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.
Dynamics of a Well-Diversiﬁed Equity Index and Martingale Inference
Space-time localisation for the dynamic |Phi^4_3 model
In this talk I will present a new method to derive a priori estimates for singular SPDE concentrating on the dynamic Φ43 equation.
Recently several methods to show non-explosion for this equation in the framework of paracontrolled distributions were put forward by several groups. Here I will show how to prove bounds in the framework of regularity structures. The main result is a space-time version of the coming down from inﬁnity property, i.e. a bound on solutions of the equation on a compact set in space-time which only depends on the stochastic data on a slightly larger set, but is uniform over all possible choices of space-time boundary data.
Our method signiﬁcantly simpliﬁes previous proofs and extensions to (much) more singular equations seem within reach.
This is joint work with Augustin Moinat.
Large urn model with local mean-field interactions
We study a large urn model connected by an underlying symmetrical graph. After some exponentially distributed amount of time all the balls of one of the urns are redistributed among the connected urns. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The degree of the graph, which is the range of interaction, is assumed large, but is not necessarily linear with respect to the number of urns. Moreover, the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. We describe the dynamic by using the local empirical distributions associated to the state of urns in the neighborhood of a given urn. Under some conditions, we are able to establish a mean-field convergence result for this measure-valued system. Convergence results of the corresponding invariant distributions are obtained for several classes of allocation policies. For the class of power of choices policies, we show that the invariant measure has an asymptotic finite support property when the average load per urn gets large. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of choices policies. This finite support property has interesting consequences in practice. (This is a joint work with Philippe Robert.)
Computation of optimal transport and related hedging problems via penalization and neural networks
We present a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual formulation and reduce it to a nite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, and bounds on the distribution of a sum of dependent random variables. As an application we focus on the problem of risk aggregation under model uncertainty. The talk is based on joint work with Stephan Eckstein and Mathias Pohl.
Optimal order placement and limit order book modelling
Optimal order placement is a key aspect of market making, and more generally, of liquidity providing strategies in electronic markets. With this motivation in mind, we study the optimal placement of limit orders from theoretical and numerical points of view, in the context of Markovian limit order book models. The theoretically optimal strategies are then backtested using real data, providing results that advocate for the design of better order book models. Some extensions are made, based either on Hawkes processes, or on processes with finite memory (joint works with C. Huré, X. Lu, H. Pham).
Robust Pricing and Hedging around the Globe
We study the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoﬀs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part. In the case of ﬁnitely supported marginal laws, solving for the static part reduces to a semi-inﬁnite linear program. This talk is based on joint work with Florian Stebegg (Columbia University).
Quantum gravity via conformal loop ensembles
I will explain how to understand the various structures that emerge in Liouville Quantum Gravity (which is the canonical way to define a random Markovian area measure in the plane) and its natural interplay with scaling limits of models from statistical mechanics, using some new special features that we derive in ongoing joint work with Jason Miller and Scott Sheeld. In particular, we will see that a CLE defined on an independent well-chosen LQG surface defines a Poisson point process of so-called quantum disks.
Optimal Portfolio under Fractional Stochastic Environment
Rough stochastic volatility models have attracted a lot of attention recently, in particular for the linear option pricing problem. In this talk, starting with power utilities, we propose to use a martingale distortion representation of the optimal value function for the nonlinear asset allocation problem in a (non-Markovian) fractional stochastic environment (for all Hurst index H ∈ (0, 1)). We rigorously establish a ﬁrst order approximation of the optimal value, when the return and volatility of the underlying asset are functions of a stationary slowly varying fractional Ornstein-Uhlenbeck process. We prove that this approximation can be also generated by the zeroth order trading strategy providing an explicit strategy which is asymptotically optimal in all admissible controls. Furthermore, we extend the discussion to general utility functions, and obtain the asymptotic optimality of this strategy in a speciﬁc family of admissible strategies. If time permits, we will also discuss the problem under fast mean-reverting fractional stochastic environment. Joint work with Ruimeng Hu (UCSB).
The conformal continuum random tree
The Continuum Random Tree, introduced by Aldous in the 90’s, arises in numerous
diﬀerent ways, for instance as the scaling limit of large uniform trees, via Brownian excursions, or
via a Poisson point processes. It carries a natural structure as a metric measure space and has
recently featured prominently in the LeGall’s and Miermont’s work on the Brownian map, as well
as Duplantier’s, Miller’s and Sheﬃeld’s Brownian sphere. Beginning from basic deﬁnitions, I will
describe some of the highlights of this emerging theory such as connections to Liouville Quantum
Gravity, and will describe how to draw the CRT in the plane in a conformally natural way.