Humboldt Universität zu Berlin Naturwissenschaftliche Fakultät II Institut für Mathematik

## Forschungsseminar Stochastische Analysis und Stochastik der Finanzmärkte

### Bereich für Stochastik P. BANK, D. BECHERER, P.K. FRIZ, H. FöLLMER, U. HORST, P. IMKELLER, M. KELLER-RESSEL, U. KüCHLER, M. KUPPER, A. PAPAPANTOLEON

Ort: HU Berlin, Institut für Mathematik, Johann von Naumann - Haus, Rudower Chaussee 25, Hörsaal 1.115
Zeit: Donnerstag, 16 Uhr/17 c.t.

25. Oktober 2012
Michal Barski (Universität Leipzig)
Monotonicity of the CDO term structure models
##### Abstract: The CDO term structure model can be described by a family of defaultable bonds. Let $P(t,T,x)$ be a price at time $t$ of a bond with maturity $T$ and risk parameter $x$. The prices should be decreasing in $T$ and increasing in $x$, but mathematical models do not preserve this monotonicity property even if they satisfy the severe no-arbitrage conditions. The talk will be devoted to the models which are arbitrage-free and monotone. We will formulate conditions for the corresponding Heath-Jarrow-Morton-Musiela equation for the forward rates which provide monotonicity. Bibliography: Barski, M., Zabczyk, J. : ''Heath-Jarrow-Morton-Musiela equation with L\'evy perturbation'', (2012), Journal of Differential Equations, 253, 9, p. 2657-2697; Barski M. : ''Monotonicity of the CDO term strcture models'', (2012), submitted; Filpovi\'c, D., Overbeck, L., Schmidt, T. : ''Dynamic CDO Term Structure Modelling'', (2011), Mathematical Finance 21, 53-71,
Mihail Zervos (London School of Economics)
Optimal liquidation with log-linear price impact and transaction costs
##### Abstract: We consider the so-called optimal liquidation problem'' in algorithmic trading, which is the problem faced by an investor who has a large number of stock shares to sell over a given time horizon and whose actions have stock price impact. In particular, we develop and study a price model that presents the stochastic dynamics of a geometric Brownian motion and incorporates a log-linear effect of the investor's transactions. We then consider the maximisation of a risk-neutral performance criterion with transaction costs. We analyse the resulting optimisation problem, which takes the form of a two-dimensional degenerate singular stochastic control problem, by means of both analytic and probabilistic techniques. We establish a simple sufficient condition for the market to allow for no arbitrage opportunities in a finite time horizon and we develop a detailed characterisation of the value function and the optimal strategy. In particular, we derive the solution to the problem in an explicit form if the investor's time horizon is infinite. Interesting features of the problem's solution include the fact that the value function may be discontinuous as a function of the time horizon and the fact that an optimal strategy may not exist. (with Xin Guo)

08. November 2012
Philipp Dörsek (ETH Zürich)
Splitting and cubature schemes for stochastic partial differential equations
##### Abstract: We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed. Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms. Parts of this work are joint with J. Teichmann and D. Veluscek.

Jose Infante Acevedo (Ecole de Pons)
Optimal execution and price manipulations in time-varying limit order books
##### Abstract: This talk focuses on an extension of the Limit Order Book (LOB) model with general shape introduced by Alfonsi, Fruth and Schied [1]. Here, the additional feature allows a time-varying LOB depth. We solve the optimal execution problem in this framework for both discrete and continuous time strategies. This gives in particular sufficient conditions to exclude Price Manipulations in the sense of Huberman and Stanzl or Transaction-Triggered Price Manipulations (see Alfonsi, Schied and Slynko [1]). These conditions give interesting qualitative insights on how market makers may create or not price manipulations. References [1] Aurlien Alfonsi, Antje Fruth, and Alexander Schied. Optimal exe- cution strategies in limit order books with general shape functions. Quantitative Finance, 10(2):143{157, 2010.

22. November 2012
Pricing by stochastic control as alternative to measure transformation
##### Abstract: Given a (multi)-factor model for the term structure of interest rates, we present an approach for the pricing of bonds and interest rate derivatives that is alternative to the classical approach involving measure transformation and it is based on the solution of a suitable stochastic control problem thereby avoiding any change of measure. It can be extended to various more general situations and here we mention as possible examples the case of forward prices and Swaptions. As example of possible ensuing computational benefits we provide an explicit formula for the pricing of bond options in a bivariate linear-quadratic factor model. (Based on joint work with A.Gombani and R.Cogo) References: - A. Gombani, W.J.Runggaldier, Arbitrage-free multifactor term structure models: a theory based on stochastic control. Mathematical Finance, published online June 19, 2012. - R. Cogo, A. Gombani, W.J.Runggaldier, Stochastic control and pricing under Swap measures. Preprint 2012. To appear in : Seminar on Stochastic Analysis, Random Fields and Applications (Ascona 2011).

Zorana Grbac (TU Berlin)
Post-crisis valuation of interest rate derivatives: multiple curves, counterparty risk and funding issues
##### Abstract: In this talk we address new issues related to the pricing of interest rate derivatives after the recent credit crisis. Firstly, we present a multiple-curve interest rate model based on the HJM methodology and driven by time-inhomogeneous Levy processes. This model is capable of capturing various spreads between quantities that had been essentially the same before the crisis and which emerged due to the crisis. Using the model a "clean value" of a portfolio of interest rate derivatives is calculated, which is a value in a hypothetical situation where the two parties would be risk-free and funded at a risk-free rate. Next, to account for counterparty risk and funding issues, various valuation adjustments have to be computed, jointly referred to as Total Value Adjustment (TVA). We streamline the reduced-form methodology through which the TVA pricing problem can be reduced to Markovian pre-default TVA BSDEs. Numerical results are presented for two different simple short rate interest rate models, thus also illustrating the related model risk issue.

