Browsing by Author "Swishchuk, Anatoliy"
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- ItemOpen AccessAffine GARCH option pricing models, stochastic interest rates, and diffusion limits(2022-09) Gu, Zhouzhou; Badescu, Alexandru; Qiu, Jinniao; Swishchuk, AnatoliyThis article proposes a derivative pricing framework when the asset returns and the short term rate process are modelled with affine GARCH models driven by correlated Gaussian innovations. The risk neutral dynamics are derived based on a co-variance dependent pricing kernel and semi-closed form solutions are derived for European style options and bond prices. We further derive the weak diffusion limits of the underlying processes under both physical and risk-neutral measure and we investigate the consistency between the proposed pricing kernel with the well-known Girsanov principle in continuoustime. A variety of numerical exercises are provided to analyze the validity of our pricing formulae, the sensitivity of the option prices relative to the pricing kernel parameters, and the convergence of option prices to those based on the limiting diffusions. Finally, we illustrate an empirical analysis based on a joint estimation using historical asset returns and short-term rates, and cross sections of options and bond prices.
- ItemOpen AccessAlpha-stable, normal inverse gaussian and multi-factor models for spot and futures modelling in natural gas(2009) Nedunthally, Thomas; Swishchuk, Anatoliy
- ItemOpen AccessConstrained LASSO for Sparse Identification of Nonlinear Dynamical Systems (SINDy)(2022-11-28) Mackie, Alexander Douglas; Sezer, Ayse Deniz; Martinuzzi, Robert; Swishchuk, AnatoliyOur work explores and expands on (Brunton, Proctor, and Kutz, 2016) with regard to bluff body vortex shedding. We have adapted the SINDy method by applying a transformation of the data to reduce the number of dimensions under investigation. We also applied Galerkin constraints associated with our transformation in order to further reduce the variables being considered when model building. Finally, by using LASSO as our method of solving the SINDy problem rather than sequential threshold least squares, we have created a much more efficient approach that attempts to discover the generating equations of the non-linear dynamical system associated with vortex shedding in the wake of a flat plate. Our approach was tested by modeling vortex shedding in the wake of a cylindrical bluff body with a low Reynolds number, and was able to extract expected elements of the governing equations. With this success, we established several models for vortex shedding in the wake of a flat plate bluff body (for both open and closed ends) at a high Reynolds number. Under our transformed data, we obtained 2, 3 and 5 mode models that may shed some light into the dynamics of the system.
- ItemOpen AccessDelta Hedging Variable Annuities under Wiener Chaos Expansion(2017) Wiredu, David; Ambagaspitiya, Rohana Shantha; Ware, Antony; Swishchuk, Anatoliy; Scollnik, DavidWe present a method for Delta-Hedging of variable annuity products under the Guaranteed Minimum withdrawal Benefit (GMWB) Rider based on Wiener Chaos expansion. We also discuss well-known Monte-Carlo methods for computing Delta for derivatives, particularly for variable annuities. A detailed discussion of Wiener Chaos expansion is then provided. A discussion on the Heath-Jarrow-Morton (HJM) framework in the context of constant and exponential volatility is given. Choosing a Geometric Brownian motion for the underlying in the variable annuity account and the Heath-Jarrow-Morton model for interest rates we present results of delta computed by the Wiener Chaos technique in MATLAB using the UQLab framework. Comparisons between this method and the Monte-Carlo benchmark are then presented.
- ItemOpen AccessDerivatives Pricing with Fractional Discrete-time Models(2022-07-07) Jayaraman, Sarath Kumar; Badescu, Alexandru; Godin, Frederic; Qiu, Jinniao; Swishchuk, Anatoliy; Ware, AntonyThis thesis studies the pricing of European style derivatives with various affine models. Most of this thesis focuses on the impact of long memory on asset return modelling and option pricing. We propose a general discrete-time pricing framework based on affine multi-component volatility models that admit ARCH(∞) representations. It not only nests a large variety of option pricing models from the literature, but also allows for the introduction of novel fractionally integrated processes for option valuation purposes. Using an infinite sum characterization of the log-asset price’s cumulant generating function, we derive semi-explicit expressions for European option prices under a variance-dependent stochastic discount factor. We carry out an extensive empirical analysis which includes estimations based on different combinations of returns and options of the S&P 500 index for a variety of short- and long-memory models. Our results indicate that the inclusion of long memory into return modelling substantially improves the option pricing performance. Using a set of out-of-sample option pricing errors, we show that long-memory models outperform richer parametrized one- and two-component models with short-memory dynamics. The last part of the thesis studies the pricing of volatility derivatives with affine models. We propose semi-closed form solutions, subject to an inversion of the Fourier transform, for the price of VIX options and target volatility options under affine GARCH models based on Gaussian and Inverse Gaussian distributions. The empirical performance of the two affine GARCH models is tested using different calibration exercises based on historical returns and market quotes on VIX and SPX options.
