arXiv subjects

Sascha Desmettre

Publications and source records attributed to Sascha Desmettre.

At least 19 recordsLinked to original sources

arXiv2026

Robust Control for Marked Point Processes under Transition-Rate Uncertainty

We consider a novel robust utility maximisation problem under bounded cumulative transition rate uncertainty within the class of non-Markovian marked point processes on a finite state-space. Utility is maximised over the class of admissible controls, while Nature chooses a worst-case biometric scenario from the class of admissible, path-dependent cumulative transition rates restricted by path-dependent upper and lower bounds. We prove a martingale optimality principle and a novel existence and uniqueness result for a non-standard worst-case backwards stochastic differential equation, which allows us to establish existence and uniqueness of worst-case and best-case prospective reserves of life and health insurance contracts with reserve-dependent payments. Finally, we find an explicit solution of a novel robust consumption-insurance problem with power utility preferences.

math.OC↗

arXiv2025

Equilibrium investment under dynamic preference uncertainty

We study a continuous-time portfolio choice problem for an investor whose state-dependent preferences are determined by an exogenous factor that evolves as an Itô diffusion process. Since risk attitudes at the end of the investment horizon are uncertain, terminal wealth is evaluated under a set of utility functions corresponding to all possible future preference states. These utilities are first converted into certainty equivalents at their respective levels of terminal risk aversion and then (nonlinearly) aggregated over the conditional distribution of future states, yielding an inherently time-inconsistent optimization criterion. We approach this problem by developing a general equilibrium framework for such state-dependent preferences and characterizing subgame-perfect equilibrium investment policies through an extended Hamilton-Jacobi-Bellman system. This system gives rise to a coupled nonlinear partial integro-differential equation for the value functions associated with each state. We then specialize the model to a tractable constant relative risk aversion specification in which the preference factor follows an arithmetic Brownian motion. In this setting, the equilibrium policy admits a semi-explicit representation that decomposes into a standard myopic demand and a novel preference-hedging component that captures incentives to hedge against anticipated changes in risk aversion. Numerical experiments illustrate how features of the preference dynamics -- most notably the drift of the preference process and the correlation between preference shocks and asset returns -- jointly determine the sign and magnitude of the hedging demand and the evolution of the equilibrium risky investment over time.

q-fin.MF↗

arXiv2025

Approximate Bayesian computation for stochastic hybrid systems with ergodic behaviour

Piecewise diffusion Markov processes (PDifMPs) form a versatile class of stochastic hybrid systems that combine continuous diffusion processes with discrete event-driven dynamics, enabling flexible modelling of complex real-world hybrid phenomena. The practical utility of PDifMP models, however, depends critically on accurate estimation of their underlying parameters. In this work, we present a novel framework for parameter inference in PDifMPs based on approximate Bayesian computation (ABC). Our contributions are threefold. First, we provide detailed simulation algorithms for PDifMP sample paths. Second, we extend existing ABC summary statistics for diffusion processes to account for the hybrid nature of PDifMPs, showing particular effectiveness for ergodic systems. Third, we demonstrate our approach on several representative example PDifMPs that empirically exhibit ergodic behaviour. Our results show that the proposed ABC method reliably recovers model parameters across all examples, even in challenging scenarios where only partial information on jumps and diffusion is available or when parameters appear in state-dependent jump rate functions. These findings highlight the potential of ABC as a practical tool for inference in various complex stochastic hybrid systems.

stat.ME↗

arXiv2025

The hybrid exact scheme for the simulation of first-passage times of jump-diffusions with time-dependent thresholds

The first-passage time is a key concept in stochastic modeling, representing the time at which a process first reaches a specified threshold. In this work, we consider a jump-diffusion (JD) model with a time-dependent threshold, providing a more flexible framework for describing stochastic dynamics. We are interested in the Exact simulation method developed for JD processes with constant thresholds, where the Exact method for pure diffusion is applied between jump intervals. An adaptation of this method to time-dependent thresholds has recently been proposed for a more general stochastic setting. We show that this adaptation can be applied to JD models by establishing a formal correspondence between the two frameworks. A comparative analysis is then performed between the proposed approach and the constant-threshold version in terms of algorithmic structure and computational efficiency. Finally, we show the applicability of the method by predicting neuronal spike times in a JD model driven by two independent Poisson jump mechanisms.

cond-mat.stat-mech↗

arXiv2025

First-passage time for PDifMPs: an Exact simulation approach for time-varying thresholds

