# Plenary Talks

## Suleyman Basak (London Business School and CEPR)

## Belief Dispersion in the Stock Market

We develop a dynamic model of belief dispersion which simultaneously explains the empirical regularities in a stock price, its mean return, volatility, and trading volume. Our model with a continuum of (possibly Bayesian) investors differing in beliefs is tractable and delivers exact closed-form solutions. Our model has the following implications. We find that the stock price is convex in cash-flow news, and it increases in belief dispersion while its mean return decreases when the view on the stock is optimistic, and vice versa when pessimistic. We also show that the presence of belief dispersion leads to a higher stock volatility, trading volume, and a positive relation between these two quantities. Furthermore, we demonstrate that otherwise identical two-investor economies with heterogeneous beliefs do not necessarily generate our main results. This is joint work with Adem Atmaz.

## RĂĽdiger Frey (WirtschaftsuniversitĂ¤t Wien)

## Optimal liquidation under partial information with Price Impact

We study the problem of a trader who wants to maximize the expected reward from liquidating a given stock position. We model the stock price dynamics as a geometric pure jump process with local characteristics driven by unobservable to finite-state Markov chain and the liquidation rate. This reflects uncertainty about the activity of other traders and price impact from trading. We use stochastic filtering to reduce the optimization problem-under partial information to at equivalent one under complete information. This leads to a control a problem for piecewise deterministic Markov processes (in short PDMP). We apply control theory for PDMPs to our trouble. In Particular, we derive the optimality equation for the value function and we characterize the value function as unique viscosity solution of the associated dynamic programming equation. The talk Concludes with a detailed (numerical) analysis of specific examples. joint work with Katia Colaneri, Zehra Eksi and Michaela SzĂ¶lgyenyi

## Mete Soner (ETH ZĂĽrich)

## Viability, Arbitrage and Preferences

Consider a financial market in which all financial instruments are contracts X of cash flows up to time T. The value of any such cash flow X at time t is the cumulative non-discounted cash payments up to time t. Agents are then presented a set of contracts that are tradable with no cost and a cloud of possible weak orders among the contracts. A natural notion of viability is the existence of a preference relation that is consistent with this plausible orders so that all contracts are weakly preferred to any position obtained by adding a replicable contract to itself. Hence in an economy populated with agents with this "viable" preference relation every agent is content to remain at her endowment. This is an equivalent statement of viability defi ned by Harrison & Kreps in 1979. We will prove in this context that a market is viable if and only if there are no free lunches with vanishing risk. However, this notion also needs to be appropriately redefined. These notions are then shown to be equivalent to the existence of a sublinear expectation consistent with the market. This is joint work with Frank Riedel from Bielefeld and Matteo Burzoni from ETH.

# Invited Talks

## Peter Bank (TU Berlin)

## Scaling limits for super-replication under friction

We compute the continuous-time scaling limits of super-replication prices in discrete-time financial models with different market frictions. A seminal result in this direction for proportional transaction costs is due to Kusuoka (1995) and shows that the trivial super-replication prices no longer apply if one rescales binomial models along with the transaction costs. We give a multivariate extension of his findings. We also consider the case of smooth convex transaction costs under model uncertainty and a first case study into the more demanding problem of scaling limits for fixed transaction costs. This is based on joint papers with Yan Dolinsky, Selim GĂ¶kay and Ari-Pekka PerkkiĂ¶.

## Christian Bayer (WIAS)

## Short dated option pricing under rough volatility

We consider rough stochastic volatility models. Specifically, volatility has fractional - worse than diffusion - scaling, a regime which recently attracted considerable attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation result of Forde-Zhang in a way that allows to zoom-in around the money while maintaining full analytical tractability. Mathematically speaking, this amounts to prove higher order moderate deviations estimates, recently introduced in the option pricing context by Friz, Gerhold and Pinter. (Joint work with Peter Friz, Archil Gulisashvili, Blanka Horvath, Benjamin Stemper)

## Patrick Beißner (UniversitĂ¤t Bielefeld)

## Financial Equilibria under Volatility Uncertainty

In diffusion models, few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow-Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow-Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow-Debreu equilibrium allocations are generically unattainable by dynamically trading few long-lived assets.

## Min Dai (National University of Singapore)

## Bridging Dynamic Mean Variance and CRRA Utility

We propose a continuous-time mean-variance model that leads to the same trading strategy as in the Merton's model with CRRA utility. We interpret the risk aversion as the trade-off between mean and variance. We also extend the result to an incomplete market with stochastic volatility. This is a joint work with Hanqing Jin, Steven Kou, and Yuhong Xu.