- ItemOpen AccessDiffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas(Hindawi Publishing Corporation, 2010-11-08) Swishchuk, Anatoliy; Islam, M. Shafiqul
- ItemOpen AccessDiffusion Approximations of the Geometric Markov Renewal Processes and Option Price Formulas(2010-12-19) Swishchuk, Anatoliy; Islam, M. ShafiqulWe consider the geometric Markov renewal processes as a model for a securitymarket and study this processes in a diffusion approximation scheme. Weak convergenceanalysis and rates of convergence of ergodic geometric Markov renewal processes in diffusionscheme are presented. We present European call option pricing formulas in the case ofergodic, double-averaged, and merged diffusion geometric Markov renewal processes.
- ItemOpen AccessEuropean and American Option Pricing with a Geometric Markov Renewal Process Underlying Asset Model(2015-02-03) Moyer, Zachary; Swishchuk, AnatoliyWe present methods for pricing of American and European options under a Geometric Markov Renewal Process (GMRP) as the underlying asset model. We provide a detailed overview of the GMRP. Discussions of Markov processes, Geometric Brownian Motion, and GMRP approximation techniques are presented. We discuss the Aase trading model, with a MATLAB implementation. We discuss the Black-Scholes and binomial Cox-Ross-Rubinstein formulas for European and American options. We present results on Fixed Time Increments GMRP, with a derivation of a method for a limiting case of Fixed Time Increments GMRP (applicable to perpetual American options), complete with MATLAB implementations. We also present a MATLAB implementation for the pricing of European options under GMRP with an arbitrary jump distribution. We discuss diffusion and normal deviated approximations of a GMRP, and present MATLAB implementations for pricing American and European options. We follow this with a discussion of a Poisson approximation of a security market. A literature review is presented, together with an appendix including our MATLAB implementations. Conclusions and recommendations for future research directions conclude the paper.
- ItemOpen AccessEuropean and swing option pricing under mean-reverting jump diffusion models(2007) Ouyang, Yuyuan; Swishchuk, Anatoliy
- ItemOpen AccessFundamental Modeling of the Alberta Power Market(2016) Elham, Negahdary; Ware, Antony Frank; Swishchuk, Anatoliy; Zinchenko, Yuriy; Davison, MattIn this project, we identify the primary price drivers and characterize their dynamics in an engineering-based bottom-up model. This fundamental model is based on the economic theory of supply and demand. The power price is determined by the intersection of demand with the dynamic characteristics of the generators’ supply functions. The dynamics of power prices will be represented naturally while satisfying operational constraints. In view of the uncertainty around the future values, we model independent exogenous variables such as fuel prices, outages, and load, as stochastic processes. A broad analysis of levels of aggregation and model simplification will be a comprehensive reference for future modelers. The downside of this methodology is the onerous data and input parameter requirements and burdensome computational costs.
- ItemOpen AccessHydrological Time Series Modelling and Applications(2016) Asadzadeh, Ilnaz; Ware, Antony; Badescu, Alexandru; Swishchuk, Anatoliy; Zinchenko, YuriyWe consider the problem of measuring reliability of a hydro reservoir over a finite horizon with a stochastic optimal control technique. To apply this technique, we need to model the underlying stochastic process which is the inflow of water to the reservoir. Typical time series models for such problems only capture linear dependency (simple correlation) in the data. Alternative approaches include artificial neural network methods but these lack a theoretical foundation and a systematic procedure for the construction of the model. To overcome both of these limitations, we propose a new framework based on the application of copulas to univariate time series modelling. Our model shows that some important statistical characteristics of hydrological time series, such as upper and lower tail dependencies, persistence, etc., can be described with the aid of copulas. In turn, this provides insight regarding the qualitative properties of the underlying time series. Our main contribution is a new method of estimation based on a semi-parametric technique. By semi-parametric we mean using empirical autocopula (copula of a time series with itself with different lags), and parametric marginal distributions. Goodness of fit analysis is carried out and numerical results are illustrated with variety of concrete examples and sample data sets. We then benchmark and compare our scheme to alternative methods such as parametric models and various other time series modelling techniques. The final part of the dissertation proposes an application of stochastic optimal control to measure the reliability function (the probability that a system will perform the required function for a specified period of time under stated conditions) of the reservoir. For this section, we work with both uncorrelated and correlated inflow time series. For the first case, we generate independent inflow series using some probability distribution and for the second assumption, correlated inflow series, we employ the values of inflow generated using the autocopula method.
