Time in an automated trading system does not move in a constant deterministic fashion. Instead, it is a random variable drawn from a distribution. This happens because messages enter and exit automated systems though different gateways and then race across a complex infrastructure of parallel cables, safeguards, throttles and routers into and out of the central limit order books. Add to it market fragmentation and you get a pretty complex picture about the effects of latency on price formation.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Using a firm-identified limit-order book dataset we show that competition among high-frequency trading firms (HFT) influences liquidity. HFT entries increase liquidity. The reverse is true for exits. Market participants’ behavioral changes are consistent with competitive pressure. HFT entries increase total HFT market share and take market share from incumbents. After HFT entry (exit), incumbent HFT spreads tighten (widen). Trading revenue suggests competition reduces HFT firm profitability. Impacts are larger in markets with fewer incumbents. The results show that part of the value of HFT comes from its competitiveness.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

''Optimal execution'' are typically derived assuming an exogenous Price process which is unaffected by the trading behaviour of market participants. On the other hand, in intraday price behavior in electronic markets reveals evidence of the price impact of algorithmic order flow, an extreme example being the 'Flash Crashes' repeatedly observed in such markets. We propose a simple model for analyzing the feedback effects which arise in a market where participants use market signals to minimize the impact of their trade execution. We show that commonly used execution algorithms which aim at reducing market impact of trades can actually lead to unintended synchronization of participants' order flows, increase their market impact and generate large « self-exciting » intraday swings in volume and volatility. We show that such bursts may occur even in absence of large orders, and lead to a systematic underperformance of 'optimal execution' strategies. These results call for a critical assessment of "optimal execution" algorithms and point to a notion of order flow toxicity distinct from information asymmetry or adverse selection.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Investors can acquire either raw or processed information about the payoff of risky assets. Information processing filters out the noise in raw information but it takes time. Hence, investors buying processed information trade with a lag relative to investors buying raw information. As the cost of raw information declines, more investors trade on it, which reduces the value of processed information, unless raw information is very unreliable. Thus, a decline in the cost of raw information can reduce the demand for processed information and, for this reason, the informativeness of asset prices in the long run.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Adverse selection in financial markets is traditionally measured by the correlation between the direction of market order trading and price movements. We show this relationship has weakened dramatically with limit orders playing a larger role in price discovery and with high-frequency traders’ (HFTs) limit orders playing the largest role. HFTs are responsible for 60–80% of price discovery, primarily through their limit orders. HFTs’ limit orders have 50% larger price impact than non-HFTs’ limit orders, and HFTs submit limit orders 50% more frequently. HFTs react more to activity by non-HFTs than the reverse. HFTs react more to messages both within and across stock exchanges.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Competitive and regulatory forces in the U.S. have resulted in almost all equity executions being handled using sophisticated electronic trading systems. Empirical evidence from Nasdaq shows that order submission patterns once restricted to proprietary trading firms, the prototypical High Frequency Trader, are now observed in orders originating from almost all types of market participants. One aspect of the widespread automation of trading is that the use of "price taker" algorithms has become increasingly prevalent. The implications for the market where each algorithm's order placement decision is dependent on other algorithms' order placement decisions is not well understood. Some consequences of widespread "price taking" behavior are seen every trading day as well as on occasional events such as the May 6, 2010 and August 24, 2015 market breaks. The public policy discussion around market structure needs a better understanding of how the automated price setting mechanism works under the current structure and would work under future alternative market structure designs.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

We will review the accumulating empirical evidence for an approximately square-root impact of a metaorder. Interestingly, this square-root law appears to be universal, i.e. to a large extent ndependent of markets (futures, equities, volatility, Bitcoin), microstructure and epochs (pre and post HFT). This suggests that this law must originate from a simple and robust statistical mechanism. We propose a dynamical theory of the latent market liquidity that predicts that the average supply/demand profile is V shaped and vanishes around the current price, leading to the square-root impact. This result only relies on mild assumptions about the order flow and on diffusive prices. We test our arguments numerically using a minimal model of order flow and provide further theoretical predictions that can be compared to further experimental observations. Our scenario suggests that markets are intrinsically prone to liquidity crises and puts in perspective the recent debate on the role of HFT liquidity.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

