Markov Chains

A Stochastic Model for Illiquid Stock Prices and its Conclusion about Correlation Measurement ArXiv ID: 2509.10553 “View on arXiv” Authors: Erina Nanyonga, Juma Kasozi, Fred Mayambala, Hassan W. Kayondo, Matt Davison Abstract This study explores the behavioral dynamics of illiquid stock prices in a listed stock market. Illiquidity, characterized by wide bid and ask spreads affects price formation by decoupling prices from standard risk and return relationships and increasing sensitivity to market sentiment. We model the prices at the Uganda Securities Exchange (USE) which is illiquid in that the prices remain constant much of the time thus complicating price modelling. We circumvent this challenge by combining the Markov model (MM) with two models; the exponential Ornstein Uhlenbeck model (XOU) and geometric Brownian motion (gBm). In the combined models, the MM was used to capture the constant prices in the stock prices while the XOU and gBm captured the stochastic price dynamics. We modelled stock prices using the combined models, as well as XOU and gBm alone. We found that USE stocks appeared to have low correlation with one another. Using theoretical analysis, simulation study and empirical analysis, we conclude that this apparent low correlation is due to illiquidity. In particular data simulated from combined MM-gBm, in which the gBm portion were highly correlated resulted in a low measured correlation when the Markov chain had a higher transition from zero state to zero state. ...

Determining a credit transition matrix from cumulative default probabilities ArXiv ID: 2503.14646 “View on arXiv” Authors: Unknown Abstract To quantify the changes in the credit rating of a bond is an important mathematical problem for the credit rating industry. To think of the credit rating as the state a Markov chain is an interesting proposal leading to challenges in mathematical modeling. Since cumulative default rates are more readily measurable than credit migrations, a natural question is whether the credit transition matrix (CTM) can be determined from the knowledge of the cumulative default probabilities. Here we use a connection between the CTM and the cumulative default probabilities to setup an ill-posed, linear inverse problem with box constraints, which we solve by an entropy minimization procedure. This approach is interesting on several counts. On the one hand, we may have less data that unknowns, and on the other hand, even when we have as much data as unknowns, the matrix connecting them may not be invertible, which makes the problem ill-posed. Besides developing the tools to solve the problem, we apply it to several test cases to check the performance of the method. The results are quite satisfactory. ...

High-Frequency Stock Market Order Transitions during the US-China Trade War 2018: A Discrete-Time Markov Chain Analysis ArXiv ID: 2405.05634 “View on arXiv” Authors: Unknown Abstract Statistical analysis of high-frequency stock market order transaction data is conducted to understand order transition dynamics. We employ a first-order time-homogeneous discrete-time Markov chain model to the sequence of orders of stocks belonging to six different sectors during the USA-China trade war of 2018. The Markov property of the order sequence is validated by the Chi-square test. We estimate the transition probability matrix of the sequence using maximum likelihood estimation. From the heat-map of these matrices, we found the presence of active participation by different types of traders during high volatility days. On such days, these traders place limit orders primarily with the intention of deleting the majority of them to influence the market. These findings are supported by high stationary distribution and low mean recurrence values of add and delete orders. Further, we found similar spectral gap and entropy rate values, which indicates that similar trading strategies are employed on both high and low volatility days during the trade war. Among all the sectors considered in this study, we observe that there is a recurring pattern of full execution orders in Finance & Banking sector. This shows that the banking stocks are resilient during the trade war. Hence, this study may be useful in understanding stock market order dynamics and devise trading strategies accordingly on high and low volatility days during extreme macroeconomic events. ...

Reconstructing cryptocurrency processes via Markov chains ArXiv ID: 2308.07626 “View on arXiv” Authors: Unknown Abstract The growing attention on cryptocurrencies has led to increasing research on digital stock markets. Approaches and tools usually applied to characterize standard stocks have been applied to the digital ones. Among these tools is the identification of processes of market fluctuations. Being interesting stochastic processes, the usual statistical methods are appropriate tools for their reconstruction. There, besides chance, the description of a behavioural component shall be present whenever a deterministic pattern is ever found. Markov approaches are at the leading edge of this endeavour. In this paper, Markov chains of orders one to eight are considered as a way to forecast the dynamics of three major cryptocurrencies. It is accomplished using an empirical basis of intra-day returns. Besides forecasting, we investigate the existence of eventual long-memory components in each of those stochastic processes. Results show that predictions obtained from using the empirical probabilities are better than random choices. ...