2023 Seminars
May 25th, 2023
Speaker: Benjamin Jourdain (Ecole des Ponts ParisTech, France)
Title: Approximation of martingale couplings on the real line and stability in robust finance
Abstract: When approximating in Wasserstein distance the two marginals of a martingale coupling on the real by probability measures in the convex order, it is possible to construct a sequence of martingale couplings between these probability measures converging in adapted Wasserstein distance to the original coupling. We deduce the stability with respect to the marginal distributions of the Weak Martingale Optimal Transport problem in dimension one. As an application, we obtain the stability of the superreplication price of the VIX future. To deal with the subreplication price, we need a generalisation of our main result to extended martingale couplings with an extra parameter which enables to take into account additional information.
Date and Time: Thursday, May 25th, 2023, 4:00pm - 5:00pm, HK time.
Location: Online via Zoom
Meeting ID: 958 9985 2810
Passcode: 3485
April 27th, 2023
Speaker: Jiequn Han (Princeton University, USA)
Title: Income and wealth distribution in macroeconomics: A continuous-time approach
Abstract: We recast the Aiyagari–Bewley–Huggett model of income and wealth distribution in continuous time. This workhorse model—as well as heterogeneous agent models more generally—then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (1) an analytic characterization of the consumption and saving behaviors of the poor, particularly their marginal propensities to consume; (2) a closed-form solution for the wealth distribution in a special case with two income types; (3) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including—but not limited to—the Aiyagari–Bewley–Huggett model.
Date and Time: Thursday, April 27, 2023, 10:30 am - 11:30 am, HK time.
Location: Online via Zoom
Meeting ID: 972 9790 6033
Passcode: 692979
March 16th, 2023
Speaker: Zhenjie Ren (Université Paris-Dauphine, France)
Title: Mean Field Optimization Problem Regularized by Fisher Information
Abstract: Recently there is a rising interest in the research of mean-field optimization, in particular because of its role in analyzing the training of neural networks. In this talk, by adding the Fisher Information (in other word, the Schrodinger kinetic energy) as the regularizer, we relate the mean-field optimization problem with a so-called mean field Schrodinger (MFS) dynamics. We develop a free energy method to show that the marginal distributions of the MFS dynamics converge exponentially quickly towards the unique minimizer of the regularized optimization problem. We shall see that the MFS is a gradient flow on the probability measure space with respect to the relative entropy. Finally, we propose a Monte Carlo method to sample the marginal distributions of the MFS dynamics.
This is an ongoing joint work with Julien Claisse, Giovanni Conforti and Songbo Wang.
Date and Time: Thursday, Mar 16th, 2023, 4:30pm-5:30pm, HK Time.
Location: Online via Zoom
Meeting ID: 932 4471 5329
Passcode: 3485
March 2nd, 2023
Speaker: Peter Tankov (ENSAE, France)
Title: Decarbonization of financial markets: a mean-field game approach
Abstract: We build a model of a financial market where a large number of firms determine their dynamic emission strategies under climate transition risk in the presence of both green-minded and neutral investors. The firms aim to achieve a trade-off between financial and environmental performance, while interacting through the stochastic discount factor, determined in equilibrium by the investors' allocations. We formalize the problem in the setting of mean-field games and prove the existence and uniqueness of a Nash equilibrium for firms. We then present a convergent numerical algorithm for computing this equilibrium and illustrate the impact of climate transition risk and the presence of greenminded investors on the market decarbonization dynamics and share prices. We show that uncertainty about future climate risks and policies leads to higher overall emissions and higher spreads between share prices of green and brown companies. This effect is partially reversed in the presence of environmentally concerned investors, whose impact on the cost of capital spurs companies to reduce emissions. However, if future climate policies are uncertain, even a large fraction of greenminded investors is unable to bring down the emission curve: clear and predictable climate policies are an essential ingredient to allow green investors to decarbonize the economy. Joint work with Pierre Lavigne (Institut Louis Bachelier).
Date and Time: Thursday, Mar 2nd, 2023, 4:30pm-5:30pm, HK Time.
Location: Online via Zoom
Meeting ID: 919 3967 5664
Passcode: 3485
February 17th, 2023
(Jointly organized with the CUHK Distinguished Lectures in Quantitative Finance)
Speaker: Rama Cont (Oxford University, UK)
Title: A model-free approach to continuous-time finance
Abstract: We present a pathwise approach to continuous-time finance based on causal functional calculus. Our framework does not rely on any probabilistic concept. We introduce a definition of continuous-time self-financing portfolios, which does not rely on any integration concept and show that the value of a self-financing portfolio belongs to a class of nonanticipative functionals, which are pathwise analogs of martingales.
