My talk is dedicated to the memory of Peter Laurence, whose untimely death has left a void in many peoples hearts. Peter was a truly great mathematician and a wonderful person. In the first part of the talk, Peter's scientific biography will be presented. I will also share personal recollections of my meetings with Peter face-to-face and in the skype world. The second part of the talk will be more mathematical. I will speak about my joint work with Peter on Riemannian geometry of the Heston model, which is one of the classical stock price models with stochastic volatility. My collaboration with Peter resulted in the paper "The Heston Riemannian distance function", which was published in 2014 by "Journal de Mathematiques Pures et Appliquees". In the paper, we found two explicit formulas for the Riemannian Heston distance, using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and a similar distance function in the Grushin plane. We also proved a partial large deviation principle for the Heston and the Grushin models. After completing our work on the paper, we started discussing future projects, but fate interfered. I will finish the talk by briefly presenting my recent results on the distance to the line in the Heston plane, and how such results can be used in nancial mathematics. Peter's scientific in fluence continues after his untimely departure from this world.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

In contrast to geometric models, additive models in energy markets, in particular in markets where forward contracts are delivered during a period like electricity and natural gas, allows easily the computation of forward prices in closed form. Moreover they naturally allow the presence of negative prices, which start to appear more and more frequently in electric markets. In this paper we present an additive multicommodity model which allows for mean-reverting dynamics consistent with no-arbitrage, based on the observed prices of forward contracts based on the mean on a period, which are the most liquid instruments in natural gas and electricity markets. This allows to compute the price of more complex derivatives and of risk measures of portfolios in a way which is consistent with market data. Joint work with Luca Latini.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indierence pricing approach in a general incomplete multivariate market model driven by nitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyers utility indierence price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

This talk gives an introduction to the area of ambit stochastics with a particular focus on applications in energy markets. In particular, we will describe models for energy spot and forward prices based on so-called ambit felds. These models are very flexible and at the same time highly analytically tractable making them interesting from a mathematical perspective, but also very useful for applications.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

Our paper aims to model and forecast the electricity price in a completely new and promising style. Instead of directly modeling the electricity price as it is usually done in time series or data mining approaches, we model and utilize its true source: the sale and purchase curves of the electricity exchange. We will refer to this new model as X-Model, as almost every deregulated electricity price is simply the result of the intersection of the electricity supply and demand curve at a certain auction. Therefore we show an approach to deal with a tremendous amount of auction data, using a subtle data processing technique as well as dimension reduction and lasso based estimation methods. We incorporate not only several known features, such as seasonal behavior or the impact of other processes like renewable energy, but also completely new elaborated stylized facts of the bidding structure. Our model is able to capture the non-linear behavior of the electricity price, which is especially useful for predicting huge price spikes. Using simulation methods we show how to 11 derive prediction intervals. We describe and show the proposed methods for the dayahead EPEX spot price of Germany and Austria. Joint work with Rick Steinert.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

In the study, the Bayesian stochastic volatility model with normal errors, a leverage effect, a jump component and exogenous variables (SVLEJX) is proposed. This Bayesian framework, founded upon the idea of latent variables is computationally facilitated with Markov Chain Monte Carlo methods. In this paper, the Gibbs sampler is employed. The SVLEJX structure is applied to model electricity spot price. The results of Bayesian estimation, jump detection and forecasting are presented and discussed. The series of waiting times between two consecutive jumps is also of interest in the paper. Periods of no jumps alternating with the ones of frequent jumps could be indicative of existence of the jump clustering phenomenon. The impact of exogenous variables on electricity spot price dynamic is explored. Moreover, the leverage eect and the stochastic volatility clustering are tested.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

We aim at modeling the prices of forward contracts on electricity, by adopting a stochastic model with two Brownian motions as stochastic factors to describe their evolution over time. In contrast to the model of (Kiesel et al., 2009), the diffusion coecients are stochastic processes; the one of the rst factor is left totally unspecified, and the other one is the product of an unspecified process and of an exponential function of time to the maturity of the forward contract, which allows to account for some short-term eect in the increase of volatility. We will consider that price processes following this model are observed simultaneously, at n observation times, over a given time interval [0; T]. The time step T=n between two observation times is small with respect to T, in the asymptotics n ! 1. We estimate some parameter of the exponential factor in volatility, with the usual rate, and we explain how it can be estimated eciently in the Cramr-Rao sense. We are also able to estimate the trajectories of the two unspecied volatility processes, using nonparametric methods, with the standard rate of convergence. Numerical tests are performed on simulated data and on real prices data, so that we may see how appropriate our two-factor model is when applied to those data. Joint work with Olivier Feron (EDF, France) and Marc Hoffmann (Universite Paris-Dauphine).

