arXiv subjects

Sven Karbach

Publications and source records attributed to Sven Karbach.

14 recordsLinked to original sources

arXiv2026

Pricing and Hedging of Discretely Monitored Asian Options in the Volterra-Heston Model

We develop semi-closed pricing formulas and lifted-model hedging methods for discretely monitored geometric and arithmetic Asian options in the Volterra-Heston stochastic volatility model. Exploiting the affine Volterra structure, we derive a tractable transform for the joint law of the terminal log-price and the discretely monitored geometric average. This transform yields semi-closed pricing formulas for geometric Asian options, which in turn provide effective control variates for Monte Carlo valuation of arithmetic Asian options. Under the stated real-moment and affine-transform hypotheses, we also derive the Galtchouk-Kunita-Watanabe decomposition for Fourier-representable payoffs and obtain a variance-optimal hedge in terms of the Riccati-Volterra equation and the forward-variance curve. Using N-factor Markovian approximations, we obtain a finite-dimensional numerical implementation for hedging Asian options. Our numerical experiments document factor convergence for a regular non-Markovian kernel and the effect of rebalancing frequency on hedging error. In the Heston benchmark, geometric Asian controls substantially reduce the variance of arithmetic-Asian price estimates and improve the finite-sample stability of regression-based hedging relative to direct regression.

q-fin.PR↗

arXiv2026

Pricing and Semi-static Hedging of Green Pay-as-produced Power Purchase Agreements

Pay-as-produced power purchase agreements (PPAs) expose buyers and sellers to the joint risk of power prices and renewable production. This paper develops a theoretical framework for hedging this exposure using a semi-static strategy: liquid futures hedge traded price risk dynamically, while a fixed portfolio of renewable-linked claims targets residual volume and covariance risk. The pricing and hedging decomposition is model-free, whereas the empirical implementation for German wind and solar generation uses a calibrated stochastic model. Conditional on a valuation measure, the fair strike is a production-weighted expected spot price. We show that it decomposes exactly into the baseload forward level, a deterministic production-profile correction, and a stochastic price-volume covariance correction, where the covariance term measures the pricing effect of renewable cannibalisation. The static hedge is selected through a finite-dimensional variance projection onto claims linked to renewable volume, delivery-period average prices, and price-volume covariance. We estimate a L\'evy-driven bivariate MCARMA state-space model with state-dependent price spikes using hourly German data for 2023-2024 and apply it to monthly PPAs over the January-December 2025 delivery horizon. The results distinguish deterministic profile risk from stochastic covariance risk and show how sparse static overlays reduce residual exposures that fixed-volume futures cannot hedge. The selected portfolios also indicate which claim types are most effective for hedging residual renewable shape risk.

q-fin.MF↗

arXiv2026

Signature-Based Optimal Execution for Statistical Arbitrage with Path-Dependent Trading Signals

We develop a signature-based framework for optimal execution in statistical arbitrage strategies with path-dependent predictive signals. Both the alpha process and the trading speed are modelled as linear functionals of the truncated signature of a time-augmented market path, placing signal generation and execution on the same truncated signature basis. This allows the trading rule to react to the realised history of the signal while accounting for temporary impact, inventory exposure, terminal liquidation, and approximate dollar neutrality. The main contribution is a quadratic reduction theorem: within the class of signature-linear trading speeds, the restricted path-dependent execution problem becomes a finite-dimensional concave quadratic programme in the policy coefficients. After running synthetic experiments under a mean-reverting log-spread model, we find that the fitted policy achieves a higher return on turnover than a classical $z$-score threshold benchmark. We show how the same workflow can be deployed on a historical equity pairs-trading backtest, where the fitted signature policy again outperforms the benchmark in accounting terms.

q-fin.TR↗

arXiv2026

Hedging Maturity-Specific Risk in Forward Curve Derivatives under Stochastic Volatility

We study the variance-optimal hedging of European contingent claims written on forwards. We assume that the dynamics of the underlying forward curves follow a Heath--Jarrow--Morton--Musiela stochastic partial differential equation modulated by an infinite-rank stochastic covariance component. The variance-optimal hedge is then given by the Galtchouk--Kunita--Watanabe projection with respect to some covariance-norm quotient generated by the forward curve martingale. We show density of finite-maturity and delivery-window strategies, convergence of spectral finite-rank hedge projections and an exact decomposition of the quadratic hedging error into bucket, rank and residual risk components. In enlarged filtrations, the residual risk is a stochastic-volatility floor for claims loading on non-traded covariance noise. We illustrate the hedging framework in affine stochastic covariance and multiplicative HJMM models, and give a concrete example of the decomposition in a CIR stochastic covariance model.

