Equities (General)

Market Making with Fads, Informed, and Uninformed Traders ArXiv ID: 2501.03658 “View on arXiv” Authors: Unknown Abstract We characterise the solutions to a continuous-time optimal liquidity provision problem in a market populated by informed and uninformed traders. In our model, the asset price exhibits fads – these are short-term deviations from the fundamental value of the asset. Conditional on the value of the fad, we model how informed traders and uninformed traders arrive in the market. The market maker knows of the two groups of traders but only observes the anonymous order arrivals. We study both, the complete information and the partial information versions of the control problem faced by the market maker. In such frameworks, we characterise the value of information, and we find the price of liquidity as a function of the proportion of informed traders in the market. Lastly, for the partial information setup, we explore how to go beyond the Kalman-Bucy filter to extract information about the fad from the market arrivals. ...

Efficient evaluation of joint pdf of a Lévy process, its extremum, and hitting time of the extremum ArXiv ID: 2312.05222 “View on arXiv” Authors: Unknown Abstract For Lévy processes with exponentially decaying tails of the Lévy density, we derive integral representations for the joint cpdf $V$ of $(X_T, \bar X_T,τ_T)$ (the process, its supremum evaluated at $T<+\infty$, and the first time at which $X$ attains its supremum). The first representation is a Riemann-Stieltjes integral in terms of the (cumulative) probability distribution of the supremum process and joint probability distribution function of the process and its supremum process. The integral is evaluated using a combination an analog of the trapezoid rule. The second representation is amenable to more accurate albeit slower calculations. We calculate explicitly the Laplace-Fourier transform of $V$ w.r.t. all arguments, apply the inverse transforms, and reduce the problem to evaluation of the sum of 5D integrals. The integrals can be evaluated using the summation by parts in the infinite trapezoid rule and simplified trapezoid rule; the inverse Laplace transforms can be calculated using the Gaver-Wynn-Rho algorithm. Under additional conditions on the domain of analyticity of the characteristic exponent, the speed of calculations is greatly increased using the conformal deformation technique. For processes of infinite variation, the program in Matlab running on a Mac with moderate characteristics achieves the precision better than E-05 in a fraction of a second; the precision better than E-10 is achievable in dozens of seconds. As the order of the process (the analog of the Blumenthal-Getoor index) decreases, the CPU time increases, and the best accuracy achievable with double precision arithmetic decreases. ...

Simulation of a Lévy process, its extremum, and hitting time of the extremum via characteristic functions ArXiv ID: 2312.03929 “View on arXiv” Authors: Unknown Abstract We suggest a general framework for simulation of the triplet $(X_T,\bar X_ T,τ_T)$ (Lévy process, its extremum, and hitting time of the extremum), and, separately, $X_T,\bar X_ T$ and pairs $(X_T,\bar X_ T)$, $(\bar X_ T,τ_T)$, $(\bar X_ T-X_T,τ_T)$, via characteristic functions and conditional characteristic functions. The conformal deformations technique allows one to evaluate probability distributions, joint probability distributions and conditional probability distributions accurately and fast. For simulations in the far tails of the distribution, we precalculate and store the values of the (conditional) characteristic functions on multi-grids on appropriate surfaces in $C^n$, and use these values to calculate the quantiles in the tails. For simulation in the central part of a distribution, we precalculate the values of the cumulative distribution at points of a non-uniform (multi-)grid, and use interpolation to calculate quantiles. ...

An explanation for the distribution characteristics of stock returns ArXiv ID: 2312.02472 “View on arXiv” Authors: Unknown Abstract Observations indicate that the distributions of stock returns in financial markets usually do not conform to normal distributions, but rather exhibit characteristics of high peaks, fat tails and biases. In this work, we assume that the effects of events or information on prices obey normal distribution, while financial markets often overreact or underreact to events or information, resulting in non normal distributions of stock returns. Based on the above assumptions, we propose a reaction function for a financial market reacting to events or information, and a model based on it to describe the distribution of real stock returns. Our analysis of the returns of China Securities Index 300 (CSI 300), the Standard & Poor’s 500 Index (SPX or S&P 500) and the Nikkei 225 Index (N225) at different time scales shows that financial markets often underreact to events or information with minor impacts, overreact to events or information with relatively significant impacts, and react slightly stronger to positive events or information than to negative ones. In addition, differences in financial markets and time scales of returns can also affect the shapes of the reaction functions. ...

Copula-based deviation measure of cointegrated financial assets ArXiv ID: 2312.02081 “View on arXiv” Authors: Unknown Abstract This study outlines a comprehensive methodology utilizing copulas to discern inconsistencies in the behavior exhibited by pairs of financial assets. It introduces a robust approach to establishing the interrelationship between the returns of these assets, exploring potential measures of dependence among the stochastic variables represented by these returns. Special emphasis is placed on scrutinizing the traditional measure of dependence, namely the correlation coefficient, delineating its limitations. Furthermore, the study articulates an alternative methodology that offers enhanced stability and informativeness in appraising the relationship between financial instrument returns. ...