In this talk we focus on the combination of dimensional analysis, leverage neutrality, and a principle of market microstructure invariance to derive scaling laws expressing transaction costs functions, bid-ask spreads, bet sizes, number of bets, and other financial variables in terms of dollar trading volume and volatility. The scaling laws are illustrated using data on bid-ask spreads and number of trades for Russian stocks. These scaling laws provide useful metrics for risk managers and traders; scientific benchmarks for evaluating controversial issues related to high frequency trading, market crashes, and liquidity measurement; and guidelines for designing policies in the aftermath of financial crisis.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Prior work of Andersen, Bondarenko, Kyle and Obizhaeva (2015) establishes that the intraday trading patterns in the E-mini S&P 500 futures contract are consistent with the following invariance relationship: The return variation per transaction is log-linearly related to trade size, with a slope coefficient of -2. This association applies both across the intraday diurnal pattern and across days in the time series. The factor of proportionality deviates sharply from prior hypotheses relating volatility to transactions count or trading volume. This paper documents that a similar invariance relation holds for foreign exchange futures. However, the log-linear association is not fixed, but shifts over time reflecting an, all else equal, declining trend in the average trade size. The findings are remarkably robust across the full set of currency contracts explored, providing challenges to market microstructure research to rationalize these tight intraday and intertemporal interactions among key market activity variables. Co-authored with Oleg Bondarenko, University of Illinois at Chicago.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

The tick value is a crucial component of market design and is often considered the most suitable tool to mitigate the effects of high frequency trading. The goal of this paper is to demonstrate that the approach introduced in Dayri and Rosenbaum (2015) allows for an ex ante assessment of the consequences of a tick value change on the microstructure of an asset. To that purpose, we analyze the pilot program on tick value modifications started in 2014 by the Tokyo Stock Exchange in light of this methodology. We focus on forecasting the future cost of market and limit orders after a tick value change and show that our predictions are very accurate. Furthermore, for each asset involved in the pilot program, we are able to de ne (ex ante) an optimal tick value. This enables us to classify the stocks according to the relevance of their tick value, before and after its modification. This is joint work with Charles-Albert Lehalle and Weibing Huang.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

The attorney general for New York State, Eric Schneiderman, said at one point that he believed that high frequency trading (in the sense of co-location, that is to say extremely high frequency trading) is used for insider trading. Inspired by his remarks we purport to indicate via a mathematical model how this could come to pass. We use the newly developed theory (by Y. Kchia and this speaker) on the enlargement of filtrations via a stochastic process to show how continual infinitesimal peaks at the order book can beget a type of insider trading, thereby explaining the casual observation of the attorney general.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

Liquidity suppliers lean against the wind. We analyze whether high-frequency traders (HFTs) lean against large institutional orders that execute through a series of child orders. The alternative is that HFTs go “with the wind” and trade in the same direction. We find that HFTs initially lean against orders but eventually turn around and go with them for long-lasting orders. This pattern explains why institutional trading cost is 46% lower when HFTs lean against the order (by one standard deviation) but 169% higher when they go with it. Further analysis supports recent theory, suggesting HFTs “back-run” on informed orders.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

In this talk we present a test for the maximal rank of the volatility process in continuous diffusion models observed with noise. Such models are typically applied in mathematical finance, where latent price processes are corrupted by microstructure noise at ultra high frequencies. Using high frequency observations we construct a test statistic for the maximal rank of the time varying stochastic volatility process. We will show the asymptotic mixed normality of the test statistic and obtain a consistent testing procedure. Finally, we demonstrate some numerical and empirical illustrations.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

High-frequency financial data is certainly a `big data' problem, with all of the associated issues: what are the stylized facts of the data? what are we trying to do with the data? what are appropriate models? Industry approaches get the first two of these questions, but do badly on the third. Most academic studies do badly on all three. For example, it is a fairy tale that we can propose a time-invariant model for the evolution of high-frequency data, estimate the parameters of this model, and then apply the conclusions of an analysis that assumes that the paramters were known with certainty. In this talk, I will try to identify what we might want to do with high-frequency data, critique some existing research agendas, and illustrate a possible way of dealing with the problem of optimally liquidating a given position before a given time.

