PRISM :: Browsing by Author "Ware, Antony"

Browsing by Author "Ware, Antony"

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Open Access
A Class of Stochastic Path-Dependent Models and Controls
(2024-09-19) Yang, Yang; Qiu, Jinniao; Ware, Antony; Sezer, Deniz; Swishchuk, Anatoliy; Liao, Wenyuan; Huang, Minyi
This thesis explores controlled stochastic path-dependent differential systems and their applications in stock and energy markets. We first focus on a class of path-dependent stochastic optimal control problems within an infinite-dimensional framework. Our primary contribution is proving that the value function serves as the unique viscosity solution to the corresponding infinite-dimensional stochastic path-dependent Hamilton-Jacobi equation. Then, we investigate the introduction of path-dependent features in stochastic volatility models. The price-storage dynamic is studied in natural gas markets. We propose a novel stochastic path-dependent volatility model that incorporates path-dependence into both price volatility and storage increments. Chapter 1 focuses on the stochastic optimal control problem for infinite-dimensional differential systems that incorporate both path-dependence and measurable randomness. Unlike deterministic path-dependent cases, the value function emerges as a random field over the path space, characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. We propose a notion of viscosity solution and demonstrate that the value function is the unique viscosity solution to the associated SPHJ equation. Chapter 2 and 3 explores price-storage dynamics in natural gas markets. We introduce a novel stochastic path-dependent volatility model that incorporates path-dependence in both price volatility and storage increments. Model calibrations are performed for both price and storage dynamics. Additionally, we address the pricing problem for discrete-time swing options using the dynamic programming principle and propose a deep learning-based method for numerical approximations. A numerical algorithm is presented, accompanied by a convergence analysis of the deep learning approach. Chapter 4 outlines future research directions in stochastic path-dependent Hamilton-Jacobi-Bellman (HJB) equations, stochastic volatility models, mean field games and stochastic path-dependent filtering problems.
Open Access
A Study in Hybrid Monte Carlo Methods in Computing Derivative Prices
(2012-12-05) Wang, Binbin; Ware, Antony
Hybrid Monte Carlo (HMC) method is defined in this thesis as Monte Carlo method that utilizes conditional expectation so that the regular Monte Carlo method and other computational methods can be combined to price financial derivatives. This thesis introduces several hybrid Monte Carlo methods and studies the algorithm and efficiency of these methods, which include three methods combining Monte Carlo with fast Fourier transform, cosine series, and Black-Scholes formula respectively. These methods can be considered as ways of variance reduction. The thesis also introduces a new variance reduction method using orthogonal transformation which further reduces the variance. It is shown in this thesis that the HMC methods can significantly improve the efficiency when compared to the regular Monte Carlo method. A basket option example is used throughout this thesis for implementation and efficiency comparison.
Open Access
A Weighted Multilevel Monte Carlo Method
(2024-11-07) Li, Yu; Ware, Antony; Qiu, Jinniao; Swishchuk, Anatoliy
This thesis begins by introducing the Weighted Multilevel Monte Carlo (WMLMC) method in a one-dimensional context, expanding upon the established Multilevel Monte Carlo (MLMC) method. The focus is on demonstrating that the WMLMC method provides even greater computational savings compared to the traditional MLMC method, which has already shown superior efficiency over the standard Monte Carlo (MC) approach under similar conditions. In the second part, we apply a mixed Partial Differential Equation (PDE)/Weighted Multilevel Monte Carlo (WMLMC) method to the pricing of Double-No-Touch options. We compare these results with those obtained using a mixed PDE/Multilevel Monte Carlo (MLMC) method. The analysis reveals a significant reduction in total computational costs when substituting the MLMC method with the more efficient WMLMC method. This finding prompts us to explore the broader applicability of the WMLMC method as a superior alternative to the MLMC method across various domains, anticipating substantial cost savings in computational tasks traditionally dominated by the MLMC approach. The adoption of the WMLMC method is expected to optimize resource utilization and enhance computational performance, making it a promising avenue for future research and application in diverse fields beyond options pricing. The third part integrates the WMLMC method with one-step Richardson Extrapolation (RE), resulting in a considerable boost in efficiency from a convergence standpoint. A comparative analysis highlights this computational advantage, demonstrating that the WMLMC method with one-step RE achieves the greatest reduction in computational cost. This approach could potentially be extended to combine WMLMC with multi-step RE for even greater efficiency. In the final section, we present the Weighted Multi-Index Multilevel Monte Carlo (WMIMLMC) method designed for a multi-dimensional setting, building on the Multi-Index Monte Carlo (MIMC) method. By utilizing high-order mixed differences instead of first-order differences, the WMIMLMC method significantly reduces the variance of hierarchical differences while adhering to similar assumptions as its predecessors. We provide a 2-dimensional example to showcase the computational advantages of the WMIMLMC method.