## Nils Detering (University of California, Santa Barbara)

## Managing Default Contagion in Financial Networks

To quantify and manage systemic risk in the interbank market, we propose a weighted, directed random network model. The vertices in the network are financial institutions and the weighted edges represent monetary exposures between them. Our model resembles the strong degree of heterogeneity observed in empirical data and the parameters of the model can easily be fitted to empirical data. We derive asymptotic results that, based on these parameters, allow to determine the impact of local shocks to the entire system and the wider economy. At this, our model captures that the impact depends on the systemic importance of the defaulted institutions. Furthermore, we characterize resilient and non- resilient cases. For networks with degree sequences without second moment, a small number of initially defaulted banks can trigger a substantial default cascade even under the absence of so-called contagious links. Paralleling regulatory discussions we determine minimal capital requirements for financial institutions sufficient to make the network resilient to small shocks. The capital requirements are robust with respect to a miss-specification of the dependency structure of in- and out-degrees in the network. It is joint work together with Thilo Meyer-Brandis, Konstantinos Panagiotou and Daniel Ritter (all LMU Munich).

## Hermann Elendner(Humboldt-UniversitĂ¤t zu Berlin)

## Risk and Resiliency across crypto currencies and Markets

Since Bitcoin's introduction of the block chain, a broad cross-section of crypto-currencies has evolved. These digital tokens trade in fragmented markets and exhibit high volatility. I analyze the properties of return the most liquid crypto-currencies and place did contrary to the notion of a "value of block chain technology," they show low dependence. I investigate Whether this reflects the diverse nature of the Crypto risks or a lack of market integration by studying the market price of risk and spill-over effects of liquidity shocks. While integration is limited, liquidity improvements are Followed by Increases in market capitalization in the long-run. More over, in the short run, the liquidity of some Cryptos displays remarkable resiliency. Overall, while crypto markets are young and volatile. They appear surprisingly mature.

## Peter Frentrup (Humboldt-UniversitĂ¤t zu Berlin)

## Modeling Multiplicative Market Impact - Stability and Examples

We discuss issues in modeling a market with a large investor Whose trading of a single risky asset causes transient price impact. We postulate the evolution of the asset's price process in a way Which guarantees positivity of prices. The main trouble is to identify the functional gains from general (càdlàg) trading strategies as a continuous extension from the set of continuous finite variation strategies. For this purpose, the Skorokhod topology M1 turns out to be suitable. Having specified our model for a general class of controls, we consider application examples of optimal liquidation problems in infinite or constrained time horizons Which benefit from the continuity property. This talk is based on joint work with Dirk Becherer and Todor Bilarev from Humboldt-UniversitĂ¤t zu Berlin.

## Guanxing Fu (Humboldt-UniversitĂ¤t zu Berlin)

## Mean Field Models for Optimal Liquidation with Price Impact

Due to the untraceable nature of optimal portfolio liquidation for large population investors, we propose a mean field game(MFG) model for it. In our model, the strategy of each player is influenced by a common fundamental price process. It is shown that this more practical feature leads to the global solvability of the MFG. In some special case, the explicit solution is expected. This talk is based on joint work with Paulwin Graewe and Ulrich Horst.

## Jacob Funk (Princeton University)

## Dynamic Monopolies Under Stochastic Demand

We consider an infinite-horizon stochastic control problem in which a commodity producer sells goods from a limited supply. These problems are well-understood in models with fixed demand, but when demand is stochastic we find that new phenomena arise. One example is the concept of demand blockading, which occurs when the producer's optimal strategy is to voluntarily halt production. We present several theoretical results relating to the existence of demand blockading in certain models, and show that the absence of demand blockading is required in order for Hotelling's rule to hold in these markets.

## Sebastian Henningsen (Humboldt-UniversitĂ¤t zu Berlin)

## Decentralizing Central Banks: The Challenges of Monetary Policies on the block Chain

Decentralized block chain-based cryptocurrencies are digital assets, designed with the aim to revolutionize centrally banked currencies. We argue did money creation and the control of money supply in most cryptocurrencies are too inflexible in order to compete with traditional currencies. THEREFORE, we propose to design Decentralized monetary policies for cryptocurrencies. We identify and discuss the key challenges one is facing in did endeavor. In Particular we present current research on a multi-agent model Which captures existing proposals for Decentralized monetary policies as special cases. This model reduces comparisons between thesis approaches to solving portfolio optimization problems.