Piecewise Diffusion Markov Processes (PDifMPs) are valuable for modelling systems where continuous dynamics are interrupted by sudden shifts and/or changes in drift and diffusion. The first-passage time (FPT) in such models plays a central role in understanding when a process first reaches a critical boundary. In many systems, time-dependent thresholds provide a flexible framework for reflecting evolving conditions, making them essential for realistic modelling. We propose a hybrid exact simulation scheme for computing the FPT of PDifMPs to time-dependent thresholds. Exact methods traditionally exist for pure diffusions, using Brownian motion as an auxiliary process and accepting sampled paths with a probability weight. Between jumps, the PDifMP evolves as a diffusion, allowing us to apply the exact method within each inter-jump interval. The main challenge arises when no threshold crossing is detected in an interval: We then need the value of the process at the jump time, and for that, we introduce an approach to simulate a conditionally constrained auxiliary process and derive the corresponding acceptance probability. Furthermore, we prove the convergence of the method and illustrate it using numerical examples.

math.PR↗

arXiv2025

Utility maximization in multivariate Volterra models

This paper is concerned with portfolio selection for an investor with power utility in multi-asset financial markets in a rough stochastic environment. We investigate Merton's portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber et al., SIAM J. Financial Math., 12, 369-409, (2021)]. Based on the classical Wishart model described in [N. Bäuerle and Li, Z., J. Appl. Probab., 50, 1025-1043 (2013)], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra-Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati-Volterra equation. We thus extend the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [B. Han and Wong, H. Y., Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.

math.PR↗

arXiv2025

Pricing of geometric Asian options in the Volterra-Heston model

Geometric Asian options are a type of options 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.

q-fin.PR↗

arXiv2024

Exact simulation of the first-passage time of SDEs to time-dependent thresholds

The first-passage time (FPT) is a fundamental concept in stochastic processes, representing the time it takes for a process to reach a specified threshold for the first time. Often, considering a time-dependent threshold is essential for accurately modeling stochastic processes, as it provides a more accurate and adaptable framework. In this paper, we extend an existing Exact simulation method developed for constant thresholds to handle time-dependent thresholds. Our proposed approach utilizes the FPT of Brownian motion and accepts it for the FPT of a given process with some probability, which is determined using Girsanov's transformation. This method eliminates the need to simulate entire paths over specific time intervals, avoids time-discretization errors, and directly simulates the first-passage time. We present results demonstrating the method's effectiveness, including the extension to time-dependent thresholds, an analysis of its time complexity, comparisons with existing methods through numerical examples, and its application to predicting spike times in a neuron.

math.PR↗

arXiv2024

Worst-Case Optimal Investment in Incomplete Markets

We study and solve the worst-case optimal portfolio problem as pioneered by Korn and Wilmott (2002) of an investor with logarithmic preferences facing the possibility of a market crash with stochastic market coefficients by enhancing the martingale approach developed by Seifried in 2010. With the help of backward stochastic differential equations (BSDEs), we are able to characterize the resulting indifference optimal strategies in a fairly general setting. We also deal with the question of existence of those indifference strategies for market models with an unbounded market price of risk. We therefore solve the corresponding BSDEs via solving their associated PDEs using a utility crash-exposure transformation. Our approach is subsequently demonstrated for Heston's stochastic volatility model, Bates' stochastic volatility model including jumps, and Kim-Omberg's model for a stochastic excess return.

q-fin.MF↗

arXiv2024

Equilibrium control theory for Kihlstrom-Mirman preferences in continuous time

In intertemporal settings, the multiattribute utility theory of Kihlstrom and Mirman suggests the application of a concave transform of the lifetime utility index. This construction, while allowing time and risk attitudes to be separated, leads to dynamically inconsistent preferences. We address this issue in a game-theoretic sense by formalizing an equilibrium control theory for continuous-time Markov processes. In these terms, we describe the equilibrium strategy and value function as the solution of an extended Hamilton-Jacobi-Bellman system of partial differential equations. We verify that (the solution of) this system is a sufficient condition for an equilibrium and examine some of its novel features. A consumption-investment problem for an agent with CRRA-CES utility showcases our approach.

q-fin.MF↗

arXiv2024

American option pricing using generalised stochastic hybrid systems

This paper presents a novel approach to pricing American options using piecewise diffusion Markov processes (PDifMPs), a type of generalised stochastic hybrid system that integrates continuous dynamics with discrete jump processes. Standard models often rely on constant drift and volatility assumptions, which limits their ability to accurately capture the complex and erratic nature of financial markets. By incorporating PDifMPs, our method accounts for sudden market fluctuations, providing a more realistic model of asset price dynamics. We benchmark our approach with the Longstaff-Schwartz algorithm, both in its original form and modified to include PDifMP asset price trajectories. Numerical simulations demonstrate that our PDifMP-based method not only provides a more accurate reflection of market behaviour but also offers practical advantages in terms of computational efficiency. The results suggest that PDifMPs can significantly improve the predictive accuracy of American options pricing by more closely aligning with the stochastic volatility and jumps observed in real financial markets.