Open Access
Advection-based spatiotemporal reconstruction techniques for turbulent velocity fields
(2023-05-01) Vocke, Maegan; Martinuzzi, Robert; Morton, Chris; Wood, David; Ware, Antony
Turbulent flows are characterized by interactions of widespread spatiotemporal scales. Experimental techniques such as Particle Image Velocimetry can provide correlated flow measurements, however, obtaining suitable resolution of temporal and spatial scales remains challenging. This work investigates the ability of an advection-based flow reconstruction technique to increase the temporal resolution of turbulent flow data. A semi-Lagrangian technique is used to obtain instantaneous fluid trajectories through a forward and backward integration of the known data. The estimates are then fused using a temporal weighting scheme to yield velocity fields at intermediate times. The performance of the method is verified against a benchmark direct numerical simulation (DNS) data set. Spectral information up to two orders of magnitude beyond the Nyquist criteria is successfully - surpassing the performance of previously introduced methods. Correlation maps are used to quantify the spatial loss of memory and define a criterion for the maximum recoverable frequency.
Open Access
Analytical Unsteady Aerodynamic Models for Horizontal Axis Wind Turbines
(2016) Hammam, Mohamed; Wood, David; Crawford, Curran; Morton, Chris; Johansen, Craig; Ware, Antony; Visser, Ken
This thesis describes the development of unsteady aerodynamic models of wind turbines within the framework of blade element momentum theory. The main purpose is to build analytical models, and test them against available wind tunnel and field test data. The unsteady wake is modeled using vortex methods. The analysis covers unsteady loading due to fast pitch change at constant wind speed, and varying wind speed at constant pitch. The analysis can be extended to more complex conditions. The work starts by modeling dynamic inflow effects as an inertia force, with a new developed expression of the apparent mass and rotational inertia of the rotor vortices. Further assumptions lead to a Ricatti differential equation for the axial induction, which has an analytical solution. First, linear pitch changes are investigated, and the unsteady thrust and torque are calculated. The model is shown to be accurate in comparison to the available experiments. The model is then extended to low tip speed ratios in the form of an Abel differential equation, which appeared to be more accurate. However, comparison with measurements has shown that the assumptions leading to the Abel equation model are inconsistent. The unsteady load is characterized by a large overshoot for high pitch rate. To alleviate this an analytical solution of exponential pitch angle is developed. %An analytical model is obtained for the exponential pitch case and validated against experiment. Next, the unsteady load due to varying wind speed at constant pitch is modeled. The dynamic inflow model is extended to include the circulatory effect of the wake. A new unsteady model is developed from first principles. The wake is modeled as an initial vortex cylinder and vortex rings released onto the wake with each revolution of the blades. The circulatory effect is described by a new function. The model is simplified further by approximating the wake effect to get an analytical solution for the case of linearly varying wind speed. The unsteady lift is modeled and combined with the dynamic inflow model to obtain a fully unsteady model in state space form. The different models are validated with experiments, and the unsteady lift effect is found to have importance for short time period transients.
Open Access
Assessing the Accuracy of Convective Heat Transfer from Overhead Conductor at Low Wind Speed Using Large Eddy Simulations (LES)
(2017-12-22) Abdelhady, Mohamed; Wood, David; Morton, Christopher; Ware, Antony
This project uses Computational Fluid Dynamics (CFD) to assess the accuracy of the forced cooling term for Real Time Thermal Rating (RTTR) of power lines in overhead conductor codes, IEEE 738 and CIGRÉ 207. The analysis is done for low wind speed, corresponding to Reynolds Number of 1,000, and 3,000. The project uses Large Eddy Simulation (LES) in the ANSYS Fluent software. The primary goal is to calculate the convective heat transfer for cylindrical and stranded conductors in non-turbulent flow and for cylindrical conductors with free-stream turbulence. The results showed that the heat transfer correlations used in the codes are accurate for low turbulent flows and that the stranded conductor causes an increase in heat transfer of ~9 % over a cylindrical conductor at low wind speed. The constant heat flux boundary condition experiences ~15 % higher Nusselt Number than uniform temperature boundary condition. The calculated increase in heat transfer due to turbulence was significant; increased heat transfer due to turbulence ~24 % at Reynolds Number of 3,000 at a turbulence intensity of 8% and length scale to diameter ratio of 0.4.
