مشخصات مدل نوسانات تصادفی همراه با عیب شناسی برای بازارهای سهام معامله کوچک

The majority of world equity markets exhibit non-synchronous- and non-trading for some quoted asset series. This investigation sets out to determine the complexity of illiquid markets applying versions of stochastic differential equation (SDE) specifications. Efficient method of moments (EMM) is used to estimate and evaluate the diffusion models. EMM estimation on SNP scores reveals that a standard SDE with first and second order drift, non-synchronous trading and constant diffusion are rejected. However, a simple two-factor stochastic volatility specification with non-synchronous trading and conditional heteroscedasticity, reports success for illiquid equity markets. Tracking portfolios may therefore not track derivative products perfectly and importantly; the stochastic volatility model seems to be the preferred volatility specification.

In this paper, we construct, estimate and test several stochastic differential equations (SDE) for a seasonal adjusted index series from a thinly traded equity market. The main objective is to find SDE characteristics in the mean and volatility specifications, due to non-synchronous- and non-trading and conditional heteroscedasticity. Non-synchronous- and non-trading may cause biases to the moments and co-moments of returns owing to irregular recording intervals. Conditional heteroscedasticity is caused by volatility clustering and suggests that volatility changes over time. The motivation for the use of stochastic calculus in estimation studies is that new (unpredictable) information is revealed continuously in an open market and decision-makers may face instantaneous changes in randomness. For example, the relevant ‘time interval’ may be different on different trading days, due to volatility changes. Changing volatility may require changing the basic observation period. Furthermore, as numerical methods used in pricing securities are costly, the pace of activity may make the analyst choose coarser or finer time intervals depending on the level of volatility. Such approximations can best be accomplished using random variables defined over continuous time. The mathematics of such random variables is known as stochastic calculus. A technical advantage of stochastic calculus is that a complicated random variable can have a very simple structure in continuous time, once the attention is focused on infinitesimal intervals. If the time interval is ‘infinitesimal’ then asset prices may safely be assumed to have two likely movements; up-tick or downtick1. Under some conditionals, such a binomial structure may be a good approximation to reality during an infinitesimal interval, but not necessarily in a large discrete time interval. Solibakke (2000) shows in an empirical study from the Norwegian thinly traded market that volatility seems to exist independently of trading and non-trading as long as the market is open. Hence, the volatility process seems to exist without any relations to the mean process. Moreover, Solibakke (2001b) finds that changing volatility models (GARCH)2 report specification errors for return series exhibiting heavy non-synchronous- and non-trading. Consequently, the stochastic volatility specification seems to be the preferred volatility specification in thinly traded markets. Finally, the main tool of stochastic calculus–namely, the Ito integral–may be more appropriate to use in financial markets than the standard Riemann integral (Neftci, 1996). Several procedures have been proposed for fitting a model based on stochastic calculus3. In this paper we employ the efficient method of moments (EMM) proposed by Bansal et al., 1993 and Bansal et al., 1995 and developed in Gallant and Tauchen (1996) and shown used in Gallant et al. (1997) to estimate and test the SDE model. EMM is a simulation-based moment matching procedure with certain advantages. The moments that get matched are the scores of an auxiliary model called the score generator (SNP). SNP is a method of non-parametric time series analysis, which employs a Hermite polynomial series expansion to approximate the conditional density of a multivariate process. An appealing feature of this expansion is that it is a non-linear non-parametric model that directly nests the independently and identically distributed (i.i.d.) Gaussian model, the Gaussian VAR model, the semi-parametric VAR model, the Gaussian ARCH model, the semi-parametric ARCH model and the non-linear non-parametric model. The SNP model is fitted using conventional maximum likelihood together with a model selection strategy that determines the appropriate degree of the polynomial. If this score generator approximates the distribution of the data well, then estimates of the parameters of the SDE are as efficient as if maximum likelihood had been employed (Gallant and Long, 1995 and Tauchen, 1996). Failure to match these moments can be used as a statistical specification test procedure and, more importantly, can be used to indicate features of series that the SDE model cannot accommodate (Tauchen, 1995). Based on the preferred score model for the index series, our objective is to establish and interpret the EMM objective function surface across a comprehensive set of specifications of the SDE model. We start with a standard one factor constant volatility specification, proceed to standard stochastic volatility specifications, introduce conditional heteroscedasticity, and if needed, introduce asymmetry and employ more elaborate specifications. The EMM effort is therefore aimed at generating a comprehensive accounting of how well the SDE model and its extensions accommodate features of thinly traded series. Our approach differs from typical practice in the stochastic volatility literature, as the EMM procedure permits an exhaustive specification analysis. Hence, by employing EMM, we can confront all of the various extensions, individually and jointly, to a judiciously chosen set of moments determined by a non-parametric specification search of the score generator (SNP). The rest of the paper is organised as follows. Section 2 defines the dynamics of SDE. Section 3 defines the series and describes adjustment procedures. Section 4 expands sequentially the score generator. Section 5.1 reports our time series Score characteristics and Section 5.2 reports the empirical stochastic volatility model results. Finally, Section 6 reports the empirical findings for thinly traded markets and Section 7 summarises and concludes.

This investigation uses efficient methods of moments estimation to estimate parameters for observed mean returns and latent stochastic volatility processes in continuous time for the Norwegian value weighted index series. Our results suggest success for a general and a simple two-equation SDE model that incorporates an induced conditional heteroscedasticity specification in the drift for the mean process and both for the drift and volatility diffusion for the volatility process. The SDE model that is employed for the successful estimation is specified as SD-CHDD in Table 5, representing a two-factor model for the Norwegian index. As also shown by Tauchen, 1997 and Andersen and Lund, 1998 a two-factor model is needed, as a single factor model cannot quite capture all the dynamics in complex equity markets. Moreover, results from the liquid US market seem to suggest a need for about the same SDE models (Gallant and Tauchen, 1997). Liquidity seems therefore not to influence the SDE complexity. As the option pricing formulas incorporate functions of the parameters of the process driving the volatility of stock returns, the result may have some practical relevance. Our results suggest that we need to replace σ with View the MathML source over the option's life in the Black and Scholes formula and even more importantly, it may not be possible to determine a tracking portfolio and therefore a price of the option by arbitrage arguments. Heuristically, because we now have two sources of uncertainty, the option is no longer ‘spanned’ by a dynamic portfolio of stocks and bonds in the tracking portfolio ( Grinblatt and Titman, 1998). Since Goldman et al. (1979) show that the option to sell at the maximum is indeed spanned, we may apply the Cox-Ross method. However, it may be more difficult to verify spanning for more complex path-dependent derivatives. Hence, we may have to embed the security in a model of economic equilibrium, with specific assumptions about investors’ preferences and their investment opportunity set. Our result induces that a stochastic volatility specification is a valid specification for thinly traded markets. The result seems to be in line with variance ratio results ( Solibakke, 2000) and ARMA-GARCH results ( Solibakke, 2001a and Solibakke, 2001b). Moreover, according to the SNP score generator, the SDE specification seems to model the return series more adequately relative to changing volatility specifications.