We find empirical evidence that mean-reverting jump processes are not statistically adequate to model electricity spot price spikes but independent, signed sums of such processes are statistically adequate. Further we demonstrate a change in the composition of these sums after a major economic event. This is achieved by developing a Markov Chain Monte Carlo (MCMC) procedure for Bayesian model calibration and a Bayesian assessment of model adequacy (posterior predictive checking). In particular we determine the number of signed mean-reverting jump components required in the APXUK and EEX markets, in time periods both before and after the recent global financial crises. Statistically, consistent structural changes occur across both markets, with a reduction of the intensity and size, or the disappearance, of positive price spikes in the later period. All code and data are provided to enable replication of results.
In related contexts, MCMC has been applied to fit continuous-time stochastic volatility models to financial time series, where the price is a diffusion process whose volatility is a latent mean reverting jump process or the sum of a number of such processes (called a superposition model). In this line of research a missing data methodology is employed whereby the observed process is augmented with one or more latent marked Poisson processes and the MCMC procedure generates posterior samples in this high dimensional augmented state space. Examples include @cite_2 , @cite_3 and @cite_5 . Since energy prices additionally exhibit jumps directly in their paths, MCMC has been applied to extensions of these models in which a diffusion process with stochastic volatility is superposed with a jump process, see in the context of electricity and for gas prices. Technically the latter two papers estimate a discrete approximation of the models whereas in this study we pursue inference for continuous time models.
@cite_5 @cite_3 @cite_2
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Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Levy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Levy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used. WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematical tools required for applications can be intimidating. Potential users often get the impression that jump and Levy processes are beyond their reach.Financial Modelling with Jump Processes shows that this is not so. It provides a self-contained overview of the theoretical, numerical, and empirical aspects involved in using jump processes in financial modelling, and it does so in terms within the grasp of nonspecialists. The introduction of new mathematical tools is motivated by their use in the modelling process, and precise mathematical statements of results are accompanied by intuitive explanations.Topics covered in this book include: jump-diffusion models, Levy processes, stochastic calculus for jump processes, pricing and hedging in incomplete markets, implied volatility smiles, time-inhomogeneous jump processes and stochastic volatility models with jumps. The authors illustrate the mathematical concepts with many numerical and empirical examples and provide the details of numerical implementation of pricing and calibration algorithms.This book demonstrates that the concepts and tools necessary for understanding and implementing models with jumps can be more intuitive that those involved in the Black Scholes and diffusion models. If you have even a basic familiarity with quantitative methods in finance, Financial Modelling with Jump Processes will give you a valuable new set of tools for modelling market fluctuations. In this paper, we adapt recently developed simulation-based sequential algorithms to the problem concerning the Bayesian analysis of discretely observed diffusion processes. The estimation framework involves the introduction of m?1 latent data points between every pair of observations. Sequential MCMC methods are then used to sample the posterior distribution of the latent data and the model parameters on-line. The method is applied to the estimation of parameters in a simple stochastic volatility model (SV) of the U.S. short-term interest rate. We also provide a simulation study to validate our method, using synthetic data generated by the SV model with parameters calibrated to match weekly observations of the U.S. short-term interest rate. Diffusion processes governed by stochastic differential equations (SDEs) are a well-established tool for modelling continuous time data from a wide range of areas. Consequently, techniques have been developed to estimate diffusion parameters from partial and discrete observations. Likelihood-based inference can be problematic as closed form transition densities are rarely available. One widely used solution involves the introduction of latent data points between every pair of observations to allow a Euler-Maruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem that worsens with the degree of augmentation. A global MCMC scheme that can be applied to a large class of diffusions and whose performance is not adversely affected by the number of latent values is therefore explored. The methodology is illustrated by estimating parameters governing an auto-regulatory gene network, using partial and discrete data that are subject to measurement error. This paper provides methods for carrying out likelihood based inference for diffusion driven models, for example discretely observed multivariate diffusions, continuous time stochastic volatility models and counting process models. The diffusions can potentially be non-stationary. Although our methods are sampling based, making use of Markov chain Monte Carlo methods to sample the posterior distribution of the relevant unknowns, our general strategies and details are different from previous work along these lines. The methods we develop are simple to implement and simulation efficient. Importantly, unlike previous methods, the performance of our technique is not worsened, in fact it improves, as the degree of latent augmentation is increased to reduce the bias of the Euler approximation. In addition, our method is not subject to a degeneracy that afflicts previous techniques when the degree of latent augmentation is increased. We also discuss issues of model choice, model checking and filtering. The techniques and ideas are applied