Title

From "simple" stochastic control problems to "more realistic" ones:
an example from portfolio theory
 

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Abstract

In this talk we take an example from life-cycle portfolio theory
(modelled as a stochastic optimal control problem)
where we see how different economic questions, possibly arising from experimental data,
bring to formulate and study more complex problems, in particular
models which display path dependency and/or Mc Kean - Vlasov type dynamics.
Ongoing results on such problems will be presented.

We will discuss, depending on the available time, the following modelling levels.
Level 1: Lifecycle portfolio with Labor Income
Level 2: Lifecycle portfolio with Path-Dependent Labor Income  (History dependent wages)
Level 3/1: Lifecycle portfolio with Path-Dependent Labor Income with uncertainty on parameters
(Taking account of estimation errors)
Level 3/2: Lifecycle portfolio with Path-Dependent Labor Income with Mc Kean-Vlasov type dynamics
(Taking account of the effect of the environment: "Keeping up with the Joneses")

Based on joint papers/work in progress with various authors
( Biagini, Biffis, Cosso, Djeiche, Kharroubi, Pham, Prosdocimi, Rosestolato, Zanco, Zanella).

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Title

BPHZ renormalisation and vanishing subcriticality limit of the fractional $\Phi^3_d$ model.

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Abstract

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In this talk, we consider the fractional $\Phi^3_d$ model which is a stochastic PDEs on the d-dimensional torus

with fractional Laplacian and quadratic nonlinearity driven by space-time white noise.

We obtain precise asymptotics on the renormalisation counterterms as the mollification parameter becomes small and the parameter of the fractional Laplacian approaches its critical value.

This is a joint work with Nils Berglund.

Title

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Multilevel Picard approximations for high-dimensional semilinear parabolic partial differential equations

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Abstract

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We present new approximation methods for high-dimensional semilinear parabolic PDEs. A key idea of our methods is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of one of the proposed methods grows polynomially both in the dimension and in the reciprocal of the required accuracy. We illustrate the efficiency of the approximation methods by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe Von Wurstemberger.

Title

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Predatory Pricing and the Value of Corporate Cash Holdings

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Abstract

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We analyze the interaction between firms' payout policies and their decisions in product markets in a continuous-time stochastic game between two firms. One of these is financially constrained, whereas the other is not. Contrary to the standard literature we allow firms to choose production and payout strategies, and focus on the effect of predation incentives on both. We find that predation induces fewer dividend payouts. Furthermore, the liquidity position of the constrained firm has an economically significant effect on the production choices of both firms and, thus, on the evolution of profits, cash holdings and stock returns.  
 

Title

On the difference between volatility swaps and the ATM implied volatility

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Abstract

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This talk focuses on the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call
option. It is well known that the difference between these two quantities converges to zero as the time to maturity decreases. In this talk, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of convergence is different in the correlated and uncorrelated cases, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H. Moreover, in the case H ≥ 1/2, we develop a model-free approximation formula for the volatility swap in terms of the ATMI and its skew.

(Joint work with Kenichiro Shiraya)

Title

Optimal Execution with Stochastic Delay

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Abstract

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We show how traders use aggressive immediate execution limit orders (IELOs) to liquidate a position when there are random delays in all the steps of a trade, i.e., there is latency in the marketplace and latency is random. We frame our model as a delayed impulse control problem in which the trader controls the times and the price limit of the IELOs she sends to the exchange.  Our paper is the first to study an optimal liquidation problem that accounts for: random delays, price impact, and transaction costs.  We introduce a new type of impulse control problem with stochastic (or deterministic) delay, not previously studied in the literature. The value functions are characterised as the solution to a coupled system of a Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) and a partial differential equation. We use a Feynman-Kac representation to reduce the system  to a HJBQVI, for which we prove existence and uniqueness in a viscosity sense. We employ foreign exchange high-frequency data to estimate model parameters and  implement the random-latency-optimal strategy,  and compare it with four benchmarks: executing the entire order at once,  optimal execution with deterministic latency, optimal execution with zero latency, and time-weighted average price. For example, in the EUR/USD currency pair, we show that the random-latency-optimal strategy outperforms the benchmarks between 4 USD per million EUR traded and 105 USD per million EUR traded, this is between 0.15 and 18.18 times the value of the transaction fees paid by liquidity takers.

