# Stochastic Analysis and Stochastic Finance Seminar

## Volatility and Arbitrage

Speaker(s):
Johannes Ruf (London School of Economics)
Date:
Thursday, October 26, 2017 - 5:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

The capitalization-weighted cumulative variation $\sum_{i=1}^d \int_0^\cdot \mu_i (t) \dx \langle \log \mu_i \rangle (t)$ in an equity market consisting of a fixed number $d$ of assets with capitalization weights $\mu_i (\cdot) ,$ is an observable and a nondecreasing function of time. If this observable of the market is not just nondecreasing but actually grows at a rate bounded away from zero, then strong arbitrage can be constructed relative to the market over sufficiently long time horizons.

Speaker(s):
Alvaro Cartea (University of Oxford)
Date:
Thursday, October 26, 2017 - 4:15pm
Location:
HU Berlin, Rudower Chaussee 25, Room 1.115

We develop the optimal trading strategy for a Foreign Exchange (FX) broker who must liquidate a large position in an illiquid currency pair. To maximise revenues, the broker considers trading in a currency triplet which consists of the illiquid pair and two other liquid currency pairs. The liquid pairs in the triplet are chosen so that one of the pairs is redundant. The broker is risk-neutral and accounts for model ambiguity in the FX rates to make her strategy robust to model misspecification.

## Partially Observable Risk-Sensitive Markov Decision Processes

Speaker(s):
Nicole Bäuerle (Karlsruher Institut für Technologie (KIT))
Date:
Thursday, July 13, 2017 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We consider the problem of minimizing a certainty equivalent of the total or discounted cost over a finite time horizon which is generated by a Partially Observable Markov Decision Process (POMDP). The certainty equivalent of a random variable $X$ is defined by $U^{-1}(EU(X))$ where $U$ is an increasing function. In contrast to a risk-neutral decision maker, this optimization criterion takes the variability of the cost into account. It contains as a special case the classical risk-sensitive optimization criterion with an exponential utility.

## Contagion and Security in Inhomogeneous Financial Networks

Speaker(s):
Hamed Amini (University of Miami)
Date:
Thursday, July 13, 2017 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We derive rigorous asymptotic results for the magnitude of contagion in a large financial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. Our results extend previous studies on contagion in random graphs to inhomogeneous directed graphs with a given degree sequence and arbitrary distribution of weights. We introduce a criterion for the resilience of a large financial network to the insolvency of a small group of financial institutions and quantify how contagion amplifies small shocks to the network.

## The Risk-Tolerance Process and the Sensitivity of Optimal Investment and Consumption

Speaker(s):
Johannes Muhle-Karbe (University of Michigan)
Date:
Thursday, June 29, 2017 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

In the perturbation analysis of various models with small frictions, a crucial role is played by the risk tolerance of the indirect utility process. Building on work of Kramkov and Sirbu, we show that this object is well defined in a general semimartingale setting. We also establish that it admits a dynamic characterisation in terms of a quadratic BSDE, which in turn allows to compute the sensitivity of optimal investment strategies and consumption plans with respect to wealth.

## Financial Asset Price Bubbles under Model Uncertainty

Speaker(s):
Francesca Biagini (LMU München)
Date:
Thursday, June 29, 2017 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We study the concept of financial bubble under model uncertainty. We suppose the agent to be endowed with a family Q of local martingale measures for the underlying discounted asset price. The priors are allowed to be mutually singular to each other. One fundamental issue is the definition of a well-posed concept of robust fundamental value of a given financial asset.

## Optimal stopping of one-dimensional diffusions with integral criteria

Speaker(s):
Carlos Oliveira (Universidade de Lisboa & HU Berlin)
Date:
Thursday, June 15, 2017 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

This work provides a full characterization of the value function and solution(s) of an optimal stopping problem for a one-dimensional diffusion with an integral criterion.
The results hold under very weak assumptions, namely, the diffusion is assumed to be a weak solution of a stochastic differential equation satisfying the Engelbert-Schmidt conditions, while the (stochastic) discount rate and the integrand are required to satisfy only general integrability conditions.

## Stochastic control under partial observation

Speaker(s):
Huyen Pham (Université Paris Diderot)
Date:
Thursday, June 15, 2017 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We study and revisit the optimal control problem of partially observed stochastic systems. By using a control randomization method, we provide a backward stochastic differential equation (BSDE) representation for the value function in a general framework including path-dependence in the coefficients (both on the state and control) and without any non degeneracy condition on the diffusion coefficient.

## Model-free bounds for multi-asset options -- improved Fréchet-Hoeffding and optimal transport approaches

Speaker(s):
Antonis Papapantoleon (TU Berlin)
Date:
Thursday, June 1, 2017 - 5:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We consider a multivariate random variable with known marginals and unknown dependence structure. In this talk, we will present several methods for sharpening the classical Fréchet-Hoeffding bounds on copulas by using additional, partial information on the dependence structure. Then we will discuss applications of these results for deriving bounds on option prices and portfolio Value-at-Risk in this setting of model / dependence uncertainty. We will also discuss the detection of arbitrage in multi-asset markets and model-free hedging of multi-asset derivatives.

## Singular Copulas

Speaker(s):
Fabrizio Durante (Università del Salento)
Date:
Thursday, June 1, 2017 - 4:00pm
Location:
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We present both old and recent results about singular copulas and copulas with a singular component by discussing their relevance in (at least) three different domains. First, singular copulas may be used to obtain specific tail behavior in a multivariate distribution, a fact that has also been exploited to obtain worst-possible scenarios for risk measures. Second, special classes of singular copulas (e.g. shuffles of Min) can be used in the approximation of various dependence structures.