We show that if the set of market scenarios is generic in the sense of being stable under certain operations, such self-financing strategies do not give rise to arbitrage. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. Applying the transition principle of Rufus Isaacs in differential games, we obtain a pathwise dynamic programming principle for the superhedging cost. We show that the superhedging cost is characterized as the solution of a path-dependent equation. For the Asian option, we obtain an explicit solution.
Date and Time: Friday, Feb 17th, 2023, 5:00pm-6:00pm, HK Time.
Location: Online via Zoom
Meeting ID: 922 0689 6799
Passcode: 156866
February 10th, 2023
Speaker: Jianfeng Zhang (University of Southern California, USA)
Title: Set Valued HJB Equations
Abstract: The set values have been introduced for many applications, such as time inconsistent stochastic optimization problems, multivariate dynamic risk measures, and nonzero sum games with multiple equilibria. Among others, one crucial property of the dynamic set value is the dynamic programming principle. In this talk we introduce a notion of set valued PDEs and show that the set value function of certain multidimensional control problem is the unique solution to the corresponding set valued HJB equation. A key tool is the set valued Ito formula, which together with the DPP induces the PDE. In the one dimensional case, the set valued PDE reduces back to the standard HJB equations. Our characterization of the set values is through their boundaries, which are manifolds. Thus our approach is intrinsically connected to the existing theory of moving/evolving manifolds, such as front propagation and mean curvature flows. Roughly speaking, those equations can be viewed as first order set valued ODEs, and we extend them to second order PDEs. Another difference is that, due to different applications, those equations are forward in time (with initial conditions), while we consider backward equations (with terminal conditions). The talk is based on an ongoing joint work with Melih Iseri.
Date and Time: Friday, Feb 10th, 2023, 10:00am-11:00am.
Location: Online via Zoom
Meeting ID: 997 4582 5198
Passcode: 797118
January 26th, 2023
Speaker: Giovanni Conforti (Ecole Polytechnique, France)
Title: A coupling approach to the turnpike phenomenon in stochastic control and McKean-Vlasov control
Abstract: Coupling methods provide a powerful toolbox for the quantitative analysis of the long-time behaviour of Markov processes. In particular, coupling by reflection allows to establish sharp exponential convergence results in Wasserstein distance for the Fokker-Planck equation without having to rely on pointwise assumptions on the confinement potential. The purpose of this talk is to illustrate the construction of a variant of coupling by reflection that applies to optimally controlled diffusion processes, including controlled McKean-Vlasov processes. Such construction opens the door for a precise study of the long-time behaviour of optimizers: in particular it provides with uniform in time gradient (and Hessian) estimates for the solution of Hamilton-Jacobi-Bellman equations that enable to prove various kind of exponential turnpike properties for the optimal processes and controls. This talk is partially based on joint work with Katharina Eichinger, Alain Durmus, and Alekos Cecchin.
Date and Time: Thursday, Jan 26th, 2023, 4:30pm-5:30pm.
Location: Online via Zoom
Meeting ID: 932 8018 0675
Passcode: 3485
January 19th, 2023
Speaker: Max Reppen (Boston University, USA)
Title: Cash Policies and Firm Size
Abstract: Understanding firms’ demand for cash is critical for answering many economic questions. We develop a model of firm dynamics allowing for heterogeneous size. The firm faces costly financing, fixed costs, and decreasing returns to scale. Surprisingly, the firm’s demand for cash is U-shaped in firm size. When the firm is small, growth lowers cash demand because the relative size of the fixed costs declines sharply. Eventually, growth increases cash demand as the scale of the cash flow shocks increases. Consequently, cash holdings and issuance amounts (payout rates) are U-shaped (hump-shaped) in firm size. We find empirical support for these predictions.
Date and Time: Thursday, Jan 19th, 2023, 9:30am-10:30am.
Location: Online via Zoom
Meeting ID: 996 9726 9848
Passcode: 255863
January 12th, 2023
Speaker: Mao Fabrice Djete (Ecole Polytechnique, France)
Title: Non–regular McKean–Vlasov equations and calibration problem in local stochastic volatility models
Abstract: In this talk, motivated by the calibration problem in local stochastic volatility models, we will investigate some McKean-Vlasov equations beyond the usual requirement of continuity of the coefficients in the measure variable for the Wasserstein topology. We will provide first an existence result for this type of McKean–Vlasov equations and explain the main idea behind the proof. In a second time, we will show an approximation by particle system for this type of equations, a result almost never rigorously proven in the literature in this context.
Date and Time: Thursday, Jan 12th, 2023, 4:00pm-5:00pm.
Location: Online via Zoom
Meeting ID: 943 1332 0040
Passcode: 3485