To Örebro University

This study investigates volatility spillovers to electric power from large exogenous shocks in the prices of gas, coal, and carbon emission allowances in the German energy market. Our sample ranges from 2008 to 2016 and covers periods of different market conditions. We use a general VAR-BEKK model and the volatility impulse response function methodology to analyze and evaluate the spillover effects. Special attention is paid to selecting an appropriate econometric volatility model. Our results show that the spillover effects often are of a significant magnitude and display considerable variation over time and across commodities. Coal and gas generate non-negligible spillovers during almost the entire sample period. Carbon has very little impact during the early and late parts of the sample, but generates significant, and highly variable, spillovers during the period from 2011 to the end of 2014.

This paper suggests a stochastic volatility term structure model applied to the pricing of electricity swaptions in the Nordic power market traded at the Nasdaq OMX Commodities exchange. The volatility structure in the model is specified as a product of a time-dependent function, which handles the maturity effect, and a Cox-Ingersoll-Ross process for the stochastic volatility. We employ a Fourier-based approach to pricing electricity swaptions and perform an empirical analysis by calibrating the model to a data set consisting of more than 12 000 pairs of implied bid-ask volatilities, corresponding to swaption prices from the Nordic power market. We show that our model outperforms the lognormal benchmark both in- and out-of-sample.

This study compares the empirical performance of different continuous time models for the dynamics of nine implied volatility indices for both stocks and commodities. The models include linear, quadratic, and non-linear drift specifications with affine, constant elasticity of variance (CEV), and stochastic elasticity of variance (SEV) diffusions. Overall, we find that a non-linear drift specification is important when imposing an affine structure on the diffusion, whereas a simple linear drift is adequate with a CEV and SEV specification, of which the SEV is dominant. For all but two of the indices we investigate, the best specification is an SEV diffusion with linear drift. For gold and the US dollar/euro exchange rate, there is little difference between a CEV and SEV diffusion with linear drift.

We explore the implications for asset prices and implied volatilities in an equilibrium model of commodity production. Production of the commodity can be carried out in one of two regimes. In the first regime the reserves are set in constant decline while in the second regime new additions to the reserve base are made. The optimal production rule is to switch regime when the stochastic revenue process of the producer hits certain barrier values. As a consequence of the optimal production rule equilibrium spot prices also become regime switching. The shapes of forward curves and implied volatilities are strongly dependent on the level of the stochastic revenue process.

This PhD thesis consists of three separate papers. The common theme is methods to calculate analytical approximations for prices of different contingent claims under various model assumptions. The first two papers deals with approximations of standard European options in stochastic volatility models. The third paper is focused on approximating prices of commodity swaptions in a general model framework.In the first paper, Dynamic Extensions and Probabilistic Expansions of the SABR model, a closed form approximation to prices of call options and implied volatilities in the so called SABR model of Hagan et al. (2002) is derived. The SABR model is one of the most frequently used stochastic volatility models used for option pricing in practice. The method relies on perturbing the model dynamics and approximations are obtained from a second order Taylor expansion. It is shown how the expansion terms can be calculated in a straightforward fashion using the flows of the perturbed model and results from the Malliavin calculus. This technique is further applied to calculate a closed form approximation of the option price in a useful dynamic extension of the original model. The dynamic model is able to match the prices of several options with different maturities and can therefore be used to price path dependents products in a consistent way. In addition, we propose an alternative model specification for the dynamic SABR model where the dynamics of the underlying asset are given by a displaced diffusion. In its non-dynamic version this model has similar properties as the original SABR model but it is more analytically tractable. A closed form approximation of the dynamic version of this model is also derived. The accuracy of the approximations is evaluated in a Monte Carlo study and the method is found to work well for many parameters of interest.The second paper, General Approximation Schemes for Option Prices in Stochastic Volatility Models, further examines the method of approximation employed in the first paper. The method is developed for a general stochastic volatility specification that can generate many commonly employed models as special cases. As an important application the method is used to calculate a second order expansion for the Heston (1993) model. The Heston model is arguably the most often used stochastic volatility model in practice and academic work. A numerical study of the approximations is performed where they are compared to prices and implied volatilities calculated from numerical Fourier transforms. It is found that for several parameters of interest the approximation is very accurate. Relating the proposed method to the existing literature we find that it generalizes the work by Lewis (2005) in several directions. In the case of the Heston model the first order expansion coincides with the approximation proposed in Alos (2006). However, an important advantage of the proposed method is that it can be used to generate higher order terms and it is verified that extending to second order substantially improves the accuracy of the approximation.The third paper, Approximative Valuation of Commodity Swaptions, is concerned with the numerical calculation of prices of European options on commodity swaps. A general approximation scheme for these claims is derived within the model framework of Heath, Jarrow and Morton (1992) extended to include commodity forwards. In models with deterministic volatilities the approach generates a closed form approximation allowing the swaptions to be conveniently priced using Blacks formula. The approximation is evaluated when applied to a Gaussian 2-factor model frequently employed in the literature. A comparison of the approximative prices to Monte Carlo simulations shows that the incurred errors are small for a large set of relevant parameters.

