Seminar on Mathematical Finance - winter term
2013
Organizer: Walter
Schachermayer
Time: Thursday, 17:00-18:30
Room: Seminarraum SR11, 2.OG,
Oskar-Morgenstern-Platz 1
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| Date: | Speaker: | Title : |
| Do, 03/10/13 17:00-18:30 |
Rasonyi Miklos | Superhedging under superlinear liquidity costs |
| Abstract: A most natural question in mathematical finance is the following: An investor is facing a payment obligation at time T. Which initial endowments (at time 0) allow him/her to meet this obligation with probability one ? The answer depends on the trading mechanism of the given market. Classical results apply if we assume that there are no market frictions. However, the picture changes when more realistic models are considered (e.g. when transaction costs are taken into account). We review the available results and then present a new theorem for a general, continuous-time model with liquidity constraints. This result has applications to optimal investment problems as well. This talk is based on joint work with Paolo Guasoni. | ||
| Do, 10/10/13 17:00-18:30 |
Sergio Pulido | Quadratic BSDEs arising from a price impact model with exponential utility |
| Abstract: We analyze a price impact model where a large investor wants to trade an illiquid asset with a market maker who quotes prices for this security. In our model, the market maker's preferences are modeled through an exponential utility function and the price impact of the trading strategy of the large investor is derived endogenously through an equilibrium mechanism. We establish a relationship between the equilibrium mechanism and a two-dimensional BSDE with quadratic growth. This allows us to show that an equilibrium exists under certain conditions on the final payoff of the traded asset, the risk aversion coefficient of the market maker and the trading strategy of the large investor. The relationship between the equilibrium mechanism and the two dimensional quadratic BSDE also allows us to study stability and asymptotic behavior with respect to the parameters of the model. This is a joint project with Dmitry Kramkov. | ||
| Do, 17/10/13 17:00-18:30 |
Yiqing Lin | Discussion on some recent work of robust superhedging |
| Abstract: In this talk on the working seminar, we introduce recent progress on the topic of robust superhedging. Precisely speaking, this talk will be mainly based on the prepublication of Possamai et al. (2013), who consider this problem in a context of model uncertainty. | ||
| Do, 24/10/13 17:00-18:30 |
Radka Pickova | Volatility-Stabilized Processes |
| Abstract:We consider systems of interacting diffusion processes which generalize the volatility-stabilized market models introduced in Fernholz and Karatzas (2005). We show how to construct a weak solution of the underlying system of stochastic differential equations. In particular, we express the solution in terms of time-changed squared-Bessel processes and discuss sufficient conditions under which one can argue that this solution is unique in distribution. Moreover, we discuss the significance of these processes in the context of arbitrage relative to the market portfolio within the framework of Stochastic Portfolio Theory. | ||
| Do, 31/10/13 17:00-18:30 |
Nicolas Perkowski | Pathwise integration in model free finance |
| Abstract: Vovk’s
game-theoretic approach to mathematical finance
allows for a qualitative description of typical asset
price trajectories. It is based on pathwise
superhedging arguments and on a model free notion of
arbitrage. I will present a "model free Itô isometry"
in this context. I will also show that it is possible
to construct a rough path above every typical price
path. This leads to a multidimensional generalization
of Föllmer's "calcul d'Itô sans probabilités".
Extending Föllmer’s ideas in another direction, I
will show that every typical price path admits a
local time, which gives us a pathwise version of the
Tanaka formula. This is joint work with David Prömel. |
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| Do, 14/11/13 17:00-18:30 |
Michaela Szölgyenyi | Bayesian dividend maximization and associated SDEs |
| Abstract:We solve
the valuation problem of an insurance portfolio by
maximizing its expected discounted future dividend
payments. Extending classical contributions we study
this optimization problem in a Bayesian framework.
Specifically, we model the surplus process as a
diffusion with an unobservable drift parameter. After
applying filtering theory to overcome the issue of
uncertainty, we are able to characterize the solution
of the optimization problem as the unique viscosity
solution to the associated Hamilton-Jacobi-Bellman
equation. A numerical treatment of the problem leads
to epsilon-optimal dividend strategies of so-called
threshold type. This raises the question of
admissibility of such strategies. In particular we
need to investigate the existence of the controlled
process. In our framework this leads to the problem
of the existence of a solution to a system of SDEs
with a discontinuous drift and singular diffusion
coefficient. Based on joint work with Gunther Leobacher (University of Linz) and Stefan Thonhauser (University of Lausanne). |
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| Do, 21/11/13 17:00-18:30 |
Stefano Pagliarani | PIDE's expansions in option pricing |
| Abstract: We
consider a defaultable asset whose risk-neutral
pricing dynamics are described by an exponential
Lévy-type martingale subject to default. This class
of models allows for local volatility, local default
intensity, and locally dependent Lévy measure.
