Post/Doctoral seminar in Mathematical Finance
taking place this semester on Wednesdays, 15:15 - 17:00, in HG G 19.1
FS 12
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23/05/2012
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Sebastian Herrmann
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A class of strict local martingales of finite variation II
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tba
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16/05/2012
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Martin Herdegen
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A class of strict local martingales of finite variation I
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We construct a class of finite variation processes of
martingale-type. The paths of these processes follow a deterministic
càdlàg function F up to some killing time at which the process jumps and
stays constant afterwards. We derive necessary and sufficient
conditions, depending only on F and the distribution function G of the
killing time, for the processes to be sigma, local,
uniformly integrable or H^1 martingales. In particular, we provide an
example of a strict sigma martingale. Finally, we characterise
integrable local martingales within this class.
(This is joint work with Sebastian Herrmann.)
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09/05/2012
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Miklos Rasonyi (University of Edinburgh)
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Optimal investment: from risk-averse to behavioural agents
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Classical investment problems assume that economic agents are
risk-averse. This corresponds to using concave utility functions to
describe agents' preferences. More recently, non-concave utilities were
proposed and distortions of the probability measure
were also considered.
We are dealing with optimal investment for an agent whose behaviour is
characterized by a possibly non-concave utility function and by
probability distortions. This new setting poses several mathematical
challenges and exhibits a number of unexpected phenomena.
In discrete-time multiperiod models we discuss the well-posedness of
this investment problem and show the existence of optimal strategies
under suitable conditions. We also have a look at what happens in
continuous-time, in particular, we provide a sufficient
and (essentially) necessary condition for the Black-Scholes model in the
case of power-like utilities and distortion functions.
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02/05/2012
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No seminar
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25/04/2012
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No seminar
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18/04/2012
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Ariel Neufeld
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A note on asymptotic exponential arbitrage with exponentially decaying failure probability
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In
Föllmer and Schachermayer (2007), a special form of long-term arbitrage
was considered for the first time. In Mbele Bidima and Rásonyi (2010),
the authors called this asymptotic exponential arbitrage with
exponentially decaying failure probability.
If the price process S has that property, we can find for any large
enough maturity T, up to an exponentially small probability of failure,
an exponentially growing profit with an exponentially decreasing
potential loss. Therefore, we get an explicit relation
between any tolerance level of failure and the necessary time to reach
this level. Furthermore, when we let T go to infinity, we get in the
limit a riskless profit. Thus, asymptotic exponential arbitrage with
exponentially decaying failure probability can be
interpreted as a strong and quantitative form of long-term arbitrage.
The goal is to prove, under a slightly stronger condition, the
conjecture in Föllmer and Schachermayer (2007) that in the case where
the price process S is a continuous semimartingale and satisfies a
(strong) large deviations estimate, S allows asymptotic exponential
arbitrage with exponentially decaying failure probability.
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04/04/2012
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Jin-Hyuk Choi (University of Texas Austin)
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begin
15:30
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Shadow prices and well posedness in the problem of optimal investment and consumption with transaction costs
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We revisit the optimal investment and consumption model of Davis and
Norman (1990) and Shreve and Soner (1994), following a shadow-price
approach similar to that of Kallsen and Muhle-Karbe (2010). Making use
of the completeness of the model without transaction costs, we
reformulate and reduce the HJB equation for this singular stochastic
control problem to a free-boundary problem for a first-order ODE with an
integral constraint. Having shown that the free boundary problem has a
twicely differenciable solution, we use it to construct the solution of
the original optimal investment/consumption problem without any recourse
to the dynamic programming principle. Furthermore, we provide an
explicit characterization of model parameters for which the value
function is finite.
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28/03/2012
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Nicoletta Gabrielli
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Path Approximation for Affine Processes by space-time Transformation
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A Markov process is usually described in terms of transition probabilities for each fix state variable and any time. Following Ito's perspective on Markov processes, we want to investigate the evolution of a Markov process not only in terms of time, but also in terms of its inital value. Ito's pathwise construction turns out to be valuable in order to construct and improve pathwise discretization schemes for affine process, such as numerical methods based on time change transformations like the Lamperti representation theorem.
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21/03/2012
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Mirjana Vukelja
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Expected Utility from Terminal Wealth in an Illiquid Market
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We
give an overview of the liquidity risk model introduced by A. Roch and
H. M Soner in 2011. Then we consider the problem of maximizing the
expected
utility from terminal wealth in the above mentioned model in discrete
and continuous time. We discuss the value function and the dynamic
programming principle. Moreover, we show how the dimension of the value
function can be reduced. In continuous time we show
that the value function is a viscosity solution of the
Hamilton-Jacobi-Bellman equation.
