating the determinants of stock price changes that uses all eligible trade data and other observable parameters of market activity. This approach backs out the unobserved continuous price change distribution from the observable discrete price changes, and does not constrain the determinants to be proportions of the traded bid-ask spread. I show that theoretically impermissible results and skewed estimates of cost components are obtained when the model used for estimating the determinants of stock price changes does not attempt to uncover the mapping between the observed price changes and the underlying unobserved continuous price change process, and does not effectively use all eligible trade data.

On the Hicks–Allen Definition of Complements and Substitutes with Discrete Changes in Prices

For discrete price changes, compensated cross‐price effects for two goods need not be equal if the household consumes three or more goods. In this paper I ask whether compensated cross‐price effects must have the same sign even if they differ in magnitude. I show that with three or more goods, the answer to this question is ‘no’. For discrete price changes, with more than two goods, the signs of the compensated cross‐price effects for two goods can differ depending on which price changes. Hence, the Hicks–Allen definition of complements and substitutes can also differ depending on which price changes.