### The 2014-15 London Mathematical Finance Seminar Series will be hosted by King's College London in the first term of the academic session.

To subscribe the seminar email list: https://mailman.kcl.ac.uk/mailman/listinfo/fm-seminars-a

**Date: 9 October 2014**

**Speaker: Sam Cohen, University of Oxford **

Time: 16:30-17:30

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: **Ergodic BSDEs with Lévy noise and time dependence**

Abstract: In many control situations, particularly over the very long term, it is sensible to consider the ergodic value of some payoff. In this talk, we shall see how this can be studied in a weak formulation, using the theory of ergodic BSDEs. In particular, we shall consider the case where the underlying stochastic system is infinite dimensional, has Lévy-type jumps, and is not autonomous. We shall also see how this type of equation naturally arises in the valuation of a power plant.

**Speaker: Sergei Levendorskiy, University of Leicester**

Time: 17:45-18:45

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: **Efficient Laplace and Fourier inversions and Wiener-Hopf factorization in financial applications**

Abstract: A family of (quasi-) parabolic contour deformations increases the speed and accuracy of calculation of fairly complicated oscillatory integrals in option pricing formulas in many cases when standard approaches are either too slow or inaccurate or both. Variations: quasi-asymptotic formulas that are simple and much faster than general formulas, and which, for typical parameter values, are fairly accurate starting from relatively small distances from the barrier and maturities more than a year. When several Laplace and Fourier inversions are needed, it is necessary to use a family of contour transformations more flexible than Talbot's deformation of the contour in the Bromwich integral. Further step in a general program of study of the efficiency of combinations of one-dimensional inverse transforms for high-dimensional inversions [Abate-Whitt, Abate-Valko and others].

Calculations of Greeks and pdf can be made much more accurate; the latter can be used for fast Monte-Carlo simulations (faster than Madan-Yor method). Examples when insufficiently accurate pricing procedures may prevent one to see a good model (“sundial calibration”) or to see a local minimum of the calibration error when there is none, and the model may be unsuitable (“ghost calibration”) will be presented.

**Date: 23 October 2014**

**Speaker: Jan Kallsen, University of Kiel**

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: **On portfolio optimization and indifference pricing with small transaction costs **

Abstract: Portfolio Optimization problems with frictions as e.g. transaction costs are hard to solve explicitly. In the limit of small friction, solutions are often of much simpler structure. In the last twenty years, considerable progress has been made both in order to derive formal asymptotics as well as rigorous proofs. However, the latter usually rely on rather strong regularity conditions, which are hard to verify in concrete models. Some effort is still needed to make the results really applicable in practice. This talk is about a step in this direction. More specifically, we discuss portfolio optimization for exponential utility under small proportional transaction costs. As an example, we reconsider the Whalley-Willmott results of utility-based pricing and hedging in the Black-Scholes model. We relax the conditions required by Bichuch who gave a rigorous proof for smooth payoffs under sufficiently small risk aversion.

**Speaker: Martijn Pistorius, Imperial College London**

Time: 17:45-18:45

Title: *Optimal time to sell a stock with a jump to default *

Abstract: We consider the problem of identifying the optimal time to sell a defaultable asset in the sense of minimizing the "prophet's drawdown" which is the ratio of the ultimate maximum (up to a random default time) and the value of the asset price at the moment of sale. We assume that default occurs at a constant rate, and that at the moment of default there is a random recovery value of $\rho(100)\%$. This problem is transformed to an optimal stopping problem, which we solve explicitly in the case that the asset price before default is modelled by a spectrally negative exponential Levy process. This is joint work with A. Mijiatovic.

**Date: 6 November 2014**

**Speaker: Martin Schweizer, ETH Zurich**

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: **A new approach for stochastic Fubini theorems**

Abstract: We prove a new stochastic Fubini theorem in a setting where we stochastically integrate a mixture of parametrised integrands, with the mixture taken with respect to a stochastic kernel instead of a fixed measure on the parameter space. To that end, we introduce a notion of measure-valued stochastic integration with respect to a multidimensional semimartingale. As an application, we show how one can handle a class of quite general stochastic Volterra semimartingales. The original question for this work came from a problem in mathematical finance, and we also briefly comment on that. The talk is based on joint work with Tahir Choulli (University of Alberta, Edmonton).

**Speaker: Knut Aase, Norwegian School of Economics**

Time: 17:45-18:45

Title: *Beyond the local mean-variance analysis in dynamic economics: Recursive utility etc *

Abstract: I derive the equilibrium interest rate and risk premiums using recursive utility for jump-diffusions. Compared to to the continuous version, including jumps allows for a separate risk aversion related to jump size risk in addition to risk aversion related to the continuous part. We consider the version of recursive utility which gives the most unambiguous separation of risk preference from time substitution, and use the stochastic maximum principle to analyze the model. This method uses forward/backward stochastic differential equations. The model with jumps is shown to have a potential to give better explanation of empirical regularities than the recursive models based on merely continuous dynamics. Deviations from normality in the conventional model are also treated.

**Date: 20 November 2014**

**Speaker: Gordan Zitkovic, University of Texas at Austin**

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: *On the dynamic programming principle for problems posed over martingale measures*

Abstract: After an overview of the existing results on dynamic programming in continuous time, a simple abstract framework in which it holds will be described, and then used to analyze a class of problems posed over the "set of martingale measures". As an application, the super-replication and utility-maximization problems in a rather general family of incomplete Markovian financial market models will be treated.

