Browsing by Author "Swishchuk, Anatoliy V."
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Item Open Access Analysis of Financial Transmission Rights Obligations and Hourly Congestion Prices in PJM Markets(2019-09-17) Arablou, Zahra; Ware, Antony Frank; Swishchuk, Anatoliy V.; Wu, JingjingIn this research we analyze patterns of FTR obligation contracts using hourly data from January 1st, 2015 through December 31st, 2018 between PJM Western and AEP Dayton hubs and concluded that there are positive net profits in FTR obligation contracts if someone were to buy and hold the contracts until they were actualized. We applied Schwartz’s one-factor log-model to changes in congestion prices, but our tests showed that residuals of the model are not normally distributed. We then applied OU process to outright congestion prices and used recursive approach proposed by Clewlow and Strickland (2000) to remove jumps from OU residuals. We concluded that non-jump and jump data sets are not normally distributed in both PJM Western and AEP Dayton hubs. Given congestion prices showed fat-tails and non-normal distributions for both jumps and non-jumps data points, we applied Johnson’s Unbounded Distribution to congestion prices and calibrated its parameters for both congestion prices. We defined a variable to filter out hourly data depending on how much of transmission interface were used in different hours and we recalibrated sets of parameters for those cases.Item Open Access Applications of Mean-reverting Processes in Alberta Energy Markets(2020-09-21) Lu, Weiliang; Swishchuk, Anatoliy V.; Goutte, Stéphane; Swishchuk, Anatoliy V.; Goutte, Stéphane; Qiu, Jinniao; Badescu, Alexandru M.This thesis introduce fuel-switching price, which designed for encouraging power plant companies to switch from coal to natural gas when they produce electricity and successfully applied on European market, to Albertan Market. Moreover, we consider a energy-switching price which consider power switch from natural gas to wind. We modeled these two prices using five mean reverting processes including Regime-switching processes, Lévy-driven Ornstein-Uhlenbeck process and Inhomogeneous Geometric Brownian Motion, and estimate them based on multiple procedures such as Maximum likelihood estimation and Expectation-Maximization algorithm. At last, this thesis prove previous result applied on Albertan Market that the jump modeling techniques is needed when modeling fuel-switching data. In addition, it not only give promising conclusion on the necessity of introducing Regime-switching models to the fuel-switching data, but also show that Regime-switching model is better fitted to the data.Item Open Access Bounds of Quantum Mixing Processes via Controlled Walks(2020-04-29) Li, Shang; Høyer, Peter; Woelfel, Philipp; Swishchuk, Anatoliy V.; Høyer, PeterQuantum mixing is a category of processes of reaching a stationary state when starting from a single initial vertex via quantum walks. In this thesis, we introduce the quantum mixing processes based on the framework of controlled quantum walks in [DH17a]. We first discuss why Szegedy's walks are not well-suited for the quantum mixing processes. We then show that the controlled quantum walk is a suitable framework for achieving a quadratic speed-up in the quantum mixing processes, because of its relationship with interpolated walks [KMOR16] and the phase estimation algorithm [CEMM98]. Accordingly, we give two definitions of quantum mixing times via controlled quantum walks: instant quantum mixing time and average quantum mixing time. We later prove the bounds of these two quantum mixing times on complete graphs. We also prove the upper bound of the average quantum mixing times on cycle graphs. To have clearer views on the bounds of instant and average quantum mixing times, we then discuss and present the simulations of quantum mixing processes, with four types of graphs (complete graphs, cycle graphs, 2-dimensional tori and dumbbell graphs) as examples. The simulations include the quantum mixing processes with the same ~θ (an important parameter in the controlled quantum walk operator) as in [DH17a], and the suitable range of ~θ that achieves the quadratic speed-up for different types of graphs. Inspired by the quantum search algorithm via controlled quantum walks in [DH17a], we adapt this algorithm to a new algorithm of quantum mixing, which generates samples from the stationary distribution of the Markov chain, with its runtime upper bounded by the quantum hitting time, which is quadratically faster than the classical hitting