to both simulated and real data. Diffusion processes are a family of continuous-time continuous-state stochastic processes that are in general only partially observed. The joint estimation of the forcing parameters and the system noise (volatility) in these dynamical systems is a crucial, but non-trivial task, especially when the system is nonlinear and multi-modal. We propose a variational treatment of diffusion processes, which allows us to compute type II maximum likelihood estimates of the parameters by simple gradient techniques and which is computationally less demanding than most MCMC approaches. We also show how a cheap estimate of the posterior over the parameters can be constructed based on the variational free energy. This paper discusses practical Bayesian estimation of stochastic volatility models based on OU processes with marginal Gamma laws. Estimation is based on a parameterization which is derived from the Rosinski representation, and has the advantage of being a non-centered parameterization. The parameterization is based on a marked point process, living on the positive real line, with uniformly distributed marks. We define a Markov chain Monte Carlo (MCMC) scheme which enables multiple updates of the latent point process, and generalizes single updating algorithm used earlier. At each MCMC draw more than one point is added or deleted from the latent point process. This is particularly useful for high intensity processes. Furthermore, the article deals with superposition models, where it discuss how the identifiability problem inherent in the superposition model may be avoided by the use of a Markov prior. Finally, applications to simulated data as well as exchange rate data are discussed. We present a novel approach to inference in conditionally Gaussian continuous time stochastic processes, where the latent process is a Markovian jump process. We first consider the case of jump-diffusion processes, where the drift of a linear stochastic differential equation can jump at arbitrary time points. We derive partial differential equations for exact inference and present a very efficient mean field approximation. By introducing a novel lower bound on the free energy, we then generalise our approach to Gaussian processes with arbitrary covariance, such as the non-Markovian RBF covariance. We present results on both simulated and real data, showing that the approach is very accurate in capturing latent dynamics and can be useful in a number of real data modelling tasks. In this paper, we introduce a new Markov chain Monte Carlo approach to Bayesian analysis of discretely observed diffusion processes. We treat the paths between any two data points as missing data. As such, we show that, because of full dependence between the missing paths and the volatility of the diffusion, the rate of convergence of basic algorithms can be arbitrarily slow if the amount of the augmentation is large. We offer a transformation of the diffusion which breaks down dependency between the transformed missing paths and the volatility of the diffusion. We then propose two efficient Markov chain Monte Carlo algorithms to sample from the posterior-distribution of the transformed missing observations and the parameters of the diffusion. We apply our results to examples involving simulated data and also to Eurodollar short-rate data. We develop Markov chain Monte Carlo methodology for Bayesian inference for non-Gaussian Ornstein-Uhlenbeck stochastic volatility processes. The approach introduced involves expressing the unobserved stochastic volatility process in terms of a suitable marked Poisson process. We introduce two specific classes of Metropolis-Hastings algorithms which correspond to different ways of jointly parameterizing the marked point process and the model parameters. The performance of the methods is investigated for different types of simulated data. The approach is extended to consider the case where the volatility process is expressed as a superposition of Ornstein-Uhlenbeck processes. We apply our methodology to the US dollar-Deutschmark exchange rate. Copyright 2004 Royal Statistical Society. We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the risk-neutral process should be free of a diffusion component. We conclude that the statistical and risk-neutral processes for equity prices are pure jump processes of infinite activity and finite variation. The analysis of correlated point process data has wide applications, ranging from biomedical research to network analysis. In this work, we model such data as generated by a latent collection of continuous-time binary semi-Markov processes, corresponding to external events appearing and disappearing. A continuous-time modeling framework is more appropriate for multichannel point process data than a binning approach requiring time discretization, and we show connections between our model and recent ideas from the discrete-time literature. We describe an efficient MCMC algorithm for posterior inference, and apply our ideas to both synthetic data and a real-world biometrics application. Stochastic partition processes for exchangeable graphs produce axis-aligned blocks on a product space. In relational modeling, the resulting blocks uncover the underlying interactions between two sets of entities of the relational data. Although some flexible axis-aligned partition processes, such as the Mondrian process, have been able to capture complex interacting patterns in a hierarchical fashion, they are still in short of capturing dependence between dimensions. To overcome this limitation, we propose the Ostomachion process (OP), which relaxes the cutting direction by allowing for oblique cuts. The partitions generated by an OP are convex polygons that can capture inter-dimensional dependence. The OP also exhibits interesting properties: 1) Along the time line the cutting times can be characterized by a homogeneous Poisson process, and 2) on the partition space the areas of the resulting components comply with a Dirichlet distribution. We can thus control the expected number of cuts and the expected areas of components through hyper-parameters. We adapt the reversible-jump MCMC algorithm for inferring OP partition structures. The experimental results on relational modeling and decision tree classification have validated the merit of the OP. A neural network model that processes financial input data is developed to estimate the market price of options at closing. The network's ability to estimate closing prices is compared to the Black-Scholes model, the most widely used model for the pricing of options. Comparisons reveal that the mean squared error for the neural network is less than that of the Black-Scholes model in about half of the cases examined. The differences and similarities in the two modeling approaches are discussed. The neural network, which uses the same financial data as the Black-Scholes model, requires no distribution assumptions and learns the relationships between the financial input data and the option price from the historical data. The option-valuation equilibrium model of Black-Scholes determines option prices under the assumptions that prices follow a continuous time path and that the instantaneous volatility is nonstochastic. Preface. Part I: Extensions of Basic Models. 