Talks in Financial and Insurance Mathematics

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This is the regular weekly research seminar on Insurance Mathematics and Stochastic Finance.

Autumn Semester 2015

Note: The highlighted event marks the next occurring event and events marked with an asterisk (*) indicate that the time and/or location are different from the usual time and/or location.

Date / Time Speaker Title Location
* 22 September 2015
Peter Bank
TU Berlin
Optimal investment with price impact  HG G 19.1 
Abstract: We consider a financial model with price impact where an large investor’s orders affect bid and ask prices. In a Brownian setting with exponential utility, this model allows for an explicit description of optimal investment strategies. In order to learn about pricing and hedging risks in such a frictional model, we consider a quadratic benchmark problem which emerges heuristically as the high-resilience limit of the original one. The benchmark problem also allows for a closed-form solution. It turns out that, rather than trading towards the currently optimal position from a frictionless reference model, it is optimal to trade towards a weighted average of this positions future expected values. This is joint work in progress with Mete Soner and Moritz Voss.
24 September 2015
Prof. Dr. Irene Klein
Universität Wien
New perspectives on the fundamental theorem of asset pricing for large financial markets  HG G 43 
Abstract: The talk is based on joint work with Christa Cuchiero and Josef Teichmann. In the context of large financial markets in continuous time we formulate the notion of no asymptotic free lunch with vanishing risk (NAFLVR), under which we prove a version of the fundamental theorem of asset pricing (FTAP) in markets with an (uncountably) infinite number of assets (as it is the case, for example, in bond markets). It turns out that the general setting of admissible portfolio wealth processes introduced by Y. Kabanov [3] under a substantially relaxed concatenation property and the proof variant of C. Cuchiero and J. Teichmann [1] can be adapted to the given setting. In the case of countably many assets, our setting includes the large financial market model considered by M. De Donno et al. [2]. NAFLVR turns out to be an economically meaningful ``no arbitrage'' condition (in particular not involving weak-star-closures), which is equivalent to the existence of an equivalent separating measure. Furthermore there is a counterexample showing that the existence of an equivalent separating measure does not lead to an equivalent sigma-martingale measure, even in a countable large financial market situation. In the context of large financial markets in discrete time we present a version of the fundamental theorem of asset pricing under restricted and delayed information which combines results of Miklos Rasonyi [5] and C. Stricker & Y. Kabanov [4]. References: [1] C. Cuchiero and J. Teichmann. A convergence result in the Emery topology and a variant of the proof of the fundamental theorem of asset pricing, to appear in Finance and Stochastics (2015) [2] M. De Donno, P. Guasoni and M. Pratelli. Super-replication and utility maximization in large financial markets. Stochastic Process. Appl., 115 (12), 2006-2022, 2005 [3] Y. Kabanov . On the FTAP of Kreps-Delbaen-Schachermayer. In Statistics and control of stochastic processes (1995/1996), 191-203, World Sci. Publ., 1997 [4] Y. Kabanov and C. Stricker. The Dalang-Morton-Willinger Theorem under delayed and restricted information. In Memoriam Paul-André Meyer - Séminaire de Probabilités XXXIX (2006). [5] Miklos Rasonyi (2003). Equivalent martingale measures for large financial markets in discrete time. Math. Meth. Oper. Res., 58, 401-415, 2003
* 29 September 2015
Nevroz Sen
McGill University, Montreal
Partially Observed Mean Field Games with a Major Player  HG G 19.1 
Abstract: In the Mean Field Games (MFG) framework where there is an agent (so-called Major) which has asymptotically non-vanishing influence on any other Minor agent, the best response control process of each Minor agent depends upon its own state, the Major agent's state and the conditional distribution of the generic minor agent, namely the system's stochastic mean field; this is in contrast to the basic MFG setup where the mean field is deterministic. The theory of MFG with a Major agent (MM-MFG) is well understood when the observations of the Minor agents are complete. In this talk we analyze the non-linear MM-MFG problem where each Minor agent partially observes the Major agent's state. We employ non-linear filtering theory derived for McKean-Vlasov type state equations and the Separation Principle in order to analyze the game in the infinite population limit. The main results are the existence and uniqueness of the solutions to the stochastic MFG system equations and the epsilon-Nash equilibrium property where the best response control process of each Minor agent depends upon the conditional density generated by that agent's non-linear filter together with the system's mean field and its own state. Work with Peter E. Caines.
1 October 2015
Peter Friz
TU Berlin
Towards Malliavin calculus on regularity structures  HG G 43 
Abstract: I will discuss some basic aspects of Malliavin's stochastic calculus of variations in the context of Hairer's regularity structures. (Joint with P. Gassiat and G. Cannizzaro.)
8 October 2015
Martin Keller-Ressel
TU Dresden
Implied Volatilities from Strict Local Martingales  HG G 43 
Abstract: Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors’ exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the “distance to a true martingale” can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.
15 October 2015
Jordan Stoyanov
Newcastle University (UK) and University of Ljubljana, Slovenia
Moment Properties of Distributions Used in Stochastic Financial Models  HG G 43 
Abstract: We deal with any kind of probability distributions assuming only that their positive integer order moments are finite. Either such a distribution is M-determinate (uniquely determined by its moments), or it is M-indeterminate (non-unique). It is known that the light tailed distributions are M-determinate which may not be true for heavy tailed distributions. And here is the point: Most of the distributions used in stochastic financial models are heavy tailed. Hence it is important to know, in general and in particular, which distributions are M-determinate and which are not. We also discuss how M-determinacy of a distribution is related to another fundamental property, the infinite divisibility. We start with a brief and clear picture of well-known classical conditions (Cramer, Carleman, Hausdorff, Krein, ...). However the emphasis will be on new developments and results obtained over the last years. Essential details will be given on the following specific topics: (a) New Hardy condition for M-determinacy. (b) Criteria based on the rate of growth of the moments. (c) Stieltjes classes for M-indeterminate distributions. Index of dissimilarity. (d) Multidimensional moment problem. (e) Nonlinear transformations of random data and their M-(in)determinacy. (f) M-determinacy of distributions of stochastic processes defined by SDEs. There will be new and well-referenced results, hints for their proof, and/or new proofs of known results. Illustrations by examples and counterexamples will involve distributions widely used in stochastic modelling. Perhaps some facts are not so well-known and even look surprising. Challenging open questions and conjectures will be outlined.