• Thematic program: Mathematical Finance (2016/2017)
• Event: High-Frequency Trading - Curse or Blessing (2016)

With the emergence of renewable energies (as wind or solar genera- tion), local generation systems are rapidly multiplying integrating renewables, batteries or more conventional plants (such as gas turbines or hydro plants). The impact of random factors (such as demand, energy prices, wind, luminosity etc.) on the management of such local generation systems are significant. Hence, an important issue is to be able to manage efficiently such microgrids in presence of uncertainties. Mathematically, the related optimization problem can be stated in terms of a stochastic control problem which can be reduced to a nonlinear Partial Differential Equation (PDE), known as Hamilton-Jacobi-Bellman (HJB) equation. The presentation focuses on recent forward numerical schemes based on generalized Fokker-Planck representations for nonlinear PDEs in high space dimension. In the specific case of mass conservative PDEs, it is well known that the solution can be probabilistically represented as the marginal densities of a Markov diffusion nonlinear in the sense of Mckean. Then one can design forward interacting particle schemes to approximate numerically the PDEs solu- tion. We present some extensions of this kind of representation and interacting particle scheme associated to a large class of PDEs including the case when they are non-conservative, non integrable with various kind of nonlinearities. (Joint work with Anthony Le Cavil, (HSBC, Paris) and Francesco Russo, (ENSTA ParisTech)

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

By means of Malliavin Calculus we construct an optimal linear strike convention for exchange options under stochastic volatility models. This convention allows us to minimize the difference between the model and implied correlations between the two underlying assets in the spread. Moreover, we show that this optimal convention does not depend on the specific stochastic volatility model. Numerical examples are given. Joint work with Elisa Alos.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

Pricing models of derivative instruments usually fail to provide reli- able results when risks rise and financial crises occur. More advanced stochastic pricing models try to improve the fitting results adding risk factors and/or pa- rameters to the models, incurring the risk of overfitted results. Drawing on these observations, it is proposed a generalisation of the Akaike information criterion suitable to evaluate forecasting power of alternative stochastic pricing models for any fixed arbitrary forecasting time-horizon. The Predictability Informa- tion Criterion (PIC) differs from the classical criteria for evaluating statistical models as it assumes that the random variable to study can ( or cannot) be par- tially predictable, which makes it particularly suitable for studying stochastic pricing models coherently with the semimartingale definition of the price pro- cess. On the basis of this assumption the criterion measures and compares the uncertainty of the predictions of two different alternative models when prices are (or are not) predictable. We conclude with a focus on Crude Oil market by comparing GBM and OU stochastic processes that are generally used for modeling West Texas Intermediate (WTI) oil spot price returns in derivative pricing models.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

This paper analyses the interaction between centralised carbon emis- sive technologies and distributed intermittent non-emissive technologies. In our model, there is a representative consumer who can satisfy her electricity demand by investing in distributed generation (solar panels) and by buying power to a centralised firm at a price he set up. Distributed generation is intermittent and induces an externality cost to the consumer. The firm provides non-random electricity generation subject to carbon price and to transmission costs. The objective of the consumer is to satisfy her demand while minimising investment costs, payment to the firm and intermittency cost. The objective of the firm is to satisfy consumer’s residual demand while minimising investment costs, de- mand deviation costs and maximising payment from the consumer. Investment decisions are formulated as McKean-Vlasov control problems with stochastic coefficients. We provide explicit, model-free solutions to the optimal decision problems faced by each player, the solution of the Pareto optimum and the Stackelberg equilibrium where the firm is the leader. We find that, from the social planner point of view, carbon price or transmission costs are necessary to justify a positive share of distributed capacity in the long-term, whatever the re- spective investment costs of both technologies are. The Stackelberg equilibrium is far from the Pareto equilibrium, leading to a much larger share of distributed energy and to a much higher price for centralised energy. Joint work with Rene Aid, Imen Ben Tahar and Huyen Pham

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

Renewable energy sources such as wind and solar have a high degree of unpredictability and time variation. As a result, balancing supply and demand in real time is becoming ever more challenging and the power grids need greater flexibility on many levels. The proposed approach addresses this challenge by harnessing the inherent flexibility in demand of many types of loads. We develop a distributed control theory and algorithms for automated demand dispatch, which can be used by grid operators as ancillary service to regulate demand- supply balance. The proposed approach uses local control solutions that a) take into account local measurements, constraints, and preferences, and b) lead to a controllable input-output model for the aggregate dynamics. The local control problem can be defined by a family of Markov decision processes, parameterized by a weighting factor that appears in the one-step reward function. This talk introduces a new methodology for solving an entire family of MDPs. In our application to demand control, the focus will be on a family of average-cost optimal control models in which the one-step reward function is defined by Kullback-Leibler divergence with respect to nominal dynamics. The proposed ODE methodology can be seen as a generalization of the linearly solvable MDP framework of Todorov to the case with exogenous disturbances, such as weather or customer behavior.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