q-fin.CP↗

arXiv2024

A Comparative Study of Factor Models for Different Periods of the Electricity Spot Price Market

Due to major shifts in European energy supply, a structural change can be observed in Austrian electricity spot price data starting from the second quarter of the year 2021 onward. In this work we study the performance of two different factor models for the electricity spot price in three different time periods. To this end, we consider three samples of EEX data for the Austrian base load electricity spot price, one from the pre-crises from 2018 to 2021, the second from the time of the crisis from 2021 to 2023 and the whole data from 2018 to 2023. For each of these samples, we investigate the fit of a classical 3-factor model with a Gaussian base signal and one positive and one negative jump signal and compare it with a 4-factor model to assess the effect of adding a second Gaussian base signal to the model. For the calibration of the models we develop a tailor-made Markov Chain Monte Carlo method based on Gibbs sampling. To evaluate the model adequacy, we provide simulations of the spot price as well as a posterior predictive check for the 3- and the 4-factor model. We find that the 4-factor model outperforms the 3-factor model in times of non-crises. In times of crises, the second Gaussian base signal does not lead to a better fit of the model. To the best of our knowledge, this is the first study regarding stochastic electricity spot price models in this new market environment. Hence, it serves as a solid base for future research.

q-fin.MF↗

arXiv2022

Supervised machine learning classification for short straddles on the S&P500

In this working paper we present our current progress in the training of machine learning models to execute short option strategies on the S&P500. As a first step, this paper is breaking this problem down to a supervised classification task to decide if a short straddle on the S&P500 should be executed or not on a daily basis. We describe our used framework and present an overview over our evaluation metrics on different classification models. In this preliminary work, using standard machine learning techniques and without hyperparameter search, we find no statistically significant outperformance to a simple "trade always" strategy, but gain additional insights on how we could proceed in further experiments.

q-fin.CP↗

arXiv2021

A mean-field extension of the LIBOR market model

We introduce a mean-field extension of the LIBOR market model (LMM) which preserves the basic features of the original model. Among others, these features are the martingale property, a directly implementable calibration and an economically reasonable parametrization of the classical LMM. At the same time, the mean-field LIBOR market model (MF-LMM) is designed to reduce the probability of exploding scenarios, arising in particular in the market-consistent valuation of long-term guarantees. To this end, we prove existence and uniqueness of the corresponding MF-LMM and investigate its practical aspects, including a Black '76-type formula. Moreover, we present an extensive numerical analysis of the MF-LMM. The corresponding Monte Carlo method is based on a suitable interacting particle system which approximates the underlying mean-field equation.

math.PR↗

arXiv2020

Change of drift in one-dimensional diffusions

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this we have to know that the change-of-measure local martingale that we write down is a true martingale; we provide a complete characterization of when this happens. This is then used to discuss absence of arbitrage in a generalized Heston model including the case where the Feller condition for the volatility process is violated.

q-fin.MF↗

arXiv2019

Change of Measure in the Heston Model given a violated Feller Condition

When dealing with Heston's stochastic volatility model, the change of measure from the subjective measure P to the objective measure Q is usually investigated under the assumption that the Feller condition is satisfied. This paper closes this gap in the literature by deriving sufficient conditions for the existence of an equivalent (local) martingale measure in the Heston model when the Feller condition is violated. We also supplement the existing literature by the case of a finite lifetime of the Laplace transform of the integrated volatility process. Moreover, we deduce conditions for the stock price process in the Heston model being a true martingale, regardless if the Feller condition is satisfied or not.

q-fin.MF↗

arXiv2019

Integral Representation of Generalized Grey Brownian Motion

In this paper we investigate the representation of a class of non Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular the underlying process can be seen as a non Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev as well as Harms and Stefanovits to the non Gaussian case.

math.PR↗

arXiv2019

Portfolio Optimization in Fractional and Rough Heston Models

We consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions. Using a suitable representation of the fractional part, followed by a reasonable approximation we show that it is possible to cast the problem into the classical stochastic control framework. This approach is generic for fractional processes. We derive explicit solutions and obtain as a by-product the Laplace transform of the integrated volatility. In order to get rid of some undesirable features we introduce a new model for the rough path scenario which is based on the Marchaud fractional derivative. We provide a numerical study to underline our results.

q-fin.PM↗