 

Authors: Alvaro Cartea and Leandro Sanchez-Betancourt

Title

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No–Arbitrage Commodity Option Pricing with Market Manipulation

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Abstract

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We design three continuous-time models in finite horizon of a 
commodity price, whose dynamics can be affected by the actions of a 
representative risk-neutral producer and a representative risk-neutral 
trader. Depending on the model, the producer can control the drift 
and/or the volatility of the price whereas the trader can at most 
affect the volatility. The producer can affect the volatility in two 
ways: either by randomizing her production rate or, as the trader, 
using other means such as spreading false information. Moreover, the 
producer contracts at time zero a fixed position in a European convex 
derivative with the trader. The trader can be price-taker, as in the 
first two models, or she can also affect the volatility of the 
commodity price, as in the third model. We solve all three models 
semi-explicitly and give closed-form expressions of the derivative 
price over a small time horizon, preventing arbitrage opportunities to 
arise. We find that when the trader is price-taker, the producer can 
always compensate the loss in expected production profit generated by 
an increase of volatility by a gain in the derivative position by 
driving the price at maturity to a suitable level. Finally, in case 
the trader is active, the model takes the form of a non-zero-sum 
linear-quadratic stochastic differential game and we find that when 
the production rate is already at its optimal stationary level, there 
is an amount of derivative position that makes both players better off 
when entering the game.

This is a joint work with René Aid and Luciano Campi.

Title

Mean field limits for interacting Hawkes processes in a diffusive regime

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Abstract

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We consider a sequence of systems of Hawkes processes having mean field interactions in a diffusive regime. The stochastic intensity of each process is a solution of a stochastic differential equation driven by N independent Poisson random measures. We show that, as the number of interacting components N tends to infinity, this intensity converges in distribution in Skorohod space to a CIR-type diffusion. Moreover, we prove the convergence in distribution of the Hawkes processes to the limit point process having the limit diffusion as intensity. To prove the convergence results, we use analytical technics based on the convergence of the associated infinitesimal generators and Markovian semigroups.

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Joint work with Xavier Erny (Evry)  et Eva Löcherbach (Paris 1).

Title

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Convergence rate of random walk approximation of Backward SDEs
 

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Abstract

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Briand, Delyon and Mémin have shown in 2001 a Donsker-type
theorem for forward-BSDE. If one approximates the Brownian motion B by a
random walk B^n, the according solutions (X^n,Y^n,Z^n) converges weakly
to (X,Y,Z). We investigate under which conditions (Y^n_t,Z^n_t)
converges to (Y_t,Z_t) in L_2 and compute the rate of convergence in
dependence of smoothness properties of the coefficients b and sigma, the
terminal condition and the generator.

The talk is based on joint results with Philippe Briand, Christel Geiss,
Stefan Geiss and Antti Luoto.
 

Title

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On nonlinear filtering of jump diffusions

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Abstract

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Partially observed jump diffusions $Z_t=(X_t,Y_t)$ on a time interval $[0,T]$ are considered. 

The classic problem of describing the mean square estimate for the `unobserved component’ $X_t$ from `past observations' $Y_s$, $s\leq t$ is discussed. Stochastic integro-differential equations for the conditional distribution $\mu_t(dx)$ of $X_t$, given the past observations, are shown, and recent results on the existence of the density $\mu_t(dx)/dx$ and its analytical properties are presented. 

The talk is based on joint work with Sizhou Wu and Fabian Germ. 

Title

A monotone scheme for G-equations with application to 

the explicit convergence rate of robust central limit theorem

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Abstract

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We propose a monotone approximation scheme for a class of fully nonlinear PDEs called G-equations. Such equations arise often in the characterization of G-distributed random variables in a sublinear expectation space. The proposed scheme is constructed recursively based on a piecewise constant approximation of the viscosity solution to the G-equation. We establish the convergence of the scheme and determine the convergence rate with an explicit error bound, using the comparison principles for both the scheme and the equation together with a mollification procedure. The first application is obtaining the convergence rate of Peng's robust central limit theorem with an explicit bound of Berry-Esseen type. The second application is an approximation scheme with its convergence rate for the Black-Scholes-Barenblatt equation

Title

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Optimal trade execution with transient relative price impact and directional views: A 2nd-order variational approach to a 3-dimensional non-convex free boundary problem

