Mathematical Finance Seminar
Date
Time
17:oo
Location:
HUB, RUD 25; 1.115
Yves Achdou (Universite de Paris)

A short-term model for the oil industry addressing commercial storage

We propose a plausible mechanism for the short-term dynamics of the oil market based on the interaction of a cartel, a fringe of competitive producers, and a crowd of capacity-constrained physical arbitrageurs that store the resource. The model leads to a system of two coupled nonlinear partial differential equations, with a new type of boundary conditions that play a key role and translate the fact that when storage is either full or empty, the cartel has enhanced strategic power. We propose a finite difference scheme and report numerical simulations. The latter result in apparently surprising facts: 1) the optimal control of the cartel (i.e., its level of production) is a discontinuous function of the state variables; 2) the optimal trajectories (in the state variables) are cycles which take place around the discontinuity line. These patterns help explain remarkable price swings in oil prices in 2015 and 2020.

The talk is based on joint work with C. Bertucci, J.M Lasry, P.L Lions, A. Rostand and J. Scheinkman.

Mathematical Finance Seminar
Date
Time
16:oo
Location:
HUB, RUD 25; 1.115
Gudmund Pammer (ETH Zürich)

Adapted transport and a link to robust hedging

Wasserstein distance induces a natural Riemannian structure for the probabilities on the Euclidean space. This insight of optimal transport theory is fundamental for tremendous applications in various fields of pure and applied mathematics. We believe that an appropriate dynamic variant of optimal transport, adapted transport, can play a similar role for stochastic processes. By imposing a causality constraint on couplings, the flow of information which is encoded in the filtration is adequately taken into account. Notably, the resulting adapted Wasserstein distance is a geodesic distance for stochastic processes and suitable for sensitivity analysis in finance as well as for dynamic stochastic optimization problems. In this talk we will explore the connection of adapted transport with a robust hedging problem, along with further recent developments.

The talk is based on joint works with Beatrice Acciaio, Daniel Bartl, Mathias Beiglbo ̈ck, Stephan Eckstein and Daniel Krˇsek.

Probability Colloqium
Date
Time
17:oo
Location:
HUB, RUD 25; 1.115
Paul Gassiat (Paris-Dauphine)

Zero noise limit for singular ODE regularized by fractional noise

A classical manifestation of regularization by noise is that adding an irregular term to an ill-posed equation may restore well-posedness (existence/uniqueness). A natural question is then, in the limit where the coefficient in front of the noise is taken to zero, whether this selects one (or several) particular solutions to the original equation (this is typically referred to as "selection by noise"). In the case of one-dimensional ODEs, perturbed by a Brownian motion, Bafico and Baldi ’82 showed that this procedure selects extremal solutions, i.e. those that exit the problematic point instantly. We extend this result to the case of fractional noise (and obtain in addition some exponential concentration estimates). The main difficulty lies in the absence of the Markov property for the system. Our proof is based on the dynamical approach of Delarue-Flandoli ’14, combined with recent progress in regularisation by fractional noise (Catellier-Gubinelli ’16), and techniques coming from the study of ergodicity of fractional SDEs (Hairer ’05, Panloup-Richard ’20). Based on a joint work with ÅĄukasz MÄĚdry (Univ. Paris-Dauphine).

Probability Colloqium
Date
Time
16:oo
Location:
HUB, RUD 25; 1.115
Anna Aksamit (Sydney)

Superhedging duality for multi-action options under model uncertainty with information delay

We consider the superhedging price of an exotic option under nondominated model uncertainty in discrete time in which the option buyer chooses some action from an (uncountable) action space at each time step. By introducing an enlarged space, we reformulate the superhedging problem for such an exotic option as a problem for a European option, which enables us to prove the pricing-hedging duality. Next, we present a duality result that, when the option buyer ́s action is observed by the seller up to k periods later, the superhedging price equals the model-based price where the option buyer has the power to look into the future for k-periods.

Mathematical Finance Seminar
Date
Time
17:oo-18:oo
Location:
TUB, MA043
Yufei Zhang (LSE)

Convergence of policy gradient methods for stochastic control problems

Policy gradient (PG) methods have demonstrated remarkable success in a wide range of sequential decision-making tasks. However, the majority of research efforts have focused on discrete pro- blems, leaving the convergence analysis of PG methods for controlled diffusions as an unresolved issue. This work proves the convergence of PG methods for finite-horizon linear-quadratic control problems. We consider a continuous-time Gaussian policy whose mean is linear in the state variable and whose covariance is state-independent. We propose geometry-aware gradient descents for the mean and covariance of the policy using the Fisher geometry and the Bures-Wasserstein geometry, respectively. The policy iterates are shown to converge globally to the optimal policy with a linear rate. We further propose a novel PG method with discrete-time policies. The algorithm leverages the continuous-time analysis, and achieves a robust linear convergence across different action frequencies. A numerical experiment confirms the convergence and robustness of the proposed algorithm. If time allows, extensions of the algorithm to nonlinear control problems will be discussed.

