Talks in Financial and Insurance Mathematics

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

Spring Semester 2013

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
14 February 2013
Prof. Dr. Mark Podolskij
University of Heidelberg
A test for the rank of the volatility process: the random perturbation approach  HG G 43 
Abstract: In this paper we present a test for the maximal rank of the matrix-valued volatility process in the continuous Ito semimartingale framework. Our idea is based upon a random perturbation of the original high frequency observations of an Ito semimartingale, which opens the way for rank testing. We develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses. Finally, we demonstrate a homoscedasticity test for the rank process. (joint work with Jean Jacod)
21 February 2013
Dr. Amel Bentata 
Short time asymptotics for semimartingales and an application for short maturity index options in a multivariate jump-diffusion model  HG G 43 
Abstract: We extend and unify the short-time asymptotics of the marginal laws of a stochastic process to the more general case when ξ is a d-dimensional discontinuous semimartingale with jumps. We compute the leading term in the asymptotics in terms of the local characteristics of the semimartingale. In contrast to previous derivations, our approach is purely based on Ito calculus, and makes no use of the Markov property or independence of increments. We derive in particular the asymptotic behavior of call options with short maturity in a semimartingale model: whereas the behavior of out-of-the-money options is found to be linear in time, the short time asymptotics of at-the-money options is shown to depend on the fine structure of the semimartingale. Our multidimensional setting allows to treat examples which are not accessible using previous results (e.g the index process). We propose an analytical approximation for short maturity index options, generalizing the approach by Avellaneda & al 03 to the multivariate jump-diffusion case.
* 21 February 2013
Dr. Leif Doering 
On Jump SDEs with singular coefficients  HG G 43 
Abstract: In his famous articles from the 50s of the last century Feller classified all Markov processes corresponding to second order differential operators with possibly singular coefficients. We briefly discuss what can be expected if the Laplace operator is replaced by a non-local operator and than discuss in detail a natural example that occurs in the theory of self-similar processes.
28 February 2013
Prof. Dr. Jan Dolinsky
Hebrew University
Robust Hedging with Proportional Transaction Costs.  HG G 43 
Abstract: Duality for robust hedging with proportional transaction costs of path dependent European options is obtained in a discrete time financial market with one risky asset. Investor's portfolio consists of a dynamically traded stock and a static position in vanilla options which can be exercised at maturity. Only stock trading is subject to proportional transaction costs. The main theorem is duality between hedging and a Monge-Kantorovich type optimization problem. In this dual transport problem the optimization is over all the probability measures which satisfy an approximate martingale condition related to consistent price systems in addition to the usual marginal constraints. Joint work with Mete Soner.
14 March 2013
Prof. Dr. Bruno Bouchard
BSDEs with weak terminal condition  HG G 43 
Abstract: We introduce a new class of Backward Stochastic Differential Equations in which the $T$-terminal value $Y_{T}$ of the solution $(Y,Z)$ is not fixed as a random variable, but only satisfies a weak constraint of the form $E[\Psi(Y_{T})]\ge m$, for some (possibly random) non-decreasing map $\Psi$ and some threshold $m$. We name them BSDEs with weak terminal condition and obtain a representation of the minimal time $t$-values $Y_{t}$ such that (Y,Z) is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the $m$-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in F\"ollmer and Leukert, and in Bouchard, Elie and Touzi. Finally, we observe a surprisingly strong connection between BSDEs with weak terminal condition and 2nd order BSDEs in the quasi linear case.
* 21 March 2013
Prof. Dr. Marcel Nutz
Columbia University
Arbitrage and Duality in Nondominated Discrete-Time Models  HG G 43 
Abstract: We study a nondominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging. In a general measure-theoretic setup, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a family of martingale measures with certain properties. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. (Joint work with Bruno Bouchard.)
* 26 March 2013
Dr. Christa Cuchiero 
Universität Wien
Fourier Transform Methods for Pathwise Covariance Estimation in the Presence of jumps  HG G 19.1 
Abstract: With a view to calibrating multivariate stochastic covariance models, we provide a nonparametric method to estimate the trajectory of the instantaneous covariance process from observations of a d-dimensional logarithmic price process. We work under the mild structural assumption of It^o- semimartingales allowing in particular for a general jump structure. Our approach combines instantaneous covariance estimation based on Fourier methods, as proposed by Malliavin and Mancino [5, 6], with jump robust estimators for integrated covariance estimation [1, 2, 3, 4, 7]. We study in particular asymptotic properties and provide a central limit theorem, showing that in comparison to classical (local) estimators of the instantaneous covariance the asymptotic estimator variance of this Fourier estimator is smaller by a factor 2/3. The procedure is robust enough to allow for an iteration and we can therefore show theoretically and empirically how to estimate the integrated realized covariance of the instantaneous stochastic covariance process itself. In view of robust calibration in nance, this can then be used to determine quantities of multivariate models which remain invariant under equivalent measure changes, such as volatility of volatility, from time series observations. The talk is based on joint work with Josef Teichmann.
