2021 Seminars

January 15, 2021: Arne Bang Huseby (University of Oslo)

Title: Optimal reinsurance contracts in the multivariate case

Abstract: An insurance contract implies that risk is ceded from ordinary policy holders to companies.  However, companies do the same thing between themselves.  This is known as reinsurance, and the ceding company is known as the cedent.  The rationale could be the same; i.e., that a financially weaker agent is passing risk to a stronger one. In reality even the largest companies do this to diversify risk, and financially the cedent may be as strong as the reinsurer.  The problem of determining re­in­su­rance contracts which are optimal with respect to some reasonable criterion has been studied extensively within actuarial science.  Different contact types are considered such as stop-loss contracts where the reinsurance company covers risk above a certain level, and insurance layer contracts where the reinsurance company covers risk within an interval.  The contracts are then optimized with respect to some risk measure, such as value-at risk (VaR) or conditional tail expectation (CTE). In this seminar we consider the problem of minimizing VaR in the case of multiple insurance layer contracts.  Such contracts are known to be optimal in the univariate case, and the optimal contract is easily determined.  In the multivariate case, however, finding the optimal set of contracts is not easy.  In fact the optimal contract is not even unique in this case.  Still by considering solutions where the risk is balanced between the contracts, a solution can be found using an iterative Monte Carlo method.


January 29, 2021: Josep Vives (University of Barcelona)

Title: Decomposition and high order approximation of option prices. Some applications to Heston, Bates, CEV and rough volatility models.

Abstract: Using Itô calculus techniques we present an option price decomposition for local and stochastic volatility jump diffusion models and we use it to obtain fast and accurate approximations of call option prices for different local or stochastic volatility models.

The main purpose is to present the ideas given in the recent papers

A. Gulisashvili, M. Lagunas, R. Merino and J. Vives (2020): “Higher order approximation of call option prices in stochastic volatility models”. Journal of Computational Finance 24 (1).

But I will also comment ideas of the papers:

E. Alòs, R. De Santiago and J. Vives (2015): “Calibration of stochastic volatility models via second order approximation: the Heston case”. International Journal of Theoretical and Applied Finance 18 (6): 1550036 (31 pages).

J. Vives (2016): “Decomposition of the pricing formula for stochastic volatility models based on Malliavin – Skorohod type calculus”. Proocedings of the Research School CIMPA-UNESCO-MSER-MINECO-MOROCCO on Statistical Methods and Applications in Actuarial Science and Finance 2013. Springer.

R. Merino and J. Vives (2017): “Option price decomposition in local volatility models and some Applications”. International Journal of Stochastic Analysis. Volume 2017, Article ID 8019498, 16 pages

R. Merino, J. Pospísil, T. Sobotka and J. Vives (2018): “Decomposition formula for jump diffusion models”. International Journal of Theoretical and Applied Finance 21 (8).

R. Merino, J. Pospisil, T. Sobotka, T. Sottinen and J. Vives (2021): “Decomposition formula for rough Volterra stochastic volatility models”. Submitted.


February 12, 2021: Emil R. Framnes (Global Head of Trading Norges Bank Investment Management)

Title: Equity trading at NBIM

Abstract: Emil will give an introduction to Norges Bank Investment Management and its trading operations. His presentation will mainly focus on trading in equity markets and feature some of the dynamics and characteristics of the equity market and explain how various participants like institutional managers, high frequency traders and retail clients trade and shape equity markets today.


February 19, 2021: Nacira Agram (Linnaeus University)

Title: Deep learning and stochastic mean-field control for a neural network model

Abstract: We study a membrane voltage potential model by means of stochastic  control of mean-field stochastic differential equations and by machine learning techniques. The mean-field stochastic control problem is a new type, involving the expected value of a combination of the state X(t) and the running control u(t) at time t. Moreover, the control is two-dimensional, involving both the initial value z of the state and the running control u(t).
We prove a necessary condition for optimality and a verification theorem of a control (u; z) for such a general stochastic mean-field problem. The results are then applied to study a particular case of a neural network problem, where the system has a drift given by E[u(t)X(t)] and the problem is to arrive at a terminal state value X(T) which is close in terms of variance to a given terminal value F under minimal costs, measured by z^2 and the integral of u^2(t).
This problem is too complicated to handle by mathematical methods alone. We solve it using deep learning techniques.
The talk is based on joint work with A. Bakdi and B. Øksendal at University of Oslo.