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Abstract

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We solve the optimal execution problem to trade a large financial asset position within finite time in an illiquid market, where price impact is transient and possibly non-linear (in log-prices), like in models by J.P.Bouchaud, F.Lillo or J.Gatheral. We derive a complete solution for the optimal control problem of finite-fuel type, where mechanical price impact is 1.) multiplicative instead of additive as in the of seminal articles by Obizhaeva/Wang (2013),  Predoiu/Shaikhet/Shreve (2011) or Schied et al. (2010), and 2.) we admit for non-vanishing drift in the fundamental price process, that is directional views about short-term price trends (as in Almgren/Chriss 2000, ch.4). Multiplicative impact in relative percentage terms is well established (cf. Bertsimas/Lo 1998, ch.3) and avoids the possibility of negative asset prices. We obtain a complete characterization of the regular three-dimensional free boundary surface which separates the no-/action regions for the non-convex three dimensional singular control problem by a family of characteristic curves, whose description is explicit up to the solution of ODEs.  While the free boundary description is almost as explicit as in Obizhaeva/Wang, our analysis is more demanding for a lack of an apparent convexity structure to exploit. Yet, a key argument to prove global optimality turns out to show at first a local optimality for a candidate free boundary under smooth perturbations by 2nd-order variational arguments.

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For the optimal trading application , the results may shed light on phenomena, like for instance
a) how to profit from directional views (signals) about price trends by optimal (non-trivial) round-trips;
b) down- or upward directional views do lead to respective front- or back-loading in the optimal execution of (sell) trading strategies;
c) optimal trading strategies are qualitatively different and their profitability can depend non-monotonically on the resilience (transience) parameters for the price impact.  

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This is joint work with Peter Frentrup.

Title

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Backward propagation of chaos and large population games asymptotics.

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Abstract

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In this talk we will present a generalization of the theory of propagation of chaos to backward (weakly) interacting diffusions. The focus will be on cases allowing for explicit convergence rates and concentration inequalities in Wasserstein distance for the empirical measures. As the main application, we derive results on the convergence of large population stochastic differential games to mean field games, both in the Markovian and the non-Markovian cases.

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The talk is based on joint works with M. Laurière and Dylan Possamaï.

Title

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Optimal trade execution in a stochastic limit order book model

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Abstract

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We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and resilience evolve randomly in time. Trading is allowed in both directions. Due to the stochastic dynamics of the order book depth and resilience, optimal execution strategies are typically of infinite variation, and the first thing to be discussed is how to extend the state dynamics and the cost functional to allow for general semimartingale strategies. We then derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs, identify conditions under which an optimal execution strategy exists and, finally, illustrate our findings in several examples.

This is a joint work with Julia Ackermann and Thomas Kruse.

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Title

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An equilibrium analysis of price impact and order flow

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Abstract

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We develop an equilibrium model for market impact of trades when investors with private signals execute via a competitive dealer. The fat tails in the signal distribution lead to a power law for price impact, while the impact is logarithmic
for lighter tails. The spread increases with the degree of information asymmetry and the number of insiders. In case of a monopolistic insider, the last slice traded against the limit order book is priced at the liquidation value of the asset reminiscent of Kyle (1985). However, competition among the insiders
leads to aggressive trading, hence vanishing profit in the limit.

 

Joint work with Henri Waelbroeck.

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Title

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Utility Indifference Pricing with Delayed Information

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Abstract

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We consider the Bachelier model with information delay where

investment decisions can be based only on observations from $H>0$ time

units before. Exponential utility indifference prices are studied for

vanilla options and we compute their non-trivial scaling limit for

vanishing delay when risk aversion is scaled liked $A/H$ for some

constant $A$. We develop discrete-time duality for this setting and

show how the relaxed form of martingale property introduced by Kabanov

and Stricker (2006) results in the scaling limit taking the form of a

volatility control problem with quadratic penalty. 

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(This talk is on joint work with Yan Dolinsky)

Title

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Probabilistic approach to free boundary problems and applications

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Abstract

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We will discuss a recently developed probabilistic approach to (singular) free boundary problems, such as the supercooled Stefan problem. The approach is based on a new notion of solution, referred to as probabilistic, which arises naturally in the context of large system limits of interacting particle systems. In the talk, I will give an example of how such interacting particle systems arise in applications (e.g., finance), then obtain a solution of a free boundary problem in the large system limit, and discuss how this solution can be analyzed mathematically (thereby answering natural questions about the systemic risk in financial systems and neural synchronization in the brain).

 

The talk is based on recent and ongoing joint works with Sergey Nadtochiy, Francois Delarue, Jiacheng Zhang and Xiling Zhang.