Filtering and forecasting commodity futures prices under an HMM framework

We propose a model for the evolution of arbitrage-free futures prices under a regime-switching framework. The estimation of model parameters is carried out using the hidden Markov filtering algorithms. Comprehensive numerical experiments on real financial market data are provided to illustrate the effectiveness of our algorithm. In particular, the model is calibrated with data from heating oil futures and its forecasting performance as well as statistical validity is investigated. The proposed model is parsimonious, self-calibrating and can be very useful in predicting futures prices. © 2013 Elsevier B.V

Stochastic models with random parameters for financial markets

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London.The aim of this thesis is a development of a new class of financial models with random parameters, which are computationally efficient and have the same level of performance as existing ones. In particular, this research is threefold. I have studied the evolution of storable commodity and commodity futures prices in time using a new random parameter model coupled with a Kalman filter. Such a combination allows one to forecast arbitrage-free futures prices and commodity spot prices one step ahead. Another direction of my research is a new volatility model, where the volatility is a random variable. The main advantage of this model is high calibration speed compared to the existing stochastic volatility models such as the Bates model or the Heston model. However, the performance of the new model is comparable to the latter. Comprehensive numerical studies demonstrate that the new model is a very competitive alternative to the Heston or the Bates model in terms of accuracy of matching option prices or computing hedging parameters. Finally, a new futures pricing model for electricity futures prices was developed. The new model has a random volatility parameter in its underlying process. The new model has less parameters, as compared to two-factor models for electricity commodity pricing with and without jumps. Numerical experiments with real data illustrate that it is quite competitive with the existing two-factor models in terms of pricing one step ahead futures prices, while being far simpler to calibrate. Further, a new heuristic for calibrating two-factor models was proposed. The new calibration procedure has two stages, offline and online. The offline stage calibrates parameters under a physical measure, while the online stage is used to calibrate the risk-neutrality parameters on each iteration of the particle filter. A particle filter was used to estimate the values of the underlying stochastic processes and to forecast futures prices one step ahead. The contributory material from two chapters of this thesis have been submitted to peer reviewed journals in terms of two papers: • Chapter 4: “A fast calibrating volatility model” has been submitted to the European Journal of Operational Research. • Chapter 5: “Electricity futures price models : calibration and forecasting” has been submitted to the European Journal of Operational Research

Essays on asset pricing

The dissertation consists of three chapters that represent separate papers in the area of asset pricing. The first chapter studies investors optimal asset allocation problem in which mean reversion in stock prices is captured by explicitly modeling transitory and permanent shocks. The second chapter focuses on option pricing with stochastic dividend yield. In this work, we present an option formula which does not depend on the dividend yield risk premium. In the final chapter, we work on commodity derivative pricing under the existence of stochastic convenience yield. In this paper, we discuss a Gaussian complete market model of commodity prices in which the stochastic convenience yield is assumed to be an affine function of a weighted average of past commodity price changes

Estimation of Hidden Markov Models and Their Applications in Finance

Movements of financial variables exhibit extreme fluctuations during periods of economic crisis and times of market uncertainty. They are also affected by institutional policies and intervention of regulatory authorities. These structural changes driving prices and other economic indicators can be captured reasonably by models featuring regime-switching capabilities. Hidden Markov models (HMM) modulating the model parameters to incorporate such regime-switching dynamics have been put forward in recent years, but many of them could still be further improved. In this research, we aim to address some of the inadequacies of previous regime-switching models in terms of their capacity to provide better forecasts and efficiency in estimating parameters. New models are developed, and their corresponding filtering results are obtained and tested on financial data sets. The contributions of this research work include the following: (i) Recursive filtering algorithms are constructed for a regime-switching financial model consistent with no-arbitrage pricing. An application to the filtering and forecasting of futures prices under a multivariate set-up is presented. (ii) The modelling of risk due to market and funding liquidity is considered by capturing the joint dynamics of three time series (Treasury-Eurodollar spread, VIX and S\&P 500 spread-derived metric), which mirror liquidity levels in the financial markets. HMM filters under a multi-regime mean- reverting model are established. (iii) Kalman filtering techniques and the change of reference probability-based filtering methods are integrated to obtain hybrid algorithms. A pairs trading investment strategy is supported by the combined power of both HMM and Kalman filters. It is shown that an investor is able to benefit from the proposed interplay of the two filtering methods. (iv) A zero-delay HMM is devised for the evolution of multivariate foreign exchange rate data under a high-frequency trading environment. Recursive filters for quantities that are functions of a Markov chain are derived, which in turn provide optimal parameter estimates. (v) An algorithm is designed for the efficient calculation of the joint probability function for the occupation time in a Markov-modulated model for asset returns under a general number of economic regimes. The algorithm is constructed with accessible implementation and practical considerations in mind