March 5, 2021: Annika Lang (Chalmers University of Technology)

Title: The stochastic wave equation on the sphere: properties and simulation

Abstract: The stochastic wave equation driven by isotropic Gaussian noise is considered on the unit sphere. We solve this stochastic partial differential equation and discuss properties of the derived solutions. These are used in the developed approximation scheme based on spectral methods and its convergence analysis. We derive strong, weak, and almost sure convergence rates for the proposed algorithm and show that these rates depend only on the smoothness of the driving noise, the initial conditions, and the test functions. Numerical experiments confirm the theoretical rates. Finally we discuss extensions to more general domains and equations that can be treated in a similar way.

This talk is based on joint work with David Cohen.


March 19, 2021: Alexander Lobbe (University of Oslo)

Title: Pathwise approximations for the solution of the non-linear filtering problem


Abstract: Stochastic Filtering deals with the recovery of the state of a signal process from noisy observations.
Filtering models are ubiquitous within science and engineering, weather prediction being only one important example. In such applications, accurate, fast, and stable algorithms for the approximation of the filtering functional are essential.
After introducing the stochastic filtering framework, we consider high order approximations of the solution of the stochastic filtering problem and derive their pathwise representation in the spirit of earlier work by Clark and Davis. The robustness property of the derived approximation is subsequently proved. Thus, we establish that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space.

Joint work with Dan Crisan and Salvador Ortiz-Latorre


April 16, 2021: Arne Løkka (London School of Economics)

Title: Foreign exchange equilibrium, international trade and trading costs

Abstract: In this paper we prove existence and uniqueness of an equilibrium for an international economy consisting of two separate economies and a complete financial market. Each economy produce a single perishable good and trade between the two economies carries proportional trading costs. In each economy there are a number of agents aiming to maximise their expected utility of consumption of the single perishable good. We draw on the methods used for the one economy case using the Negishi argument, and obtain semi-explicit formulas for the equilibrium solutions. In order to prove uniqueness, we establish that for any equilibrium, the consumptions must be Pareto optimal. To account for the costs of trading between the economies, this requires a modification of the standard notion of feasible allocations and Pareto optimality. Our results therefore generalise the theory for the one economy in a number of interesting ways that offer new insights and perspectives. Models of international economies with proportional trading costs have received a lot of attention in economics, but as far as we know, existence and uniqueness of an equilibrium have not rigorously been established.


April 30, 2021: Lyudmila Grigoryeva (University of Konstanz)

Title: Discrete-time signatures and randomness in reservoir computing
Abstract: A new explanation of geometric nature of the reservoir computing phenomenon is presented. Reservoir computing is understood in the literature as the possibility of approximating input/output systems with randomly chosen recurrent neural systems and a trained linear readout layer. Light is shed on this phenomenon by constructing what is called strongly universal reservoir systems as random projections of a family of state-space systems that generate Volterra series expansions. This procedure yields a state-affine reservoir system with randomly generated coefficients in a dimension that is logarithmically reduced with respect to the original system. This reservoir system is able to approximate any element in the fading memory filters class just by training a different linear readout for each different filter. Explicit expressions for the probability distributions needed in the generation of the projected reservoir system are stated and bounds for the committed approximation error are provided.


May 7, 2021: Dan Crisan (Imperial College London)

Title: Well-posedness Properties for a Stochastic Rotating Shallow Water Model 

Abstract: The rotating shallow water (RSW) equations describe the evolution of a compressible rotating fluid below a free surface. The typical vertical length scale is assumed to be much smaller than the horizontal one, hence the shallow aspect. The RSW equations are a simplification of the primitive equations which are the equations of choice for modelling atmospheric and oceanic dynamics. In this talk, I will present some  well-posedness properties of a viscous rotating shallow water system. The system is stochastically perturbed in such a way that two key properties of its deterministic counterpart are preserved. First, it retains the characterisation of its dynamics as the critical path of a variational problem. In this case, the corresponding action function is stochastically perturbed. Secondly, it satisfies the classical Kelvin circulation theorem.  The introduction of stochasticity replaces the effects of the unresolved scales.  The stochastic RSW equations are shown to admit a unique maximal strong solution in a suitably chosen Sobolev space which depends continuously on the initial datum. The maximal stopping time up to which the solution exist is shown to be strictly positive and,  for sufficiently small initial datum, the solution is shown global in time with positive probability. This is joint work with Dr Oana Lang (Imperial College London) and forms part of the ERC Synergy project “Stochastic transport in upper ocean dynamics” (https://www.imperial.ac.uk/ocean-dynamics-synergy/)


May 21, 2021: Gabriel Lord (Radboud University)

Title: Adaptive time-stepping for S(P)DEs​

Abstract: We present how adaptive time-stepping might be used to solve SDEs with non-Lipschitz drift (and potentially diffusion) combined with a tamed or similar method. We illustrate how to pick the timestep and look at strong convergence.  We then consider the extension to stochastic PDEs and will mention the two cases of additive and multiplicative noise and illustrate the results numerically.