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

In this paper we assess the real option value of operating reserve pro- vided by an electricity storage unit. The contractual arrangement is a series of American call options in an energy imbalance market (EIM), physically covered and delivered by the store. The EIM price is a general regular one-dimensional Diffusion. Necessary and sucient conditions are provided for a unique optimal strategy and value. We provide a straightforward procedure for numerical solution and several examples. Joint work with John Moriarty.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

 Benets of a unied modeling framework  The non-Markovian approach as a unied framework for the consistent modeling of power spots, forwards and swaps  Extracting forward-looking market-implied risk-neutral probabilities for the intraday hourly and intra-hourly power spots from a single or multiple market forward curves  Taking into account: { daily, weekly, annual and meta-annual cyclical patterns, { linear and nonlinear trends, { upwards and downwards spikes, { positive and negative prices  Interpolating and extrapolating power market forward curves: { intra-hourly, hourly, daily, weekly and monthly power forward curves, { extending power market forward curves beyond their liquidity hori- zons  Modeling the German Intraday Cap Week Futures as an hourly strip of Asian call options on forwards on the intraday hourly power spots

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

The Heath Jarrow Morton (HJM) approach treats the family of futures - written on a commodity as primary assets and models them directly. This approach has been used for the modelling of future prices in various markets by several authors and it has found its use by practitioners. We derive several representations of possible future dynamics and implications on futures and the spot from an innite dimensional point of view. To be more specically, let us denote the spot price by St and the future prices by ft(x) := E(St+xjFt); x; t  0. Due to the well-known Heath Jarrow Morton Musiela drift condition the dy- namics of ft cannot be specied arbitrarily under the pricing measure. We model it by dft = @xftdt + tdLt in a suitable function space where L is some Levy process. Then we derive a series representation for the futures in terms of the spot price process and Ornstein-Uhlenbeck type processes, we represent the spot as a Levy-semistationary process and nd formulae for the correlation between the spot and futures.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

This paper considers two elements of the oil-futures markets: Ex- pected return and risk. 3 With respect to expected return, the paper presents a parsimonious and theoretically-sound basis for extracting forward-looking measures of equity and commodity betas, and the risk-premium on crude-oil futures contracts. Dening forward-looking betas as perturbations of historical estimates, we use the mar- ket prices of equity, index and commodity options under a single-factor market model to estimate the appropriate forward-looking perturbation to apply to the historical beta. This permits us to compute forward-looking term structures of equity and commodity betas. In the commodity arena, we use both one- and two-factor models to obtain estimates of a forward-looking measure of the correlation between crude-oil and the S&P 500. Combining these with forward- looking (i.e., implied) volatilities on commodities and stock-market indices, we utilize these forward-looking betas and correlations to provide an ex-ante esti- mate of the expected future crude-oil spot price through the use of an equity ex-ante risk premium and the conditional CAPM. With respect to risk, we use the market prices for crude-oil futures options and the prices of their underlying futures contracts to calibrate the volatility skew using the Merton (1976) jump-diusion option-pricing model. We demon- strate the jump-diusion parameters bear a close relationship to concurrent eco- nomic, nancial and geopolitical events. This produces an informationally-rich structure covering the time period of the turbulent post-2007 time period.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

