In the realm of Martin Hairer's regularity structures we aim to introduce topologies on spaces of modelled distributions, which enable on the one hand reconstruction and which allow on the other hand a rich class of modelled distribution valued semi-martingales. This is done to have tools from regularity structures and semi-martingale theory at hand. Examples from the theory of term structures in mathematical Finance are shown. Joint work with David Prömel, ETH Zürich.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

This talk will introduce a class of game-theoretic supermartingales, whose main advantage over their measure-theoretic counterparts is that they do not presuppose a given probability measure; instead, they can be used to define an outer measure motivated by economic considerations combined only with topological (but not statistical) assumptions. Under the continuity assumption, it is possible to show that a typical continuous price path "looks like Brownian motion" with a possibly deformed time axis. A weaker assumption of boundedness of jumps still implies the almost sure existence of pathwise stochastic integrals of functions with finite p-variation for some p with respect to cadlag price paths with bounded jumps.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Vovk recently introduced a pathwise approach to continuous time mathematical finance which does not require any measure-theoretic foundation and allows us to describe properties of “typical price paths” or “game-theoretic martingales" by only relying on superhedging arguments. I will show how to construct a model free Itô integral in this setting. We will also see that every typical price paths a rough path in the sense of Lyons. Based on joint work with David Prömel.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We present a pathwise Tanaka formula for absolutely continuous functions with weak derivative of finite q-variation provided the local time is of finite p-variation with 1/p + 1/q >1. To justify the assumption on the local time, we follow Vovk's hedging based approach to model free financial mathematics. We prove that it is possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times fulfilling the aforementioned assumptions. This talk is based on a joint work with Nicolas Perkowski.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Since Hobson's seminal paper the connection between model-independent pricing and the skorokhod embedding problem has been a driving force in robust finance. We establish a general pricing-hedging duality for financial derivatives which are susceptible to the Skorokhod approach. Using Vovk's approach to mathematical finance we derive a model-independent super-replication theorem in continuous time, given information on finitely many marginals. Our result covers a broad range of exotic derivatives, including lookback options, discretely monitored Asian options, and options on realized variance.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We study a notion of pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums, as defined by Follmer (1979) and extended by Cont & Fournie (2010). We prove a pathwise isometry property for this integral, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition, which is the pathwise analog of the semimartingale decomposition, for a large class of irregular paths obtained through functional transformations of a reference path with non-vanishing quadratic variation. The relation with controlled rough paths is discussed.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In his seminal paper "Calcul d'Ito sans probabilités", Hans Föllmer proposed a non-probabilistic version of the Itô formula, which was recently generalized by Rama Cont and David-Antoine Fournié in a functional framework. Using the notion of pathwise quadratic variation, we derive first a pathwise isometry formula for functionals of a given path. This formula allows to generalize the notion of vertical derivatives and allows to define a weak version of vertical derivatives for functionals which are not necessarily smooth in the classical sense. The whole approach involves only pathwise arguments and does not rely on any probability notions. Nevertheless, we show that when applying to a stochastic process, this notion of weak derivatives coincides with the weak derivatives proposed by Cont and Fournié in a probabilistic framework.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We develop a pathwise calculus for functionals of integer-valued measures. We show that smooth functionals in the sense of this pathwise calculus are dense in the space of square-integrable (compensated) integrals with respect to a large class of integer-valued random measures. Using these results, we extend the framework of Functional Itô Calculus to functionals of integer-valued random measures. We construct a 'stochastic derivative' operator with respect to such integer-valued random measures and obtain an explicit martingale representation formula for square-integrable martingales with respect to the filtration generated by such integer-valued random measures. Our results hold beyond the class of Poisson random measures and allow for random and time-dependent compensators. This is joint work with R. Cont.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Davis, Obloj and Raval (2013) developed a theory of robust pricing and hedging of weighted variance swaps given market prices of co-maturing put options. They make use of Föllmer’s quadratic variation for continuous paths, and of an analogous notion of local time. Here we develop a theory of pathwise local time, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. We provide equivalent conditions for the existence of pathwise local time. Our approach agrees with the usual (stochastic) local times for a.e. path of a continuous semimartingale. We establish pathwise versions of the Itô-Tanaka, change of variables and change of time formulae. Finally, we study in detail how the limiting objects, the quadratic variation and the local time, depend on the choice of partitions. In particular, we show that an arbitrary given non-decreasing process can be achieved a.s. by the pathwise quadratic variation of a standard Brownian motion for a suitable sequence of (random) partitions; however, such degenerate behavior is excluded when the partitions are constructed from stopping times.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We explore the robust replication of forward-start straddles given quoted (Call and Put options) market data. One approach to this problem classically follows semi-infinite linear programming arguments, and we propose a discretisation scheme to reduce its dimensionality and hence its complexity. Alternatively, one can consider the dual problem, consisting in finding optimal martingale measures under which the upper and the lower bounds are attained. Semi-analytical solutions to this dual problem were proposed by Hobson and Klimmek (2013) and by Hobson and Neuberger (2008). We recast this dual approach as a finite dimensional linear programme, and reconcile numerically, in the Black-Scholes and in the Heston model, the two approaches.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We study the Monge-Kantorovich transport between two probability measures, where the transport plans are subject to a probabilistic constraint. For instance, in the martingale optimal transport problem, the transports are laws of martingales. Interesting new couplings emerge as optimizers in such problems. Constrained transport arises in the context of robust hedging in mathematical finance via linear programming duality. We formulate a complete duality theory for general performance functions, including the existence of optimal hedges. This duality leads to an analytic monotonicity principle which describes the geometry of optimal transports. Joint work with Mathias Beiglböck, Florian Stebegg and Nizar Touzi.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider robust (pathwise) approach to pricing and hedging. Motivated by the notion of prediction set in Mykland (2003), we include in our setup modelling beliefs by allowing to specify a set of paths to be considered, e.g. super-replication of a contingent claim is required only for paths falling in the given set. The framework interpolates between model--independent and model--specific settings. We establish a general pricing--hedging duality. The setup is parsimonious and includes the case of no traded options as well as the so-called martingale optimal transport duality of Dolinsky and Soner (2013) which we extend to multiple dimensions and multiple maturities. In presence of non-trivial beliefs, the equality is obtained between limiting values of perturbed problems indicating that the duality holds only if the market is stable under small perturbations of the inputs. Our framework allows to quantify the impact of making assumptions or gaining information. We focus in particular on the latter and study if the pricing-hedging duality is preserved under additional information. Joint work with Zhaoxu Hou and Anna Aksamit.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider a continuous-time financial market that consists of securities available for dynamic trading, and securities only available for static trading. We work in a robust framework and discuss two different ways of including additional information. In the first case, the informed agent's information flow is modeled by a filtration which is finer that the one of the uninformed agent. This clearly leads to a richer family of trading strategies, and to a smaller set of pricing measures. In the second case, we assume that the additional information consists in being able to exclude some evolution of the asset price process. In particular, super-replication of a contingent claim is required only along paths falling in the smaller set of admissible paths, and the pricing measures to be considered are only those supported on this set. The talk is based on joint works with Martin Larsson, Alex Cox and Martin Huesmann.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