29 October 2015
Torsten Schöneborn
TU Berlin
Optimal Trade Execution In Order Books With Time-Varying Liquidity  HG G 43 
Abstract: - Introduction of a limit order book model that captures seasonal patterns of market liquidity (depth and resilience of the book) - Derivation of optimal trading strategies that follow an intuitive wait region / trade region pattern - If a zero market spread is assumed irrespective of trading activity, then profitable roundtrip trading strategies can occur in the model. These opportunities disappear if the market spread is assumed to be influenced by trading activity
5 November 2015
Frank Riedel
Universität Bielefeld
Incompleteness of Financial Markets under Knightian Uncertainty — the Non-Implementability of Arrow-Debreu Equilibria under Volatility Uncertainty  HG G 43 
Abstract: In diffusion models, few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow-Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow-Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow-Debreu equilibrium allocations are generically unattainable by dynamically trading few long-lived assets.
* 10 November 2015
Johannes Ruf
University College London, Oxford Man Institute
Some remarks on functionally generated portfolios  HG G 19.1 
Abstract: In the first part of the talk I will review Bob Fernholz' theory of functionally generated portfolios. In the second part I will discuss questions related to the existence of short-term arbitrage opportunities. This is joint work with Ioannis Karatzas
* 12 November 2015
Philipp Doersek
BNP Paribas
Complexity and the approximation of stochastic differential equations  HG G 3 
Abstract: We consider the approximation of the solution of stochastic differential equations by numerical schemes. There are fundamental results (Creutzig, Dereich, Müller-Gronbach, and Ritter 2009, Giles 2008 and Giles and Szpruch 2014) giving precise bounds on the best possible rate of convergence in terms of numerical cost, i.e., algorithmic complexity, under very general circumstances. However, in many practical situations, it is possible to achieve better complexity because additional assumptions can be made. We discuss both classical and novel methods that overcome the order barrier.
12 November 2015
Ryan Donnelly
EPF Lausanne
Enhancing Trading Strategies with Order Book Signals  HG G 43 
Abstract: We use high-frequency data from the Nasdaq exchange to build a measure of volume imbalance in the limit order book (LOB). We show that our measure is a good predictor of the sign of the next market order (MO), i.e. buy or sell, and also helps to predict price changes immediately after the arrival of an MO. Based on these empirical findings, we introduce and calibrate a Markov chain modulated pure jump model of price, spread, LO and MO arrivals, and volume imbalance. As an application of the model, we pose and solve a stochastic control problem for an agent who maximizes terminal wealth, subject to inventory penalties, by executing trades using LOs. We use in-sample-data (January to June 2014) to calibrate the model to ten equities traded in the Nasdaq exchange, and use out-of-sample data (July to December 2014) to test the performance of the strategy. We show that introducing our volume imbalance measure into the optimization problem considerably boosts the profits of the strategy. Profits increase because employing our imbalance measure reduces adverse selection costs and positions LOs in the book to take advantage of favorable price movements.
19 November 2015
Miriam Isabel Seifert
Helmut Schmidt Universität Hamburg
Weakening the independence assumption on polar components: Limit theorems for generalized elliptical distributions.  HG G 43 
Abstract: Elliptical distributions and their asymptotics have been intensively investigated. In the bivariate case, they possess a natural extension to random vectors with a polar representation (X,Y)=R(u(T),v(T)) where u and v are quite arbitrary functions. Thereby it is commonly assumed that the radial component R and the angular component T are stochastically independent. In my talk, I will present to which extent this rigid independence assumption could be relaxed, such that conditional limit theorems still can be deduced. For this purpose, I propose a novel measure for the dependence structure and present convenient criteria for validity of limit theorems. The criteria and results are interpreted geometrically in terms of the level curves of the joint density for (X,Y) and elucidated by several figures. Thus, a stability of the available limit results is shown for a certain degree of dependence.
26 November 2015
David Hobson
University of Warwick
Prospect Theory in a Dynamic Context  HG G 43 
Abstract: Prospect theory introduces several innovations when compared with expected utility. Outcomes are measured relative to a reference point; agents are risk-seeking on losses; probabilities associated with gains and losses are re-weighted. Applying prospect theory (PT) in a dynamic context brings new challenges. We study PT agents facing optimal timing decisions and consider the impact of allowing them to follow randomized strategies. In the discrete model of casino gambling of Barberis (2012) we show that allowing randomization leads to gains in PT value. In the continuous analog (Ebert and Strack (2015)) we show that allowing randomization can significantly alter the predictions of the model. Ebert and Strack show that a naive investor never stops. We show that allowing naive PT agents to use randomized strategies leads to predictions which are closer to reality and include voluntary cessation of gambling.
3 December 2015
David Dickson
Melbourne University
Analysis of a risk model with capital injections  HG G 43 
Abstract: We consider a risk model under which the insurer purchases a reinsurance contract that provides capital injections. These capital injections restore the surplus process to a fixed level, k, every time the surplus falls between 0 and k. In the first part of the talk we briefly discuss the effect of capital injections on the insurer's ultimate ruin probability. In the second part, we discuss calculation of the finite time ruin probability under this model, and show how to find the density of the time of ruin when the claim size distribution is exponential. We then construct a Gerber-Shiu function and find an interesting formula for the finite time survival probability.
* 10 December 2015
Frank Seifried
Universität Trier
Backward Nonlinear Expectation Equations and Continuous-Time Recursive Utility  HG G 19.2 