Energy markets, and in particular, electricity markets, exhibit very peculiar features. The historical series of both futures and spot prices include seasonality, mean reversion, spikes and small uctuations. Very often a stochastic volatility dynamics is postulated in order to explain their high degree of variability. Moreover, as it also appears in other kind of markets, they exhibit also the USV (Unspanned Stochastic Volatility) phaenomenon [7]. After the pioneering paper by Schwartz, where an Ornstein-Uhlenbeck dy- namics is assumed to describe the spot price behavior, several different approaches have been investigated in order to describe the price evolution. A comprehensive presentation of the literature until 2008 is oered in the book by F.E. Benth, J. Saltyte-Benth and S. Koekebakker [4]. High frequency trading, on the other hand, introduced some new features in com- modity prices dynamics: in the paper by V. Filimonov, D. Bicchetti, N. Maystre and D. Sornette [5] evidence is shown of endogeneity and structural regime shift, and in order to quantify this level the branching ratio is adopted as a measure of this endoge- nous impact and a Hawkes processes dynamics is assumed as a reasonable modelling framework taking into account the self- exciting properties [1]. The purpose of the present paper is to propose a new modeling framework including all the above mentioned features, still keeping a high level of tractability. The model considered allows to obtain the most common derivatives prices in closed or semi-closed form. Here with semi-closed we mean that the Laplace transform of the derivative price admits an explicit expression. The models we are going to introduce can describe the prices dynamics in two dierent forms, that can be proved to be equivalent: the rst is a representation based on random elds, the second is based on Continuous Branching Processes with Immigration (CBI in the following). The idea of adopting a random felds framework for power prices description is not new: O.E. Barndor-Nielsen, F.E. Benth and A. Veraart introduced the Ambit Fields to this end, showing how this approach can provide a very exible and still tractable setting for derivatives pricing [2], [3]. A model based on CBI has been proposed recently by Y. Jiao, C. Ma and S. Scotti in view of short interest rate modelling, and in that paper it was shown that, with a suitable choice of the Levy process driving the CBI dynamics, the model can oer a signicant extension of the poular CIR model [6]. We shall propose two dierent types of dynamics for the prices evolution. The rst class will be named the Arithmetic models class, and the second will be named the Geometric model class; in adopting the present terminology we are following the classication proposed in [4]. We shall compare the Advantages and the limitations implied by each model class and we shall investigate the risk premium behavior for each of the classes considered. The paper will be organized as follows: in the rst Section we introduce the stochastic processes we are going to consider, while in the second Section we discuss how these pro- cesses can be successfully applied to power markets description. In the third Section we derive some closed formulas for Futures and Option prices when the underlying dynamics is assumed to be given by the model introduced. In the fourth Section we shall investigate the risk premium term structure for the models under consideration. In the fth Section, we provide some suggestions about estimation and/or calibration methods for the same model. We complete our presentation with a statistical analysis on the two cases and some numerical illustrations of the results obtained. In the final section we provide some concluding remarks and discuss futures extensions of the present work. Joint work with Ying Jiao, Chunhua Ma and Simone Scotti. References: [1] Bacry, E., Mastromatteo, J., Muzy, J.-F. Hawkes Processes in Finance, PREPRINT(2015). [2] Barndor-Nielsen, O.E., Benth, F.E., Veraart, A. Modelling energy spot prices by volatil- ity modulated Levy driven Volterra processes, Bernoulli, 19, 803-845 (2013). [3] Barndor-Nielsen, O.E., Benth, F.E., Veraart, A. Modelling Electricity Futures by Am- bit Fields, Advances in Applied Probability, 46 (3), 719-745 (2014). [4] Benth, F.E., Saltyte-Benth J., Koekebakker S. Stochastic Modelling of Elec- tricity and Related Markets , World Scientic, Singapore (2008). [5] Filimonov, V., Bicchetti, D., Maystre, N., Sornette, D. Quantication of the High Level of Endogeneity and Structural Regime Shifts in Commodity Markets, PREPRINT (2015). [6] Jiao, Y., Ma, C., Scotti, S. Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling, PREPRINT (2016). [7] Schwarz, A.B., Trolle, E.S. Unspanned Stochastic Volatility and the Pricing of Com- modity Derivatives, PREPRINT (2014).

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equa- tion. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deter- ministic approximation of the related HJB variational inequality is provided. We nally show that this numerical approximation converges to the value function. This allows us to describe the investment and dividend optimal policies. Joint work with Stephane Villeneuve and Xavier Warin.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

We propose new copulae to model the dependence between two Brow- nian motions and to control the distribution of their dierence. Our approach is based on the copula between the Brownian motion and its re ection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copu- lae. These copulae allow for the survival function of the dierence between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. We derive two models based on the structure of the Re ection Brownian Copula which present two states of correlation ; one is directly based on the re ection of the Brownian motion and the other is a local correlation model. These models can be used for risk management and option pricing in commodity energy markets.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

This paper studies the risk assessment of alternative methods for a wide variety of Commodity ETFs. We implement well-known as well as and recently proposed backtesting techniques for both value-at-risk (VaR) and ex- pected shortfall (ES) under extreme value theory (EVT), parametric, and semi- nonparametric techniques. The application of the latter to ES was introduced in this paper and for this purpose we derive a straightforward closed form of ES. We show that, for the condence levels recommended by Basel Accords, EVT and Gram-Charlier expansions have the best coverage and skewed-t and Gram-Charlier the best relative performance. Hence, we recommend the ap- plication of the above mentioned distributions to mitigate regulation concerns about global nancial stability and commodities risk assessment. Joint work with Esther Del Brio and Javier Perote.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