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14/03/2012
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Mete Soner
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Optimal investment with small transaction costs
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In this talk, I will revisit the classical problem of proportional
transaction costs. Since the initial paper of Constantinides, small
transaction cost asymptotics have been studied by several people with
different tools. Since the convergence
of the value function to the value function of the Merton problem is
clear, the next term in the asymptotic expansion is the focus of these
research activities. Recently, we have devised an approach using the
techniques from the theory of homogenization. This
method together is flexible enough to handle the multi-dimensional cases
as well.
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07/03/2012
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Dejan Veluscek
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Extrapolation methods for weak approximation schemes
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We will give an quick
overview of the semigroup perspective on splitting schemes for S(P)DEs
which present a robust, "easy to implement" numerical method for
calculating the expected value of a certain payoff of a stochastic
process driven by a S(P)DE.
Having a high numerical order of convergence enables us to replace the
Monte Carlo integration technique by alternative, faster techniques. The
numerical order of splitting schemes for S(P)DEs is bounded by 2. The
technique of combining several splittings using
linear combinations which kills some additional terms in the error
expansion and thus raises the order of the numerical method is called
the extrapolation. In the presentation we will focus on a special
extrapolation of the Lie-Trotter splitting: the symmetrically
weighted sequential splitting, and its subsequent extrapolations. Using
the semigroup technique their convergence will be investigated. At the
end several applications to the S(P)DEs will be given.
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29/02/2012
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No seminar
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22/02/2012
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No seminar
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HS 11
16/12/2011
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Sebastian Herrmann
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On the martingale property of stochastic integrals
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We construct a uniformly integrable continuous martingale X and a bounded predictable process H such that the stochastic integral H·X is not a uniformly integrable martingale. Applying a deterministic change of time, we obtain a continuous martingale X' and a bounded integrand H'
such that the stochastic integral H'·X' is a strict local martingale.
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09/12/2011
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Martin Schweizer
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A continuous-time DMW result, with an infinite-dimensional proof
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One
aspect of the classical Dalang-Morton-Willinger result is that for a
stochastic process in finite discrete time, the existence of an
equivalent martingale measure automatically implies the existence of an
equivalent martingale measure with bounded
density. We prove a continuous-time variant of this result. The proof is
also of interest since it involves a nontrivial use of a separation
theorem in an infinite-dimensional space.
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25/11/2011
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Philipp Dörsek
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begin 15:30
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Splitting method for stochastic (partial) differential equations
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We give an overview of the semigroup perspective on splitting schemes for S(P)DEs. We explain the basic methodology using the method of stochastic characteristics. The Feller and generalised Feller properties are introduced and analysed. Applications to S(P)DEs are given.
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18/11/2011
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Anja Richter
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Explicit solutions to quadratic BSDEs and applications to utility maximization
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We want to give explicit solutions to a class of quadratic growth Backward Stochastic Differential Equations (BSDEs). Due to its analytical tractability we consider a setting involving affine processes on the cone of positive semidefinite matrices. It is then shown that the solution of a class of BSDEs can be reduced to solving a system of generalized Riccati ordinary differential equations for which we give existence and uniqueness results.
We apply our results to the problem of maximizing expected utility of terminal wealth in multivariate affine stochastic volatility models.
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11/11/2011
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No seminar
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04/11/2011
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No seminar
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28/10/2011
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No seminar
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21/10/2011
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Winslow Strong
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Arbitrage in Market Models Possessing a Stochastic
Number of Assets
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I
will discuss financial market models in which the number of assets is a
finite but unbounded stochastic process. The focus will be on the
possibility of various forms of arbitrage in these models. The framework
permits modeling of equity markets where companies
may enter, leave, merge, and split. The asset price process is taken to
be a piecewise semimartingale of stochastic dimension,
which I will define. Stochastic integration with respect to piecewise
semimartingales is extended from the usual case by use of localization
and partitioning. The “No free lunch with vanishing risk” equivalence to
the existence of an equivalent sigma-martingale
measure for the class of nonnegative wealth processes, and the “No
arbitrage of the first kind” equivalence to the existence of an
equivalent local martingale deflator for the price process are extended
to this setting. In the case where the price process is
a piecewise Itô process, functionally generated relative arbitrage of
stochastic portfolio theory is found to be less readily available than
in Itô process models.