This presentation is on the simulation of the hour-based German day-ahead power auction, where I apply vector autoregressive (VAR) models, in order to capture the effects of the market infrastructure of the day-ahead auction. This approach ensures that the correct intraday correlation structure is simulated, which will be important for valuing assets with production timing issues (e.g. pumped storages and batteries), thereby creating a more suitable simulation alternative to classic Brownian motion based stochastic simulation for these flexible assets. In order to handle the large dimensionality of the data created by the VAR approach, lasso and elasticnet shrinkages are applied, as well as their adaptive versions. The assessment of these methods is done by performing a classic forecast quality assessment, combined with an evaluation of the (often asymptotic) simulation relevant properties of each model. After estimating the model parameters, simulation from the fitted model is carried out using a block bootstrap. Sanity checks of the appropriateness of the forecasting approach are presented, highlighting both the advantages of the model and the points where future work is necessary.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In energy markets forward contracts can be of two types: in our ter- minology, forwards and swaps. Who sells a swap contract commits to deliver over a certain period, for instance, power, while by forward we mean the classi- cal financial agreement settled on a maturity date. Our purpose is to design a Heath-Jarrow-Morton framework for an additive, mean-reverting, multidimen- sional market consisting of forward contracts of any maturity date or delivery period. The main assumption is that forward prices can be represented as affine functions of a universal source of randomness. In a Brownian setting, we are able to completely characterize the models which do not allow for arbitrage opportunities. We study the possibility of introducing more general L´evy com- ponents either driving the dynamics of prices or in the context of a stochastic volatility model. Joint work with Fred Espen Benth and Tiziano Vargiolu.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The use of energy storage to balance electric grids is increasing and, with it, the importance of operational optimisation from the twin viewpoints of cost and system stability. In this paper we assess the real option value of balancing reserve provided by an energy-limited storage unit. The contractual arrangement is a series of American-style call options in an energy imbalance market (EIM), physically covered and delivered by the store, and purchased by the power system operator. We take the EIM price as a general regular one- dimensional diffusion and impose natural economic conditions on the option parameters. In this framework we derive the operational strategy of the storage operator by solving two timing problems: when to purchase energy to load the store (to provide physical cover for the option) and when to sell the option to the system operator. We give necessary and sufficient conditions for the finiteness and positivity of the value function – the total discounted cash flows generated by operation of the storage unit. We also provide a straightforward procedure for the numerical evaluation of the optimal operational strategy (EIM prices at which power should be purchased) and the value function. This is illustrated with an operational and economic analysis using data from the German Amprion EIM. (Joint work with Jan Palczewsk (University of Leeds)).