This talk presents two recent papers on backward stochastic differential equations and the foundations of continuous-time recursive utility.

In the first part, we establish existence, uniqueness, monotonicity, concavity, and a utility gradient inequality for continuous-time recursive utility in the Epstein-Zin parametrization with relative risk aversion and elasticity of intertemporal substitution greater than unity. This is done in a fully general semimartingale setting, extending existing results for Brownian filtrations as in Schroder and Skiadas (1999) and Xing (2015).

In the second part, building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations (BNEEs). Such equations can be thought of as backward stochastic differential equations under nonlinear expectations. We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. In the context of recursive preferences, BNEEs emerge naturally when ambiguity is taken into account; we apply our results to show that discrete-time recursive utility with ambiguity converges to the nonlinear stochastic differential utility introduced formally by Chen and Epstein (2002) and Epstein and Ji (2014).

10 December 2015
Johannes Muhle-Karbe
ETH Zürich
Equilibrium Models with Small Frictions  HG G 43 

How would the introduction of a small trading friction such as a transaction tax affect financial markets? To answer questions of this kind, one needs to consider equilibrium models, where prices are determined endogenously. Indeed, taxes change agents' individual decision making, which in turn affects the market prices determined by their interactions. The new market environment then again alters the agents' behavior, leading to a notoriously intractable fixed point problem.

In this talk we report on recent progress using asymptotic techniques for small trading frictions. In this practically relevant limiting regime, explicit solutions become available for many of the arising singular control problems, bringing analytical results for the equilibrium problem within reach.

(The talk is based on joint works with Martin Herdegen and Jan Kallsen)

17 December 2015
Vladimir Ulyanov
Moscow State University
Cornish-Fisher expansions and their generalizations for quantiles  HG G 43 

In statistical inference it is of fundamental importance to obtain the sampling distribution of statistics. However, we often encounter situations where the exact distribution cannot be obtained in closed form, or even if it is obtained, it might be of little use because of its complexity. One practical way of getting around the problem is to provide reasonable approximations of the distribution function and its quantiles, along with extra information on their possible errors. It can be made with help of Edgeworth and Cornish–Fisher expansions (EE and CFE resp.) (see, e.g., Ulyanov V.V., Cornish-Fisher Expansions, International Encyclopedia of Statistical Science (Ed. M.Lovric), Springer-Verlag, Berlin–Heidelberg, 2011, p.312–315). Starting from 90-s the interest for CFE stirred up because of intensive study of risk measures (Value at Risk, Expected Shortfall etc.)

In the talk we discuss new approaches to CFE appeared in last years, in particular generalized CFE and rearrangements to monotonize CFE. Generalized CFE (non-normal limit distributions) appear naturally for statistics based on samples of random size. We consider as well the computable error bounds for CFE in the case when there are error bounds for EE of the distributions of statistics. The different applications for risk measurement and construction of the confidence intervals will be also discussed.

Organizers: Paul Embrechts , Martin Schweizer , Halil Mete Soner , Josef Teichmann , Mario Valentin Wüthrich 

Archive: SS 17  AS 16  SS 16  AS 15  SS 15  AS 14  SS 14  AS 13  SS 13  AS 12  SS 12  AS 11  SS 11  AS 10  SS 10  AS 09 

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