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 2011

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
29 September 2011
Stephan Denkl
University of Kiel
Asymptotics of hedging errors and option prices in Lévy models with small jumps   HG G 43 
Abstract: It is well-known that the Black-Scholes model exhibits certain drawbacks, which can be partly overcome by considering, e.g., exponentials of general Lévy processes with jumps. However, this leads to incomplete markets, where replication of options payoffs is not possible in general. One way out is variance-optimal hedging, where one seeks to minimize the second moment of the hedging error. Specifically in the framework of exponential Lévy models, Hubalek et al. (2006) and Černý (2007) have derived semi-explicit formulas for the hedging error of the optimal strategy by the use of Fourier-Laplace techniques. In contrast, we ask for an approximate formula of the hedging error which essentially depends only on the higher moments of the stock returns and not on more specific properties of the Lévy model. More precisely, we use the exact results on the hedging error mentioned above to study its asymptotics when the driving Lévy process tends to a Brownian motion. The resulting approximation of the error is expressed in terms of the first four moments of log-returns of the Lévy model as well as sensitivities of the option price in the limiting Black-Scholes model. Adopting the same methodology, we obtain approximate formulas for option prices in exponential Lévy models. The talk is based on joint work with Aleš Černý and Jan Kallsen.
6 October 2011
Takuji Arai
Keio University, Japan
Convex risk measures for good deal bounds  HG G 43 
Abstract: We discuss convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further we investigate conditions under which any good deal valuation is relevant. This is joint work with Masaaki Fukasawa (Osaka Univ.)
* 19 October 2011
Prof. Dr. Ioannis Karatzas
Columbia University
Stable diffusions with rank-based interactions, and models of large equity markets  HG G 43 
Abstract: We introduce and study ergodic multidimensional diffusion processes interacting through their ranks. These interactions give rise to invariant measures which are in broad agreement with stability properties observed in large equity markets over long time-periods. The models we develop assign growth rates and variances that depend on both the name (identity) and the rank (according to capitalization) of each individual asset. Such models are able realistically to capture critical features of the observed stability of capital distribution over the past century, all the while being simple enough to allow for rather detailed analytical study. The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) collisions to occur. We show, however, that such multiple collisions have no effect on any of the stability properties of the resulting system. This is accomplished through a detailed analysis of collision local times. The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing particle systems in Statistical Mechanics such as the Sherrington-Kirkpatrick model for spin-glasses.
10 November 2011
Prof. Dr. Damir Filipovic
The Term Structure of Interbank Risk  HG G 43 
Abstract: We use the term structure of spreads between rates on interest rate swaps indexed to LIBOR and overnight indexed swaps to infer a term structure of interbank risk. Using a dynamic term structure model, we decompose the term structure of interbank risk into default and non-default components. We find that, on average, from August 2007 to January 2011, the fraction of total interbank risk due to default risk increases with maturity. At the short end of the term structure, the non-default component is important in the first half of the sample period and is correlated with various measures of funding liquidity and market liquidity. Further out the term structure, the default component is the dominant driver of interbank risk throughout the sample period. These results hold true in both the USD and EUR markets and are robust to different model parameterizations and measures of interbank default risk. The analysis has implications for monetary and regulatory policy as well as for pricing, hedging, and risk-management in the interest rate swap market.
17 November 2011
Dr. Thorsten Rheinländer
Self-dual stochastic processes  HG G 43 
Abstract: We introduce the notion of conditionally symmetric processes and discuss their relation to the self-duality of their stochastic exponentials. This is carried out for continuous Ocone martingales. We provide an example of a non-Ocone martingale which is process, but not conditionally symmetric. In the Levy world, the notion of quasi self-duality is linked to an exponentially decaying factor when comparing the right- and left tails of the distribution. Lifting symmetry to the stochastic exponential, a certain Moebius transform is involved, and for the right choice of subordinator, the quasi self-duality property translates into the functional equation of the Riemann Zeta-function. These studies are motivated by financial engineering, in particular the semi-static hedging of path-dependent barrier options by non-vanilla European ones which only depend on the payoff at maturity. In fact, the common practice is questionable since it would only work out in case the stochastic log-returns are conditionally symmetric which is unrealistic. We show in the context of continuous stochastic volatility models how to modify the known strategy via a change to an 'asymmetric risk measure', under which all characteristics can be obtained easily. The talk is based on joint work with Michael Schmutz, University of Bern, and on ongoing work with Zhanyu Chen, London School of Economics.
