Quantitative Finance

The dissertation investigates the application of Probabilistic Graphical Models (PGMs) in forecasting the price of Crude Oil. This research is important because crude oil plays a very pivotal role in the global economy hence is a very critical macroeconomic indicator of the industrial growth. Given the vast amount of macroeconomic factors affecting the price of crude oil such as supply of oil from OPEC countries, demand of oil from OECD countries, geopolitical and geoeconomic changes among many other variables - probabilistic graphical models (PGMs) allow us to understand by learning the graphical structure. This dissertation proposes condensing data numerous Crude Oil factors into a graphical model in the attempt of creating a accurate forecast of the price of crude oil. The research project experiments with using different libraries in Python in order to construct models of the crude oil market. The experiments in this thesis investigate three main challenges commonly presented while trading oil in the financial markets. The first challenge it investigates is the process of learning the structure of the oil markets; thus allowing crude oil traders to understand the different physical market factors and macroeconomic indicators affecting crude oil markets and how they are \textit{causally} related. The second challenge it solves is the exploration and exploitation of the available data and the learnt structure in predicting the behaviour of the oil markets. The third challenge it investigates is how to validate the performance and reliability of the constructed model in order for it to be deployed in the financial markets. A design and implementation of a probabilistic framework for forecasting the price of crude oil is also presented as part of the research.

The goal of this note is to illustrate the impact of a self-financing condition recently introduced by the authors. We present the analyses of two specific applications usually considered in more traditional models in financial mathematics. They include hedging European options with limit orders and the optimal behavior of market makers.

We revisit the trading invariance hypothesis recently proposed by Kyle and Obizhaeva by empirically investigating a large dataset of bets, or metaorders, provided by ANcerno. The hypothesis predicts that the quantity $I:=\ri/N^{3/2}$, where $\ri$ is the exchanged risk (volatility × volume × price) and N is the number of bets, is invariant. We find that the 3/2 scaling between $\ri$ and N works well and is robust against changes of year, market capitalisation and economic sector. However our analysis clearly shows that I is not invariant. We find a very high correlation R 2 >0.8 between I and the total trading cost (spread and market impact) of the bet. We propose new invariants defined as a ratio of I and costs and find a large decrease in variance. We show that the small dispersion of the new invariants is mainly driven by (i) the scaling of the spread with the volatility per transaction, (ii) the near invariance of the distribution of metaorder size and of the volume and number fractions of bets across stocks.

This thesis serves three primary purposes, first of which is to forecast two stocks, i.e. Goldman Sachs (GS) and General Electric (GE). In order to forecast stock prices, we used a long short-term memory (LSTM) model in which we inputted the prices of two other stocks that lie in rather close correlation with GS. Other models such as ARIMA were used as benchmark. Empirical results manifest the practical challenges when using LSTM for forecasting stocks. One of the main upheavals was a recurring lag which we called "forecasting lag". The second purpose is to develop a more general and objective perspective on the task of time series forecasting so that it could be applied to assist in an arbitrary that of forecasting by ANNs. Thus, attempts are made for distinguishing previous works by certain criteria (introduced by a review paper written by Ahmed Tealab) so as to summarise those including effective information. The summarised information is then unified and expressed through a common terminology that can be applied to different steps of a time series forecasting task. The last but not least purpose of this thesis is to elaborate on a mathematical framework on which ANNs are based. We are going to use the framework introduced in the book "Neural Networks in Mathematical Framework" by Anthony L. Caterini in which the structure of a generic neural network is introduced and the gradient descent algorithm (which incorporates backpropagation) is introduced in terms of their described framework. In the end, we use this framework for a specific architecture, which is recurrent neural networks on which we concentrated and our implementations are based. The book proves its theorems mostly for classification case. Instead, we proved theorems for regression case, which is the case of our problem.

We propose a model for equity trading in a population of agents where each agent acts to achieve his or her target stock-to-bond ratio, and, as a feedback mechanism, follows a market adaptive strategy. In this model only a fraction of agents participates in buying and selling stock during a trading period, while the rest of the group accepts the newly set price. Using numerical simulations we show that the stochastic process settles on a stationary regime for the returns. The mean return can be greater or less than the return on the bond and it is determined by the parameters of the adaptive mechanism. When the number of interacting agents is fixed, the distribution of the returns follows the log-normal density. In this case, we give an analytic formula for the mean rate of return in terms of the rate of change of agents' risk levels and confirm the formula by numerical simulations. However, when the number of interacting agents per period is random, the distribution of returns can significantly deviate from the log-normal, especially as the variance of the distribution for the number of interacting agents increases.

