# Price Formation Mechanism

The aim to understand the price formation mechanism is not novel. It is well known that price process of any financial instrument follows a stochastic-like path: a price path can include or not a deterministic trend; but in any case the price process is smeared by noise movements. The noise movements are known as market volatility, and they make the price unpredictable. These noise movements can be decomposed into two components: the first component is called regular noise, it represents noise that is frequent but does not bring any abrupt changes, the second component is known as price jumps, it designates rare but very abrupt price movements. The origin of regular noise is in the statistical nature of the markets: any market is a result of the interplay between many different market participants with different incentives, purposes and financial constraints. This interaction of many different agents can be mathematically described as the standard Gaussian distribution (Merton 1976), this assumption allows dealing easy with in mathematical models of the price processes of financial instruments, calculating expectations and establishing various characteristics of financial instruments. The discontinuities in price evolution (price jumps) have been recognized as an essential part of the price time series generated on financial markets. Price jumps can’t be fitted by the description of the first noise component and thus have to be modeled

M. Frolova (H)

Department of Economics, Prognoz Risk Lab, Perm State University, Perm, Russia e-mail: frolovam@prognoz. ru

© Springer International Publishing Switzerland 2015

A. K. Bera et al. (eds.), Financial Econometrics and Empirical Market

Microstructure, DOI 10.1007/978-3-319-09946-0_____ 7 on their own, (Merton 1976). But it is worth noting, that the unpredictability of the price movements is not a negative feature, it is rather the nature of financial markets.

Many studies (Andersen et al. 2002; Gatheral 2006) demonstrate that continuoustime models have to incorporate the discontinuous component. Andersen et al. (2002) extend the class of stochastic volatility diffusions by allowing for Poisson jumps of time-varying intensity in returns. However, the problem is the mathematical description of price jumps cannot be easily handled (Pan 2002; Broadie and Jain 2008). The serious problems in the mathematical description of price jumps are very often the reason why price jumps are wrongly neglected. However, the nonGaussian price movements influence the models employed in finance to estimate the performance of various financial vehicles (Heston 1993; Gatheral 2006). Andersen et al. (2007) conclude that most of the standard approaches in the financial literature on pricing assets assume a continuous price path. Since this assumption is clearly violated in most cases the results tend to be heavily biased.

The literature contains a broad range of ways to classify volatility. Each classification is suitable for an explanation of a different aspect of volatility or an explanation of volatility from a different point of view (see e. g. Harris 2003, where the volatility is discussed from the financial practitioners’ points of view). The most important aspect is to separate the Gaussian-like component from price jumps (Merton 1976; Gatheral 2006).

Mathematical finance has developed a class of models that make use of jump processes (Cont and Tankov 2004) and that are used for pricing derivatives and for modeling volatility. Financial econometrics has developed several methods to disentangle the continuous part of the price path from the discontinuous one (Lee and Mykland 2008; Barndorff-Nielsen and Shephard 2006), and the latter is modeled as jumps.

Bormetti et al. (2013) found that, as far as individual stocks are concerned, jumps are clearly not described by a Poisson process, the evidence of time clustering can be accounted for and modelled by means of linear self-exciting Hawkes processes. Clustering of jumps means that the intensity of the point process describing jumps depends on the past history of jumps, and a recent jump increases the probability that another jump occurs. The second deviation from the Poisson model is probably more important in a systemic context. Bormetti et al. find a strong evidence of a high level of synchronization between the jumping times of a portfolio of stocks. They find a large number of instances where several stocks (up to 20) jump at the same time. This evidence is absolutely incompatible with the hypothesis of independence of the jump processes across assets. Authors use Hawkes processes for modeling the dynamics of jumps of individual assets and they show that these models describe well the time clustering of jumps. However they also show that the direct extension of the application of Hawkes processes to describe the dynamics of jumps in a multiasset framework is highly problematic and inconsistent with data. For this reason, Bormetti et al. introduce Hawkes factor models to describe systemic cojumps. They postulate the presence of an unobservable point process describing a market factor, when this factor jumps, each asset jumps with a given probability, which is different for each stock. In general, an asset can jump also by following an idiosyncratic point process. In order to capture also the time clustering of jumps, they model the point processes as Hawkes processes. Authors show how to estimate this model and discriminate between systemic and idiosyncratic jumps and they claims that the model is able to reproduce both the longitudinal and the cross sectional properties of the multi-asset jump process.

On the opposite, tests applied by Bajgrowicz and Scaillet (2011) do not detect time clustering phenomena of jumps arrivals, and, hence, do not reject the hypothesis that jump arrivals are driven by a simple Poisson process.

The presence of price jumps has serious consequences for financial risk management and pricing. Thus, it is of great interest to describe the noise movements as accurately as possible. Nyberg and Wilhelmsson (2009) discuss the importance of including event risk as recommended by the Basel II accord, which suggests employing a VAR model with a continuous component and price jumps representing event risks.

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