## Wei Jiang (National University of Singapore)

## Simulating Risk Measures

Risk measures, such as value-at-risk and expected shortfall, are widely used in risk management, as exemplified in the Basel Accords proposed by Bank of International Settlements. We propose a simple general amework, allowing dependent samples, to compute these risk measures via simulation. The framework consists of two steps: in the S-step, risk measure is estimated by using selected sorting algorithm; in R-step, necessary sample size is computed based on newly derived asymptotic expansions of relative error for dependent samples, and the S-step is repeated until requirement on relative error is met. We systematically investigate various sorting methods in the S-step. Numerical experiments indicate that the algorithm is easy to implement and fast, compared to existing methods, even at the 0.001 quantile level. We also give a comparison of the relative errors of value-at-risk and expected shortfall. This is a joint work with Steven Kou.

## Ester Mariucci (Humboldt-UniversitĂ¤t zu Berlin)

## Compound Poisson approximation to estimate the LĂ©vy density

We construct an estimator of the LĂ©vy density, with respect to the Lebesgue measure, of a pure jump LĂ©vy process from high frequency observations: we observe one trajectory of the LĂ©vy process over $[0,T]$ at the sampling rate $\Delta$, where $\Delta\to0$ as $T\to\infty$. The main novelty of our result is that we directly estimate the LĂ©vy density in cases where the process may present infinite activity. Moreover, we study the risk of the estimator with respect to $L_p$ loss functions, $1\leq p<\infty$, whereas existing results only focus on $p\in\{2,\infty\}$. The main idea behind the estimation procedure that we propose is to use that "*every infinitely divisible distribution is the limit of a sequence of compound Poisson distributions*'' (see e.g. Corollary 8.8 in Sato (1999)) and to take advantage of the fact that it is well known how to estimate the LĂ©vy density of a compound Poisson process in the high frequency setting. We consider linear wavelet estimators and the performance of our procedure is studied in term of $L_{p}$ loss
functions, $p\geq 1$, over Besov balls. The results are illustrated on several examples.
This is a joint work with CĂ©line Duval.

## Nicolas Perkowski (Humboldt-UniversitĂ¤t zu Berlin)

## Recent results in game theoretic mathematical finance

Vovk's game theoretic, hedging based approach provides an alternative view on model free financial mathematics and allows for example to derive sample path properties of "typical price paths" and to set up a model free stochastic calculus. Also more quantitative results exist, such as Vovk's pathwise Dambis Dubins-Schwarz theorem or pathwise pricing-hedging dualities. I will present the main ideas and results of the approach, with an emphasis on applications to model free stochastic calculus and time permitting to model free pricing. The talk is based on joint works with R. Lochowski and D. Prömel.

## BjĂ¶rn Scheuermann (Humboldt-UniversitĂ¤t zu Berlin)

## Bitcoin and the block chain - a (not too) Technical Introduction

Block Chain-based, decentral digital currencies, and in Particular Bitcoin, have raised significant interest in recent years. In computer science, Because the idea of block chains for the first time realized a fully distributed and yet resilient consensus mechanism. And far beyond computer science, because a tamper-proof payment mechanism without any central authority or trust anchor opened up a vast field of opportunities Previously inaccessible for building system (and businesses) and THEREFORE sparked the creativity of many. The talk will introduce the key technical ideas and concepts behind the block chain implemented as in Bitcoin, with a focus on the question how and why it is able to guarantee attacker-resilient digital asset ownership in a fully Decentralized setting.

## Hui Shao (National University of Singapore)

## Gini Curve and Top Incomes

Given that top incomes are increasing significantly in recent years and traditional inequality measures failed to distinguish between differences between those at the top end and the overall income distribution, this paper suggests using Gini curve that consists of the truncated Gini coefficients excluding the information of top incomes, instead of using single inequality measure. The Gini curve turns out to be able to present the full information of an income distribution and what is more important, this paper provides an axiomatic framework based on weighted expected utility theory to support using such an inequality curve. The properties of Gini curves are examined. In terms of empirical study we investigate the evolutions of the Gini curves for annual individual incomes of United States and Australia, in particular we find the Gini coefficients excluding the top incomes are decreasing significantly for both America and Australia, while the overall Gini coefficients of them remains relative stable or even shows increase tendency, we also demonstrate that such a phenomenon is consistent with the key conclusion that the top income shares are increasing from 1970 in the recent top income research. This is a joint work with Min Dai and Steven Kou.

