**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

View Larger
Map

Date: |
Speaker: |
Title : |

Do, 03/10/1317:00-18:30 |
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, 10/10/1317:00-18:30 |
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, 17/10/1317:00-18:30 |
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, 24/10/1317:00-18:30 |
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, 31/10/1317:00-18:30 |
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. |
||

Do, 14/11/1317:00-18:30 |
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). |
||

Do, 21/11/1317:00-18:30 |
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. |
||

Do, 28/11/1317:00-18:30 |
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, 09/01/1417:00-18:30 |
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 Z^{d} 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. |
||

Do, 16/01/1417:00-18:30 |
Mathieu Rosenbaum |
Unfortunately this talk has been CANCELLED |

Do, 23/01/1417:00-18:30 |
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. |
||

Do, 30/01/1417:00-18:30 |
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. |
||