We consider a market including a traded asset whose forward price St is unambiguously defined and on which put options are traded with maturity/strike pairs {(Tj,Kji), i = 1, . . . , ij, j = 1, . . . , n}. The prices of these options, and the underlying asset price, are known at the current time t = 0, and are assumed to satisfy the Davis-Hobson (2007) conditions for consistency with an arbitrage-free model. Given a path-dependent contingent claim with exercise value ö(ST1, . . . , STn) we look for the cheapest semi-static superhedging portfolio, consisting of static positions in the traded options together with dynamic trading in the underlying where rebalancing takes place only at the option exercise times Tj. This problem is naturally formulated as an infinite-dimensional linear program (LP) and (under stated conditions) we can apply interior point conditions to show that there is no duality gap, the dual problem being maximization of expectation over martingale measures. One advantage of this approach is that computations can be done by finite-dimensional LP algorithms, following a 2-stage discretization process where we firstly restrict the dynamic trading integrands to finite linear combinations of basis functions, and then discretize the state space; we present some examples. Finally, we comment on possible extensions of these results to models with transaction costs. This is joint work with Sergey Badikov and Antoine Jacquier.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

Path-dependent Kolmogorov equations are a class of infinite dimensional partial differential equations on the space of cadlag functions which extend Kolmogorov's backward equation to path-dependent functionals of stochastic processes. Solutions of such equations are non-anticipative functionals which extend the notion of harmonic function to a non-Markovian, path-dependent setting. We discuss existence, uniqueness and properties of weak and strong solutions of path-dependent Kolmogorov equations using the Functional Ito calculus. Time permitting, some applications to mathematical finance and non-Markovian stochastic control will be discussed.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In the first part of the talk we revisit the basic theory of functional Ito calculus, using the regularization approach. This allows us to explore its relations with the corresponding Banach space stochastic calculus. In the second part of the talk, we introduce a viscosity type solution for path-depenendent partial differential equations, called strong-viscosity solution, with the peculiarity that it is a purely analytic object. We discuss its properties and we present an existence and uniqueness result for strong-viscosity solutions to semilinear parabolic path-dependent partial differential equations.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

In this talk, we consider a class of parabolic semilinear path-dependent PDEs that can be associated with a class of stochastic integral equations, which may depend on the entire sample paths of a time-inhomogeneous diffusion process. For instance, such integral equations can determine the log-Laplace functionals of historical superprocesses. By exploiting this relationship, we show uniqueness, existence and non-extendibility of mild solutions, and verify that every mild solution turns out to be a viscosity solution of the path-dependent PDE in question.

• Thematic program: Mathematics for Risk in Finance and Energy (2015/2016)
• Event: Workshop on "Pathwise methods, Functional Calculus and applications in Mathematical Finance" (2016)

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)

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 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)

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)