We consider a typical optimal control problem from the viewpoint of an energy utility company. The company faces a varying energy demand of its associated consumers, modelled by a stochastic process. Demands can be satised by either buying energy at an exchange or the utilisation of an energy storage system. Furthermore the company is able to buy energy on a larger scale - than needed to satisfy demands - and enlarge the storage level or respectively sell energy from the storage directly to the market. In contrast to previous lit- erature the storing facility therefore serves as a hedge against market price and demand volume risks and is not considered isolated from other market activities of the operator. Therefor the value function - which can be interpreted as a real option value of the storage - diers from classical optimal storage control prob- lems and delivers a better quantication of the storage value for a specic user. We formulate a stochastic control problem including these features and pay par- ticular attention to the operational constraints of the storage. Furthermore we will introduce methods to model the energy spot price and the consumption rate stochastically. Subsequently we will derive a candidate for the optimal policy, verify its optimality and solve the arising Hamilton-Jacobi-Bellman equation for the value function numerically using a novel nite elements discretization.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

In recent years, for a variety of reasons, it has become popular in North American to produce Ethanol (for blending with gasoline) from Corn. The resulting industrial process can be modelled as an option on the "crush spread" between Ethanol and Corn. Under a price - taker assumption, real options models of ethanol production can be made incorporating random corn and ethanol prices. In the rst part of my talk I will report work done in my group, together with Natasha Burke and Christian Maxwell, on creating and solving real options models of the corn-ethanol industry. These models provide interesting insights about the relationship between corn prices, ethanol prices, and their correlation with valuations and operational decisions. Using a jump process, we are also able to incorporate the impact of random changes in government subsidies on the valuation and operation of ethanol facilities. However, while in the relatively fragmented US corn ethanol market it might be (just) reasonable to model any given ethanol producer as a price taker, all producers taken together do have market impact. In the second part of my talk I report work, joint with Nicolas Merener (Universidad Torcuata di Tella, Buenos Aires) on creating tractable models for this price impact. I will also sketch our progress toward solving the models and confronting them with data.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

The paper studies the time-varying correlation between oil prices and exchange rates and their volatilities. Generally, when the value of the dollar weakens against other major currencies, the prices of commodities tend move higher. The signicance of this relationship has increased since 2000 with indications of structural breaks around the beginning of the so-called nancialization of commodity markets-regime and again around the beginning of the nancial crisis. Also the correlation between the volatility of oil prices and the volatility of exchange rates seems to experience the same behaviour as the returns correlation. This paper introduces and estimates a term structure model for futures contracts and option contracts on WTI crude oil and EURUSD. The model is tted a panel data of futures prices covering 2000-2013. The model allows for stochastic volatility and correlation and identies how the number of joint factors increases over time.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

It is typical for electricity contracts, that the time of concluding the contract and the time of delivery are quite different. For this reason, these contracts are subject to risk and risk premia are and must be part of the pricing rules. In the rst part of the talk, we investigate electricity futures to nd out pricing rules, which the market is applying, such as the distortion priciple, the certainty equivalence priciple or the ambiguity priciple. We then investigate a no-arbitrage principle in the presence of capacity contraints on production and storage. We review then the idea of acceptance pricing and indierence pricing using a concrete model. Finally we present a bilevel problem, where the pricing decision depends on the behavioral pattern of the counterparty. Some algorithmic aspects will be discussed as well. Joint work with Raimund Kovacevic

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

The price spikes observed in electricity spot markets may be understood to arise from fundamental drivers on both the supply and demand sides. Each driver can potentially create spikes with dierent frequencies, height distributions and rates of decay. This behaviour can be accounted for in models with multiple superposed components, however their calibration is challenging. Given a price history we apply a Markov Chain Monte Carlo (MCMC) based procedure to generate posterior samples from an augmented state space comprising parameters and multiple driving jump processes. This also enables posterior predictive checking to assess model adequacy. The procedure is used to determine the number of signed jump components required in two dierent markets, in time periods both before and after the recent global financial crises. Joint work with John Moriarty and Jan Palczewski.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)

We address the valuation of an energy storage facility in the presence of stochastic energy prices as it arises in the case of a hydro-electric pump station. The valuation problem is related to the problem of determining the optimal charging/discharging strategy that maximizes the expected value of the resulting discounted cash ows over the life- time of the storage. We use a regime switching model for the energy price which allows for a changing economic Environment described by a non-observable Markov chain. The valuation problem is formulated as a stochastic control problem under partial information in continuous time. Applying ltering theory we and an alternative state process containing the lter of the Markov chain, which is adapted to the observable ltration. For this alternative control problem we derive the associated Hamilton- Jacobi-Bellman (HJB) equation which is not strictly elliptic. Therefore we study the HJB equation using regularization arguments. We use numerical methods for computing approximations of the value function and the optimal strategy. Finally, we present some numerical results. Joint work with Anton Shardin.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Conference on the Mathematics of Energy Markets (2016)