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

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 fluctuations. After the pioneer- ing paper by Schwartz, where an Ornstein-Uhlenbeck dynamics is assumed to describe the spot price behavior, several different approaches have been inves- tigated in order to describe the price evolution. A comprehensive presentation of the literature until 2008 is offered in the book by Benth, Saltyte-Benth and Koekebakker [8]. High frequency trading, on the other hand, introduced some new features in commodity prices dynamics: in the paper by Filimonov, Bic- chetti, Maystre and Sornette [11] evidence is shown of endogeneity and struc- tural regime shift, and in order to quantify this level the branching ratio is adopted as a measure of this endogenous impact and a Hawkes processes dy- namics is assumed as a reasonable modeling framework taking into account the self-exciting properties [1]. The purpose of the present paper is to pro- pose 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 different forms, that can be proved to be equivalent: the first is a representation based on random fields, the second is based on Continuous Branching Processes with Immigration (CBI in the following). The idea of adopting a random fields framework for power prices description is not new: O.E. Barndorff-Nielsen, F.E. Benth and A. Veraart introduced the Ambit Fields to this end, showing how this approach can provide a very flexible 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 modeling, and in that paper it was shown that, with a suitable choice of the L´evy process driving the CBI dynamics, the model can offer a significant extension of the popular CIR model [12]. The model we propose extends in different ways some relevant models al- ready available in the literature. It belongs to the class of arithmetic models (following the classification proposed by F.E. Benth, J. Salthythe-Benth and S. Koekebakker), and the driving processes are L´evy processes with positive jumps, i.e. subordinators, so it extends the model proposed by F.E. Benth, J. Kallsen and T. Meyer-Brandis [6] by formulating the dynamics via a random field ap- proach, which allows to include some self-exciting features. On the other hand, the random field approach highlights some similarities with the Ambit Field- based models introduced by O.E. Barnorff-Nielsen, F.E. Benth and A. Veraart [3]; the main difference between the model proposed in this paper and the Ambit Field-based models consists in the character of the extra dimension appearing in the random field adopted: while in the Ambit Field setting the parameter of this dimension is a time parameter, in the present setting this will be a pa- rameter of space type. This main difference will be reflected moreover in the integration domain of the integrals defining the dynamics. The features of our modeling approach just outlined, allow to introduce the so- called self-exciting properties in a simple and natural way and, although the pricing formulas for basic contracts like forward will exhibit very small changes with respect to those obtained for the previous models, the present model will exhibit a substantially different risk premium term structure. The presentation will be organized as follows: in Section 2 we’ll introduce the market model we are going to consider, while in Section 3 we shall discuss the relations between our model and the CBI processes. In Section 4 we’ll present some closed formulas for Futures and Option prices when the underlying dynamics is assumed to be given by the model introduced. Section 5 includes a theoretical analysis of the jumps behavior and the self-exciting property. In Section 6 we’ll provide some suggestions about estimation methods for the same model. In this last section, in particular, we are going to highlight the main issues and to propose a theoretical statistical approach. In particular, we are going to derive the maximum likelihood estimator for the parameters of the intensity process. By following the ideas presented in [7] and in [13], the first step to perform will be to de-seasonalise the data. The second step, definitely less trivial, is to split the components Y1 and Y2 emerging from the data. This issue is well analyzed in [7] and [13] and their approach is directly applicable to our framework. Then, we first focus on the process Y1, sometimes called the base signal. Following [7], we look for the ergodic distribution of Y1 fitting the data. By recalling that the ergodic distribution of a CIR diffusion is of Gamma type [10], our model is in agreement with the previous literature (see subsection 5.4.2 in[7]) and we obtain the estimated parameters values for the driving processes. Joint work with Ying Jiao, Chunhua Ma and Simone Scotti. References [1] Bacry, E., Mastromatteo, J. and Muzy, J.-F. Hawkes Processes in Finance, PREPRINT (2015). [2] Barndorff-Nielsen, O.E., Benth, F.E. and Veraart, A. (2013): Modelling en- ergy spot prices by volatility modulated L´evy driven Volterra processes, Bernoulli, 19, 803-845. [3] Barndorff-Nielsen, O.E., Benth, F.E. and Veraart, A. (2014): Modelling Electricity Futures by Ambit Fields, Advances in Applied Probability, 46 (3), 719-745. [4] Barndorff-Nielsen, O.E. and Shephard, N. (2000): Modelling by L´evy Pro- cesses for Financial Econometrics, in L´evy Processes Theory and Applications, eds. Barndorff- Nielsen, Mikosch and Resnick, Boston, Birkhauser. [5] Benth F. E., Cartea A. and Kiesel R. (2008): Pricing forward contracts in power markets by the certainty equivalence principle: explaining the sign of the market risk premium, Journal of Banking and Finance, 32, 2006-2021. [6] Benth, F. E., Kallsen J. and Meyer-Brandis T. (2007): A Non-Gaussian Ornstein- Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing, Appl. Math. Finance, 14(2), 153-169. [7] Benth, F. E., Kiesel, R. and Nazarova A. (2012): A critical empirical study of three electricity price models, Energy Economics, 34, 1589-1616. [8] Benth, F. E., Salthyte-Benth J. and Koekebakker S. (2008): Stochastic Mod- elling of Electricity and Related Markets , World Scientific, Singapore. [9] Benth, F. E. and Sgarra C. (2012): The Risk Premium and the Esscher Transform in Power Markets, Stoch. Anal. Appl., 30(1), 20-43. [10] Cox, J., Ingersoll, J. and Ross, S. (1985): A theory of the term structure of interest rate. Econometrica 53, 385-408. [11] Filimonov, V., Bicchetti, D., Maystre, N., Sornette, D. (2015):Quantifica- tion of the High Level of Endogeneity and Structural Regime Shifts in Com- modity Markets, preprint. [12] Jiao, Y., Ma, C., Scotti, S. (2016): Alpha-CIR Model with Branching Processes in Sovereign Interest Rate Modelling, preprint, hal-01275397v2. [13] Meyer-Brandis, T. and Tankov, P. (2008): Multi-factor jump-diffusion mod- els of electricity prices. International Journal of Theoretical and Applied Fi- nance, 11(5), 503-528.