Stochastic Analysis and Stochastic Finance Seminar

Pricing under Rough Volatility

Speaker(s): 
Christian Bayer (WIAS Berlin)
Date: 
Thursday, April 16, 2015 - 5:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041

From an analysis of the time series of volatility using recent high frequency data, Gatheral, Jaisson and Rosenbaum [SSRN 2509457, 2014] showed that log-volatility behaves

Sensitivity of Optimal Comsumption Streams

Speaker(s): 
Martin Herdegen (ETH Zürich)
Date: 
Thursday, April 16, 2015 - 4:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 041

We study the sensitivity of optimal consumption streams with respect to perturbations of the random endowment. We show that to the leading order, any consumption correction for the perturbed endowment is still optimal as long as the budget constraint is binding. More importantly, we also establish the optimal correction at the next-to-leading order. This can be computed in two steps. First, one has to find the optimal correction for a deterministic perturbation. This only involves the risk-tolerance process of the unperturbed problem and yields a "risk-tolerance martingale".

Asymptotic indifference pricing in Lévy models

Speaker(s): 
Peter Tankov (Université Paris Diderot)
Date: 
Thursday, February 12, 2015 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Financial markets based on Lévy processes are typically incomplete and option prices depend on risk preferences of individual agents. In this context, the notion of utility indifference price has gained a certain popularity in the academic circles. Although theoretically very appealing, this pricing method remains difficult to apply in practice, due to the high computational cost of solving the non-linear partial integro-differential equation associated to the indifference price.

Markovian stochastic control with f-expectation

Speaker(s): 
Marie-Claire Quenez (University Paris-Diderot)
Date: 
Thursday, February 12, 2015 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We study mixed optimal control/stopping problems for f-expectations in the Markovian framework. We first establish a dynamic programming principle. This requires some special techniques of stochastic analysis and backward stochastic differential equations to handle the difficulties arising from the nonlinearity of the expectation. Using this result and properties of reflected backward stochastic differential equations, we prove that the value function of our mixed control problem is a viscosity solution of a nonlinear Hamilton-Jacobi-Bellman variational inequality.

Multi-dimensional quadratic BSDEs

Speaker(s): 
Shanjian Tang (Fudan University)
Date: 
Thursday, January 29, 2015 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Quadratic BSDEs refer to those BSDEs whose generators grow quadratically in the second unkown variable. In this talk, I will start with recalling J. M. Bismut's Ph. D work on the linear quadratic optimal stochastic control problem and the introduction of backward stochastic Riccati equations, which motivated the study of general quadratic BSDEs. Then I review the theory of one-dimensional quadratic BSDEs and show the difficulty in a general solution of multi-dimensional quadratic BSDEs even when the terminal value is essentially bounded.

Second order Pontriagin's principle for stochastic control problems - CANCELLED!

Speaker(s): 
F. Frédéric Bonnans (INRIA Saclay, Ecole Polytechnique)
Date: 
Thursday, January 29, 2015 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We discuss stochastic optimal control problems whose volatility does not depend on the control, and which have finitely many equality and inequality constraints on the expected value of function of the final state, as well as control constraints. The main result is a proof of necessity of some second order optimality conditions involving Pontryagin multipliers.

Non-Implementability of Arrow-Debreu Equilibria by Continuous Trading under Knightian Uncertainty

Speaker(s): 
Frank Riedel (Universität Bielefeld)
Date: 
Thursday, January 15, 2015 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Under risk, Arrow–Debreu equilibria can be implemented as Radner equilibria by continuous trading of few long–lived securities. We show that this result generically fails if there is Knightian uncertainty in the volatility. Implementation is only possible if all discounted net trades of the equilibrium allocation are mean ambiguity–free.

This is a joint work with Patrick Beissner.

Hawkes processes, microstructure and market impact

Speaker(s): 
Mark Hoffmann (Université Paris Dauphine)
Date: 
Thursday, December 18, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

I will first shortly review the issue of obtaining simple lattice price models for assets observed at fine temporal scales that are 1) able to reproduce microstructure effects like variance noise or the Epps effect and 2) behave like continuous semimartingales compatible with the theory of arbitrage on large diffusive scales. The use of mutually exciting point processes enable to track such microstruture effects across scales and I will present some recent (and less recent) models based on Hawkes processes.

Arbitrage-Free Pricing of XVA

Speaker(s): 
Agostino Capponi (John Hopkins University)
Date: 
Thursday, December 18, 2014 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We introduce a framework for computing the total valuation adjustment (XVA) of an European claim accounting for funding spreads, counterparty risk, and collateral mitigation. We use no-arbitrage arguments to derive the nonlinear backward stochastic differential equations (BSDEs) associated with the portfolios which replicate long and short positions in the claim. This leads to defining buyer and sellers? XVAs which in turn identify a no-arbitrage band. When borrowing and lending rates coincide, our framework reduces to a generalized Piterbarg's model.

Dealing with partial hedging or risk management constraints via BSDEs with weak reflections

Speaker(s): 
Romuald Elie (Université Paris-Est Marne-la-Vallée)
Date: 
Thursday, December 4, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In many incomplete markets, the super replication price of a given (eventually non Markovian) claim rewrites in terms of the minimal super-solution of a well chosen Backward stochastic differential equation (BSDE). The price obtained is often numerically very high and hereby useless in practice. In order to lower the price, one must accept to take some risk and this can be formalized via the use of quantile hedging type objectives, where the agent only wishes to upper hedge the claim of interest with a given a priori probability of success p.

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