Prof. Dr. Dirk Becherer
Profil
Forschungsthemen6
AvH Becherer
Quelle ↗Förderer: Alexander von Humboldt-Stiftung Zeitraum: 10/2012 - 12/2014 Projektleitung: Prof. Dr. Dirk Becherer
Berlin-AIMS-Netzwerk auf dem Gebiet der Stochastischen Analysis
Quelle ↗Förderer: Forschungsverbund Berlin e.V. Zeitraum: 07/2018 - 12/2022 Projektleitung: Prof. Dr. Dirk Becherer
DFG-Forschungszentrum: "Mathematik für Schlüsseltechnologien - MATHEON"
Quelle ↗Förderer: DFG sonstige Programme Zeitraum: 01/2008 - 05/2014 Projektleitung: Prof. Dr. Dirk Becherer
IGRK 2544/1: Stochastische Analysis in Interaktion
Quelle ↗Förderer: DFG Graduiertenkolleg Zeitraum: 04/2020 - 09/2024 Projektleitung: Prof. Dr. Ulrich Horst, Prof. Dr. Markus Reiß, Prof. Dr. Dirk Becherer, Prof. Dr. Dörte Kreher
IGRK 2544: Stochastische Analysis in Interaktion
Quelle ↗Förderer: DFG Graduiertenkolleg Zeitraum: 04/2020 - 03/2029 Projektleitung: Prof. Dr. Peter Bank, Terry Lyons Ph.D.
Multidimensional Portfolio Optimization"
Quelle ↗Förderer: DFG sonstige Programme Zeitraum: 01/2012 - 05/2014 Projektleitung: Prof. Dr. Dirk Becherer
Mögliche Industrie-Partner10
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Publikationen25
Top 25 nach Zitationen — Quelle: OpenAlex (BAAI/bge-m3 embedded für Matching).
Finance and Stochastics · 146 Zitationen · DOI
Insurance Mathematics and Economics · 116 Zitationen · DOI
Finance and Stochastics · 110 Zitationen · DOI
DepositOnce · 50 Zitationen · DOI
Die vorliegende Arbeit behandelt stochastische Optimierungsprobleme, in denen ein konkaves Funktional über einem Raum von stochastischen Integralen maximiert wird. In der Finanzmathematik treten derartige Probleme bei der Behandlung von Bewertungs-, Absicherungs-, und Anlageproblemen in unvollständigen Finanzmärkten auf. Wir beschäftigen uns vornehmlich mit nutzenbasierten Methoden zur Bewertung und Absicherung von zufallsbehafteten Finanzpositionen, welche unvermeidbare intrinsische Risiken beinhalten. Wir betrachten das Problem aus der Perspektive eines rationalen Investors, dessen Ziel die Maximierung seines erwarteten exponentiellen Nutzens ist. Ausgehend von diesen Präferenzen, definieren wir mittels Nutzenindifferenz-Argumenten seinen Bewertungsprozess und eine Absicherungsstrategie. In einem Semimartingalmodell kann die Lösung durch ein stochastisches Darstellungsproblem charakterisiert werden. Um das Problem zu lösen, gilt es ein Martingalmaß zu finden, dessen Dichteprozess eine bestimmte Form hat. Im Weiteren untersuchen wir zwei Modelle, welche gewissen strukturellen Bedingungen genügen. In einem halbvollständigen Produktmodell wird gezeigt, dass die Nutzenindifferenz-Bewertungs- und Absicherungsmethode additiv ist, wenn sie auf ein Aggregat von ''genügend unabhängigen'' Positionen angewandt wird. Wir untersuchen Diversifikationseffekte und leiten ein Berechnungsschema her. Für das zweite Modell betrachten wir ein Markovsches System stochastischer Differentialgleichungen, welches einen Ito-Prozess und einen weiteren Prozess mit endlichem Zustandsraum beschreibt und verschiedene wechselseitige Abhängigkeiten zulässt. In unserem Marktmodell stellt der Ito-Prozess die Preise der riskanten Anlagen dar während der zweite Prozess irgendwelche nicht handelbaren Risikofaktoren repräsentiert. Die Lösung des Bewertungs- und Absicherungsproblems wird durch ein wechselwirkenendes System semilinearer partieller Differentialgleichungen, eine so genannte Reaktions-Diffusions Gleichung, beschrieben. Mittels Feynman-Kac Resultaten und der Iterationstechnik von Picard zeigen wir Existenz und Eindeutigkeit einer klassischen Lösung. Ergänzend zu unserem Hauptthema nutzen wir ein ähnliches Indifferenzargument um den Wert von zusätzlichen Anlage-Informationen zu quantifizieren. Die wesentlichen Mittel sind eine Martingal erhaltende Maß transformation und Martingal-Darstellungsresultate für anfangsvergröß erte Filtrationen. Schließlich zeigen wir, dass das so genannte Numeraire-Portfolio mit einer weiteren nutzenbasierten Bewertungsmethode zusammenhängt, welche sich auf ein Grenznutzenargument stützt und als Grenzfall der Indifferenz-Methode angesehen werden kann.
Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 43 Zitationen · DOI
This article studies the exponential utility–indifference approach to the valuation and hedging problem in incomplete markets. We consider a financial model, which is driven by a system of interacting Itô and point processes. The model allows for a variety of mutual stochastic dependencies between the tradable and non–tradable factors of risk, but still permits a constructive and fairly explicit solution. In analogy to the Black–Scholes model, the utility–based price and the hedging strategy can be described by a partial differential equation (PDE). But the non–tradable factors of risk in our model demand an interacting semi–linear system of parabolic PDEs. To obtain the solution for the underlying utility–maximization problem, we use a verification theorem to identify the optimal martingale measure for the corresponding dual problem.
Finance and Stochastics · 41 Zitationen · DOI
16 Zitationen · DOI
Good-deal bounds have been introduced as a way to obtain valuation bounds for deriva- tive assets which are tighter than the arbitrage bounds. This is achieved by ruling out not only those prices that violate no-arbitrage restrictions but also trading opportunities that are 'too good'. We study dynamic good-deal valuation bounds that are derived from bounds on optimal expected growth rates. This leads naturally to restrictions on the set of pricing measure which are local in time, thereby inducing good dynamic properties for the good-deal valuation bounds. We study good-deal bounds by duality arguments in a general semimartingale setting. In a Wiener space setting where asset prices evolve as Itˆ o-processes, good-deal bounds are then conveniently described by back- ward SDEs. We show how the good-deal bounds arise as the value function for an optimal control problem, where a dynamic coherent a priori risk measure is minimized by the choice of a suitable hedging strategy. This demonstrates how the theory of no-good-deal valuations can be associated to an established concept of dynamic hedging in continuous time.
Applied Mathematics & Optimization · 15 Zitationen · DOI
SIAM Journal on Financial Mathematics · 11 Zitationen · DOI
This paper presents an application of the recently developed method for simultaneous dimension reduction and metastability analysis of high-dimensional time series in the context of computational finance. Further extensions are included to combine state-specific principal component analysis (PCA) and state-specific regressive trend models to handle the high-dimensional, nonstationary data. The identification of market phases allows one to control the involved phase-specific risk for futures portfolios. The numerical optimization strategy for futures portfolios based on Tikhonov-type regularization is presented. The application of proposed strategies to online detection of the market phases is exemplified first on the simulated data and then on historical futures prices for oil and wheat from 2005–2008. Numerical tests demonstrate the comparison of the presented methods with existing approaches.
Probability Uncertainty and Quantitative Risk · 8 Zitationen · DOI
We study robust notions of good-deal hedging and valuation under combined uncertainty about the drifts and volatilities of asset prices. Good-deal bounds are determined by a subset of risk-neutral pricing measures such that not only opportunities for arbitrage are excluded but also deals that are too good, by restricting instantaneous Sharpe ratios. A non-dominated multiple priors approach to model uncertainty (ambiguity) leads to worst-case good-deal bounds. Corresponding hedging strategies arise as minimizers of a suitable coherent risk measure. Good-deal bounds and hedges for measurable claims are characterized by solutions to second-order backward stochastic differential equations whose generators are non-convex in the volatility. These hedging strategies are robust with respect to uncertainty in the sense that their tracking errors satisfy a supermartingale property under all a-priori valuation measures, uniformly over all priors.