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In recent years, renewable energy has gained importance in producing power in many markets. The aim of this article is to model photovoltaic (PV) production for three transmission operators in Germany. PV power can only be generated during sun hours and the cloud cover will determine its overall production. Therefore, we propose a model that takes into account the sun intensity as a seasonal function. We model the deseasonalized data by an au- toregressive process to capture the stochastic dynamics in the data. We present two applications based on our suggested model. First, we build a relationship between electricity spot prices and PV production where the higher the volume of PV production, the lower the power prices. As a further application, we discuss virtual power plant derivatives and energy quanto options. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this paper we develop a statistical arbitrage trading strategy with two key elements in high frequency trading: stop-loss and leverage. We con- sider, as in Bertram (2009), a mean-reverting process for the security price with proportional transaction costs; we show how to introduce stop-loss and lever- age in an optimal trading strategy. We focus on repeated strategies using a self-financing portfolio. For every given stop-loss level we derive analytically the optimal investment strategy consisting of optimal leverage and market en- try/exit levels. First we show that the optimal strategy a la Bertram depends on the probabilities to reach entry/exit levels, on average First-Passage-Times and on average First-Exit-Times from an interval. Then, when the underlying log- price follows an Ornstein-Uhlenbeck process, we deduce analytical expressions for average First-Exit-Times and we write the long-run return of the strategy as an elementary function of the stop-loss. Finally we describe how to apply the strategy to a generic continuous mean-reverting process. Following industry practice of pairs trading we consider two examples of pairs in the energy futures’ market. We report in detail the analysis for two spreads on Heating-Oil and Gas-Oil futures in a year and a half sample of half-hour market prices. Joint work with Tommaso Santagostino Baldi.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The growing penetration of non-programmable renewable sources, like solar and wind, introduced in the latest years market uncertainties in the quan- tity of electricity produced, which can possibly originate price spikes. Capacity markets have exactly the purpose of providing new potential capacity when that present in the market is already allocated and there is a sudden drop in supply (due for example to unexpected adverse weather events). In this talk we will present the different capacity remuneration mechanisms, and analyze in more detail the so-called reliability option, which is a call option sold by producers to transmit system operators. This option has the important advantage of shaving possible price peaks, but its correct pricing require non-trivial techniques.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The recent price coupling of many European electricity markets has triggered a fundamental change in the interaction of day-ahead prices, challeng- ing additionally the modeling of the joint behavior of prices in interconnected markets. We propose a regime-switching AR-GARCH copula to model pairs of day-ahead electricity prices in coupled European markets. While capturing key stylized facts empirically substantiated in the literature, this model easily allows us to 1) deviate from the assumption of normal margins and 2) include a more detailed description of the dependence between prices. We base our empirical study on four pairs of prices, namely Germany-France, Germany- Netherlands, Netherlands-Belgium and Germany-Western Denmark. We find that the marginal dynamics are better described by the flexible skew t distribu- tion than the benchmark normal distribution. Also, we find significant evidence of tail dependence in all pairs of interconnected areas we consider. As appli- cations of the proposed empirical model, we consider the pricing of financial transmission rights and the forecasting of tail quantiles. In both applications, we highlight the effects of the distributional assumptions for the margins and the tail dependence. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this paper we find an approximation to a non-semimartingale Volterra-type process by semimartingales, and furthermore, in the setting of gen- eralized Lebesgue-Stieltjes integration, we find an approximation to the pathwise stochastic integral with this non-semimartingale process as noise. A link to the Itˆo integral and an algorithm for numerical simulation are presented. Joint work with Giulia Di Nunno.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We give new insights into the theory of the dynamic behavior of com- modity prices with an infinite horizon rational expectations equilibrium model for spot and futures commodity prices. Numerical simulations of the model emphasize the heterogeneity that exists in the behavior of commodity prices by showing the link between the physical characteristics of a market and some stylized facts of commodity futures prices. They show the impact of storage costs on both the variability of the basis and on the Samuelson effect. Finally, the simulations of the model show that an increase in the speculative activity on commodity futures markets has an overall positive effect on risk premia. However, not all of the agents benefit from it.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