June 11, 2021: Andrey A. Dorogovtsev (Institute of Mathematics, Ukrainian Academy of Sciences)

Title: Occupation and evolutionary measure-valued processes

Abstract: In the talk we consider two types of measure-valued processes constructed from the processes on the phase space. These are visitation processes and solutions to equations with interactions. We will discuss questions of stability and stochastic calculus for such processes. Applications to construction of loop eraised random walks are presented.

The talk is based on the joint work with Iryna Nishchenko and Jasmina Đorđević.


August 20, 2021: Ralf Korn (Department of Mathematics, TU Kaiserslautern)

Title: Least-Squares MC for Proxy Modeling in Life Insurance: Linear Regression and Neural Networks

Abstract: The Solvency Capital Requirement (SCR) is the amount of Available Capital that an insurer has to provide to be solvent by the end of the year with a probability of (at least) 99.5%. Due to regulations, the SCR should be calculated from the distribution of the one-year loss  if the insurer uses an interal model. Given the complicated cash flow projections of a life insurer, this calculation is a tremendous task and cannot be performed by a crude Monte Carlo approach. In this talk, we show how to overcome computational complexity by using the so called least-squares Monte Carlo approach in combination with both linear regression and a feedforward neural network. Here, it is particularly challenging to obtain the so-called ground truth to calibrate our models.


September 3, 2021: Blanka Horvath (King's College London)

Title: Data -Driven Market Simulators some simple applicatons of signature kernel methods in mathematical finance

Abstract: Techniques that address sequential data have been a central theme in machine learning research in the past years. More recently, such considerations have entered the field of finance-related ML applications in several areas where we face inherently path dependent problems: from (deep) pricing and hedging (of path-dependent options) to generative modelling of synthetic market data, which we refer to as market generation.
We revisit Deep Hedging from the perspective of the role of the data streams used for training and highlight how this perspective motivates the use of highly accurate generative models for synthetic data generation. From this, we draw conclusions regarding the implications for risk management and model governance of these applications, in contrast torisk-management in classical quantitative finance approaches.
Indeed, financial ML applications and their risk-management heavily rely on a solid means of measuring and efficiently computing (smilarity-)metrics between datasets consisting of sample paths of stochastic processes. Stochastic processes are at their core random variables with values on path space. However, while the distance between two (finite dimensional) distributions was historically well understood, the extension of this notion to the level of stochastic processes remained a challenge until recently. We discuss the effect of different choices of such metrics while revisiting some topics that are central to ML-augmented quantitative finance applications (such as the synthetic generation and the evaluation of similarity of data streams) from a regulatory (and model governance) perpective. Finally, we discuss the effect of considering refined metrics which respect and preserve the information structure (the filtration) of the marketand the implications and relevance of such metrics on financial results.


September 17, 2021: Stefano De Marco (CMAP, Ecole Polytechnique)

Title: On the implied and local volatility surfaces generated by rough volatility

Abstract: Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by such models, and supporting their calibration power to SP500 option data.

Rough volatility models also generate a local volatility surface, via the Markovian projection of the stochastic volatility (equivalently, via Dupire's formula applied to the model's option price surface). We complement the existing results with the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models encompassing the rough Bergomi model.

Notably, we observe that the celebrated "1/2 skew rule" linking the short-term at-the-money (ATM) skew of the implied volatility to the short-term ATM skew of the local volatility, a consequence of the celebrated "harmonic mean formula" of [Berestycki, Busca, and Florent, QF 2002], is replaced by a new rule: the ratio of the implied volatility and local volatility ATM skews tends to the constant 1/(H + 3/2) (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

Joint work with  Florian Bourgey, Peter Friz, and Paolo Pigato.


October 1, 2021: Mathieu Rosenbaum (CMAP, Ecole Polytechnique)

Title: A rough volatility tour from market microstructure to VIX options via Heston and Zumbach

Abstract: In this talk, we present an overview of recent results related to the rough volatility paradigm. We consider both statistical and option pricing issues in this framework. We notably connect the behaviour of high frequency prices to that of implied volatility surfaces, even for complex products such as the VIX.