**Speaker: **Sergio Pulido, **Swiss Finance Institute**

Time: 17:45-18:45

Title: *Existence and uniqueness results for multi-dimensional quadratic BSDEs arising from a price impact model with exponential utility*

Abstract: In this work we study multi-dimensional systems of quadratic BSDEs arising from a price impact model where an influential investor trades illiquid assets with a representative market maker with exponential preferences. The impact of the strategy of the investor on the prices of the illiquid assets is derived endogenously through an equilibrium mechanism. We show that a relationship exists between this equilibrium mechanism and a multi-dimensional system of quadratic BSDEs. We also specify conditions on the parameters of the model that guarantee that the system of BSDEs has a unique solution, which corresponds to a family of unique equilibrium prices for the illiquid assets. The proof relies on estimates that exploit the structure of the equilibrium problem. Finally, we provide examples of parameters for which the corresponding system of BSDEs in not well-posed.

Joint work with Dmitry Kramkov.

**Date: 4 December 2014**

**Speaker: Ragnar Norberg, University Lyon 1**

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: **On Marked Point Processes: Modelling, Stochastic Calculus, and Computational Issues**

Abstract: The talk starts with a friendly introduction to marked point processes and their associated counting processes and martingales. Then it proceeds to three distinct, still intertwined, aspects of the theory: *Modelling* is a matter of specifying the intensities, which are the fundamental model entities with a clear interpretation as instantaneous transition probabilities; *Prediction* is a matter of calculating conditional expected values of functionals of the process, which involves stochastic calculus (can be made simple); *Computation* is a matter of solving Ordinary or Partial Integral-Differential Equations, looking for shortcuts (ODEs replacing PDEs) and looking out for pitfalls (non-smoothness points that cannot be detected by inspection of the equations). The unifying powers and the versatility of the model framework are demonstrated with examples from risk theory, life insurance, and non-life insurance.

**Speaker: ****Michael Kupper, University of Konstanz**

Time: 17:45-18:45

Title: *Robust Pricing Dualities*

**Date: 11 December 2014**

**Speaker: ****Peter Imkeller, Humboldt University Berlin**

Time: 16:30-17:30

Place: Strand Campus, S-1.27 (1st basement, Strand Building)

Title: *Cross hedging, (F)BSDE of quadratic growth and convex duality*

Abstract: A financial market model is considered on which agents (e.g. insurers) are subject to an exogenous financial risk, which they trade by issuing a risk bond. They are able to invest in a market asset correlated with the exogenous risk. We investigate their utility maximization problem, and calculate bond prices using utility indifference. In the case of exponential utility, this hedging concept is interpreted by means of martingale optimality, and solved - even for non-convex constraints - with BSDE with drivers of quadratic growth. For more general utility functions defined on the whole or nonnegative real linewe show that if an optimal strategy exists then it is given in terms of the solution (X; Y;Z) of a fully coupled FBSDE. Conversely if the FBSDE admits a solution (X; Y;Z) then an optimal strategy can be obtained.

On a general stochastic basis, and with liability 0, we finally combine this FBSDE approach with the duality approach by Kramkov-Schachermayer who provide an abstract existence and uniqueness result for the optimal hedging strategy. Under some regularity conditions on the utility functions we associate to their solution a constructive one given by a numerically accessible FBSDE system describing the optimal investment process as the forward component, and a functional of the dual optimizer as the backward one. This is joint work with U.Horst, Y. Hu, V. Nzengang, A. Reveillac, and J. Zhang.

**Speaker: **Dylan Possamaï**,****Université Paris Dauphine**

Time: 17:45-18:45

Title: *BSDEs, existence of densities and Malliavin calculus*

Abstract: In recent years the field of Backward Stochastic Differential Equations (BSDEs) has been a subject of growing interest in stochastic calculus as these equations naturally arise in stochastic control problems in Finance, and as they provide Feynman-Kac type formulae for semi-linear PDEs. Since it is not generally possible to provide an explicit solution to these equations, one of the main issues especially regarding the applications is to provide a numerical analysis for the solution of a BSDE. This calls for a deep understanding of the regularity of the solution processes Y and Z. Here, we focus on the marginal laws of the random variables Yt, Zt at a given time t in (0,T). More precisely, we are interested in providing sufficient conditions ensuring the existence of a density (with respect to the Lebesgue measure) for these marginals on the one hand, and in deriving some estimates on these densities on the other hand. This type of information on the solution is of theoretical and of practical interest since the description of the tails of the (possible) density of Zt would provide more accurate estimates on the convergence rates of numerical schemes for quadratic growth BSDEs. This issue has been pretty few studied in the literature, since up to our knowledge only references [2, 1] address this question. The first results about this problem have been derived in [2], where the authors provide existence of densities for the marginals of the Y component only and when the driver h is Lipschitz continuous in (y,z), and some smoothness properties of this density. Concerning the Z component, much less is known since existence of a density for Z has been established in [1] only under the condition that the driver is linear in z. We revisit and extend the results of [2, 1] by providing sufficient conditions for the existence of densities for the marginal laws of the solution Yt,Zt (with t an arbitrary time in (0,T)) of a qgBSDE with a terminal condition ξ in (1) given as a deterministic mapping of the value at time T of the solution to a one- dimensional SDE, together with estimates on these densities. En route to these results, we provide new conditions for the Malliavin differentiability of solutions of Lipschitz or quadratic BSDEs. These results rely on the interpretation of the Malliavin derivative as a Gâteaux derivative in the directions of the Cameron-Martin space. Incidentally, we provide a new formulation for the characterization of the Malliavin-Sobolev type spaces D1,p.

The talk is based on joint works with Peter Imkeller, Thibaut Mastrollia and Anthony Reveillac.