Encyclopedia of Quantitative Finance · 7 Zitationen · DOI
Abstract Under market frictions like illiquidity or transaction costs, contingent claims can incorporate some inevitable intrinsic risk that cannot be completely hedged away but remains with the holder. In general, they cannot be synthesized by dynamical trading in liquid assets and hence cannot be priced by no‐arbitrage arguments alone. Still, an agent (she) can determine a valuation with respect to her preferences toward risk. The utility indifference value for a variation in the quantity of illiquid assets held by the agent is defined as the compensating variation of wealth, under which her maximal expected utility remains unchanged.
Encyclopedia of Quantitative Finance · 6 Zitationen · DOI
Abstract Arrow–Debreu prices are the prices of atomic time and state‐contingent claims, which deliver one unit of a specific consumption good if a specific state realizes at a specific future date. Such claims were introduced by Arrow and Debreu in their work on general equilibrium theory under uncertainty to allow agents to exchange state and time contingent claims, reducing the equilibrium problem with uncertainty to a conventional one without uncertainty. In finite‐state models, Arrow–Debreu securities delivering one unit of the numeraire good can be viewed as atomic building blocks for all other state–time contingent financial claims. Their prices determine a unique arbitrage‐free price system.
arXiv (Cornell University) · 5 Zitationen
We study a multiplicative limit order book model for an illiquid market, where price impact by large orders is multiplicative in relation to the current price, transient over time, and non-linear in volume (market) impact. Order book shapes are specified by general density functions with respect to relative price perturbations. Market impact is mean reverting with possibly non-linear resilience. We derive optimal execution strategies that maximize expected discounted proceeds for a large trader over an infinite horizon in one- and also in two-sided order book models, where buying as well as selling is admitted at zero bid-ask spread. Such markets are shown to be free of arbitrage. Market impact as well as liquidation proceeds are stable under continuous Wong-Zakai-type approximations of strategies.
On the Monotone Stability Approach to BSDEs with Jumps: Extensions, Concrete Criteria and Examples
2019Springer proceedings in mathematics & statistics · 4 Zitationen · DOI
Finance and Stochastics · 2 Zitationen · DOI
Abstract We solve the superhedging problem for European options in an illiquid extension of the Black–Scholes model, in which transactions have transient price impact and the costs and strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring nonnegativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black–Scholes model with a relative price impact proportional to the volume of shares traded, where the transience for impact on log-prices is modelled like in Obizhaeva and Wang (J. Financ. Mark. 16:1–32, 2013) for nominal prices. More generally, we allow nonlinear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. The pricing equations under illiquidity extend no-arbitrage pricing à la Black–Scholes for complete markets in a non-paradoxical way (cf. Çetin et al. (Finance Stoch. 14:317–341, 2010)) even without additional frictions, and can recover it in base cases.
On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples
2016RePEc: Research Papers in Economics · 2 Zitationen · DOI
We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.
arXiv (Cornell University) · 2 Zitationen · DOI
We develop a multilevel approach to compute approximate solutions to backward differential equations (BSDEs). The fully implementable algorithm of our multilevel scheme constructs sequential martingale control variates along a sequence of refining time-grids to reduce statistical approximation errors in an adaptive and generic way. We provide an error analysis with explicit and non-asymptotic error estimates for the multilevel scheme under general conditions on the forward process and the BSDE data. It is shown that the multilevel approach can reduce the computational complexity to achieve precision $ε$, ensured by error estimates, essentially by one order (in $ε^{-1}$) in comparison to established methods, which is substantial. Computational examples support the validity of the theoretical analysis, demonstrating efficiency improvements in practice.
Stochastics · 2 Zitationen · DOI
Mark Davis has been associated with numerous major advances, and has been influential to several generations of researchers working in the fields of stochastic analysis, stochastic control and fina...
