Conference Agenda

Geometric Asian options are a type of option where the payoff depends on the geometric mean of the underlying asset over a certain period of time. This paper is concerned with the pricing of such options for the class of Volterra-Heston models, covering the rough Heston model. We are able to derive semi-closed formulas for the prices of geometric Asian options with fixed and floating strikes for this class of stochastic volatility models. These formulas require the explicit calculation of the conditional joint Fourier transform of the logarithm of the stock price and the logarithm of the geometric mean of the stock price over time. Linking our problem to the theory of affine Volterra processes, we find a representation of this Fourier transform as a suitably constructed stochastic exponential, which depends on the solution of a Riccati-Volterra equation. Finally we provide a numerical study for our results in the rough Heston model.

In this talk, we present a new perspective on the comparison principle for viscosity solutions of Hamilton-Jacobi (HJ), HJ-Bellman, and HJ-Isaacs equations.

Our approach innovates in three ways: (1) We reinterpret the classical doubling-of-variables method in the context of second-order equations by casting the Ishii-Crandall Lemma into a test function framework. This adaptation allows us to effectively handle non-local integral operators, such as those associated with L\'{e}vy processes. (2) We translate the key estimate on the difference of Hamiltonians in terms of an adaptation of the probabilistic notion of couplings, providing a unified approach that applies to both continuous and discrete operators. (3) We strengthen the sup-norm contractivity resulting from the comparison principle to one that encodes continuity in the strict topology.

We apply our theory to derive well-posedness results for partial integro-differential operators. In the context of spatially dependent L\'{e}vy operators, we show that the comparison principle is implied by a Wasserstein-contractivity property on the L\'{e}vy jump measures.

The aim of this work is to study a hedging problem in an incomplete market model in which the underlying log-asset price is driven by a diffusion process with self-exciting jumps of Hawkes type.

We aim at hedging a variance swap (target claim) at time $T > 0$, using a basket of European options (contingent claims). We investigate a semi-static variance-optimal hedging strategy, combining dynamic (i.e., continuously rebalanced) and static (i.e., buy-and-hold) positions to minimize the residual error variance at time $T$. The semi-static strategy has already been computed in literature for different models of the asset price $S$. The purpose of our work is to solve the hedging problem for an unexplored model featuring self-exciting jumps of Hawkes type. The key aspect of our work is the generality of our framework, both from the perspective of the hedging and the model investigated. Moreover, research into models with self-exciting jumps is significant as it has been observed that prices in the financial market (e.g. commodity markets) exhibit spikes having clustered behavior.

In our work, we establish and analyze our model, studying its properties as an affine semimartingale. We characterize its Laplace transform to rewrite contingent claims using a Fourier transform representation. We finally obtain a semi-explicit expression for the hedging strategy. A possible further development might regard the problem of optimal selection of static hedging assets and potential applications in energy markets.

This research develops theoretical tools for the risk management and optimization of intermittent renewable energy in short-term electricity markets. The first part introduces a stochastic model based on a mutually exciting marked Hawkes process to capture key empirical characteristics of Germany's intraday energy market prices. The model effectively reflects the increasing market activity, volatility patterns, and the Samuelson effect observed in the realized mid-price process as time to delivery approaches. By fitting the empirical signature plot of the mid-price process, the model provides a robust calibration method through a closed-form solution, using least squares to match the empirical data.

Building on this foundation, the second part of this research explores optimal execution strategies for energy companies managing large trading volumes, whether from outages, renewable generation, or trading decisions. The study employs a linear transient price impact model combined with a bivariate Hawkes process, which models the flow of market orders, to solve a meta-order execution problem. The optimal execution problem is solved explicitly in this context, due to the affine structure of the state space dynamics. The model determines an optimal liquidation strategy which minimizes the expected costs, allowing traders to react to the actions of other market participants. The research concludes with a back-testing transaction cost analysis for the German intraday energy market, comparing the proposed optimal strategy against benchmark execution strategies like Time Weighted Average Price (TWAP) and Volume Weighted Average Price (VWAP). The results confirmed that the optimal strategy is cost-efficient, significantly reducing transaction costs compared to the benchmark strategies. Further analysis of individual hourly products revealed that cost reductions were particularly substantial for early trading hourly products and stabilized thereafter, with a slight decrease in the mid-day products. This indicates that cost savings are negatively correlated with the average traded volume, suggesting that less volatile, more liquid products might benefit less from the optimal strategy compared to less liquid ones, where improvements are more pronounced.