15th Winter school on Mathematical Finance


Dirk Becherer: Model uncertainty and market impact

Model uncertainty (ambiguity) and market impact are relevant for many applications of mathematical finance. Moving beyond classical models, to devise optimal trading strategies for suitably refined models also leads to interesting problems in the theory of stochastic optimal control, and to new questions in modelling.
In the first part of the course, we formulate a robust approach to hedging and valuation in incomplete markets. To this end, we introduce a no good deal approach to optimal (partial) hedging, which corresponds to the valuation approach of good deal bounds. Hedging strategies are minimizers of some a-priori coherent dynamic risk measure. Tracking errors of hedging strategies satisfy a uniform supermartingale property. The approach permits for concise description and analysis of hedging strategies by backward SDE theory, and for constructive or explicit solutions in several examples. We discuss extensions beyond Itô processes to models with jumps from Poisson random measures (e.g. from Lévy processes or semi-Markov chains). Since common notions for good deals (like Sharpe ratios, expected utilities or else) are defined with respect to one given probability, ambiguity about the probabilistic model becomes a relevant problem for good deal theory. We investigate robust extensions for good deal hedging and valuation under model uncertainty about the market prices of risk or about the volatilities of asset prices.
In the second part of the course, we present a multiplicative model of market impact, where portfolio strategies of a large trader have an impact on the evolution of asset prices. The price impact is multiplicative and transient over time. This corresponds to a (shadow) limit order book with intertemporal resilience, whose shape is specified in terms of relative (instead of absolute) price changes, ensuring that asset prices remain positive. We discuss solutions for variations of the optimal trade execution problem (e.g. in/finite horizon, one/two-sided order books, aspects of risk averse preferences), explore properties of intertemporal stability and absence of market manipulation strategies. Some comparison is provided with properties of related pioneering models for additive impact in the literature. We explore advanced further questions, like e.g. stochastic resilience of market impact.
This series of talks is based on joint projects with Todor Bilarev, Peter Frentrup and Klebert Kentia.

Dirk Becherer, Klebert Kentia: Hedging Under Generalized Good-Deal Bounds and Model Uncertainty.
Dirk Becherer, Todor Bilarev, Peter Frentrup: Multiplicative Limit Order Markets with Transient Impact and Zero Spread.
More references on Dirk Becherer's homepage.

Fred Espen Benth: Analysis of futures price models in commodity and energy markets

We formulate the stochastic dynamics of futures prices as the solution of a stochastic partial differential equation, inspired by the HJM-framework in fixed-income theory. Due to the high degree of idiosyncratic risk in some commodity markets like gas, weather and power, the noise driving the dynamics is conveniently modelled as an infinite-dimensional Lévy process. We define various Lévy processes of interest for power markets, and follow up with a detailed analysis of the futures price and its properties. In particular, we show that the futures price dynamics essentially is an infinite-factor Ornstein-Uhlenbeck process. Moreover, we may use this insight to approximate in an arbitrage-free manner the futures price dynamics.
We define power futures via integral operators and discuss pricing and hedging of some commonly traded energy options. Also, cross-commodity futures price models are introduced, leading to a discussion of covariance operators of "bivariate" Wiener processes in Hilbert space.
A certain class of stochastic volatility models is introduced for the futures price dynamics. We extend the Barndorff-Nielsen and Shephard stochastic volatility process to operator-valued processes, and analyse its properties for models of the futures price dynamics. We show a connection to ambit fields, which is a different class of models for energy futures prices.
The series of lectures are based on work in collaboration with Paul Krühner (Vienna), Barbara Rüdiger (Wuppertal) and André Süß (Zurich). (Slides of lecture I, lecture II, lecture III, lecture IV, lecture V)

Fred E. Benth, Paul Krühner: Subordination of Hilbert space valued Lévy processes, Fred Espen Benth, Paul Krühner: Derivatives pricing in energy markets: an infinite dimensional approach, Fred Espen Benth, Paul Krühner: Representation of infinite dimensional forward price models in commodity markets (Lectures I-III).
Fred Espen Benth, Barbara Ruediger, Andre Suess: Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility (Lecture IV)
Fred Espen Benth, Heidar Eyjolfsson: Representation and approximation of ambit fields in Hilbert space (Lecture V)
Ole E. Barndorff-Nielsen, Fred Espen Benth, Almut E. D. Veraart: Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency (Lecture V)

Special invited lectures

Nicole Bäuerle: Markov decision processes with applications to finance

Markov Decision Processes are controlled Markov processes in discrete time. They appear in various fields of applications like e.g. economics, finance, operations research, engineering and biology. The aim is to maximize the expected (discounted) reward of the process over a given time horizon. We consider problems with arbitrary (Borel) state and action space with a finite and an infinite time horizon. Solution methods and the Bellman equation are discussed as well as the existence of optimal policies. For problems with infinite horizon we give convergence conditions and present solution algorithms like Howard's policy improvement or linear programming. The statements and results are illustrated by examples from finance and insurance like consumption-investment problems and dividend pay-out problems. (slides)

N. Bäuerle, U. Rieder: Markov Decision Processes with Applications to Finance. Springer, Universitext, 2011.
N. Bäuerle, U. Rieder: Markov Decision Processes. Jahresbericht der DMV 112(4), 217-243, 2010.