24 November 2011
Prof. Dr. Suresh Sethi
University of Dallas
Co-op Advertising and Pricing in a Stochastic Supply Chain: Feedback Stackelberg Strategies  (CANCELLED) HG G 43 
Abstract: Cooperative (co-op) advertising is an important instrument for aligning manufacturer and retailer decisions in supply chains. In this, the manufacturer announces a co-op advertising policy, i.e., a participation or subsidy rate that specifies the percentage of the retailer’s advertising expenditure that it will provide. In addition, it also announces the wholesale price. In response, the retailer chooses its optimal advertising and pricing policies. We model this supply chain problem as a stochastic Stackelberg differential game whose dynamics follows Sethi’s stochastic sales-advertising model of 1983. We obtain the condition under which offering co-op advertising is optimal for the manufacturer. We provide, in feedback form, the optimal advertising and pricing policies for the manufacturer and the retailer. We extend the analysis to a (deterministic) case of two retailers competing with one another, and solve the resulting Stackelberg-Nash differential game, in which the manufacturer announces his subsidy rates for the two retailers, and the retailers in response play a Nash differential game in choosing their optimal advertising efforts over time. We identify the key drivers that influence the optimal subsidy rates and obtain the conditions under which the manufacturer will support one or both of the retailers. We analyze the impact of co-op advertising on the profits of the channel members and the extent to which it can coordinate the channel. We investigate the case of an anti-discriminatory act which restricts the manufacturer to offering equal subsidy rates to the two retailers.
8 December 2011
Dr. Rudolf Riedi
EIA Fribourg
Infinitely Divisible Shotnoise: local scaling and moments  HG G 43 
Abstract: One of the reasons for the popularity of fractional Brownian motion and self-similar processes lies in their simple scaling properties. Real world signals in computational finance, networking and other fields, however, show often more complex behavior. Classical multiplicative cascades can accommodate complexity in scaling but lack basic statistical properties such as stationarity of increments. In this talk we present Infinitely Divisible Shotnoise, a novel class of processes in form of infinite products of pulses which combine stationarity of increments with rich scaling properties. We establish convergence and scaling conditions in terms of moments and the shape of the pulses.
15 December 2011
Kasper Larsen
Carnegie Mellon University
Unspanned endowment and face-lifting  HG G 43 
Abstract: We study the relation between optimizing over finite additive and countable additive probability measures. Standard arguments show that the optimizer is always attained in the finite additive class. We will then discuss and derive the influence a possible singular component has on the value function close to maturity. More specifically, in general incomplete Brownian settings, we explicitly identify the face-lift (boundary layer) in the value function. This is joint with Gordan Zitkovic.
19 January 2012
Dr. Stephan Sturm
Princeton University
Portfolio Optimization under Convex Incentive Schemes  HG G 43 
Abstract: We consider the utility maximization problem from the point of view of a portfolio manager paid by a convex incentive scheme $g$. The manager's own utility function $U$ is assumed to be smooth and strictly concave, however his implied utility function $U \circ g$ fails to be concave. As a consequence, this problem does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal wealth in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has no atom with respect to the Lebesgue measure. In many cases, this fact is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As example we discuss stochastic volatility models and show that existence and uniqueness of an optimizer are guaranteed as long as the market price of risk satisfies a certain (Malliavin-)smoothness condition. We provide also a detailed analysis of the case when this criterium fails, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor. This is joint work with Maxim Bichuch.

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