Econstor (Econstor) · 2 Zitationen · DOI
We consider an investor maximizing his expected utility from terminal wealth with portfolio decisions based on the available information flow. This investor faces the opportunity to acquire some additional initial information G.. The subjective fair value of this information for the investor is defined as the amount of money that he can pay for G such that this cost is balanced out by the informational advantage in terms of maximal expected utility. We calculate this value for common utility functions in the setting of a complete market modeled by general semimartingales. The main tools are results of independent interest, namely a martingale preserving change of measure and a martingale representation theorem for initially enlarged filtrations.
SSRN Electronic Journal · 1 Zitationen · DOI
arXiv (Cornell University) · 1 Zitationen · DOI
We solve the superhedging problem for European options in an illiquid extension of the Black-Scholes model, in which transactions have transient price impact and the costs and the strategies for hedging are affected by physical or cash settlement requirements at maturity. Our analysis is based on a convenient choice of reduced effective coordinates of magnitudes at liquidation for geometric dynamic programming. The price impact is transient over time and multiplicative, ensuring non-negativity of underlying asset prices while maintaining an arbitrage-free model. The basic (log-)linear example is a Black-Scholes model with relative price impact being proportional to the volume of shares traded, where the transience for impact on log-prices is being modelled like in Obizhaeva-Wang \cite{ObizhaevaWang13} for nominal prices. More generally, we allow for non-linear price impact and resilience functions. The viscosity solutions describing the minimal superhedging price are governed by the transient character of the price impact and by the physical or cash settlement specifications. Pricing equations under illiquidity extend no-arbitrage pricing a la Black-Scholes for complete markets in a non-paradoxical way (cf.\ {Ç}etin, Soner and Touzi \cite{CetinSonerTouzi10}) even without additional frictions, and can recover it in base cases.
RePEc: Research Papers in Economics · 1 Zitationen · DOI
We prove continuity of a controlled SDE solution in Skorokhod's $M_1$ and $J_1$ topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that $M_1$-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.
arXiv (Cornell University) · 1 Zitationen
We solve explicitly a two-dimensional singular control problem of finite fuel type in infinite time horizon. The problem stems from the optimal liquidation of an asset position in a financial market with multiplicative price impact with stochastic resilience. The optimal control is obtained as a diffusion process reflected at a non-constant free boundary. To solve the variational inequality and prove optimality, we show new results of independent interest on constructive approximations and Laplace transforms of the inverse local times for diffusions reflected at elastic boundaries.
Applied Mathematical Finance · 1 Zitationen · DOI
Abstract We develop a generic method for constructing a weak static minimum variance hedge for a wide range of derivatives that may involve optimal exercise features or contingent cash flow streams to provide a hedge along a sequence of future hedging dates. The optimal hedge is constructed using a portfolio of pre-selected hedge instruments, which could be derivatives with different maturities. The hedge portfolio is weakly static in that it is initiated at time zero, does not involve intermediate re-balancing, but hedges may be gradually unwound over time. We study the static hedging of a convertible bond to demonstrate the method by an example that involves equity and credit risk. We investigate the robustness of the hedge performance with respect to parameter and model risk by numerical experiments.
ArXiv.org · DOI
We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path $W$ of finite $q$-variation for $q \in [1, 2)$ and by Brownian motion $B$. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale $B$ and an independent discontinuous stochastic process $L$ of finite $q$-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of $L$.
Kooperationen5
Bestätigte Forscher↔Partner-Paare aus HU-FIS — Gold-Standard-Positive für das Matching.
Berlin-AIMS-Netzwerk auf dem Gebiet der Stochastischen Analysis
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IGRK 2544/1: Stochastische Analysis in Interaktion
university
IGRK 2544: Stochastische Analysis in Interaktion
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IGRK 2544: Stochastische Analysis in Interaktion
university
IGRK 2544/1: Stochastische Analysis in Interaktion
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Stammdaten
Identität, Organisation und Kontakt aus HU-FIS.
- Name
- Prof. Dr. Dirk Becherer
- Titel
- Prof. Dr.
- Fakultät
- Mathematisch-Naturwissenschaftliche Fakultät
- Institut
- Institut für Mathematik
- Arbeitsgruppe
- Stochastische Analysis und Stochastik der Finanzmärkte
- Telefon
- +49 30 2093-45461
- HU-FIS-Profil
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- Zuletzt gescrapt
- 26.4.2026, 01:02:26