Pricing derivatives with stochastic volatility

This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy

Stochastic modelling in volatility and its applications in derivatives

This thesis consists of three articles concentrating on modelling stochastic volatility in commodity as well as equity and applying stochastic volatility models to evaluate financial derivatives and real options. Firstly, we introduce the general background and the incentive of considering stochastic volatility models. In Chapter 2 we derive tractable analytic solutions for futures and options prices for a linear-quadratic jump-diffusion model with seasonal adjustments in stochastic volatility and convenience yield. We then calibrate our model to data from the fish pool futures market, using the extended Kalman filter and a quasi-maximum likelihood estimator and alternatively using an implied-state quasi-maximum likelihood estimator. We find no statistical evidence of jumps. However, we do find evidence for the positive correlation between salmon spot prices and volatility, seasonality in volatility and convenience yield. In addition we observe a positive relationship between seasonal risk premium and uncertainty within the EU salmon demand. We further show that our model produces option prices that are conform with the observation of implied volatility smiles and skews. In Chapter 3, we introduce a linear quadratic volatility model with co-jumps and show how to calibrate this model to a rich dataset. We apply general method of moments (GMM) and more specifically match the moments of realized power and multi-power variations, which are obtained from high-frequency stock market data. Our model incorporates two salient features: the setting of simultaneous jumps in both return process and volatility process and the superposition structure of a continuous linear quadratic volatility process and a Lévy-driven Ornstein-Uhlenbeck process. We compare the quality of fit for several mod- els, and show that our model outperforms the conventional jump diffusion or Bates model. Besides that, we find evidence that the jump sizes are not normal distributed and that our model performs best when the distribution of jump-sizes is only specified through certain (co-) moment conditions. A Monte Carlo experiments is employed to confirm this. Finally, in Chapter 4, we study the optimal stopping problems in the context of American options with stochastic volatility models and the optimal fish harvesting decision with stochastic convenience yield models, in the presence of drift ambiguity. From the perspective of an ambiguity averse agent, we transfer the problem to the solution of a reflected backward stochastic differential equation (RBSDE) and prove the uniform Lipschitz continuity of the generator. We then propose a numerical algorithm with the theory of RBSDEs and a general stratification technique, and an alternative algorithm without using the theory of RBSDEs. We test the accuracy and convergence of the numerical schemes. By comparing to the one dimensional case, we highlight the importance of the dynamic structure of the agent’s worst case belief. Results also show that the numerical RBSDE algorithm with stratification is more efficient when the optimal generator is attainable