28 March 2013
No Talk - ETH Closes Early
4 April 2013
No Talk - Easter Holiday
11 April 2013
Dr. Alessandro Gnoatto 
Ludwig Maximilian University (LMU) of Munich Mathematics Institute
Coherent foreign exchange market models  HG G 43 
Abstract: The first part of the talk provides an introduction to the valuation problem of foreign exchange (FX) options and focuses on the particular features of FX rates as opposed to other asset classes. More specifically, we concentrate on the foreign-domestic parity, which provides a no-arbitrage relationship between call options on the e.g. EURUSD exchange rate and puts on USDEUR. This no arbitrage requirement implies a set of restrictions on the parameters of the model under different pricing measures. We generalize the results in (Del Baño Rollin, 2008) to the class of exponential Lévy processes and show that these models price call and puts coherently, i.e. in line with the foreign domestic parity. We then extend the result to the class of affine stochastic volatility models. The foreign-domestic parity is however, only a first requirement that an FX market model should satisfy, since arbitrary products or ratios of FX rates are still FX rates. In the second part of the talk, we use the idea of coherency in order to build multi-factor stochastic volatility models which are coherent w.r.t triangles of currencies, while being able to provide a satisfactory fit to market data. Literature: De Col, A., Gnoatto, A. Grasselli, M.(2012) Smiles all around: FX joint calibration in a multi-Heston model Gnoatto, A., (2013) Coherent foreign exchange market models. Gnoatto, A. and Grasselli, M. (2013) An analytic multi-currency model with stochastic volatiltiy and stochastic interest.
18 April 2013
Dr. Bastian Pentenrieder
Amplitude Capital
HFT: one acronym - many flavors  HG G 43 
Abstract: The talk tries to cast some light onto the diverse field of high-frequency trading (HFT). Its introductory part presents the concept of the central limit orderbook by means of a little toy simulation; different order types and the matching of orders at the exchange will be explained. In particular, I will highlight the problem of slippage and its implications for any form of active trading. The main part of the presentation then gives an overview over some common HFT strategies such as market-making, pure/statistical arbitrage and trend-following. At the end of the talk, I want to use the opportunity and give you an idea of my work as a researcher with a hedge fund.
* 18 April 2013
Prof. Dr. Marco Frittelli
University of Milano
Title T.B.A. (CANCELLED) HG G 43 
25 April 2013
Prof. Dr. Hao Xing 
London School of Economics (LSE)
Large time behavior of solutions to HJB equations and multivariate portfolio turnpikes  HG G 43 
Abstract: We study the large time limiting behavior of solutions to a Cauchy problem for semilinear parabolic equations with quadratic nonlinearity in gradients. The spatial domain for the equation can either be R^n or the space of positive definite matrice. When a Lyapunov function exists, as time tends to infinity, the solution (and its gradient) to the Cauchy problem converges to a solution (and its gradient) of the associated ergodic problem. Applications to long term portfolio choice problems and their turnpike properties will be discussed. When the investment opportunities are driven by a multivariate factor process with the state space R^n or the space of positive definite matrice, the convergence of solutions implies the optimal investment strategy for the power utility agent converges to its long run optimal analogue, when the investment horizon tends to infinity. This is a joint work with Scott Robertson.
2 May 2013
Prof. Dr. Christoph Frei
ETH Zurich, Switzerland
Optimal Execution of a VWAP Order: a Stochastic Control Approach  HG G 43 
Abstract: We consider the optimal liquidation of a position of stock (long or short) where trading has a temporary market impact on the price. The aim is to minimize both the mean and variance of the order slippage with respect to a benchmark given by the market VWAP (volume weighted average price). In this setting, we introduce a new model for the relative volume curve which allows simultaneously for accurate data fit, economic justification and mathematical tractability. Tackling the resulting optimization problem using a stochastic control approach, we derive and solve the corresponding Hamilton-Jacobi-Bellman equation to give an explicit characterization of the optimal trading rate and liquidation trajectory. The talk is based on joint work with Nicholas Westray (Deutsche Bank AG).
9 May 2013
ETH Closed - Ascension Day
16 May 2013
Prof. Dr. Archil Gulisashvili
Ohio University
Asymptotic behavior of distribution densities of sums of log-normal random variables with applications to risk management and finance  HG G 43 
Abstract: This is a joint work with P. Tankov. The talk concerns asymptotic approximations of distribution functions and distribution densities of sums of correlated log-normal random variables. We obtain sharp asymptotic formulas with relative error estimates for such densities in the case where the value of the independent variable is small. Applications to risk management and finance will be discussed. For example, for a log-normal portfolio, we characterize the asymptotic behavior of the Value at Risk. We will also discuss certain results concerning stress testing of log-normal portfolios. Applications to finance will be represented by asymptotic formulas characterizing the left-wing behavior of the implied volatility for basket options associated with the multidimensional Black-Scholes model.
23 May 2013
Prof. Dr. Pierre Collin-Dufresne 
EPF Lausanne
Insider Trading, Stochastic Liquidity and Equilibrium Prices  HG G 43 
