We propose two probabilistic numerical methods for mean field type problems based on deep learning. The first method amounts to solve mean field control problems (i.e., problems of optimal control of McKean-Vlasov dynamics) by learning the optimal control using Monte-Carlo samples and stochastic gradient descent. This can be done in a somewhat brute force fashion thanks to deep learning. The second method deals with forward-backward stochastic differential equation (FBSDE) systems of mean field type. As such, this method can be applied to both mean field control problems and mean field games. We rephrase the problem of finding a solution to a generic mean field FBSDE system as a certain mean field control problem, and we then apply a variant of the first method proposed. Several numerical examples will be provided. This is joint work with René Carmona (Princeton University).

## Probabilistic numerical methods for MFC and MFG based on deep learning

## Financial Contagion in a Generalized Stochastic Block Model

One of the most defining features of modern financial networks is their inherent complex and intertwined structure. In particular the often observed core-periphery structure plays a prominent role. Here we study and quantify the impact that the complexity of networks has on contagion effects and system stability, and our focus is on the channel of default contagion that describes the spread of initial distress via direct balance sheet exposures. We present a general approach describing the financial network by a random graph, where we distinguish vertices (institutions) of different types – for example core/periphery – and let edge proba- bilities and weights (exposures) depend on the types of both the receiving and the sending vertex. Our main result allows to compute explicitly the systemic damage caused by some initial local shock event, and we derive a complete characterization of resilient respectively non-resilient financial systems. Due to the random graphs approach these results bear a considerable robustness to local uncertainties and small changes of the network structure over time. In particular, it is possible to condense the precise micro-structure of the network to macroscopic statistics. Applications of our theory demonstrate that indeed the features captured by our model can have significant impact on system stability; we derive resilience conditions for the global network based on subnetwork conditions only. (Joint with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter (LMU)

## Epstein-Zin utility and its utility maximization

Epstein-Zin utility is widely used in many macro-economics and asset pricing models because it decouples risk aversion and elasticity of intertemporal substitution. In the first part of the talk, we will review results on existence and uniqueness of finite horizon Epstein-Zin utilities. Then we will present new results on infinite horizon Epstein-Zin utilities. In the second part of the talk, we will consider an optimal consumption and investment problem for Epstein-Zin utilities from two approaches: control of BSDEs and convex duality.

## Option Pricing under Jump Uncertainty

We study the problem of European and American option pricing in the presence of uncertainty about the timing and the size of a jump in the price of the underlying. In a non-Markovian market setting, we characterize the worst-case option price as the minimal solution of a constrained backward stochastic differential equation and derive a pricing PDE in the special case of a Markovian market model. In a Black-Scholes market, explicit pricing formulae for European call and put options are obtained, and we study properties of the American put option price numerically.

## Dealer Funding and Market Liquidity

When a client wants to exit a position, dealers can provide immediacy by taking over the position.We consider a model in which dealers need to raise external nance to do so, and can subsequently exert unobservable effort to improve the chance of closing the positions that they take over at a profit. This moral hazard problem affects how and how much external finance dealers can raise. Therefore, it limits intermediation volume, soften competition between dealers, and widens bid-ask spreads. When dealers suffer losses, the problem becomes worse. Effects are stronger for riskier assets. Endogenous correlation and contagion in liquidity arise between otherwise unrelated assets. As the optimal financing arrangement involves debt, regulations that limits the leverage of bank-affiliated dealers can have adverse effects on market liquidity.

## Recurrent and transient transformations for one-dimensional di

I will present a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. It turns out that these transformations are very useful in Euler schemes for killed diffusions, simplifying the solutions of optimal stopping problems with discounting, and characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well-known for not having unique solutions. I will give a description of these transformations and discuss their connections with h-transforms and Schroedinger semigroups, and how one can use them to solve the above problems.

## Approximate Local Volatility Model for Vanilla Rates Options

In this presentation we analyse a model for the pricing of vanilla interest rate options (e.g caps/floors and European swaptions). Within that model we specify a parametric form of the terminal distribution of the underlying rate. The driver of the distribution is a Brownian motion and the parametric form is closely linked to local volatility models. We choose the local volatility function such that the model allows analytic pricing of vanilla and CMS options.

The parametrisation in terms of a local volatility function provides transparent intuition of the model parameters as well as high flexibility for smile calibration. Moreover, the linkage to an underlying Brownian motion may be used as a normalising basis for interpolation between expiries and swap terms.

## Lifting the Heston model

How to reconcile the classical Heston model with its rough counterpart? We introduce a lifted version of the Heston model with n multifactors sharing the same Brownian motion but mean reverting at different speeds. Our model nests as extreme cases the classical Heston model (when n=1) and the rough Heston model (when n goes to infinity). We show that the lifted model enjoys the best of both worlds: Markovianity and satisfactory fits of implied volatility smiles for short maturities. Further, our approach speeds up the calibration time and opens the door to time-efficient simulation schemes.

## The Deutsche Bank Risk Center

**Bernadette Walter:**

The Quant Institute at Deutsche Bank Risk Center: A center for development and validation of models for Risk Management across major risk types

**TBD:**

Model risk in the model landscape for the trading book

We give an overview of the challenges banks face in identifying and measuring model risk in the complex model landscape required to run a trading operation. We explain the changes in the assessment and regulation of model risk since the Global Financial Crisis, and the consequences this has had on modelling and model governance in banks. We will use the example of pricing models to illustrate this and to articulate the problems that remain to be solved.

**Michael Kalkbrener:**

Credit rating migration processes based on economic-state-dependent transition matrices

We develop a model for rating migration. The objective is to study rating migration processes and corresponding default rates. It is generally accepted that rating migrations depend on economic factors. We use the theory of time-homogeneous Markov chains to jointly model the rating process and the state of the economy. Although the rating process itself is neither Markovian nor time-homogeneous in general, we show that sequence of the rating process’ transition matrices converges to a limit. We further analyse the properties of different rating methods, namely point-in-time (PIT) ratings and through-the-cycle (TTC) ratings. Although these rating philosophies have become important from a regulatory perspective, to the best of our knowledge, no formal definition exists yet. We further discuss if and how a rating philosophy can be detected from given rating transition time series.

## Dynamics of a Well-Diversiﬁed Equity Index and Martingale Inference

The paper derives an endogenous model for the long-term dynamics of a well-diversiﬁed equity index with rough volatility, the S&P500. It assumes that the index is a proxy of the respective growth optimal portfolio, the variance of its increments evolves in some market time proportionally to the index value and the derivative of market time is a linear function of the squared derivative of a smoothed proxy of the single driving Brownian motion. The resulting model is highly tractable, allows almost exact simulation and leads beyond classical ﬁnance theory. Its parameters are estimated via a novel martingale inference method, which employs higher-strong order, implicit approximations of the increments of the system of stochastic diﬀerential equations.