روش نوسانات احتمالی سانسور شده برای برآورد حرکت های حد مجاز قیمت

A censored stochastic volatility model is developed to reconstruct a return series censored by price limits, one popular form of market stabilization mechanisms. When price limits are reached, the observed prices are truncated and the equilibrium prices are unobservable, which makes further financial analyses difficult. The model offers theoretically sound estimates of censored returns and is demonstrated via simulations to outperform existing approaches with respect to the estimates of model parameters, unconditional means, and standard deviations. The algorithm is applied to model stock and futures returns and results are consistent with the simulation outcomes.

One of the major challenges faced by all empirical price limit researchers is the appropriate handling of price limit moves. In essence, none of the empirical studies that examine financial data subject to price limits are immune from this challenging task. When price limits are hit, the observed prices are truncated and the equilibrium prices are unobservable, which makes it difficult to calculate true returns and volatilities for further financial analyses.3 Existing studies do not provide a generally accepted solution. We contribute to the literature by developing a censored stochastic volatility (CSV) model to capture important features of a return series censored by price limits. Price limits are imposed in many financial markets to prevent excessive price fluctuation.4 Although price limit rules vary from country to country and market to market, broadly speaking, they may be defined as boundaries set by market regulators to restrict daily security price changes. Because prices are constrained, well-established financial analytical methods and models, such as a mean-variance analysis or GARCH (generalized autoregressive conditional heteroscedasticity) modeling, may create biased results. In their comprehensive literature review, Kim and Yang (2004) conclude that it is necessary for future studies to develop models that might overcome potential biases in the statistical inference that are due to the constrained return-generating process. Price-limited data share similar characteristics—such as heteroscedasticity and volatility clustering—with regular asset return data, but differ in two important ways. First, equilibrium prices are unobservable when price limits are hit, and observable prices are truncated at the limit. Second, when price limits are hit, the return beyond price limits must be reflected on the next trading day; thus, the unrealized return spills over to the following day, creating the spillover effect.5Fig. 1 provides a simple illustration of these two distinct features. At time t, the observed return rto equals the equilibrium return rt because it is within the price limit boundary. 6 At the next time period t + 1, the equilibrium return rt + 1 exceeds the upper price limit, but due to the price limit mechanism, we can only observe rt + 1o. Because of the price limit constraint, the amount of the unrealized return Et + 1 spills over to time t + 2. Hence, though the equilibrium return rt + 2 at time t + 2 falls below the lower limit, the observed return rt + 2o lies between the limits if the spillover portion Et + 1 is included. Full-size image (33 K) Fig. 1. Unobservability of price-limited data and the spillover effect. Figure options Our CSV model accounts for the unobservability and spillover effect associated with price limits.7 Although several models have been developed to recognize the unobservable feature of price-limited data, to our knowledge, only Wei's (2002) censored-GARCH model has incorporated the spillover effect. However, as Wei (2002) acknowledges, the disadvantage of the censored-GARCH model is the long running time of its procedure, which makes it impractical to design a Monte Carlo simulation to verify its claimed performance. We are able to run simulations to show the superior performance of our CSV model because of the simple conditional likelihood function of its parameters and the resulting efficient algorithm for parameter estimation. We use a theoretically sound imputation method to generate fill-ins for censored observations and thereby create an imputed return series that is appropriate for subsequent financial analyses, such as option pricing or asset allocations. As suggested by Jacquier et al. (1994) and Andersen et al. (1999), we use the Markov chain Monte Carlo (MCMC) technique for efficient parameter estimation. Because our model addresses the spillover effect explicitly, a complete return series can be imputed even if price limits are reached on consecutive days, a situation that occurs in practice but cannot be handled appropriately by existing models. The stochastic volatility (SV) models have recently gained popularity in modeling financial asset returns. Durham (2007) shows that SV models do a good job capturing the dynamic of volatility and the shape of the conditional distribution of financial asset returns. Nardari and Scruggs (2007) also utilize the SV process to capture the time-varying covariance matrix of returns in evaluating empirical implications of asset pricing theory. Our CSV model provides a realistic and flexible modeling of financial time series data in that it involves two noise processes and allows error terms to be correlated. Given the increasing adoption of SV models in empirical financial studies, our CSV model provides researchers an appropriate and efficient approach to handle price-limited data. We compare our model with several existing approaches through a Monte Carlo simulation. The results suggest that our model outperforms other approaches with respect to the estimation of model parameters, the unconditional means, and the standard deviations. When the price limits are set at ± 7%, our CSV model recovers censored returns and gives an estimate of standard deviation with less than 1% error. However, the standard deviation is underestimated by 5% when observed prices are used to calculate returns and by 14% when limited prices are deleted from the sample. The consequences of such underestimation can be substantial. First, an underestimation of volatility generally translates into the underpricing of an option, which can be costly to option traders. Second, the underestimation of risk affects asset allocation decisions by portfolio managers and capital budgeting decisions by corporate managers. In addition, results from empirical studies that use the underestimated standard deviation are likely to be biased. We demonstrate the usefulness of our algorithm by modeling the returns of two actively traded stocks on the Taiwan Stock Exchange (TSE) and two U.S. futures contracts on the Chicago Board of Trade (CBOT) during volatile periods when price limit moves are more likely to occur. To examine the robustness of our results, we compare the performance of our CSV approach with that of other existing models using the S&P 500 Index returns and the GARCH-generated returns. Overall, our result that the CSV approach outperforms other existing models is robust. We organize the remainder of this article as follows: In the next section, we review existing methods of handling and modeling price-limited data in the literature. In Section 3, we describe our model and estimation approach. Then in Section 4, we present the simulation design and results, followed by some empirical results for stock and futures returns in Section 5. We perform robustness check in Section 6 and conclude in Section 7.