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14/10/2011
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Roman Muraviev
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Natural selection with habits and learning in heterogeneous economies
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We study natural selection in
complete financial markets, populated by heterogeneous agents. We allow
for a rich structure of heterogeneity: Individuals may differ in their
beliefs concerning the economy, information
and learning mechanism, risk aversion, impatience (time preference rate)
and degree of habits. We develop new techniques for studying long run
behavior of such economies, based on the Strassen's functional law of
iterated logarithm. In particular, we explicitly
determine an agent's survival index and show how the latter depends on
the agent's characteristics. We use these results to study the long run
behavior of the equilibrium interest rate and the market price of risk.
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07/10/2011
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No seminar
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30/09/2011
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No seminar
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23/09/2011
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No seminar
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FS 11
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10/05/2011
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Main talk: Marcel Nutz
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Weak Dynamic Programming for Generalized State Constraints
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We provide a dynamic programming principle for stochastic optimal control problems with expectation constraints.
A weak formulation, using test functions and a probabilistic relaxation
of the constraint, avoids restrictions related to a measurable selection
but still implies the Hamilton-Jacobi-Bellman equation in the viscosity
sense. Moreover, we show how to treat state
constraints as a special case of expectation constraints.
(Joint work with Bruno Bouchard, Paris)
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ETHZ HG G 19.2, 15:15 - 17:00
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12/04/2011
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Main talk: Santiago Moreno (UZH)
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Pollution Permits, Strategic Trading and Dynamic Technology Adoption
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In
this talk I will present a model of a cap-and-trade system as observed
in the European market for C02 allowances. I will analyze, from the
point of view of a regulator, the design of dynamic incentives for
technology adoption under a transferable
permits system, which allows for strategic trading on the permit market.
Initially, firms can invest both in low-emitting production
technologies and trade permits. In the model, technology adoption and
allowance price are generated endogenously and are inter-dependent.
I will show that the non-cooperative permit trading game possesses a
pure-strategy Nash equilibrium, where the allowance value reflects the
level of uncovered pollution (demand), the level of unused allowances
(supply), and the technological status. Further
I will introduce a price support instrument (EC4P), which contingent on
the adoption of the new technology, allows firms to sell permits back to
the regulator. Numerical investigation confirms that the policy with
EC4Ps generates a floating price floor for
the allowances, and it restores the dynamic incentives to invest. Given
that this policy comes at a cost, I will propose and implement a
criterion for the selection of a self-financing policy (based on convex
risk measures).
(Work in collaboration with Luca Taschini.)
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ETHZ HG G 19.2, 15:15 - 17:00
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05/04/2011
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Main talk: Johannes Muhle-Karbe
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Transaction costs made tractable
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In a market with one safe and one risky asset, an investor with a
long horizon and constant relative risk aversion trades with constant
investment opportunities and proportional transaction costs. We derive
the optimal investment policy, its
welfare, and the resulting trading volume, as explicit functions of
market and preference parameters, and of the implied liquidity gap,
identified as the solution of a scalar equation. For small transaction
costs, all these quantities admit asymptotic expansions
of arbitrary order.
The results exploit the equivalence of the transaction cost market
to another frictionless market, with a shadow risky asset, in which
investment opportunities are stochastic. The shadow price also has an
explicit expression.
(Joint work with with Stefan Gerhold, Paolo Guasoni and Walter Schachermayer)
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Appetiser talk: Santiago Moreno (UZH)
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ETHZ HG G 19.2, 15:15 - 17:00
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22/03/2011
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Main talk: Martin Herdegen
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Numéraire independent characterisation of no-arbitrage markets
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We introduce a numéraire independent modelling framework for semimartingale markets and define arbitrage therein via a robust notion of superreplication prices. Moreover, we study the question how to define "good" investment strategies in such markets. We then link our framework to the classical framework of Mathematical Finance and survey different notions of no-arbitrage including (NA) and (NFLVR). Finally, we prove the existence of "good" investment strategies in no-arbitrage markets in the numéraire independent sense and give a martingale characterisation of such markets.
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Appetiser talk: Johannes Muhle-Karbe
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ETHZ HG G 19.2, 15:15 - 17:00
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08/03/2011
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Main talk: Yan Dolinsky
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Hedging of game options in the presence of transaction costs (Part II)
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We
study the problem of superreplication for game options under
proportional transaction costs. We consider a multidimensional model
which is an extension of the usual Black-Scholes (BS) model, in a sense
that the volatility is a progressively measurable function of the stock.
