# Graduate Seminar Financial- and Actuarial Mathematics LMU and TUM (SoSe 2019)

## Organisers

Prof. F. Biagini, Prof. C. Czado, Prof. K. Glau, Prof. C. Klüppelberg, Prof. T. Meyer-Brandis, Prof. M. Scherer, Prof. R. Zagst

## Dates - May

### Monday, 13.05.2019 - 1. Speech

**Speaker:**Rama Cont, Oxford University

**Topic:** Rough calculus: pathwise calculus for functionals of irregular paths

**Time:** 14:15 o'clock

**Place: **LMU Mathematics Institute, Theresienstraße 39-B (Room B 349)

**Description: **Hans Foellmer showed that the Ito formula holds pathwise, for functions paths with finite quadratic variation along a sequence of partitions. We build on Foellmer's insight to construct a pathwise calculus for smooth functionals of continuous paths with regularity defined in terms of the p-th variation along a sequence of time partitions for arbitrary large p >0. We construct a pathwise integral, defined as a pointwise limit of compensated Riemann sums and show that it satisfies a change of variable formula and an isometry formula. Results for functions are extended to path-dependent functionals using the concept of vertical derivative of a functional. Finally, we obtain a "signal plus noise" decomposition for regular functionals of paths with strictly increasing p-th variation. Our results apply to sample paths of semi-martingales as well as fractional Brownian motion with arbitrary Hurst parameter H>0.

Based on joint work with: Anna Ananova (Oxford), Henry Chiu (Imperial College London) and Nicholas Perkowski (Humboldt).

### Monday, 13.05.2019 - 2. Speech

**Speaker:**Russell Gerrard, CASS Business School

**Topic:** An optimal investment strategy for future pension plans

**Time:** 15:00 o'clock

**Place: **LMU Mathematics Institute, Theresienstraße 39-B (Room B 349)

**Description: **Since the work of Merton (1969) the construction of optimal investment portfolios in a time-homogeneous market has been well understood. We review the situation when market parameters are allowed to vary stochastically and apply this to the problem of constructing a pension investment scheme which provides a guaranteed minimum sum on retirement at the expense of imposing an upper limit. We limit ourselves to the accumulation phase, during which payments are made into the fund. The aim of the overall project is to devise pension investment strategies whose behaviour is close to optimal but which present investors with easily explained choices.

### Monday, 13.05.2019 - 3. Speech

**Speaker:**Alessandra Cretarola, Perugia University

**Topic:** Indifference Pricing of Pure Endowments via BSDEs under Partial Information

**Time:** 16:00 o'clock

**Place: **LMU Mathematics Institute, Theresienstraße 39-B (Room B 349)

**Description: **We investigate the pricing problem of a pure endowment contract when the insurer has a limited information on the mortality intensity of the policyholder. The payoff of this kind of policies depends on the residual life time of the insured as well as the trend of a portfolio traded in the financial market, where investments in a riskless asset, a risky asset and a longevity bond are allowed. We propose a modeling framework that takes into account mutual dependence between the financial and the insurance markets via an observable stochastic process, which affects the risky asset and the mortality index dynamics. Since the market is incomplete due to the presence of basis risk, in alternative to arbitrage pricing we use expected utility maximization under exponential preferences as evaluation approach, which leads to the so-called indifference price. Under partial information this methodology requires filtering techniques that can reduce the original control problem to an equivalent problem in complete information. Using stochastic dynamics techniques, we characterize the indifference price of the insurance derivative via the solutions of suitable backward stochastic differential equations.

## Dates - June (24.06.2019, Speakers and titles TBD)

## Dates - July (15.07.2019)

### Monday, 15.07.2019 - 1. Speech

**Speaker:**Matteo Buzoni, ETH Zürich

**Topic:** Risk measures and benchmark distributions

**Time:** 14:15 - 15:00

**Place: **LMU Mathematics Institute, Theresienstraße 39-B (Room B 349)

**Description: **In this talk we discuss how the commonly used risk measures fail to control the probability of exceeding given level of losses. In order to address this aspect of tail risk we propose new classes of law invariant risk measures that generalize Value at Risk and Expected Shortfall. The key ingredient is a benchmark function which allows to specify admissibility constraints based on the whole tail of the distribution. As an interesting particular case, stochastic dominance constraints can be used in order to define acceptability. We present the main finance theoretical and statistical properties of this new class of risk measures and for the convex case we show its dual representation. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented. Based on joint works with Cosimo Munari, Valeria Bignozzi and Ruodu Wang.

### Monday, 15.07.2019 - 2. Speech

**Speaker:**Lukas Gonon, University of St. Gallen

**Topic:** Random neural networks, reservoir computing and hedging by deep learning techniques

**Time:** 15:00 - 15:45

**Place: **LMU Mathematics Institute, Theresienstraße 39-B (Room B 349)

**Description: T**his talk presents our recent work on hedging derivatives under market frictions by deep learning techniques and approximation error bounds for random recurrent neural networks. We first consider the problem of optimally hedging a portfolio of derivatives in a scenario based discrete-time market with transaction costs. Risk-preferences are specified in terms of a convex risk-measure. Such a framework has suffered from numerical intractability up until recently, but this has changed thanks to technological advances: using hedging strategies built from neural networks and machine learning optimization techniques, optimal hedging strategies can be approximated efficiently. In order to optimize the choice of the network architecture we then study approximation properties of reservoir computing systems, putting particular emphasis on random recurrent neural networks. We prove that only a small proportion of the parameters need to be trained and obtain high-probability bounds on the generalization error in terms of the network parameters. The talk is based on joint work with Hans Bühler, Josef Teichmann and Ben Wood as well as joint work in progress with Lyudmila Grigoryeva and Juan-Pablo Ortega.