- ItemOpen AccessLevy driven Markov-modulated Ornstein-Uhlenbeck processes: application to Alberta electricity market(2012-10-03) Zhao, Ke; Swishchuk, AnatoliyThis thesis is a study of the distinctive stochastic properties exhibited in Alberta's electricity market. Electricity spot prices are notoriously difficult to model, which motivate us to develop new models. Our model combines the Ornstein-Uhlenbeck process for the spot dynamics of electricity with Markov-Modulated parameters. In this way, the model allows for Markov-Modulated mean-reversion rates and volatilities. Compared with the classical finance model, Markov-Modulated model or Markovian regime-switching models, by construction, should provide a better fit to volatile electricity spot prices. A brief overview of the history of Markov-Modulated models in finance theory, as well as the main contributions and contents of this thesis, is given in Chapter 1. Chapter 2 is the theoretical foundation of this thesis; in this chapter, we review some basic definitions and results on Markov process, Semi-Markov process, continuous time Markov chain, Levy processes and Ito's formula. In Chapter 3, we develop a new process called the Markov-Modulated Ornstein-Uhlenbeck process. We study all properties of this Markov-Modulated Ornstein-Uhlenbeck process. We build two models, geometric and arithmetic, for electricity spot price dynamics. In Chapter 4, using the models we described in Chapter 3, we give the pricing formulas for forwards and swaps contracts of electricity. Two approaches, geometric and arithmetic, are used to derive different types of pricing formulas using different properties of geometric and arithmetic models. Two specific Levy processes, NIG and CGMY are studied in detail for both geometric and arithmetic cases. Another main contribution of this thesis is given in Chapter 5, where we invoke the Markov-Modulated and Semi-Markov-Modulated volatilities together with electricity forward prices to get a generalization of the Black-76 formula to price European call options, for the cases with symmetric and non-symmetric transition rates between states, and with and without jumps in the forward dynamics. Simulation results for the models described in Chapter 3 are given in Chapter 6. Finally, daily average electricity spot market data from the Alberta electricity market for the period of January 1, 2000 to December 31, 2011 are studied in detail in Chapter 7. In Chapter 8, we conclude and propose future work.
- ItemOpen AccessLocational Spread Options with Stochastic Correlation(2023-05-05) Ali, Syeda Fareeha; Ware, Antony; Swishchuk, Anatoliy; Zinchenko, YuriyContrary to the common assumption, the correlation between financial derivatives may not be constant across time. This thesis analyses the role of stochastic correlation in modeling for locational spread options for natural gas. We first derive a model with Ornstein–Uhlenbeck process between two spread assets with constant correlation and then a combination of the Ornstein–Uhlenbeck and Jacobi process is used to model a stochastic correlation. The Margrabe formula is employed to evaluate options prices with constant correlation, the solution for which is used to compare with Monte Carlo simulations for stochasticity. Comparing the results, we find out why stochastic correlation is more important in real markets.
- ItemOpen AccessMerton Investment Problem for the Hawkes-based Risk Model(2022-09-21) Nova, Mushfika Hossain; Qiu, Jinniao; Swishchuk, Anatoliy; Badescu, Alexandru; Jiang, WenjunWe study the Merton investment problem in insurance where the risk process is based on the general compound Hawkes process. That means the arrival of claims modeled with a Hawkes process and the modeled claim sizes follow a finite number of fixed jump sizes governed by a Markov chain evolution. The Merton investment problem in insurance is an optimal control problem and we use the dynamic programming method to derive the stochastic Hamilton-Jacobi-Bellman (SHJB) equation satisfied by the value function. The stochastic HJB equation yields a means to obtain the optimal control and thus the optimally controlled stochastic differential equation. Finally, using the claim size from the empirical data set, we simulate the optimal investment portfolio and risk process.