process.Item Open Access Credit Risk Pricing based on Epstein-Zin Preference(2019-12-20) Ma, Junchi; Sezer, Deniz; Qiu, Jinniao; Swishchuk, Anatoliy V.; Liao, WenyuanWe present a consumption-based equilibrium framework for credit risk pricing in an Epstein-Zin setting. The default time is modeled as the first hitting time of a default boundary. Bond investors have imperfect information about the firm value which is unobservable. The state variables, consumption and volatility are modeled as affine diffusion processes. Using the Epstein-Zin equilibrium solution as the pricing kernel, the price of a zero-coupon bond is expressed as the solution of a system of a two-dimensional parabolic partial differential equation (PDE) which is solved numerically. The price under the imperfect information is derived based on the solution of a stochastic partial differential equation (SPDE). Finally, We analyze the implications of imperfect information and firm parameters on the yield spreads.Item Open Access Numerical approximations of coupled forward-backward SPDEs with applications(2020-09-10) Molla, Md Hasib Uddin; Qiu, Jinniao; Ware, Antony Frank; Swishchuk, Anatoliy V.We introduce a new scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. For the FBSPDE, the finite element method in the spatial domain leads to approximations by finite-dimensional forward-backward stochastic differential equations (FBSDEs) in the temporal domain. We then approximate the solution of FBSDE by some existing machine learning schemes. Strong convergence results for spatial discretization of FBSPDEs are addressed.Item Open Access Optimal Portfolios of Natural Gas Futures(2021-01-28) Li, Xiang; Ware, Antony Frank; Swishchuk, Anatoliy V.; Qiu, JinniaoIn this thesis we investigate portfolios consisting of a collection of positions in natural gas futures. The logarithms of the futures prices follow correlated Ornstein-Uhlenbeck processes which are mean-reverting. Under the assumption that the portfolios are constant and short-selling is allowed, formulae for the portfolio that is optimal in terms of minimizing Capital at Risk (CaR) in a continuous-time context are obtained following an adapted version of the theory in [14]. We show that the results in this paper hold with small modifications in our situation and we apply them to portfolios of positions in AB NIT natural gas futures contracts. Furthermore, we allow the portfolio to be reoptimized periodically, showing dramatically better performance of portfolio with regard to the final wealth. And finally we consider the contract size limitation so that the portfolio is tradeable, and demonstrate the unsurprising underperformance of the portfolio that results.Item Open Access Option Pricing Using Neural Networks(2019-08-30) Que, Danfeng; Badescu, Alexandru M.; Ware, Antony Frank; Swishchuk, Anatoliy V.Due to the properties of large transaction volumes, high innovation and positive promotion to the financial market development, options play an essential role. However, the flexible design of options leads to complicated pricing, which makes accurate option pricing a challenging task for a long time. In this paper, two types of neural networks - feed-forward networks such as the radial basis function (RBF), multilayer perceptron (MLP), Modular network and a recurrent deep learning network Long short-term memory (LSTM) - and three stochastic process pricing models (Black-Scholes-Merton model, Heston stochastic volatility model and Merton Jump-diffusion model) are proposed so as to predict European call option prices. Firstly, it generates simulated option prices data from three stochastic process models to test the effectiveness and approximation ability of the neural networks. Secondly, effective factors such as moneyness, time to maturity and greeks via the Black-Scholes-Merton formula are used as input variables for neural networks. Historical data of S&P 500 European option prices empirically analyzes validity and stability of the neural networks. The performance measures R2, statistical error of the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) are used to evaluate the performance of two types of pricing models. It shows that the MLP network with two hidden layers performs best. In addition, neural networks do outperform the pricing ability of stochastic process pricing models.Item Open Access Quantum walk on two-dimensional torus and grid(2018-05-28) Komeili, Mojtaba; Høyer, Peter; Woelfel, Philipp; Swishchuk, Anatoliy V.Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.Item Open Access Semi-Markov Switching Lévy Processes and their Applications in Finance(2020-07-23) Zhang, Yi (Ivy); Swishchuk, Anatoliy V.; Ambagaspitiya, Rohana S.; Badescu, Alexandru M.; Choi, Kyoungjin; Simard, ClarenceWe, as humans, learn from our mistakes. Ultimately, we progress and grow, both as an individual and as a society. After the catastrophic recession period due to the sub-prime mortgage crisis in 2007, researchers and mathematicians began to look for answers about the massive damages done by the crisis. These events suggested that there must be some other factors that had hidden and needed to be accounted for in the previous prevailing models of our financial markets. As many different explanations arose, the main focus was to find a way to account for these more significant crises with some economically interpretable assumptions. The existence of different states or regimes in our financial market is one of the most acceptable ideas so far, as we tend to notice that the market has been switching between cycles of booms and recessions. This idea has triggered many studies on regime-switching models, but mostly with Markov regimes, as they are mathematically simple and relatively easy to solve. However, some of the assumptions made behind the Markov switching models have been questionable and unrealistic. For instance, Markov regimes imply that each one of them can switch to any other state no matter how long we have stayed in the current state. This is deviating from what we would expect, as we usually observe that the likelihood of recovery has some correlation with the recession’s age; the longer we stay in a recession, the harder for the market to recover. Thus, the primary purpose of this thesis is to find a better model with a better explanation. The introduction of semi-Markov markets intuitively has a time-varying propensity of regime changes using the conditional intensity matrix. In general, the semi-Markov switching models should be more in line with our financial market and generate better results when simulating financial derivatives prices. In this thesis, we will start by introducing some related definitions and theorems first. We will develop a semi-martingale representation for both the discrete-time semi-Markov chains and continuous-time semi-Markov processes, with some examples and applications. Then, we will construct the theoretical framework of a stochastic model under a semi-Markov regime-switching process driven by Lévy processes. The first step is to derive its Itô’s formula, as we need it to find the semi-closed form formulas for the characteristic function of log prices. Then, we will be developing the risk-neutral measure specific to our semi-Markov switching models. As some of us may already know, the Lévy driven regime-switching markets are incomplete, which means that there is more than one risk-neutral measure when pricing financial derivatives. When pricing a European-style option, since we already have the semi-closed form of the characteristic function for log asset prices, it allows us to use a Fourier transform method, first derived by Carr and Madan, namely the Fractional Fast Fourier Transform (FRFT) algorithm to obtain the estimated option prices. When comparing with Markov switching models, estimations and simulations show that the semi-Markov model performs better. It also offers more insight into the dynamics of market regimes, providing us with a better explanation of where the financial market is headed to next.Item Open Access VIX-linked GMMB under affine GARCH models and its Diffusion Limits(2018-07-06) Chen, Yuyu; Badescu, Alexandru M.; Ware, Antony F.; Swishchuk, Anatoliy V.In variable annuity (VA) industry, to compensate for the liability coming from embedded riders in VA, insurer usually charge a fixed percentage of investment fund as the riders fee. However, the traditional fixed-fee structure would misalign insurer’s income and liability and in consequence cause risk management challenges for insurer. In 2013, the Chicago Board of Options Exchange (CBOE) suggests linking riders fee in variable annuity with VIX index in a white paper and shows that VIX-linked fee structure can help to re-align insurer’s income and liability using non-parametric models. Affine GARCH models are used in this work to analyze VIX-linked fee structure for VA with guarantee minimum maturity benefit (GMMB). A closed-form solution to GMMB has been derived and is used to determine a fair fee structure. Comparison between fixed-fee structure and VIX-linked fee structure has been been shown by numerical examples.