This paper proposes an auction type resolution for smart derivatives. It has been discussed to migrate derivatives contracts to smart contracts (smart derivatives). Automation is often discussed in this context. It is also important to prepare to avoid disputes from practical perspectives. There are controversial issues to terminate the relationship at defaults. In OTC derivative markets, master agreements define a basic policy for the liquidation process but there happened some disputes over these processes. We propose to define an auction type resolution in smart derivatives, which each participant would find beneficial.

We present results demonstrating that an appropriately configured deep learning neural network (DLNN) can automatically learn to be a high-performing algorithmic trading system, operating purely from training-data inputs generated by passive observation of an existing successful trader T. That is, we can point our black-box DLNN system at trader T and successfully have it learn from T's trading activity, such that it trades at least as well as T. Our system, called DeepTrader, takes inputs derived from Level-2 market data, i.e. the market's Limit Order Book (LOB) or Ladder for a tradeable asset. Unusually, DeepTrader makes no explicit prediction of future prices. Instead, we train it purely on input-output pairs where in each pair the input is a snapshot S of Level-2 LOB data taken at the time when T issued a quote Q (i.e. a bid or an ask order) to the market; and DeepTrader's desired output is to produce Q when it is shown S. That is, we train our DLNN by showing it the LOB data S that T saw at the time when T issued quote Q, and in doing so our system comes to behave like T, acting as an algorithmic trader issuing specific quotes in response to specific LOB conditions. We train DeepTrader on large numbers of these S/Q snapshot/quote pairs, and then test it in a variety of market scenarios, evaluating it against other algorithmic trading systems in the public-domain literature, including two that have repeatedly been shown to outperform human traders. Our results demonstrate that DeepTrader learns to match or outperform such existing algorithmic trading systems. We analyse the successful DeepTrader network to identify what features it is relying on, and which features can be ignored. We propose that our methods can in principle create an explainable copy of an arbitrary trader T via "black-box" deep learning methods.

This paper compares mathematical models for automated market makers including logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR), constant product/mean/sum, and others. It is shown that though LMSR may not be a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several advantages over constant product/mean based automated market makers. However, LS-LMSR requires complicated computation (i.e., logarithm and exponentiation) and the cost function curve is concave. In certain DeFi applications, it is preferred to have computationally efficient cost functions with convex curves to conform with the principle of supply and demand. This paper proposes and analyzes constant circle/ellipse based cost functions for automated market makers. The proposed cost functions are computationally efficient (only requires multiplication and square root calculation) and have several advantages over widely deployed constant product cost functions. For example, the proposed market makers are more robust against front-runner (slippage) attacks.

The autonomous trading agent is one of the most actively studied areas of artificial intelligence to solve the capital market portfolio management problem. The two primary goals of the portfolio management problem are maximizing profit and restrainting risk. However, most approaches to this problem solely take account of maximizing returns. Therefore, this paper proposes a deep reinforcement learning based trading agent that can manage the portfolio considering not only profit maximization but also risk restraint. We also propose a new target policy to allow the trading agent to learn to prefer low-risk actions. The new target policy can be reflected in the update by adjusting the greediness for the optimal action through the hyper parameter. The proposed trading agent verifies the performance through the data of the cryptocurrency market. The Cryptocurrency market is the best test-ground for testing our trading agents because of the huge amount of data accumulated every minute and the market volatility is extremely large. As a experimental result, during the test period, our agents achieved a return of 1800% and provided the least risky investment strategy among the existing methods. And, another experiment shows that the agent can maintain robust generalized performance even if market volatility is large or training period is short.

We present a formulation of the transaction cost analysis (TCA) in the Bayesian framework for the primary purpose of comparing broker algorithms using standardized benchmarks. Our formulation allows effective calculation of the expected value of trading benchmarks with only a finite sample of data relevant to practical applications. We discuss the nature of distribution of implementation shortfall, volume-weighted average price, participation-weighted price and short-term reversion benchmarks. Our model takes into account fat tails, skewness of the distributions and heteroscedasticity of benchmarks. The proposed framework allows the use of hierarchical models to transfer approximate knowledge from a large aggregated sample of observations to a smaller sample of a particular algorithm.