q-fin.MF↗

arXiv2026

Semi-Static Variance-Optimal Hedging of Covariance Risk in Multi-Asset Derivatives

We develop a semi-static framework for the variance-optimal hedging of multi-asset derivatives exposed to correlation and covariance risk. The approach combines continuous-time dynamic trading in the underlying assets with a static portfolio of auxiliary contingent claims. Using a multivariate Galtchouk--Kunita--Watanabe decomposition, we show that the resulting global mean-variance problem decouples naturally into an inner continuous-time projection onto the space spanned by the underlying assets and an outer finite-dimensional quadratic optimization over the static hedging instruments. To systematically select suitable auxiliary claims, we leverage multidimensional functional spanning theory, establishing that otherwise unhedgeable cross-gamma exposures can be structurally mitigated through static strips of vanilla, product, and spread options. As a central application, we derive explicit semi-static replication formulas for covariance swaps and geometric dispersion trades. Our framework accommodates a broad class of asset dynamics, including quadratic and stochastic Volterra covariance models, as well as affine stochastic covariance models with jumps, yielding tractable semi-closed-form solutions via Fourier transform techniques. Extensive numerical experiments demonstrate that incorporating optimally weighted static strips of cross-asset instruments substantially reduces the mean-squared hedging error relative to purely dynamic benchmark strategies across various model classes.

q-fin.MF↗

arXiv2025

Optimal Execution in Intraday Energy Markets under Hawkes Processes with Transient Impact

This paper investigates optimal execution strategies in intraday energy markets through a mutually exciting Hawkes process model. Calibrated to data from the German intraday electricity market, the model effectively captures key empirical features, including intra-session volatility, distinct intraday market activity patterns, and the Samuelson effect as gate closure approaches. By integrating a transient price impact model with a bivariate Hawkes process to model the market order flow, we derive an optimal trading trajectory for energy companies managing large volumes, accounting for the specific trading patterns in these markets. A back-testing analysis compares the proposed strategy against standard benchmarks such as Time-Weighted Average Price (TWAP) and Volume-Weighted Average Price (VWAP), demonstrating substantial cost reductions across various hourly trading products in intraday energy markets.

q-fin.TR↗

arXiv2025

Pricing Options on Forwards in Function-Valued Affine Stochastic Volatility Models

We study the pricing of European-style options written on forward contracts within function-valued infinite-dimensional affine stochastic volatility models. The dynamics of the underlying forward price curves are modeled within the Heath-Jarrow-Morton-Musiela framework as solution to a stochastic partial differential equation modulated by a stochastic volatility process. We analyze two classes of affine stochastic volatility models: (i) a Gaussian model governed by a finite-rank Wishart process, and (ii) a pure-jump affine model extending the Barndorff--Nielsen--Shephard framework with state-dependent jumps in the covariance component. For both models, we derive conditions for the existence of exponential moments and develop semi-closed Fourier-based pricing formulas for vanilla call and put options written on forward price curves. Our approach allows for tractable pricing in models with infinitely many risk factors, thereby capturing maturity-specific and term structure risk essential in forward markets.

q-fin.MF↗

arXiv2025

Measure-Valued CARMA Processes

In this paper, we examine continuous-time autoregressive moving-average (CARMA) processes on Banach spaces driven by Lévy subordinators. We show their existence and cone-invariance, investigate their first and second order moment structure, and derive explicit conditions for their stationarity. Specifically, we define a measure-valued CARMA process as the analytically weak solution of a linear state-space model in the Banach space of finite signed measures. By selecting suitable input, transition, and output operators in the linear state-space model, we show that the resulting solution possesses CARMA dynamics and remains in the cone of positive measures defined on some spatial domain. We also illustrate how positive measure-valued CARMA processes can be used to model the dynamics of functionals of spatio-temporal random fields and connect our framework to existing CARMA-type models from the literature, highlighting its flexibility and broader applicability.