October 15, 2021: Luca Galimberti (NTNU)

Title: Neural Networks in Fréchet spaces

Abstract: In this talk we present some novel results obtained by Fred Espen Benth (UiO), Nils Detering (University of California Santa Barbara) and myself on abstract neural networks and deep learning. More precisely, we derive an approximation result for continuous functions from a Fréchet space XX into its field F,(F∈{R,C})F,(F∈{R,C}). The approximation is similar to the well known universal approximation theorems for continuous functions from RnRn to RR with (multilayer) neural networks [1, 2, 3, 4]. Similar to classical neural networks, the approximating function is easy to implement and allows for fast computation and fitting.

Few applications geared toward derivative pricing and numerical solutions of parabolic partial differential equations will be outlined.

[1] G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4):303–314, 1989.
[2] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359–366, 1989.
[3] K.-I. Funahashi. On the approximate realization of continuous mappings by neural networks. NeuralNetworks, 2(3):183–192, 1989.
[4] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861–867, 1993.


November 5, 2021: 

Asma Khedher (University of Amsterdam)

Title: An infinite-dimensional affine stochastic volatility model

Abstract: We introduce a flexible and tractable infinite-dimensional stochastic volatility model. More specifically, we consider a Hilbert space valued Ornstein–Uhlenbeck-type process, whose instantaneous covariance is given by a pure-jump stochastic process taking values in the cone of positive self-adjoint Hilbert-Schmidt operators. The tractability of our model lies in the fact that the two processes involved are jointly affine, i.e., we show that their characteristic function can be given explicitly in terms of the solutions to a set of generalised Riccati equations. The flexibility lies in the fact that we allow multiple modeling options for the instantaneous covariance process, including state-dependent jump intensity.

Infinite dimensional volatility models arise e.g. when considering the dynamics of forward rate functions in the Heath-Jarrow-Morton-Musiela modeling framework using the Filipović space. In this setting we discuss various examples: an infinite-dimensional version of the Barndorff-Nielsen–Shephard stochastic volatility model, as well as a model involving self-exciting volatility.

Michèle Vanmaele (Ghent University)

Title: Mortality/Longevity Risk-Minimization with or without Securitization

Abstract: In this talk we will address the risk-minimization problem, with and without mortality securitization, à la Föllmer–Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes situations in which the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information—the public information, which is generated by the financial assets, and a larger flow of information that contains additional knowledge about the death time of an insured. We will derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. Next, we will quantify, as explicitly as possible, the effect of mortality on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. We will obtain risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds.
The talk is based on joint work with Tahir Choull (University of Alberta)i and Catherine Daveloose (Ghent University).


November 9, 2021: Julian Tugaut (Université Jean Monnet, Saint-Etienne)

Title: Exit-problem for Self-Stabilizing Diffusions

Abstract: In this talk, we will mainly be focused on the exit-problem (exit-time and exit-location) for McKean-Vlasov diffusions of self-stabilizing form. First, I will present the questions related to the exit-problem. Then, I will give some classical results about the exit-time namely Kramers'type law, by using Freidlin-Wentzell theory. In the second part of the talk, I will introduce the self-stabilizing processes by the mean-field system of interacting particles. Then, I will give classical results when the potentials (confining and interacting) are both convexes. Also, I will present some results when the external force corresponds to a non-convex confining potential. The last part of the talk will deal with the exit-time for the McKean-Vlasov diffusion: first case when both potentials are convexes and second case (more challenging) when we do not assume uniform convexity property.


December 10, 2021:

Carlo Sgarra (Politecnico di Milano)

Title: Optimal Reinsurance Strategies in a Partially Observable Contagion Model

Abstract: We investigate the optimal reinsurance problem when the loss process exhibits jump clustering features and the insurance company has restricted information about the loss process. We maximize expected exponential utility and show that an optimal solution exists. We provide the equation governing the dynamics of the (infinite-dimensional) filter and characterize the solution of the stochastic optimization problem as the solution of a BSDE.

Sven Karbach (University of Amsterdam)

Title: Positive multivariate CARMA processe

Abstract: In this talk we discuss positivity of multivariate continuous-time autoregressive moving-average (MCARMA) processes. In particular, we introduce matrix valued MCARMA processes and derive sufficient and necessary conditions such that the processes leave the cone of positive semi-definite matrices invariant. MCARMA processes on the cone of positive semi-definite matrices can be used to model e.g. the instantaneous covariance process in multivariate stochastic volatility models.

 

Published Jan. 17, 2022 11:59 AM - Last modified Jan. 17, 2022 11:59 AM