Models for the price of a storable commodity

The current literature does not provide efficient models for commodity prices and futures valuation. This inadequacy is partly due to the fact that the two main streams of the literature - structural models and reduced form models - are largely disjoint. In particular, existing structural models are developed under rigid discrete time framework that does not take into account the mean-reverting properties of commodity prices. Furthermore, most of the literature within this class does not analyze the properties of the futures prices. Current reduced-form models allow cash-and-carry arbitrage possibilities and do not take into account the dependence between the spot price volatility and the inventory levels. This thesis investigates three new models for the price of a storable commodity and futures valuation. Specifically, we develop a structural model and two reduced-form models. In doing so, we expand the leading models within each of the two streams of the literature, by establishing a link between them. Each of these models provide an advance of their type. This study makes several contributions to the literature. We provide a new structural model in continuous time that takes into account the mean reversion of commodity prices. This model is formulated as a stochastic dynamic control problem. The formulation provided is flexible and can easily be extended to encompass alternative microeconomic specifications of the market. The results provide an optimal storage policy, the equilibrium prices and the spot price variability. We also develop a numerical method that allows the construction and analysis of the forward curves implied by this model. We provide a separate analysis considering a competitive storage and considering a monopolistic storage. The results are consistent with the theory of storage. Furthermore, the comparison between monopoly and competition confirm the economic theory. We developed a simple reduced-form model that focuses both on the mean reverting properties of commodity prices and excludes cash-and-carry arbitrage possibilities. This model is compared with a standard single-factor model in the literature. This new model adds two important features to the standard model and motivates the development of a more sophisticated reduced-form model. Accordingly, the last model developed in this thesis is a reduced-form model. It is a two-factor model that represents the spot price and the convenience yield as two correlated stochastic factors. This model excludes cash-and-carry arbitrage possibilities and takes into account the relationship between the spot price volatility and the inventory level. We find an analytical solution for the futures prices. This model is tested empirically using crude oil futures data and it Is compared with one of the leading models in the literature. Both models are calibrated using Kalman filter techniques. The empirical results suggest that both models need to be improved in order to better fit the long-term volatility structure of futures contracts

Spotpreis-Modelle für Erdgas - Robustheit des Convenience Yield-Ansatzes

This thesis investigates spot price models for natural gas and develops a new model, which incorporates both the simplicity of the set-up of reduced-form models and economic insights into the most important drivers of gas prices. The model responds to common criticism on existing reduced-form models for energy prices, especially to certain misspecifications identified. More precisely, the stochastic convenience yield model by Gibson and Schwartz (1990) and Schwartz (1997), which has gained notable attention in practice, is extended by using a two-component convenience yield. The first component mirrors fundamental convenience yield dynamics, which arise from changes in air temperature and national gas inventories, whereas the second component represents changes in the risk attitude of market participants. With empirical data from the UK and US gas markets it is shown that the extended model significantly improves the out-of-sample price forecast with regard to the stochastic convenience yield model when the forecast horizon is increased beyond one day. At the same time, the in-sample and the cross-sectional fit to quoted futures prices along the term structure are at least as good as for the mentioned benchmark model. For both conceptual and numerical reasons, some further room for an amendment of the cross-sectional fit remains. Its realization would, however, necessitate a far more complex model set-up.Diese Dissertation untersucht Spotpreis-Modelle für Erdgas und entwickelt ein neues Modell, welches die gute Handhabbarkeit von „reduced form“-Modellen einerseits und fundamentale ökonomische Erkenntnisse über die Treiber von Erdgaspreisen andererseits zusammenführt. Das Modell begegnet der häufig geäußerten Kritik an existierenden „reduced form“-Modellen für Energiepreise, insbesondere bestimmten konstatierten Fehlspezifikationen. Hierzu wird das stochastische Convenience Yield-Modell von Gibson und Schwartz (1990) und Schwartz (1997), welches in der Praxis weit verbreitet ist, durch eine Zwei-Komponenten-Modellierung der Convenience Yield erweitert. Die erste Komponente bildet fundamental begründete Convenience Yield-Änderungen durch Veränderung von Lufttemperatur und nationalen Gasspeicherständen ab, während die zweite Komponente Veränderungen der Risikobereitschaft der Investoren anzeigt. Mit empirischen Daten aus dem UK- und dem US-Gasmarkt kann gezeigt werden, dass das erweiterte Modell signifikant genauere „out-of-sample“-Preisprognosen erzeugen kann als das stochastische Convenience Yield-Modell, wenn der Prognosehorizont länger als einen Tag beträgt. Gleichzeitig bleibt die Qualität der Modellanpassung innerhalb der Stichprobe und im Querschnitt über verschiedene Fristen von Futures-Kontrakten erhalten. Sowohl aus konzeptionellen als auch numerischen Gründen verbleibt bei der Querschnittsanpassung noch weiteres Verbesserungspotenzial, dessen Realisierung jedoch einen deutlich komplexeren Modellaufbau voraussetzt