For this case we show that the superreplication price is the cheapest
cost of a trivial superreplication strategy. These results are
extensions of previous papers in which only European options with
Markovian structure were considered.
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Appetiser talk: Martin Herdegen
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ETHZ HG G 19.2, 15:15 - 17:00
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01/03/2011
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Main talk: Yan Dolinsky
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Hedging of game options in the presence of transaction costs (Part I)
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We study the problem of superreplication for game options under proportional transaction costs. We consider a multidimensional model which is an extension of the usual Black-Scholes (BS) model, in a sense that the volatility is a progressively measurable function of the stock. For this case we show that the superreplication price is the cheapest cost of a trivial superreplication strategy. These results are extensions of previous papers in which only European options with Markovian structure were considered.
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Appetiser talk: Martin Schweizer
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ETHZ HG G 19.2, 15:15 - 17:00
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HS 10
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24.9.2010
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Gilles-Edouard Espinosa
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Optimal Investment under Relative Performance Concerns I
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ETHZ ML J 34.1, 14:00-16:00
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Since
the seminal papers of Merton in the late 60's-early 70's, the problem
of optimal investment has been extensively studied in order to
generalize in many directions the initial framework. However, in all
these works, the investor only takes into account his own wealth (or
consumption) in his optimization criterion, whereas in the real world,
people tend to compare themselves with their peers. For example, a 5%
return during a crisis is not at all the same as a 5% return during a
financial bubble. Our aim is to derive an optimal investment theory with
such relative concerns.
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08.10.2010
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Gilles-Edouard Espinosa
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Optimal Investment under Relative Performance Concerns II
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ETHZ ML J 34.1, 14:00-16:00
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After recalling the main results from the previous talk we start explaining the details and giving the proofs.
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15.10.2010
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Gilles-Edouard Espinosa
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Optimal Investment under Relative Performance Concerns III
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ETHZ ML F 34, 14:00-16:00
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We continue explaining the details and giving the proofs of the results in the previous talks.
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22.10.2010
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Gilles-Edouard Espinosa
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Optimal Investment under Relative Performance Concerns IV
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ETHZ ML F 34, 14:00-16:00
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We continue explaining the details and giving the proofs of the results in the previous talks.
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05.11.2010
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Keita Owari
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On the Optimal Strategy in Robust Utility Maximization with Unbounded Random Endowment
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ETHZ ML F 34, 14:00-16:00
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We consider the robust utility maximization with random endowment. After recalling a duality result, we discuss the question of the existence of an optimal strategy in a "natural admissible class'' which is defined to be the set of predictable integrands whose stochastic integrals with respect to the underlying price process are supermartingales under certain "reasonable local martingale measures''. We provide a partial result that the "natural class'' indeed admits an optimizer under three additional assumptions.
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12.11.2010
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Martin Herdegen
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Numéraire independent properties of general semimartingale markets
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ETHZ ML F 34, 14:00-16:00
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We develop a numéraire independent description of general semimartingale markets and study numéraire independent properties thereof. Primarily, we define arbitrage in a numéraire independent and economically intuitive way via a new notion of superreplication prices. We study this concept of arbitrage in detail, introduce stopping times describing finer features of arbitrage markets and show that any so-called partial arbitrage market can be decomposed into a no-arbitrage market and a so-called full arbitrage market under two measures that are mutually singular and absolutely continuous with respect to the physical measure. Finally, we link our results to the standard model of Delbaen and Schachermayer.
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26.11.2010
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Martin Herdegen
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Numéraire independent properties of general semimartingale markets (part II)
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ETHZ ML F 34, 14:00-16:00
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03.12.2010
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Yan Dolinsky
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Limit Theorems for Liquidity in Binomial Markets.
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We study super--replication prices for European options with path dependent payoffs in the Binomial version of the illiquid market model. First we derive a dual representation for the super--replication prices in the Binomial models. Next, we use these duality results to prove limit theorems for the super--replication prices in the constructed Binomial models. In particular, we prove the existence of the liquidity premium for the continuous time limit of the above. This paper extends previous work which was done only for the Markovian case. Our approach is purely probabilistic and allows to deal with path dependent payoffs and path dependent supply curves. (Joint work with Mete Soner)
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ETHZ ML F 34, 14:00-16:00
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10.12.2010
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Yan Dolinsky
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Limit Theorems for Liquidity in Binomial Markets (part II).