Mathematical Finance Seminar
Date
Time
16:oo-17:oo
Location:
TUB, MA043
Mehdi Talbi (ETH)

Mean field games of optimal stopping

We are interested in the study of stochastic games for which each player faces an optimal stopping problem. In our setting, the players may interact through the criterion to optimize as well as through their dynamics. After briefly discussing the N-players game, we formulate the corresponding mean field problem. In particular, we introduce a weak formulation of the game for which we are able to prove existence of Nash equilibria for a large class of criteria. We also prove that equilibria for the mean field problem provide approximated Nash equilibria for the N-players game, and we formally derive the master equation associated with our mean field game.

This talk is based on joint work with D. Possamai.

Probability Colloqium
Date
Time
17:oo-18:oo
Location:
TUB; MA 041
Yufei Zhang (LSE)

Exploration-exploitation trade-off for continuous-time reinforcement learning

Recently, reinforcement learning (RL) has attracted substantial research interests. Much of the attention and success, however, has been for the discrete-time setting. Continuous-time RL, despite its natural analytical connection to stochastic controls, has been largely unexplored and with limited progress. In particular, characterising sample efficiency for continuous-time RL algorithms remains a challenging and open problem.

In this talk, we develop a framework to analyse model-based reinforcement learning in the episodic setting. We then apply it to optimise exploration-exploitation trade-off for linear-convex RL problems, and report sublinear (or even logarithmic) regret bounds for a class of learning algorithms inspired by filtering theory. The approach is probabilistic, involving analysing learning efficiency using concentration inequalities for correlated continuous-time observations, and applying stochastic control theory to quantify the performance gap between applying greedy policies derived from estimated and true models.

Probability Colloqium
Date
Time
16:oo-17:oo
Location:
TUB MA 041
Guanxing Fu (HK PolyU)

Mean field portfolio games

First, I will discuss a mean field portfolio game in a general framework. Using a dynamic programming principle and a martingale optimality principle, I establish a one-to-one correspondence between the Nash equilibrium and some BSDE. Such a correspondence is key to the uniqueness result of Nash equilibria. Generally, this BSDE can be solved under a weak interaction assumption. Motivated by this assumption, I will introduce an asymptotic expansion result of the game value in terms of the interaction parameter. Second, I will incorporate consumption into the portfolio game and show that the equilibrium investment and consumption can be fully characterized by one BSDE. 

Mathematical Finance Seminar
Date
Time
17:00-18:oo
Location:
TUB, MA043
Sigrid Källblad (KTH Stockholm)

Adapted Wasserstein distance between the laws of SDEs

We consider an adapted optimal transport problem between the laws of Markovian stochastic differential equations (SDEs) and establish optimality of the so-called synchronous coupling between the given laws. The proof of this result is based on time-discretisation methods and reveals an interesting connection between the synchronous coupling and the celebrated discrete-time Knothe–Rosenblatt rear- rangement. We also provide a related result on equality of various topologies when restricted to certain laws of continuous-time stochastic processes. The result is of relevance for the study of stability with respect to model specification in mathematical finance.

The talk is based on joint work with Julio Backhoff-Veraguas and Ben Robinson.

Probability Colloqium
Date
Time
17:00-18:oo
Location:
TUB, MA041
Giorgia Callegaro (University of Padova)

A McKean-Vlasov game of commodity production, consumption and trading

We propose a model where a producer and a consumer can affect the price dynamics of some commodity controlling drift and volatility of, respectively, the production rate and the consumption rate. We assume that the producer has a short position in a forward contract on λ units of the underlying at a fixed price F, while the consumer has the corresponding long position. Moreover, both players are risk-averse with respect to their financial position and their risk aversions are modelled through an integrated-variance penalization. We study the impact of risk aversion on the interaction between the producer and the consumer as well as on the derivative price. In mathematical terms, we are dealing with a two-player linear-quadratic McKean–Vlasov stochastic differential game. Using methods based on the martingale optimality principle and BSDEs, we find a Nash equilibrium and characterize the corresponding strategies and payoffs in semi-explicit form. Furthermore, we compute the two indifference prices (one for the producer and one for the consumer) induced by that equilibrium and we determine the quantity λ such that the players agree on the price. Finally, we illustrate our results with some numerics. In particular, we focus on how the risk aversions and the volatility control costs of the players affect the derivative price.

This is a joint paper with R. Aid, O. Bonesini and L. Campi.