1. The Solidarity of Markov Renewal Processes R. Pyke. 2. A Generalization of Semi-Markov Processes M. Iosifescu. 3. Quasi-stationary Phenomena for Semi-Markov Processes M. Gyllenberg, D.S. Silvestrov. 4. Semi-Markov Random Walks V.S. Korolyuk. 5. Diffusion Approximation for Processes with Semi-Markov Switches V.V. Anisimov. 6. Approximations for Semi-Markov Single Ion Channel Models S.M. Pitts. Part II: Statistical Estimation. 7. Log-likelihood in Stochastic Processes G.G. Rousas, D. Bhattacharya. 8. Some Asymptotic Results and Exponential Approximation in Semi-Markov Models G.G. Roussas, D. Bhattacharya. 9. Markov Renewal Processes and Exponential Families V.T. Stefanov. 10. On Homogeneity of Two Semi-Markov Samples L. Afanasyeva, P. Radchenko. 11. Product-Type Estimator of Convolutions I. Gertsbakh, I. Spungin. 12. Failure Rate Estimation of Semi-Markov Systems B. Ouhbi, N. Limnios. 13. Estimation for Semi-Markov Manpower Models in a Stochastic Environment S. McClean, E. Montgomery. 14. Semi-Markov Models for Lifetime Data Analysis R. Perez-Ocon, et al Part III: Non-Homogeneous Models. 15. Continuous Time Non Homogeneous Semi-Markov Systems A.A. Papadopoulou, P.C.G. Vassiliou. 16. The Perturbed Non-Homogeneous Semi-Markov System P.C.G. Vassiliou, H. Tsakiridou. Part IV: Queueing Systems Theory. 17. Semi-Markov Queues with Heavy Tails S. Asmussen.18. MR Modelling of Poisson Traffic at Intersections Having Separate Turn Lanes R. Gideon, R. Pyke. Part V: Financial Models. 19. Stochastic Stability and Optimal Control in Insurance Mathematics A. Swishchuk. 20. Option Pricing with Semi-Markov Volatility J. Janssen, et al Part VI: Controlled Processes & Maintenance. 21. Applications of Semi-Markov Processes in Reliability and Maintenance M. Abdel-Hameed. 22. Controlled Queueing Systems with Recovery Functions T. Dohi, et al Part VII: Chromatography & Fluid Mechanics. 23. Continuous Semi-Markov Models for Chromatography B.P. Harlamov. 24. The Stress Tensor of the Closed Semi-Markov System. Energy and Entropy G.M. Tsaklidis. Index. In many stochastic models, particularly in queueing theory, Poisson arrivals both observe (see) a stochastic process and interact with it. In particular cases and or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state. In this paper, we present a proof of this result under one basic assumption: the process being observed cannot anticipate the future jumps of the Poisson process. I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spot-asset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset's price is important for explaining return skewness and strike-price biases in the Black-Scholes (1973) model. The solution technique is based on characteristic functions and can be applied to other problems. Article published by Oxford University Press on behalf of the Society for Financial Studies in its journal, The Review of Financial Studies. Earlier we proposed the stochastic point process model, which reproduces a variety of self-affine time series exhibiting power spectral density S(f) scaling as power of the frequency f and derived a stochastic differential equation with the same long range memory properties. Here we present a stochastic differential equation as a dynamical model of the observed memory in the financial time series. The continuous stochastic process reproduces the statistical properties of the trading activity and serves as a background model for the modeling waiting time, return and volatility. Empirically observed statistical properties: exponents of the power-law probability distributions and power spectral density of the long-range memory financial variables are reproduced with the same values of few model parameters. The validity of the classic Black-Scholes option pricing formula depends on the capability of investors to follow a dynamic portfolio strategy in the stock that replicates the payoff structure to the option. The critical assumption required for such a strategy to be feasible, is that the underlying stock return dynamics can be described by a stochastic process with a continuous sample path. In this paper, an option pricing formula is derived for the more-general case when the underlying stock returns are generated by a mixture of both continuous and jump processes. The derived formula has most of the attractive features of the original Black-Scholes formula in that it does not depend on investor preferences or knowledge of the expected return on the underlying stock. Moreover, the same analysis applied to the options can be extended to the pricing of corporate liabilities. Brownian motion and normal distribution have been widely used in the Black--Scholes option-pricing framework to model the return of assets. However, two puzzles emerge from many empirical investigations: the leptokurtic feature that the return distribution of assets may have a higher peak and two (asymmetric) heavier tails than those of the normal distribution, and an empirical phenomenon called "volatility smile" in option markets. To incorporate both of them and to strike a balance between reality and tractability, this paper proposes, for the purpose of option pricing, a double exponential jump-diffusion model. In particular, the model is simple enough to produce analytical solutions for a variety of option-pricing problems, including call and put options, interest rate derivatives, and path-dependent options. Equilibrium analysis and a psychological interpretation of the model are also presented.