Abstract: We extend Kyle’s (1985) model of insider trading to the case where liquidity provided by noise traders follows a general stochastic process. Even though the level of noise trading volatility is observable, in equilibrium, measured price impact is stochastic. If noise trading volatility is mean-reverting, then the equilibrium price follows a multivariate ‘stochastic bridge’ process, which displays stochastic volatility. This is because insiders choose to optimally wait to trade more aggressively when noise trading activity is higher. In equilibrium, market makers anticipate this, and adjust prices accordingly. More private information is revealed when volatility is higher. In time series, insiders trade more aggressively, when measured price impact is lower. Therefore, execution costs to uninformed traders can be higher when price impact is lower.
* 28 May 2013
Dr. Richard Martin
Longwood Credit Partners LLP, Imperial College London, and University College London
Optimal Trading under Proportional Transaction Costs  HG G 19.1 
Abstract: Proportional transaction costs present difficult theoretical problems in trading algorithm design, on account of their lack of analytical tractability. The author derives a solution of DT-NT-DT form for an arbitrary model in which the the traded asset has diffusive dynamics described by one or more stochastic risk factors. The width of the NT zone is found to be, as expected, proportional to the cube root of the transaction cost. It is also proportional to the 2/3 power of the volatility of the target position, thereby causing a faster trading strategy to be buffered more than a slower one. The displacement of the middle of the buffer from the costfree position is found to be proportional to the square of the width, and hence to the 2/3 power of the transaction cost; the proportionality constant depends on the expected short-term change in position.
30 May 2013
Dr. Valeria Bignozzi
ETH Zurich, Switzerland
Dynamic risk measurement and expectiles  HG G 43 
Abstract: There is a growing interest in risk measures that can be used to calculate solvency capital requirements for financial institutions in a dynamic framework. The present study discusses a mild notion of time-consistency between risk measurements of the same financial position at different time points in the future, called sequential consistency (Roorda & Schumacher, 2007). Such consistency is meaningful from both the investor’s and the regulator’s point of view, as it requires that current capital allocations reflect the acceptability or unacceptability of the position in future times. Sufficient conditions are provided for conditional coherent risk measures, in order that the requirements of acceptance-, rejection- and sequential consistency are satisfied. It is shown that these conditions are often violated for standard methods of updating. A method is consequently proposed for constructing a sequentially consistent risk measure, which entails the modification of the set of probability measures used to obtain the risk assessment at an initial time. This is demonstrated for the coherent entropic risk measure and for the class of Choquet risk measures, which generalizes the well-known TVaR. Finally we introduce a dynamic version of the expectiles. These are coherent risk measures that emerge as a valid alternative to TVaR and that are receiving major attention in the recent academic literature. We present preliminary results on their dynamic properties.
20 June 2013
Prof. Dr. Sergey Levendorskii
University of Leicester
Efficient Fourier and Laplace transforms and Wiener-Hopf factorization in applications to option pricing and calibration, or Computational Finance as a branch of Complex Analysis  HG G 43 
Abstract: We construct efficient methods of approximate Laplace and Fourier inversion and calculation of the Wiener-Hopf factors for wide classes of L\'evy processes with exponentially decaying L\'evy densities and affine jump-diffusions. As applications, we consider pricing European options in L\'evy and Heston model, and pricing options with lookback and barrier features and CDS in L\'evy models. In all cases, we use appropriate conformal changes of variables, which allow us to apply the simplified trapezoid rule with moderate or even small number of terms. In the case of options with barrier features, we apply the Gaver-Stehfest method as well, and design an algorithm which is very convenient for parallel computations. For barrier options and CDS of long maturity, we derive fast asymptotic formulas, which are very accurate even fairly close to the barrier. The same technique is applicable for calculation of pdf's of the supremum and infimum processes, joint pdf of a L\'evy process and its extremum or its supremum and infimum, hence, for Monte Carlo simulations.
4 July 2013
Prof. Dr. Eckhard Platen
University of Technology Sydney
The Affine Nature of Aggregate Wealth Dynamics  HG G 43 
Abstract: The presentation derives a parsimonious two-component affine diffusion model for a world stock index to capture the dynamics of aggregate wealth. The observable state variables of the model are the normalized index and the inverse of the stochastic market activity, both modelled as square root processes. The square root process in market activity time for the normalized aggregate wealth emerges from the affine nature of aggregate wealth dynamics, which will be derived under basic assumptions and does not contain any parameters that have to be estimated. The proposed model employs only three well interpretable structural parameters, which determine the market activity dynamics, and three initial parameters. It is driven by the continuous, nondiversifiable uncertainty of the market and no other source of uncertainty. The model, to be valid over long time periods, needs to be formulated in a general financial modelling framework beyond the classical no-arbitrage paradigm. It reproduces a list of major stylized empirical facts, including Student-t distributed log-returns and typical volatility properties. Robust methods for fitting and simulating this model are demonstrated. The model can be applied in various areas where long term real world index dynamics are relevant, including actuarial studies, as well as, derivative pricing and hedging.

Organizers: Winslow Strong

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