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ETHZ ML F 34, 14:00-16:00
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17.12.2010
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Christian Reichlin
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Binomial approximation of non-concave utility maximization.
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ETHZ ML F 34, 14:00-16:00
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FS 10
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2.3.2010
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Martin Herdegen
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Markets without a Numeraire
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ETHZ HG G 19.1, 15:15-17:00
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16.3.2010
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Marcel Nutz
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Risk Aversion Asymptotics for Power Utility Maximization
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ETHZ HG G 19.1, 15:15-17:00
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We consider optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to infinity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The limits are related to exponential and logarithmic utility. We use convex duality and BSDEs to derive these results.
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23.3.2010
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Idris Kharroubi
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Discrete time approximation of BSDEs with oblique reflections
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ETHZ HG G 19.1, 15:15-17:00
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20.4.2010
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Yan Dolinsky
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Limit Theorems for Partial Hedging Under Transaction Costs
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ETHZ
HG G 19.1, 15:15-17:00
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We study shortfall risk minimization for American options with path dependent payoffs under proportional transaction costs in the Black-Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time BS model for a given initial capital there exists a portfolio strategy which minimizes the shortfall risk. In the absence of transactions costs (complete markets) similar limit theorems were obtained in Dolinsky and Kifer (2008, 2010) for game options. In the presence of transaction costs the markets are no longer complete and additional machinery is required. Shortfall risk minimization for American options under transaction costs was not studied before.
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27.4.2010
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Roman Muraviev
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Characterizing The Optimal Consumption and Investment Policies With Stochastic Habits
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ETHZ
HG G 19.1, 15:15-17:00
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11.5.2010
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Martin Keller-Ressel
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Asymptotics and Exact Pricing of Options on Variance
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ETHZ
HG G 19.1, 15:15-17:00
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We consider the pricing of derivatives written on the discrete realized variance of an underlying security. In the literature, the realized variance is usually approximated by its continuous-time limit, the quadratic variation of the underlying log-price. Here, we characterize the short-time limits of call options on both objects. We find that the difference strongly depends on whether or not the stock price process has jumps. To study the exact valuation of options on the discrete realized variance itself, we then propose a novel approach that allows to apply Fourier-Laplace techniques to price European-style options efficiently. To illustrate our results, we also present some numerical examples. (Joint work with Johannes Muhle-Karbe)
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18.5.2010
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Matteo Casserini
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A functional differential equation approach to the numerical simulation of BSDEs
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ETHZ
HG G 19.1, 15:15-17:00
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Backward stochastic dynamics have been introduced by Liang, Lyons and Qian in order to provide a generalization of BSDEs for non-Brownian filtrations by the use of a Doob-Meyer decomposition argument. In particular, this approach allows a representation of the solution of the classical BSDEs introduced by Pardoux and Peng in terms of the solution of a functional differential equation (which completely describes the finite variation part arising in the Doob-Meyer decomposition for the first component of the solution of the BSDE).
In this work, this functional differential equation approach is applied in order to numerically simulate solutions of BSDEs with Lipschitz driver. (Joint work with Gechun Liang)
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25.5.2010
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Christian Reichlin
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Risk-seeking behaviour in complete markets: existence, anti-comonotonicity and consequences for financial market equilibria
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ETHZ
HG G 19.2, 15:15-17:00
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HS 09
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15.10.2009
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Marcel Nutz
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Bellman Equation for Power Utility
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ETHZ HG G 19.2, 13:15-15:00
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Optimal consumption and investment with power utility: Derivation of the Bellman Equation in semimartingale setting under closed constraints. Uniqueness/verification results. Solution of the exponential Lévy case for convex constraints.
Keywords: dynamic programming - semimartingale characteristics - measurable selection - BSDE.
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22.10.2009
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Martin Schweizer
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Horizon Dependence in Optimal Portfolio Choice I
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ETHZ HG G 19.2, 13:15-15:00
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29.10.2009
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Martin Schweizer
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Horizon Dependence in Optimal Portfolio Choice II
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ETHZ HG G 19.2, 13:15-15:00
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05.11.2009
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Christoph Czichowsky
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Dynamic Mean-Variance Portfolio Selection
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ETHZ HG G 19.2, 13:15-15:00
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19.11.2009
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Roman Muraviev
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Consumption, Concavity and Habit Formation
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ETHZ HG G 19.2, 13:15-15:00
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10.12.2009
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Selim Gökay
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Cetin-Jarrow-Protter model of liquidity in a binomial market
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ETHZ HG G 19.2, 13:15-15:00
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