In this paper,we propose a CSV approach to model the return process of assets that are subject to price limits. When price limits are reached, the observed prices are truncated and the equilibrium prices are unobservable, which makes it difficult to calculate the true returns and volatilities for any further financial nalyses. Although several approaches and models have been suggested and applied, they are either unable to account for both the unobservability and the spillover effect associated with price limits or too computationally complicated to be feasible for simulation studies. Because we use a theoretically sound imputation method to generate fill-ins for censored observations, we are able to simulate the distribution of each unobserved return in a return series. These imputed returns and distributions are then appropriate for subsequent financial analyses, such as option pricing, risk management, and asset allocations, as we have demonstrated in calculating the preceding unconditional standard deviations. To assess the performance of our model, we compare it with the ostrich, discarding, and GMM approaches through an MCMC simulation. The results of our simulation analysis suggest that our model outperforms other approaches with respect to the estimates of model parameters, unconditional means, and standard deviations. We further apply our algorithm to model the returns of two stocks and two futures contracts that are subject to price limits and find that the results are consistent with our simulation outcomes. Overall, our model is not only theoretically sound but also empirically applicable. Because of the better performance of our model, we suggest it should be used to model return series that are subject to price limits. Many financial markets in the world impose price limits; therefore, international portfolio managers might use our model to obtain better estimates of risk for financial assets. Ignoring the existence of price limits or deleting limit-hit observations leads to an underestimation of asset risks, which may incur unnecessary losses due to mismanaged portfolios and asset allocations. Corporate managers also might use our model to create a full return series for their company stocks to obtain appropriate estimates of the cost of capital for their capital budgeting analyses. To study the impact of price limit rules, regulators and researchers could use our model to conduct extensive simulation studies. Although it is relatively difficult to test the effectiveness of price limits, because we simply do not know what would have happened without price limits, the estimation from our model at least provides researchers with a benchmark that can be used to evaluate the performance of price limits. It should be noted that our CSV model has its limitation. For example, in a market where price limits may be widen on a given day, our model is not appropriate. A good example is the Spanish Stock Exchange where price limits trigger trading halts. Regulators constantly monitor the market and decide whether to widen the price limits after trading halts. Our model treats price limits as given on any given day and thus is unable to capture the widened price limits when that happens.