Open Access
Continuous Time Portfolio Selection under Conditional Capital at Risk
(2010-06-17) Dmitrasinovic-Vidovic, Gordana; Lari-Lavassani, Ali; Li, Xun; Ware, Antony
Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well-defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black-Scholes continuous time setting, with time dependent coefficients.
Open Access
Continuous Time Portfolio Selection under Conditional Capital at Risk
(Hindawi Publishing Corporation, 2010) Dmitrasinovic-Vidovic, Gordana; Lari-Lavassani, Ali; Li, Xun; Ware, Antony
Open Access
Credit Risk Pricing via Epstein-Zin Pricing Kernel
(2017) Ogunsolu, Mobolaji; Sezer, Deniz; Frei, Christoph; Badescu, Alexandru; Ware, Antony; David, Alexander; Liao, Wenyuan
We present an equilibrium framework for pricing corporate bonds with information delay in an Epstein-Zin setting. As in structural models of credit risk, the default time is modeled as the first hitting time of a default boundary by the unobservable process; the firm's asset value. The observable state variables; log consumption and volatility are affine processes which drive the unobservable firm's value process. The stochastic pricing kernel is expressed in terms of the state variables. The price of a zero-coupon bond is expressed as the solution of a multidimensional partial differential equation which is solved numerically. Our equilibrium price model is also calibrated to fit available corporate bond and consumption data. Finally, we analyze the implications of investor’s preferences and information delay on the credit yield spreads.
Open Access
Deep Learning-based Numerical Methods for Stochastic Partial Differential Equations and Applications
(2021-03-14) Yao, Yao; Qiu, Jinniao; Ware, Antony; Badescu, Alexandru
In this thesis, we are concerned with approximating solutions of stochastic partial differential equations (SPDEs) and their applications. Inspired by Huré, Pham, and Warin [15], we propose and study the deep learning-based methods for both the forward and backward SPDEs. In particular, the forward SPDEs may allow for Neumann boundary conditions. We also prove the convergence analysis of the proposed algorithms. The numerical results indicate that the performance of the algorithm is quite effective for solving the SPDEs, even in high-dimensional cases. The applications include various pricing problems under exchange rate target zone models as well as under rough volatility models.
Open Access
Delta Hedging Variable Annuities under Wiener Chaos Expansion
(2017) Wiredu, David; Ambagaspitiya, Rohana Shantha; Ware, Antony; Swishchuk, Anatoliy; Scollnik, David
We 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.
Open Access
Derivatives Pricing with Fractional Discrete-time Models
(2022-07-07) Jayaraman, Sarath Kumar; Badescu, Alexandru; Godin, Frederic; Qiu, Jinniao; Swishchuk, Anatoliy; Ware, Antony
This 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.
Open Access
Estimating Spot Price and Smooth Forward Curve in Electricity Markets with Bayesian Penalized Spline
(2014-05-05) Gezahagne, Azamed; Ware, Antony
The first part of this thesis presents a Bayesian penalized spline approach to constructing smooth forward curves in electricity markets. Since electricity should be delivered as a continuous flow, power contracts have settlement periods rather than a fixed delivery time. In addition, electricity forward curves have strong seasonal shape. Our approach provides a method for estimating a continuous forward price curve from market forward prices quoted over a period. The approach is illustrated using observed market data from the Mid-Columbia (Mid-C) and California Oregon Border (COB) pricing hubs. Since Mid-C is a liquid market where forward contracts are quoted every day and COB is an illiquid hub, a two step estimation procedure is developed from Bayesian perspective.First, the Mid-C smooth curve is constructed using Bayesian penalized spline. Next, the COB smooth curve is estimated by adding a spread to the constructed Mid-C smooth curve and incorporating the positive spread between the two hubs as an informative prior. In the second part, we present a mean reverting model for the electricity spot price and we employ a Bayesian penalized spline approach to model the deterministic seasonal function exhibited in the monthly averages. Based on the historical spot price from Alberta power market, we calibrated the model parameters, also by implementing Bayesian estimation techniques.