math.PR↗

arXiv2024

Heat modulated affine stochastic volatility models for forward curve dynamics

We present a function-valued stochastic volatility model designed to capture the continuous-time evolution of forward curves in fixed-income or commodity markets. The dynamics of the (logarithmic) forward curves are defined by a Heath-Jarrow-Morton-Musiela stochastic partial differential equation modulated by an instantaneous volatility process that describes the second-order moment structure of forwards with different time-to-maturity. We propose to model the operator-valued instantaneous covariance by an affine process on the cone of positive trace-class operators with drift given by the Lyapunov operator of the Laplacian. The so defined infinite-rank stochastic volatility model is analytically tractable due to its affine structure and allows to model maturity specific risk and volatility clustering in forward markets. Furthermore, we introduce a numerically feasible spectral Galerkin approximation of the associated operator-valued generalized Riccati equations and study the robustness of the model with respect to finite-rank approximations by providing explicit error bounds on the approximation error.

q-fin.MF↗

arXiv2023

Multivariate continuous-time autoregressive moving-average processes on cones

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on $\mathbb{R}_{d}$ with values in the positive orthant $\mathbb{R}_{d}^{+}$. Second, for MCARMA processes on real square matrices taking values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order structure of positive semi-definite well-balanced Ornstein-Uhlenbeck based models.

math.PR↗

arXiv2023

Finite-rank approximation of affine processes on positive Hilbert-Schmidt operators

In this article, we present a method for approximating affine processes on the cone of positive Hilbert-Schmidt operators using matrix-valued affine processes. By leveraging results from the theory on affine processes with values in the cone of symmetric and positive semi-definite matrices, we construct sequences of finite-rank operator-valued affine processes that converge weakly to the target processes and provide convergence rates for their Laplace transforms using Galerkin approximations of the associated operator-valued generalized Riccati equations. This article not only offers a practical approximation scheme for operator-valued affine processes with error bounds that hold uniformly in time, but also provides a novel existence proof for this class of affine processes with càdlàg paths, including affine pure-jump processes with infinite variation and state-dependent jump intensities. In addition to the theoretical significance, the results of this paper provide useful tools for analyzing and approximating infinite-dimensional affine stochastic covariance models that were recently introduced in mathematical finance.

math.PR↗

arXiv2022

Stationary Covariance Regime for Affine Stochastic Covariance Models in Hilbert Spaces

We study the long-time behavior of affine processes on positive self-adjoiont Hilbert-Schmidt operators which are of pure-jump type, conservative and have finite second moment. For subcritical processes we prove the existence of a unique limit distribution and construct the corresponding stationary affine process. Moreover, we obtain an explicit convergence rate of the underlying transition kernels to the limit distribution in the Wasserstein distance of order $p\in [1, 2]$ and provide explicit formulas for the first two moments of the limit distribution. We apply our results to the study of infinite-dimensional affine stochastic covariance models in the stationary covariance regime, where the stationary affine process models the instantaneous covariance process. In this context we investigate the behavior of the implied forward volatility smile for large forward dates in a geometric affine forward curve model used for the modeling of forward curve dynamics in fixed income or commodity markets formulated in the Heath-Jarrow-Morton-Musiela framework.

math.PR↗

arXiv2022

Affine pure-jump processes on positive Hilbert-Schmidt operators

We show the existence of a broad class of affine Markov processes in the cone of positive self-adjoint Hilbert-Schmidt operators. Such processes are well-suited as infinite dimensional stochastic volatility models. The class of processes we consider is an infinite dimensional analogue of the affine processes in the space of positive semi-definite and symmetric matrices studied in Cuchiero et al. [Ann. Appl. Probab. 21 (2011) 397-463]. As in the finite dimensional case, the processes we construct allow for a drift depending affine linearly on the state, as well jumps governed by a jump measure that depends affine linearly on the state. However, because the infinite-dimensional cone of positive self-adjoint Hilbert-Schmidt operators has empty interior, we do not consider a diffusion term. This empty interior also demands a new approach to proving existence: instead of using standard localisation techniques, we employ the theory on generalized Feller semigroups introduced in Dörsek and Teichmann [arXiv 2010] and further developed in Cuchiero and Teichmann [Journal of Evolution Equations (2020)]. Our approach requires a second moment condition on the jump measures involved, consequently, we obtain explicit formulas for the first and second moments of the affine process.

math.PR↗

arXiv2021

An infinite-dimensional affine stochastic volatility model

We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein-Uhlenbeck-type process, whose instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert-Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, i.e., we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalised Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity. Infinite dimensional volatility models arise e.g. when considering the dynamics of forward rate functions in the Heath-Jarrow-Morton-Musiela modeling framework using the Filipovic space. In this setting we discuss various examples: an infinite-dimensional version of the Barndorff-Nielsen-Shephard stochastic volatility model, as well as a model involving self-exciting volatility.

math.PR↗