In this work, we propose a model of electricity demand management through a principal-agent problem, allowing to obtain almost explicit optimal compensations for the consumer. We then illustrate our findings through several numerical experiments, putting the emphasis on the practical implementation of the contracts. (Joint work with Rene Aid and Nizar Touzi).

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We develop a Monte-Carlo based numerical method for solving discrete- time stochastic optimal control problems with inventory. These are optimal control problems in which the control affects only a deterministically evolving inventory process on a compact state space while the random underlying pro- cess manifests itself through the objective functional. We propose a Regress Later modification of the traditional Regression Monte Carlo which allows to decouple inventory levels in two successive time steps and to include in the basis functions of the regression the dependence on the inventory levels. We develop a backward construction of trajectories for the inventory which enables us to use policy iteration of Longstaff-Schwartz type avoiding nested simulations.Our al- gorithm improves on the grid discretisation procedure largely used in literature and practice, and on the recently proposed control randomisation by Kharroubi et al. (2014). We validate our approach on two numerical examples: one is a benchmark problem of energy arbitrage used to compare different methods available in literature, the other is a high-dimensional problem of the manage- ment of a battery with the purpose of assisting the operations of a wind turbine in providing electricity to a group of buildings in a cost effective way. Joint work with Alessandro Balata.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We will give an introduction to short-term electricity markets. We will start with the relation of day-ahead and intraday prices on the EPEX for deliveries in Germany/Austria. In the sequel we will focus on analyzing the intraday market. We will discuss empirical properties of intraday power markets and point out development in recent years. Furthermore, we study the optimal liquidation problem for traders in intraday power markets.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

The newly introduced wind power futures on the European Energy Exchange have brought interesting opportunities for energy market players in Germany. In this paper, we analyze the benefits of wind power futures in the context of both the buyer’s and the seller’s side. From the buyer’s side, we con- sider gas-fired power plants. To increase the competitiveness of such plants, we propose a simple yet powerful spot-based trading strategy taking advantage of wind power futures. The purpose of the trading strategy is two-fold: 1) increase the revenue of running the gas-fired power plant, and 2) minimize the variance of the revenue generated from the strategy using wind power futures. To fa- cilitate optimal hedging decisions, we employ ARMA-GARCH models for the marginal behavior of electricity price, gas price, and wind power production, and a mixed vine copula for the dependency between the variables. We find that significant benefits can be achieved by employing a spot-trading strategy as opposed to a strategy acting in the forward market (conditional on the for- ward spark spread being positive). More importantly, using wind power futures reduces the variance of the spot-trading strategy significantly. From the seller’s side, we have the wind mill owners who are facing a quite volatile revenue due to their exposure to joint price and volumetric risk, which they wish to minimize. By performing a similar analysis as in the case of the gas-fired power plants, we again find that wind power futures are beneficial. Joint work with Anca Pircalabu.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We extend the Heston stochastic volatility model to a Hilbert space framework. The stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. We compute the dynam- ics of this process under certain conditions, and project it down to the real line to compare it with the one-dimensional Heston variance process. The stochastic volatility process is defined by a Cholesky decomposition of the variance process. We define another Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional of this process. Joint work with Fred Espen Benth.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)

We investigate the risk premium in cash settled forward contracts on the Baltic Exchange Indices – the so-called Forward Freight Agreements – in the dry bulk shipping markets. We estimate multiple spot price models using Markov Chain Monte Carlo. Using a structure-preserving measure change, we then calibrate the risk premium of traded FFA contracts. Finally we link the risk premium to explanatory variables like e.g., oil prices, demand and supply for shipping and the state of the global economy. Joint work with Jonas Lager and Nikos Nomikos.

• Thematic program: Mathematical Finance (2016/2017)
• Event: Second Conference on "The Mathematics of Energy Markets" (2017)