In this paper we study the pricing of commodity swaptions in a Heath-Jarrow-Morton framework based on stochastic spot prices, interest rates and convenience yields. We develop a complementary framework for deriving approximations of swaption prices. In the class of Gaussian models the method gives a closed form approximation via Black's formula. We provide an explicit formula for a two-factor Gaussian model and show how to extend the result to incorporate stochastic volatility. Numerical results show that the approximation produces small errors for a wide range of parameters and contract specifications. The approximation is very fast compared to Monte Carlo simulations.

In this paper we derive closed form approximations of European option prices in different versions of the SABR model of Hagan et al. (2002). Our approach is based on perturbing the model dynamics and approximations of call prices are obtained from a second order Taylor expansion. The method is applied to approximate prices in a dynamic and time consistent extensions of the original SABR model. In addition, a flexible approach for obtaining analytical expressions for implied volatilities is also developed. We find the results for a time consistent displaced diffusion version of the SABR model to be particularly useful.

In this paper we develop a general method for deriving closed-form approximations of European option prices and equivalent implied volatilities in stochastic volatility models. Our method relies on perturbations of the model dynamics and we show how the expansion terms can be calculated using purely probabilistic methods. A flexible way of approximating the equivalent implied volatility from the basic price expansion is also introduced. As an application of our method we derive closed-form approximations for call prices and implied volatilities in the Heston [Rev. Financial Stud., 1993, 6, 327–343] model. The accuracy of these approximations is studied and compared with numerically obtained values.

This paper investigates model dynamics and risk premia in the short term market for crude oil futures. Stochastic volatility models, with and without jumps, are estimated using data on both futures and option prices. As an economic application we apply the estimated models to the pricing of crude oil variance swaps and an evaluation of the associated variance risk premium. The empirical results point to a positive return risk premium attached to diffusive stochastic volatility while there is not strong evidence of jump risk being priced in the market. Negative volatility and variance risk premia stand out as a robust and significant feature of the data. Jumps play a minor role for representing data and the jump risk component in both variance swaps and variance risk premia is small. Finally, a non-affine model that allows for level dependent volatility of volatility is found to have the best fit to data.

This paper proposes a flexible framework for structuring and pricing parametric heat wave insurance. The framework is based on a general heat wave definition formulated in terms of an underlying temperature index. The definition can be varied in terms of the heat wave duration, intensity, measurement period and underlying index. This construction makes it straightforward to create contracts tailored to insure against heat events of many different types. A single stochastic model for the underlying index can be used to price all contracts. We consider contracts with payments that depend on the number of heat waves of a certain type occurring in the measurement period and derive the necessary pricing relations based on a general model structure encompassing several popular temperature models in the literature. An empirical case study is performed using data for Berlin where the daily maximum temperature is used as the underlying index. Model implied heat wave probabilities are consistent with historical patterns, point to high likelihoods for short duration heat events of different threshold temperatures and non-negligible risks for future heat waves of extreme temperatures and durations never before observed. 

