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Forecasting Methods in Extreme Scenarios and Advanced Data Analytics for Improved Risk Estimation

After extensive investigation on the statistical properties of financial returns, a discrete nature has surfaced when low price effect is present. This is rather logical since every market operates on a specific accuracy. In order for our models to take into consideration this discrete nature of returns, the discretization of the tail density function is applied. As a result of this discretization process, it is now possible to improve the percentage value at risk (PVaR) and expected percentage shortfall (EPS) estimations on which we are focusing in this work. Finally, in order to evaluate the improvement provided by our proposed methodology, adjusted evaluation measures are presented, capable of evaluating percentile estimations like PVaR. These adjusted evaluation measures are not only limited to evaluating percentiles, but in any scenario where data does not bare the same amount of information and consequently, does not all carry the same degree of importance, like in the case of risk analysis.

1.1. Introduction

The quantification of risk is an important issue in finance that becomes even more important during periods of financial crises. The estimation of volatility is the main financial characteristic associated with risk analysis and management since in financial modeling, it has been consolidated the view that the asset returns are preferred over prices due to their stationarity [MEU 09, SHI 99]. Moreover, the volatility of returns is the one that can be successfully forecasted and that is essential for risk management [POO 03, AND 01]).

The increase in complexity of both the financial system and the nature of the financial risk over the last decades, results in models with limited reliability. Hence, in extreme economic events, such models become less reliable and in some instances, fully fail to measure the underlying risk. Consequently, they are producing inaccurate risk measurements which has a great impact on many financial applications, such as asset allocation and portfolio management in general, derivatives pricing, risk management, economic capital and financial stability (based on Basel III accords).

To make thinks even worse, stock exchange markets (and in general every financial market) operate with certain accuracy. In most European markets the accuracy is 0.1 cents (0.001 euros), while US markets operate with 1 cent (USD 0.01) accuracy except for securities that are priced less than USD 1.00 for which the market accuracy or minimum price variation (MPV) is USD 0.0001. Despite the readjustment for extremely low price assets, the associated fluctuation (variation) is considerable and the corresponding volatility is automatically increased. The phenomenon is magnified considerably in periods of extreme economic events (economic collapses, bankruptcies, depressions, etc.) and as a result typical market accuracy fails to handle smoothly assets of extremely low price.

However, any attempt to increase the reliability of the model identification procedure for volatility estimation results unavoidably in even more complex models, which still will be unable to identify and fully capture the governing set of rules of the global economy. The more complicated the model we use for volatility estimation, the larger the number of parameters that needs to be estimated. Hence, in order to have larger data sets, we go further back in the past to collect and use for the analysis older observations, which though may be less representative of reality. Lastly, risk models use only the returns of an asset while ignoring the prices.

In order to answer all the above, we discuss, in this chapter, the concept of low price effect (lpe) [FRI 36] and recommend a low price correction (lpc) for improved forecasts. The low price effect is the increase in variation for stocks with low price due to the existence of a minimum possible return produced when the asset price changes by the MPV. Lpe is frequently overlooked and the main reason for that is the lack of theoretical background related to the reasons resulting in this phenomenon, which surfaces primarily in periods of economic instability or extreme economic events. The pioneering of the proposed correction is that it does not require any additional parameters and takes into account the asset price. The proposed correction is associated with the rationalization of the estimated asset returns, since it is rounded to the next integer multiple of the minimum possible return. Except for the proposed correction, in this work we also provide a mathematical reasoning for the increase in volatility.

Inspired from the above, we came to the same conclusion as many before, that in risk analysis the returns of an asset do not all bare the same amount of information and do not all carry the same degree of significance [SOK 09, ASA 17, ALH 08, ALA 10]. In the absence of a formal mathematical notion, the term asymmetry in the importance describes relatively satisfactory the above phenomenon which is also apparent in other scientific areas, related to medical, epidemiological, climatological, geophysical or meteorological phenomenon. For a mathematical interpretation, we may consider a proper cost function that takes different values, depending on different regions of the dataset or the entire period of observation. For instance, in biosurveillance, the importance (and the cost) associated with an illness incidence rate is much higher in the so-called epidemic periods associated with an extreme rate increase. In the case of risk analysis, risk measures like the value at risk (VaR) and expected shortfall (ES), concentrate only on the left tail of the distribution of returns with the attention being paid to the days of violation rather than to those of no violation. Consequently, failures in fitting a model on the right tail are not considered to be important. Thus, the asymmetry in the importance of information is crucial in both choosing the most appropriate model by assigning a proper weight to each region of the dataset and evaluating its forecasting ability.

In order to judge the forecasting quality of the proposed methodology applied to typical risk measurements like VaR and ES, we have appropriately adjusted a number of popular evaluation measures that take into account the asymmetry in the importance of information bared in the data. Since the proposed low price correction is applied in a subset of the dataset with a specific property (in this case a very low price), it is logical for the adjusted evaluation measures to be computed on the same subset. Thus, the risk estimations can be evaluated with and without the implementation of low price correction and then, compared. The decrease in the values of the adjusted evaluation measures will show the improved forecasting quality of the proposed methodology.

In addition, for the evaluation of the proposed correction, backtesting methods such as the violation ratio (VR) for value at risk and normalized shortfall (NS) for expected shortfall can also be used (see, for example, [BRO 11]). For real-life examples and applications, see [SIO 17, SIO 19a, SIO 19b].

1.2. The low price effect and correction

The rules that govern the accuracy of financial markets around the world fail to smoothly handle assets of extremely low price resulting in a considerable fluctuation (variation) and increased volatility. As expected, the phenomenon is magnified considerably in periods of extreme economic events (economic collapses, bankruptcies, depressions, etc.). Indeed, since all possible (logarithmic) returns on a specific day are integer multiples of the minimum possible return, the stock movement will fluctuate more nervously, the lower the prices. As a result, violations will occur more frequently and forecasts will turn out to be irrational in the sense that such returns cannot be materialized. The resulting volatility increase is quite often overlooked, primarily due to the fact that we take into account only the returns of an asset, neglecting entirely the prices.

In order to accommodate different accuracies, we introduce below a broad definition of the minimum possible return.

DEFINITION 1.1.- Let pt be the asset value at time t and c(pt) be the minimum price variation (market accuracy) associated with the value of the asset at time t. Then the minimum possible return (mpr) of an asset at time t denoted by mprt is the logarithmic return that the asset will produce if its value changes by c(pt) and is given by:

Note that mprt is the same for both upward and downward movements due to the symmetry of logarithmic returns. For the special case, that a market has a constant accuracy, say c, irrespectively of the stock price, Definition 1 can be simplified to mprt = log ((pt + c)/pt) (see [SIO 17]).

EXAMPLE 1.1.- Let us assume that the value of an asset is equal to ?0.19 and the market operates under ?0.001 accuracy. Then the minimum possible return for the asset is 0.5%. Also, all possible (logarithmic) returns for the asset in this specific day are the integer multiples of the minimum possible return. This will have as a result, the stock movement to become even more nervous and models' failures to increase. Consequently, PVaR violations will occur more frequently and our model will almost always derive forecasts that are irrational since the stock...