This paper reviews some applications of continuous time random walks (CTRWs) to Finance and Economics. It is divided into two parts. The first part deals with the connection between CTRWs and ...anomalous diffusion. In particular, a simplified version of the well-scaled transition of CTRWs to the diffusive or hydrodynamic limit is presented. In the second part, applications of CTRWs to the ruin theory of insurance companies, to growth and inequality processes and to the dynamics of prices in financial markets are outlined and briefly discussed.
This study empirically re-examines fat tails in stock return distributions by applying statistical methods to an extensive dataset taken from the Korean stock market. The tails of the return ...distributions are shown to be much fatter in recent periods than in past periods and much fatter for small-capitalization stocks than for large-capitalization stocks. After controlling for the 1997 Korean foreign currency crisis and using the GARCH filter models to control for volatility clustering in the returns, the fat tails in the distribution of residuals are found to persist. We show that market crashes and volatility clustering may not sufficiently account for the existence of fat tails in return distributions. These findings are robust regardless of period or type of stock group.
•The existence of fat-tails in stock return distribution is investigated.•Influential factors of stylized facts (e.g., market crash, volatility clustering) are considered.•These factors have a significant influence on the fat-tails in return distribution.•After controlling for these factors, evidence supporting the existence of fat-tails is still verified.•The fat-tails in return distributions cannot be sufficiently explained by the stylized facts.
In financial time series there are time periods in which market indices values or assets prices increase or decrease monotonically. We call those events "price runs", "elementary uninterrupted ...trends" or just "uninterrupted trends". In this paper we study the distribution of the duration of uninterrupted trends for the daily indices DJIA, NASDAQ, IPC and Nikkei 225 during the period of time from 10/30/1978 to 08/07/2020 and we compare the simple geometric statistical model with p = 1 2 consistent with the EMH to the empirical data. By a fitting procedure, it is found that the geometric distribution with parameter p = 1 2 provides a good model for uninterrupted trends of short and medium duration for the more mature markets; however, longest duration events still need to be statistically characterized. Estimated values of the parameter p were also obtained and confirmed by calculating the mean value of p fluctuations from empirical data. Additionally, the observed trend duration distributions for the different studied markets are compared over time by means of the Anderson-Darling (AD) test, to the expected geometric distribution with parameter p = 1 2 and to a geometric distribution with a free parameter p, making possible to assess and compare different market geometric behavior for different dates as well as to measure the fraction of time runs duration from studied markets are consistent with the geometric distribution with p = 1 2 and in parametric free way.
•This study investigates the risk property in fat tails of the return distribution.•Fat-tails are not eliminated through portfolio diversification.•Fat-tails are highly related to properties of ...systematic risks.•Negative tail has raising fatness, and declining fatness for positive tail.•Portfolio diversification requires more profit sacrifice for loss avoidance.
This study investigates the level of risk due to fat tails of the return distribution and the changes of tail fatness (TF) through portfolio diversification. TF is not eliminated through portfolio diversification, and, interestingly, the positive tail has declining fatness until a certain level is reached, while the negative tail has rising fatness. This indicates that fat tails are highly relevant to common factors on systematic risk and that the relevance of common factors is higher for the negative tail compared to the positive tail. In the portfolio diversification effect, the declining fatness of the positive tail further reduces risk, but the rising fatness of the negative tail does not contribute to this effect. The asymmetry between the fatness of the positive and negative tails in the return distribution corresponds to the asymmetry of the trade-off relationship between loss avoidance and profit sacrifice that is expected as a consequence of portfolio diversification. Investors use portfolio diversification to reduce their risk of suffering high losses, but following this strategy means sacrificing high-profit potential. Our study provides empirical confirmation for the practical limitation of portfolio diversification and explains why investors with diversified portfolios suffer high losses from market crashes. An examination of the Northeast Asian stock markets of China, Japan, Korea, and Taiwan show identical results.
We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues. The continuous double auction defines a ...continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen.
A model for the phenomenological description of tick-by-tick share prices in a stock exchange is introduced. It is based on mixtures of compound Poisson processes. Preliminary results based on Monte ...Carlo simulation show that this model can reproduce various stylized facts.
In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a ...semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.
Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is ...illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic.
A simulation of high-frequency market data is performed with the Genoa Artificial Stock Market. Heterogeneous agents trade a risky asset in exchange for cash. Agents have zero intelligence and issue ...random limit or market orders depending on their budget constraints. The price is cleared by means of a limit order book. A renewal order-generation process is used having a waiting-time distribution between consecutive orders that follows a Weibull law, in line with previous studies. The simulation results show that this mechanism can reproduce fat-tailed distributions of returns without ad-hoc behavioral assumptions on agents. In the simulated trade process, when the order waiting-times are exponentially distributed, trade waiting times are exponentially distributed. However, if order waiting times follow a Weibull law, analogous results do not hold. These findings are interpreted in terms of a random thinning of the order renewal process. This behavior is compared with order and trade durations taken from real financial data.
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