Artur Kotlicki

DPhil student at the Oxford Mathematical Institute, affiliated with the Insitute for New Economic Thinking at the Oxford Martin School.

# Liquidity at Risk

Joint work with Rama Cont and Laura Valderrama · University of Oxford

The traditional approach to the stress testing of financial institutions focuses on capital adequacy and solvency. Liquidity stress tests have been applied in parallel to and independently from solvency stress tests, based on scenarios which may not be consistent with those used in solvency stress tests.

We propose a structural framework for the joint stress testing of solvency and liquidity: our approach exploits the mechanisms underlying the solvency-liquidity nexus to derive relations between solvency shocks and liquidity shocks. These relations are then used to model liquidity and solvency risk in a coherent framework, involving external shocks to solvency and endogenous liquidity shocks.

We define the concept of 'Liquidity at Risk', which quantifies the liquidity resources required for a financial institution facing a stress scenario. Finally, we show that the interaction of liquidity and solvency may lead to the amplification of equity losses due to funding costs which arise from liquidity needs.

# Fast KAMH

MSc dissertation under supervision of Dino Sejdinovic · University of Oxford

We propose Fast Kernel Adaptive Metropolis-Hastings (F-KAMH), a gradient-free adaptive MCMC algorithm that is highly suitable for contexts such as Pseudo-Marginal MCMC.

Our procedure bases on the Kernel Adaptive Metropolis-Hastings (KAMH) sampler of Sejdinovic et al. (2014) that offers a novel approach to sampling from multivariate target distributions with non-linear dependencies between dimensions. KAMH bases on the mapping of the samples to a reproducing kernel Hilbert space, where the choice of a proposal distribution is adaptively dictated by the estimated sample covariance in the feature space. Flexibility of the algorithm in Sejdinovic et al. (2014) comes with an increased computational cost, however.

In F-KAMH, we use a large-scale approximation of the kernel methods framework based on random Fourier features of Rahimi and Recht (2007), which leads to a significant reduction in the algorithm’s complexity. Moreover, our asymptotically exact procedure adapts to the local covariance structure of the target distribution based on the entire chain history, in contrast to KAMH's suboptimal approach which uses only a subsample of the chain history. Consequently, our newly proposed sampler offers substantial improvements in terms of effective sample size per computation unit time. Our claims are supported through experimental study on synthetic examples of highly non-linear target distributions.

# Random matrix theory

BSc dissertation under supervision of Rama Cont · Imperial College London

This projects aims to present significant results of random matrix theory in regards to the principal component analysis, including Wigner's semicircular law and Marcenko-Pastur law describing limiting distribution of large dimensional random matrices. The work bases on the large dimensional data assumptions, where both the number of variables and sample size tends to infinity, while their ratio tends to a finite limit.

Random matrix theory, over the past decade has been a fast growing area of mathematics, due to the advancements in technology and data collection methods. Treated as a tool to solve large dimensional problems, it has found its application in many research areas, such as signal processing, network security, image processing, genetic statistics, stock market analysis, and other finance or economic problems.

In this project, key results enabling to establish a low dimensional factor model from a large noisy data will be stated, as well as a general way of proving them will be given. A significant portion of the proofs relies on the Stieltjes transform, a common tool used for studying the convergence of spectral distribution of large matrices, which is also discussed in this project. An algorithm suggested by Karoui (2008) will be presented, giving a method of estimating the true population covariance.

Empirical verification of main theorems is conducted, showing fast convergence rate in case of the Marcenko-Pastur law, and slower rate for the Wigner's semicircular law. Also, the established theory is applied to a real-life financial data, based on the S&P 500 index, for which 12 principal components have been identified when time horizon is equal to 10 years, and 10 principal components for data set over 5 years.

I am Artur, a DPhil student at the Mathematical Institute, University of Oxford.

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