Open Access
Hydrological Time Series Modelling and Applications
(2016) Asadzadeh, Ilnaz; Ware, Antony; Badescu, Alexandru; Swishchuk, Anatoliy; Zinchenko, Yuriy
We 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.
Open Access
Large-scale coherent structures and three-dimensional velocity estimation in the turbulent wake of a low aspect ratio surface-mounted cone
(2017) Chen, Zixiang; Martinuzzi, Robert; Piomelli, Ugo; Ware, Antony; Wood, David; Sudak, Leszek; Morton, Christopher
The turbulent wake of a low-aspect-ratio right angle cone mounted on a flat surface, partially submerged in a steady turbulent boundary layer, is investigated using planar Stereoscopic Particle Image Velocimetry (SPIV) in a wind tunnel and Large Eddy Simulation (LES). The description of the dynamics of the large-scale quasi-periodic flow structures and their estimations using remote pressure sensors are considered in this study. The dominant vortex structures are first extracted from SPIV data using the traditional phase average method, which yields an average coherent velocity field that represents a typical shedding cycle. The formation and evolution of these large structures are visualized using volume rendering techniques and analyzed in the framework of vorticity dynamics. To go beyond the typical shedding cycle description, a technique to estimate the three-dimensional flow field from uncorrelated velocity measurements is presented. The method utilizes simultaneous pressure sensors mounted on the solid wall boundary in conjunction with the planar SPIV measurements. Compared to the typical shedding cycle representation, the technique can also characterize the low frequency modulation of the vortex shedding process. The capability of this technique to estimate the three-dimensional flow field is then assessed us- ing LES by applying the estimation techniques to planar LES velocity fields and comparing the results to the actual three-dimensional velocity field available from the simulation. The velocity estimation is found to be consistent with the actual three-dimensional velocity field. Additional coherent motions that are not captured in the experiment are also identified with the LES results, and strategies to capture these motions in SPIV measurements are developed based on the numerical data.
Open Access
Locational Spread Options with Stochastic Correlation
(2023-05-05) Ali, Syeda Fareeha; Ware, Antony; Swishchuk, Anatoliy; Zinchenko, Yuriy
Contrary 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.
Open Access
Low Order Representations in the Turbulent Wake of a Normal Flat Plate
(2015-12-22) Jochaud du Plessix, Phillip Alexandre; Martinuzzi, Robert; Morton, Chris; Ware, Antony; Wood, David
The turbulent wake of a normal flat plate, characterized by quasi-periodic vortex-shedding dynamics, is investigated in the context of low-order modeling, non-linear dynamics and kinetic energy transfer. The flow is characterized, coherent-structures are educed through the implementation of three models, and the transfer of kinetic energy between the most energetic structures and to the unresolved scales of motion is investigated. A low-order kinematic model (LOM) is constructed via the truncation of a Galerkin expansion using the energetically-optimal proper orthogonal decomposition (POD). The harmonic and shift modes of the LOM are coupled via the mean-field paraboloid in phase space, enabling the construction of a generalized phase-average (GPA). The GPA is shown to capture key flow behaviour and physics that traditional phase-averaging (TPA) cannot, including variation in the base flow and modulation of the vortex-shedding amplitude. A clearer understanding of energy transfer is made possible through the simplified representations of the models.