Sara Biagini: Robust portfolio selection

We derive a closed form portfolio optimization rule for an investor who is diffident about mean return and volatility estimates, and has a CRRA utility. The novelty is that confidence is here represented using ellipsoidal uncertainty sets for the drift, given a volatility realization. This specification affords a simple and concise analysis, as the optimal portfolio allocation policy is shaped by a rescaled market Sharpe ratio, computed under the worst case volatility. The result is based on a max-min Hamilton-Jacobi-Bellman-Isaacs PDE, which extends the classical Merton problem and reverts to it for an ambiguity-neutral investor. (slides of the lecture)

Sara Biagini and Mustafa Pınar: The Robust Merton Problem of an Ambiguity Averse Investor.

Thorsten Schmidt: Dynamic term structure theory

We visit dynamic term structure theory from a theoretical viewpoint allowing for infinitely many traded assets. Taking this seriously, a number of interesting mathematical questions arise: when is this market free of arbitrage and in what sense ? This leads to surprisingly nice conditions which point towards a class of term-structure models having cadlag properties in the additional parameter, maturity. Taking this as starting point we develop a generalized forward rate approach where the introduced codebook is designed in such a flexible way, that this class is nicely parametrized. For practical applications we study a new class of affine models, also sharing the cadlag property and hence allowing for stochastic discontinuities. This has a number of surprising applications where we study the credit risk case as prime example.

Short lectures

Hailong Bao: Multi-period risk sharing under financial fairness

We work with a multi-period system where a finite number of agents need to share multiple monetary risks. We look for the solutions that are both Pareto efficient utility-wise and financially fair value-wise. A buffer enables the inter-temporal capital transfer. A risk-neutral measure is essential for determining the risk sharing rules. It can be shown that in the model setting there always exists a unique risk sharing rule that is both Pareto efficient and financially fair. An iterative algorithm is introduced to calculate this rule numerically. (slides)

H. Bao, E.H.M. Ponds, J.M. Schumacher: Multi-Period Risk Sharing under Financial Fairness.

Tim Boonen: Pareto optima and competitive equilibria in markets with expected and dual utility

This paper analyzes optimal risk sharing between agents that are endowed with either expected utility preferences or with dual utility preferences. We find that Pareto optimal risk redistributions and the competitive equilibria are obtained via bargaining with a hypothetical representative agent of expected utility maximizers and a hypothetical representative agent of dual utility maximizers. The representative agent of expected utility maximizers resembles an average risk-averse agent, whereas representative agent of dual utility maximizers resembles a least risk-averse agent. This leads to an allocation of the aggregate risk to both groups of agents. The optimal contract for the expected utility maximizers is proportional to their allocation, and the optimal contract for the dual utility maximizing agents is given by ''tranching'' of their allocation. We show a method to derive the equilibrium prices. We identify a condition under which prices are locally independent of the expected utility functions, and given in closed form. Moreover, we characterize uniqueness of the competitive equilibrium via one condition on the dual utility preferences.

Fei Cong: Multi-period mean-variance portfolio optimization based on Monte-Carlo simulation

We propose a simulation-based approach for solving the constrained dynamic mean-variance portfolio management problem. For this dynamic optimization problem, we first consider a sub-optimal strategy, called the multi-stage strategy, which can be utilized in a forward fashion. Then, based on this fast yet sub-optimal strategy, we pose two approaches to improve the solution. The first one depends on breaking the ''self-financing'' rule and allows withdrawing money from the portfolio in some cases. The other one is to involve backward recursive programming. We design the backward recursion algorithm specially such that the result is guaranteed to converge to a solution, which is at least not worse than the one generated by the multi-stage strategy. In our numeric tests, highly satisfactory asset allocations can be achieved for the dynamic portfolio management problem in case various constraints are cast on the control variables.

F. Cong and C.W. Oosterlee: Multi-period Mean-Variance Portfolio Optimization based on Monte-Carlo Simulation. To appear in Journal of Economics Dynamics and Control.

Asma Khedher: Model risk and robustness of quadratic hedging strategies

In incomplete markets, there is no self- financing hedging strategy which allows to attain the contingent claim at maturity. In other words, one cannot eliminate the risk completely. However it is possible to find 'partial' hedging strategies which minimize some risk. One way to determine these 'partial' hedging strategies is to introduce a subjective criterion according to which strategies are optimized. We consider two types of quadratic hedging strategies. In the first approach, called mean-variance hedging (MVH), the strategy is self-financing and one minimizes the quadratic hedging error at maturity in mean square sense. The second approach is called local risk-minimization (LRM). These strategies replicate the option's payoff, but they are not self-financing and the risk is minimized in a 'local' sense. We study the relation of such strategies with the theory of backward stochastic differential equations and we apply this to the approximation and simulation of MVH and LRM strategies. (this mini lecture is based on a work in collaboration with Michèle Vanmaele)

Asma Khedher and Michèle Vanmaele: Discretisation of FBSDEs driven by càdlàg martingales, Journal of Mathematical Analysis and Applications 435, 508-531.
Giulia Di Nunno, Asma Khedher and Michèle Vanmaele: Robustness of Quadratic Hedging Strategies in Finance via Backward Stochastic Differential Equations with Jumps, Applied Mathematics & Optimization 72, 353-389.

Poster presentations

  • Alessandro Balata (University of Leeds): Battery storage arbitrage
  • Anna Sulima (Jagiellonian University): Completeness and arbitrage-free Markov regime-switching Lévy Black-Scholes-Merton market
  • Dávid Szábo (University of Manchester): American put options for power system balancing

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