This paper proposes a flexible framework for structuring and pricing parametric heat wave insurance. The framework is based on a general heat wave definition formulated in terms of an underlying temperature index. The definition can be varied in terms of the heat wave duration, intensity, measurement period and underlying index. This construction makes it straightforward to create contracts tailored to insure against heat events of many different types. A single stochastic model for the underlying index can be used to price all contracts. We consider contracts with payments that depend on the number of heat waves of a certain type occurring in the measurement period and derive the necessary pricing relations based on a general model structure encompassing several popular temperature models in the literature. An empirical case study is performed using data for Berlin where the daily maximum temperature is used as the underlying index. Model implied heat wave probabilities are consistent with historical patterns, point to high likelihoods for short duration heat events of different threshold temperatures and non-negligible risks for future heat waves of extreme temperatures and durations never before observed.

The menu of swaps available in the market today ranges far beyond the basic fixed-for-floating interest rateswap. Commodity swaps specifying a series of future cash flows, whereby one counterparty pays a fixed dollar amount and the other pays the price appreciation on some underlying commodity, are a prime example. And if commodity swaps become popular, commodity swaptions will naturally be introduced. But valuing commodity swaptions involves modeling the behavior of both future interest rates and future commodity prices for all of the payment dates. The standard approach is Monte Carlo simulation, which is complicated and time consuming. Here, Larsson introduces an approximation that is quite mild in terms of the error it introduces and for which the execution time for a given level of accuracy is improved by several orders of magnitude.

This paper studies the implications for reserve lifetime and related quantities in a continuous time model of resource extraction under uncertainty. Both the resource price and the extracted amount are assumed to follow stochastic processes. Reserve lifetime is determined by the profit-maximizing firms optimal decision of when to close down production. We derive closed form expressions for the expected value and the probability distribution of reserve lifetime, the expected level of depletion, and the expected streams of discounted revenues and costs until closing.

In this paper we explore stochastic modeling of bounded processes in continuous time using time-inhomogeneous Jacobi diffusions. We present some basic general results and introduce a subclass of models with seasonal time variation. In the seasonal models we derive the conditional mean and variance in closed form and propose a strategy for estimation based on quasi maximum likelihood. An empirical applications is carried out to daily time series data on relative humidity. Simulation methods are used to investigate properties of the resulting parameter estimators. The results show that the proposed seasonal Jacobi model gives a very satisfactory fit to data and that the estimation procedure works well. 

We propose a novel stochastic time series model able to explain the stylized features of daily irradation level data in 5 cities in Germany. The model is suitable for applications to risk management of photovoltaic power production in renewable energy markets. The suggested dynamics is a low order autoregressive time series with seasonal level given by an atmosphericclear-sky model. Moreover, we detect a skewness property in the residuals which we explain by a winter-summer regime switch. The stochastic variance is modelled by a seasonally varying GARCH-dynamics. The winter and summer standardized residuals are proposed to be a Gaussian mixture model to capture the bimodal distributions. We estimate the model on the observed data, and perform a validation study. An application to energy markets studying the production at risk for a PV-producer is presented

We propose a novel stochastic time series model able to explain the stylized features of daily irradiation level data in 5 cities in Germany. The model is suitable for applications to risk management of photovoltaic power production in renewable energy markets. The suggested dynamics is a low-order autoregressive time series with seasonal level given by an atmospheric clear-sky model. Moreover, we detect a skewness property in the residuals which we explain by a winter–summer regime switch. The stochastic variance is modeled by a seasonally varying GARCH-dynamics. The winter and summer standardized residuals are proposed to be a Gaussian mixture model to capture the bimodal distributions. We estimate the model on the observed data, and perform a validation study. An application to energy markets studying the production at risk for a PV-producer is presented.

In this paper we examine the empirical performance of affine jump diffusion models with stochastic volatility in a time series study of crude oil prices. We compare four different models and estimate them using the Markov Chain Monte Carlo method. The support for a stochastic volatility model including jumps in both prices and volatility is strong and the model clearly outperforms the others in terms of a superior fit to data. Our estimation method allows us to obtain a detailed study of oil prices during two periods of extreme market stress included in our sample; the Gulf war and the recent financial crisis. We also address the economic significance of model choice in two option pricing applications. The implied volatilities generated by the different estimated models are compared and we price a real option to develop an oil field. Our findings indicate that model choice can have a material effect on the option values.