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This paper addresses the following question: How can a financial institution, which has issued a European option, optimally hedge the payoff of this option by investing into the underlying stock and into the option itself? Here, optimality is measured in terms of minimal variance and the associated optimal hedging portfolio is derived by a sufficient stochastic maximum principle. We further obtain a pricing formula for general European options by an application of Fourier transform methods and deduce the time dynamics of the stochastic option price process. We finally apply our theoretical results to several practical examples.

In this paper, we investigate the following problem: How can a financial institution, which has sold an option to a client, optimally hedge the payoff of this option by investing into a stock and into the option itself? Optimality is measured in terms of minimal variance and the associated optimal hedging portfolio is derived by a stochastic maximum principle. Moreover, we deduce the time dynamics of the stochastic option price process by Malliavin calculus methods, particularly by an application of the Clark-Ocone formula. We finally apply our theoretical results to several examples.

In this paper, we prove a sufficient stochastic maximum principle for continuous-state branching processes with immigration (so-called CBI processes). We apply the result to several stochastic control problems stemming from finance and epidemiology.

In this paper, we derive optimal hedging strategies for options in electricity futures markets. Optimality is measured in terms of minimal variance and the associated minimal variance hedging portfolios are obtained by a stochastic maximum principle. Our explicit results are particularly useful for electricity retailers, who have sold an option to a client, and now want to hedge the payoff of this option by investing into an electricity futures and into the issued option itself. Another innovative aspect of the paper lies in the derivation of the time dynamics of the stochastic option price process by Malliavin calculus methods and the Clark-Ocone formula. We also apply our theoretical results to several practical examples. Our investigations are based upon the popular arithmetic multi-factor electricity spot price model proposed by Benth, Kallsen & Meyer-Brandis (Appl. Math. Finance 14(2):153-169, 2007).

In this paper, we propose an innovative VIX model which takes future market information available to the traders into account. The future information is modeled by an initially enlarged filtration in our setup. We derive an explicit representation for the anticipative VIX process and obtain the associated time dynamics. We also investigate the pricing of variance swaps under both backward- and forward-looking information. We finally deduce the optimal mean variance hedging portfolio in a financial market consisting of a bank account and a VIX futures. In order to have some benchmark model available, we introduce a non-anticipative stochastic volatility stock price model right at the beginning and infer representations for the related VIX index, the VIX futures and a VIX call option.

In this paper, we present several optimal control approaches which are designed to control and regulate the evolution of infection numbers in the COVID-19 disease. In our setting, the number of infected people at a certain location is modeled by a continuous-time stochastic process which can be affected by a related stochastic control process. We use mathematical tools from stochastic analysis and optimal control theory. In particular, we prove an innovative stochastic maximum principle for continuous-state branching processes with immigration (so-called CBI processes) and apply the result to a stochastic control problem stemming from epidemiology.

We propose a new model for the pricing of wind power futures written on the wind power production index. Our approach is based on an arithmetic multi-factor pure-jump Ornstein-Uhlenbeck setup with time-dependent coefficients. We express the wind power production index and the corresponding futures price in terms of Fourier integrals and derive the related time dynamics. We conclude the paper by an investigation of the so-called risk premium associated with our wind power model.

We extend the arithmetic multi-factor electricity spot price model proposed by Benth et al. (Appl Math Finance 14(2):153–169, 2007) by adding stochastic mean-level processes to their model and by taking additional information on the future behavior of these mean-level processes into account. The available anticipative information is modeled by an initially enlarged filtration in our paper. We further derive pricing formulas for electricity forwards under future information and investigate the associated information premium.

We derive stochastic Leibniz formulas for Volterra integrals under enlarged filtrations. We investigate both pure-jump and Brownian Volterra processes under diverse initially enlarged filtration approaches. In these settings, we compare the ordinary with the stochastic (Doleans-Dade) exponential of a Volterra process and provide the corresponding martingale conditions. We also consider backward stochastic Volterra integral equations (BSVIEs) under enlarged filtrations and obtain the related solution formulas. We finally propose an anticipative Heath Jarrow Morton (HJM) forward rate model of Volterra-type. In an introductory section, we compile various facts on deterministic and stochastic Leibniz formulas for parameter integrals.

In a general jump-diffusion Radon–Nikodym setup with stochastic Girsanov processes, we derive optimal equivalent probability measures. Optimality is measured in terms of minimum relative entropy and also by more general divergence concepts. We further provide an anticipative sufficient stochastic minimum principle and derive optimal equivalent probability measures under various enlarged filtration approaches.

We provide solution formulas for (linear and non-linear) jump-diffusion backward stochastic differential equations (BSDEs) under diverse enlarged filtration approaches. We also derive a comparison theorem for BSDEs in an enlarged filtration framework and present several applications of our theoretical results. For example, we study temperature derivatives under weather forecasts modeled by an enlarged filtration BSDE, investigate portfolio selection and hedging for an insider with additional information on future price behavior and solve a particular predictive mean-field BSDE in an enlarged filtration setup.

We investigate the pricing of cliquet options in a jump-diffusion model. The considered option is of monthly sum cap style while the underlying stock price model is driven by a drifted Levy process entailing a Brownian diffusion component as well as compound Poisson jumps. We also derive representations for the density and distribution function of the emerging Levy process. In this setting, we infer semi-analytic expressions for the cliquet option price by two different approaches. The first one involves the probability distribution function of the driving Levy process whereas the second draws upon Fourier transform techniques. With view on sensitivity analysis and hedging purposes, we eventually deduce representations for several greeks while putting emphasis on the vega.