Open Access
Multivariate General Compound Hawkes and Point Processes with Financial Applications
(2022-10-20) Guo, Qi; Swishchuk, Anatoliy; Qiu, Jinniao; Badescu, Alexandru; Ware, Antony; Hyndman, Cody; Swishchuk, Anatoliy
The Hawkes process (HP) significantly affected the financial modeling area in the past 15 years, especially high-frequency trading. This thesis focuses on various new Hawkes processes and considers their applications in the limit order book (LOB). Preexisting studies of the HP in the LOB showed that the arrivals of orders could be modeled by univariate or multivariate HP because of its long memory property and clustering effect. Therefore, we propose the multivariate general compound Hawkes process (MGCHP), a stochastic model for the mid-price in the LOB. For the MGCHP, we prove the Law of Large Numbers (LLN) and two Functional Central Limit Theorems (FCLT); the latter provides insights into the link between price volatilities and order flows in limit order books with several assets. The parameter estimation for the high-dimensional Hawkes process is always time-consuming. This motivates us to consider a generalization of the MGCHP. We replace the multivariate HP with a more general point process, and we call it the multivariate general compound point process (MGCPP). We also prove limit theorems for the MGCPP and compared numerical simulations for the MGCPP with the MGCHP. The MGCHP model provides us with a perfect framework for the stock price dynamics in the LOB. It’s natural to apply it to other financial applications. We extend the MGCHP to the exponential MGCHP (EMGCHP) and consider the corresponding asset-liability management problem. Risky assets are molded by the EMGCHP while the liability follows a Brownian motion with drift. We derive the Hamilton–Jacobi–Bellman equation and transformed it into a system of PDEs. With the FCLT, we can approximate the EMGCHP to a geometric Brownian motion in the LOB and apply Xie et al.’s results. Numerical simulations for the Hawkes-based model and comparisons with the Poisson-based model are also provided. In the last part of the thesis, we give an option pricing formula under the EMGCHP framework. We believe our study can provide a strong tool for not only researchers but also traders in the high-frequency market.
Open Access
Optimal Control of a Semi-Markov Limit Order Book
(2017) Novak, Joshua; Swishchuk, Anatoliy; Ware, Antony; Choi, Kyoung Jin
A Limit Order Book (LOB) is a financial exchange that automatically matches the orders of buyers and sellers according to a predefined priority rule. As the popularity of Limit Order markets has increased, more attention has been paid to the optimal strategies for an agent who wishes to buy or sell an equity on the order book. In this thesis, we first introduce challenges of market microstructure and concepts of stochastic optimal control such as the Dynamic Programming Principle, the Hamilton-Jacobi-Bellman Equation, and tools such as the Feynman-Kac Formula. Then, we present Fodra and Pham's model for the optimal risk-neutral market making strategy in a limit order book that follows a Semi-Markov stochastic process, and use numerical techniques in MATLAB to apply the theory to real-word Limit Order data. Statistical techniques such as Maximum Likelihood Estimation, the Kolmogorov-Smirnov Test and Akaike Information Criterion are employed to distributions such as the Generalized Inverse Gaussian. Finally, we find the optimal risk-neutral strategy in an extension of the model when the price process contains a stochastic bid-ask spread following a Markov Chain.
Open Access
Polynomial Maps of Polynomial Processes for Energy Markets
(2024-12-05) Sun, Zuming; Ware, Antony; Ware, Antony; Aïd, René; Shaffer, Blake; Swishchuk, Anatoliy; Qiu, Jinniao
The empirical evidence reveals that energy prices are different from other commodity prices, exhibiting seasonality, mean reversion and extremely high volatility. These features arise from the interaction between supply and demand, with each subject to seasonal variations, and in particular from constraints on storage, transmission or transportation, perhaps exacerbated by sudden changes due to unforeseen infrastructure breakdowns. Pricing energy assets and energy derivatives is a significant challenge. In this thesis, we present a modeling framework based on the mathematical foundations for polynomial diffusions, involving polynomial maps of polynomial processes (PMPP models) for various energy prices such as oil, natural gas and electricity prices. Under the PMPP framework, underlying factors are modeled by polynomial processes such as the Ornstein-Uhlenbeck process, geometric Brownian motion, inhomogeneous geometric Brownian motion and etc. Energy prices are generated by polynomial maps of these underlying processes. The polynomial maps can be determined either by monotone polynomial maps or regression, revealing the relationship between underlying factors and energy prices. We will show that PMPP models are able to describe the dynamics of spot and forward prices in energy markets, providing the advantage of cheap and convenient computation of forward prices because of the property of polynomial processes that the conditional expectations of polynomial functions of future states, conditional on current states, are given by polynomials of the current states. We also introduce a potential extension of the PMPP framework where the underlying factor process and the map may not be polynomial. Under this extended framework, the PMPP model will still reserve a quasi-polynomial property whereby conditional expectation of polynomial functions of the future state are given by functions of the current state in a space spanned by a finite set of basis functions. The optimal PMPP model is determined by the statistical criteria. The maximum likelihood estimation and Kalman filters are applied for model calibration. Preliminary results indicate that the PMPP models are able to capture the behavior of energy prices well and can be useful for risk management purposes.