- ItemOpen AccessMerton Problem in Insurance(2022-03) Fooladamoli, Ehsan; Swishchuk, Anatoliy; Swishchuk, Anatoliy; Liao, Wenyuan; Ambagaspitiya, RohanaThe goal of Insurance companies, like that of any other financial institution, is to maximize their wealth. In doing so, there are different parameters they have to consider, such as premium rate, number of claim arrivals, size of claim arrival, etc. Moreover, they can invest their money in risk-free and risky asset to earn some income from those resources as well. This thesis discusses the application of Merton problem in insurance and risk and how to solve it. That is, we design a trading strategy for an insurance company such that its utility is maximized over a given time horizon. We use General Compound Hawkes Process to model the insurance’s risk and use the corresponding diffusion approximation to approximate the risk using a diffusion process. Then, we proceed with solving the problem by Hamilton- Jacobi-Bellman equation. Finally, we show some simulation results based on the calibration on data from insurance companies in Germany and their interpretations.
- ItemOpen AccessModeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering(2010-11-21) Swishchuk, Anatoliy; Manca, RaimondoWe consider a semi-Markov modulated security market consisting of a riskless asset or bond with constant interest rate and risky asset or stock, whose dynamics follow gemoetric Brownian motion with volatility that depends on semi-Markov process. Two cases for semi-Markov volatilities are studied: local current and local semi-Markov volatilities. Using the martingale characterization of semi-Markov processes, we find the minimal martingale measure for this incomplete market. Then we model and price variance and volatility swaps for local semi-Markov stochastic volatilities.
- ItemOpen AccessModeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering(Hindawi Publishing Corporation, 2010-10-14) Swishchuk, Anatoliy; Manca, Raimondo
- ItemOpen AccessModeling and pricing variance and volatility swaps for stochastic volatility models with jumps(2007) Zhao, Lu; Swishchuk, Anatoliy
- ItemOpen AccessModeling of Currency Trading Markets and Pricing Their Derivatives in a Markov Modulated Environment(2014-05-08) Tertychnyi, Maksym; Swishchuk, Anatoliy; Elliott, RobertUsing a Levy process we generalize formulas in Bo et al. (2010) to the Esscher transform parameters for the log-normal distribution which ensures the martingale condition holds for the discounted foreign exchange rate. We also derive similar results, but in the case when the dynamics of the FX rate is driven by a general Merton jump-diffusion process. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential and exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.
- ItemOpen AccessModeling Stimulated Rock Volumes Using DPDK Approach Coupled With Rock Mechanics(2016) Deng, Hui; Chen, Zhangxing (John); Aguilera, Roberto; Chen, Shengnan; Swishchuk, Anatoliy; Gildin, EduardoModeling a hydraulically fractured unconventional tight sand reservoir is a coupled hydro-mechanical problem associated with complex interactions between dynamical fracture deformation under a loading condition and multiphase flow inside a fracture network. First, a DPDK model with numerical MINC (multiple interacting continua) algorithms is utilized to represent the pressure transient within the tight matrix and interporosity flow from the matrix into a hybrid system consisting of a complex fracture network. The flow model is coupled with a FEM stress code to model the dynamical changing of fracture aperture associated with increasing of the effective normal stress during the pressure depletion. Second, the choice of a coupling approach is critical because of the slow nonlinear convergence due to a significant increase in unknowns associated with poroelasticity equations. In this thesis, the iterative coupling is adopted to solve the multiphase flow and stress equations using parallel computations because of its flexibility and efficiency compared to the fully coupled approach. A hydraulic fracture deformation mechanical model is developed and implemented into the PRSI framework. The fracture network closure is approximated by the Barton-Bandis hyperbolic deformation model, coupled with a modified Cubic Law based on a contact theory and validated by an API proppant test on proppant conductivity under loading stress. Finally, numerical examples on hydraulic fracture deformation will be presented. The coupled fluid flow-rock mechanics will illustrate the degradation of the fracture conductivity due to the increasing of normal stresses for a variety of proppant types.
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