We prove an anticipative sufficient stochastic minimum principle in a jump process setup with initially enlarged filtrations. We apply the result to several portfolio selection problems like mean and minimal variance hedging under enlarged filtrations. We also investigate utility maximizing portfolio selection under future information. Contrarily to classical optimization methods like dynamic programming, our stochastic minimum principle likewise applies to non-Markovian setups. On the mathematical side, we are concerned with jump processes, forward and backward stochastic differential equations and forward integrals.

We consider minimal variance hedging in a pure-jump multi-curve interest rate model. In a first part, we derive arithmetic multi-factor martingale representations for the spread, OIS and LIBOR rate which are bounded from below by a real-valued constant. In a second part, we investigate minimal variance hedging and provide a closed-form formula for the related minimal variance portfolio. We apply this result to several examples covering both replicable and non-replicable claims. We conclude the paper with a consideration of delta hedging.

We propose a pure jump precipitation model embedded in an enlarged filtration framework accounting for weather forecasts. Under different anticipative approaches, we define precipitation swap/futures prices and also introduce the notion of an “information premium”. In contrast to some other models in the literature, our forward-looking swap price representations admit time-varying stochastic dynamics. In these setups, swap price processes under the physical and risk-neutral measure turn out to be indistinguishable. We also consider an extended multi-location model measuring precipitation in several locations. In order to price options on precipitation derivatives under weather forecasts modeled by enlarged filtrations, we develop customized approximation procedures involving complex power series expansions and wavelet transform techniques.

We propose a mean-reverting electricity spot price model of arithmetic jump-diffusion type yielding positive prices. Based on this approach, we derive the corresponding forward and futures price representations. We further discuss different choices for the stochastic mean level process and investigate the long-term behavior of the spot price. In the second part, we take future information available to the traders into account. The latter is modeled by initially enlarged filtrations with respect to (a) the mean level of the spot, (b) the driving diffusion component and (c) the jump term. We also derive forward and futures price representations under these enlarged filtrations. Finally, we consider the evaluation of options in the proposed models.

We derive risk-neutral option price formulas for plain-vanilla and exotic electricity futures derivatives on the basis of diverse arithmetic multi-factor Ornstein-Uhlenbeck spot price models admitting seasonality. In these setups, we take additional forward-looking knowledge on future price behavior into account via multiple initially enlargements of the underlying information filtration. In this insider trading context, we also correlate electricity prices with outdoor temperature and treat a related pricing problem under supplementary temperature forecasts. Meanwhile, we use Fourier transform techniques and results from complex analysis to handle the emerging anticipating conditional expectations. As a by-product we derive related risk and information premia. The paper can be regarded as an extension of [4] and [5], since in [4] no future information is involved while in [5] no option pricing is considered.

We derive utility maximizing portfolios and consumption rates in electricity futures markets under anticipative information modeled by enlarged filtrations. The emerging optimization exercises are solved by point-wise maximization and a sufficient stochastic maximum principle. We provide rigorous economical interpretation of our mathematical results and distinguish optimal trading behavior for small and large investors with insider knowledge. On the mathematical side we are concerned with jump processes, (B)SDEs, enlarged filtrations, forward integrals, stochastic maximum principles and Malliavin calculus.

Electricity markets commonly exhibit an oligopolistic structure with market participants whose individual trading activities may shift prices essentially. In this context, the question of how to optimally liquidate an existing electricity futures portfolio over a fixed time horizon under the constraint of minimizing unfavorable price impact effects is of striking relevance for energy risk management. In this article, we thus invent a tractable market impact model for electricity futures prices. More precisely, we derive optimal liquidation strategies with respect to different target functionals of ‘conditional expected trading cost’ type. Moreover, we take forward-looking information about future electricity price behavior into account by exploiting enlarged filtration methods. Finally, we derive optimal liquidation strategies for electricity futures portfolios under this insider trading machinery as well.

We derive risk-neutral option price formulas for plain-vanilla temperature futures derivatives on the basis of several multi-factor Ornstein-Uhlenbeck temperature models which allow for seasonality in the mean level and volatility. Our main innovation consists in an incorporation of omnipresent weather forecasts via numerous tailor-made enlargements of the underlying historical information filtration. In this insider trading framework, we obtain forward-looking price representations for cumulative average temperature (CAT) and cooling degree day (CDD) futures whereas we provide anticipative CAT option price formulas. On this occasion, the evaluation of conditional expectations under enlarged filtrations comprises the major mathematical challenge which we handle by applying suitable transformation concepts from complex analysis. Ultimately, we construct optimal positions in a temperature futures portfolio under future weather information to hedge against both temporal and spatial temperature risk simultaneously.

In this article we derive risk-neutral option price formulas for both plain-vanilla and exotic electricity futures derivatives on the basis of diverse arithmetic multi-factor Ornstein-Uhlenbeck spot price models admitting seasonality, while – in order to avoid “information miss-specification” – we take additional forward-looking knowledge on future price behavior into account via tailor-made enlargements of the underlying historical information filtrations. In this insider trading context, we also correlate electricity prices with outdoor temperature and treat a related pricing problem under supplementary temperature forecasts. Meanwhile, we use Fourier transform techniques and results from complex analysis to handle the emerging anticipating conditional expectations. As a by-product we derive related risk and information premia. Finally, we investigate utility-maximizing portfolio selection and optimal consumption rates in electricity futures markets even under forward-looking information.