06. Dezember 2012
Frank Seifried (Universität Kaiserslautern)
Post-Crisis Fixed Income Pricing: LIBOR Mechanics and Spreads
##### Abstract: In the wake of the financial crisis new pricing standards have emerged in the fixed income market that are inconsistent with classical interest rate models. In this talk I present a framework for pricing interest rate derivatives given the new market situation. The model is able to generate the most important stylized market features: LIBOR-OIS Speads, Tenor Basis Spreads, and the discrepancy between FRA rates and those implied by spot LIBOR quotes. In contrast to existing multi-curve approaches, the relevant spreads are generated endogenously from the underlying risk factors; i.e. credit and liquidity risk. We explicitly model the mechanism determining the LIBOR rate from spot quotes of individual panel banks using a reduced-form framework for the credit component. In addition, we capture liquidity risk in a structured liquidity model. Our framework thus provides a theoretical underpinning of the multi-curve approach. Finally I show how our model can be used in pricing applications.

Simone Cerreia Vioglio (Università Bocconi)
Put-–Call Parity and Market Frictions
##### Abstract: We extend the Fundamental Theorem of Finance and the Pricing Rule Representation Theorem of Cox and Ross (see Ross 1973 and 1976 and Cox and Ross 1976) to the case in which market frictions are taken into account but the Put-–Call Parity is still assumed to hold. In turn, we obtain a representation of the pricing rule as a discounted expectation with respect to a nonadditive risk neutral probability. As a further contribution, in so doing we endogenize the state space structure and the contingent claim representation usually assumed to represent assets and markets.

20. Dezember 2012
Christel Geiss (Universität Insbruck)
Approximation of BSDEs with non-smooth terminal conditions driven by Levy noise
##### Abstract: We consider {\it backward stochastic differential equations} (BSDEs) of the form \begin{eqnarray*} Y_t&=& H + \int_t^T f \bigg (s,Y_s, \int_{{\rm I\!R}} Z_{s,x} \kappa(dx) \bigg )ds - \sigma\int_{(t,T]} Z_{s,0} d W_s \\ & &- \int_{(t,T] \times {\rm I\!R}} Z_{s,x} \, x \tilde{N}(dt,dx), \quad 0 \le t \le T, \end{eqnarray*} where $(\sigma W_s)$ is the Brownian part and $\tilde{N}$ denotes the compensated Poisson random measure associated with a square integrable L\'evy process $(X_t).$ We assume that the generator $f$ is Lipschitz and $\kappa$ is a certain measure on $\R.$ \bigskip \\ The convergence rate of the discrete-time approximation of this BSDE depends on the $L_2$-regularity of the exact solution $(Y,Z)$ and on the convergence of the discretized terminal condition. The topic of the talk is the relation between the $L_2$-regularity of $(Y,Z)$ and fractional smoothness properties in the Malliavin sense of the terminal condition $H \in L_2.$ The approach is possible, for example, if $H \in L_2$ can be described by a Borel function of finitely many increments of $(X_t).$ \\ This is joint work with A. Steinicke.

Stefan Geiss (Universität Insbruck)
BMO-estimates for BSDEs
##### Abstract: We consider a backward stochastic differential equation $$dY_t = -f(t,Y_t,Z_t) dt + Z_t dW_t$$ with a random generator $f$ and a standard Brownian motion $W$ and prove exponential tail estimates for the variation of the $Y$-process and the $Z$-process under certain conditions on the generator and the terminal condition. This is done by a weighted John-Nirenberg theorem and using a decoupling technique for the Wiener space.

17. Januar 2013
Frank Riedel (Universität Bielefeld)
Intertemporal Equilibria with Knightian Uncertainty: Insurance and Market Breakdown
##### Abstract:We study a dynamic and infinite-dimensional model with Knightian uncertainty modeled by incomplete multiple prior preferences. In interior efficient allocations, agents share a common risk-adjusted prior and use the same subjective interest rate. Interior efficient allocations and equilibria coincide with those of economies with subjective expected utility and priors from the agents' multiple prior sets. We show that the set of equilibria with inertia contains the equilibria of the economy with variational preferences anchored at the initial endowments. A case study in an economy without aggregate uncertainty shows that risk is fully insured, while uncertainty can remain fully uninsured. Pessimistic agents with Gilboa-Schmeidler's max-min preferences would fully insure risk and uncertainty.

31. Januar 2013
Paolo Guasoni (Dublin City University)

N.N.
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14. Februar 2013
ACHTUNG: Der Vortrag beginnt 16 Uhr s.t.!
Emmanuel Gobet (CMAP-Ecole Polytechnique, Paris)
Preliminary control variates for improving empirical regression methods
##### Abstract: Empirical regression methods to compute E(H|X=x) are widely used in many Monte-Carlo algorithms to solve optimal stopping problems, Backward Stochastic Differential Equations and various stochastic control problems: these algorithms are often referred to as Least Squares Monte-Carlo methods. However in some situations, the number of simulations is constrained to be relatively small (due to restriction on the computational time). That case may cause significant inaccuracy in the least square regression method, because the statistical part of the global error can be dominant, in particular if the conditional variance Var(H|X) is large. The purpose of this work is to design a flexible method to significantly reduce it: this is based on Preliminary Control Variates. We present theoretical and numerical results.

Cody Hyndman ( Concordia University)
A Convolution Method for Backward Stochastic Differential Equations

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