## Ronnie Sircar (Princeton University)

## Energy Prices & Dynamic Games with Stochastic Demand

The dramatic decline in oil prices, from around USD 110 per barrel in June 2014 to around USD 30 in January 2016 highlights the importance of competition between different energy producers. Indeed, the price drop has been primarily attributed to OPEC's strategic decision (until very recently) not to curb its oil production in the face of increased supply of shale gas and oil in the US, which was spurred by the development of fracking technology. Most dynamic Cournot models focus on supply-side factors, such as increased shale oil, and random discoveries. However declining and uncertain demand from China is a major factor driving oil price volatility. We study Cournot games in a stochastic demand environment, and present asymptotic and numerical results, as well as a modified Hotelling's rule for games with stochastic demand.

## Benjamin Stemper (TU Berlin and WIAS)

## Pricing in rough fractional volatility models via regularity structures

By now, there is strong evidence both from a statistical as well as
option pricing point of view that volatility is rough, i.e. the
driving noise of instantaneous volatility has fractional - worse than
diffusion - Hölder regularity, typically modelled as a fractional
Brownian motion (fBM) with Hurst exponent $H<1/2$. Using Willard's
conditioning method, we reduce the task of European option pricing in
a rough volatility model of the form suggested by Forde-Zhang (2017)
to the simulation of $\mathcal{I} = \int f(\hat{W})dW$ where
$\hat{W} = \int K dW$. A Wong-Zakai-style approximation
$\mathcal{I}^\epsilon = \int f(\hat{W}^\epsilon)dW^\epsilon$ of
$\mathcal{I}$ however blows up as $\epsilon \rightarrow 0$ since even
for $f(x)=x$, we have an (infinite) Ito-Stratonovich correction. Using
Hairer's theory of *regularity structures*, we specify a
*renormalization* object and prove that the renormalized integral
does indeed converge to $\mathcal{I}$ in $L^p(\mathbb{P})$ for all $p
\in [1,\infty)$.

## Jingrui Sun (National University of Singapore)

## Regulator and Systemic Risk via an Interactive Control System

We propose a model of systemic risk where the evolution of the reserves of banks is described by a system of coupled diffusion processes driven by controls in the drifts, and the regulator either taxes or injects money to each bank subject to a quadratic cost. A unique stochastic control policy is obtained using a system of two coupled differential Riccati equations, in which the regulator plays two roles: First it either taxes or injects the money to banks based on the monetary reserve of each bank rising above or falling below the average, respectively; secondly, the regulator imposes a small tax on all banks to ensure sufficient funding. This work is jointly with Steven Kou.

## Wei Xu (Humboldt-UniversitĂ¤t zu Berlin)

## Limit order books driven by Hawkes processes

In this technical talk, we first give the definition of the infinite-dimensional Hawkes processes. Motivated by the non-negligible cross-dependencies (self and mutual excitation), then we introduce a new kind of continuous-time order book models driven by the infinite-dimensional Hawkes processes. Finally, we show that their rescaled limits can be described by a SDE-ODE system. Adjoint work with Prof. Ulrich Horst.

## Chen Yang (National University of Singapore and ETH ZĂĽrich)

## The Overpricing of Leveraged Products: A Case Study of Dual-Purpose Funds in China

We examine the pricing of dual-purpose funds, an important source of leveraged investment in China. We find that the B shares of all 115 dual-purpose funds are overpriced in the market as compared to their theoretical value. The magnitude of the overpricing can be explained by the leverage ratio offered by B shares, as well as the risk of investment as measured by the variance of the underlying fund. This is a joint work with Min Dai, Steven Kou and Zhenfei Ye.

## Chao Zhou (National University of Singapore)

## The Sustainable Black-Scholes Equations

In incomplete markets, a basic Black-Scholes perspective has to be complemented by the valuation of market imperfections. In this paper we consider the sustainable Black-Scholes equations that arise for a portfolio of options if one adds to their trade additive Black-Scholes price, on top of a nonlinear funding cost, the cost of remunerating at a hurdle rate the residual risk left by imperfect hedging. We assess the impact of model uncertainty in this setup. This is a joint work with Yannick Armenti and StĂ©phane CrĂ©pey.