Generalizing and extending the novel adjoint
expansion technique of Riga, Pagliarani, Pascucci
(2013), we derive a family of asymptotic expansions
for the transition density of the underlying as well
as for European-style option prices and defaultable
bond prices. For some specific choices of the Lévy measure, global asymptotic error bounds for short maturities are provided for the transition density, as well for option prices. Furthermore, the precision can be improved for medium-long maturities by means of a bootstrapping algorithm, leading to a convergence result for any given time to maturity. In the purely diffusive case, we present an extension of our technique to the n-dimensional case in order to include multi-factor stochastic-local volatility models. In this framework we also derive an asymptotic expansion for the implied volatility induced by European calls/puts options. |
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| Do, 28/11/13 17:00-18:30 |
Christian Bayer | Asymptotics beats Monte Carlo: The case of correlated localvol baskets |
| Abstract:We consider a basket of options with both positive and negative weights, in the case where each asset has a smile, e.g. evolves according to its own local volatility and the driving Brownian motions are correlated. In the case of positive weights, the model has been considered in a previous work by Avellaneda, Boyer-Olson, Busca and Friz.We derive highly accurate analytic formulas for the prices and the implied volatilities of such baskets. The computational time required to implement these formulas is under two seconds even in the case of a basket on 100 assets. The combination of accuracy and speed makes these formulas potentially attractive both for calibration and for pricing. In comparison, simulation based techniques are prohibitively slow in achieving a comparable degree of accuracy. Thus the present work opens up a new paradigm in which asymptotics may arguably be used for pricing as well as for calibration. (Joint work with Peter Laurence) | ||
| Do, 09/01/14 17:00-18:30 |
Sebastian Andres (University of Bonn) | Invariance Principle for the Random Conductance Model in a degenerate ergodic environment |
| Abstract: In this talk we consider a continuous time random walk X on Zd in an environment of random conductances taking values in [0,∞). We will discuss recent results on a quenched functional central limit theorem for this random walk. Assuming that the law of the conductances is i.i.d. or - more general - stationary ergodic with respect to space shifts, we present such an invariance principle for X under some moment conditions on the environment. Under the same conditions we also obtain a local limit theorem. This is joint work with J.-D. Deuschel and M. Slowik. |
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| Do, 16/01/14 17:00-18:30 |
Mathieu Rosenbaum | Unfortunately this talk has been CANCELLED |
| Do, 23/01/14 17:00-18:30 |
Lingqi Gu | Discussion on the article of Giuseppe Benedetti and Luciano Campi (2011) 'Multivariate utility maximization with proportional transaction costs and random endowment' |
| Abstract: In this presentation, we introduce the article of Giuseppe Benedetti and Luciano Campi (2011) 'Multivariate utility maximization with proportional transaction costs and random endowment '. In this article, the authors deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to modeling a currency market with proportional transaction costs). In particular, they extend the results in [L. Campi and M. Owen, Finance Stoch., 15 (2011), pp. 461–499] to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. They start by studying some basic properties of the value function (which is now defined on a space of random variables),and then dualize the problem following some convex analysis techniques which have proven very useful in this field of research. They finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. |
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| Do, 30/01/14 17:00-18:30 |
Damir
Filipovic (Ecole Polytechnique Fédérale de Lausanne) |
This talk is jointly organised
with TU Wien and WU Wien: Linear-Rational Term Structure Models |
| Abstract: We
introduce the class of linear-rational term structure
models, where the state price density is modeled such
that bond prices become linear-rational functions of
the current state. This class is highly tractable
with several distinct advantages: i) ensures
non-negative interest rates, ii) easily accommodates
unspanned factors affecting volatility and risk
premia, and iii) admits analytical solutions to
swaptions. For comparison, affine term structure
models can match either i) or ii), but not both
simultaneously, and never iii). A parsimonious
specification of the model with three term structure
factors and one, or possibly two, unspanned factors
has a very good fit to both interest rate swaps and
swaptions since 1997. In particular, the model
captures well the dynamics of the term structure and
volatility during the recent period of near-zero
interest rates. |
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