**London–Oxford–Warwick Mathematical Finance Workshop**

**Oxford–Man Institute, 5-6 September 2022**

Sponsored by Standard Chartered, the Heilbronn Institute, and

the Oxford–Man Institute

Schedule

**Monday 5th September**

**0930 – Welcome and Registration (with coffee)**

**1025 – Opening remarks**

**1030 – Camilo Garcias Trillos (UCL)**

Wasserstein Ascent-Descent dynamics and applications in finance

In the talk, we discuss how some problems in finance can be interpreted as solving for Nash equilibria of zero-sum games in the space of measures. These include problems like portfolio selection under non-linear pricing, or construction of stress testing scenarios for solvency. Then, we discuss an iterative algorithm to solve such problems. The algorithm is based on a system of interacting particles that can be interpreted as following ascent-descent dynamics in a lifted space. We show, under some regularity conditions, that the interacting particle dynamics converge toward appropriate mean field limit equations as the number of particles grows. We also deduce that, in the large time limit, the mean-field dynamics achieve ϵ-Nash equilibria of the original adversarial learning problems. Includes joint work with Nicolas Garcia Trillos.

**1130– Coffee Break**

**1200– Filippo De Angelis (Oxford)**

Multilevel Function Approximator

We consider function approximations for which the synthetic training set is generated by means of expensive numerical methods and is, thus, the dominant part of the computational cost. We show that multilevel ideas can reduce the computational cost by generating most samples with low accuracy at a corresponding low cost, with relatively few high-accuracy samples at a high cost. As an application of the multilevel approach, we consider learning the function that maps parameters of the model and of the financial product to the price of the financial product. In the simple case of one-layer neural networks and second-order accurate finite difference methods, the computational cost to achieve accuracy O(ϵ) is reduced from O(ϵ−4−dX/2)to O(ϵ−4), where dX is the dimension of the underlying pricing PDE. The analysis is supported by numerical results showing significant computational savings. Joint work with Mike Giles and Christoph Reisinger.

**1230– Nazem Khan (Warwick)**

Pricing with Convex Risk

The aim of this talk is to provide as explicit as possible the set of rational prices for a financial contract outside a given market. Rationality is determined through a convex risk measure ρ. More precisely we replace the term ”no-arbitrage” in classical mathematical finance with ”no-(strong)-ρ-arbitrage”. The model we consider is static and frictionless, and the obtained price bounds can be sharp enough to be useful in the practise of pricing. This is joint work with Martin Herdegen.

**1300– Lunch**

**1430– Gechun Liang (Warwick)**

Robust limit theorem for nonlinear Levy processes under sublinear expectation

We introduce a universal robust limit theorem under a sublinear expectation framework.

It covers both Peng’s robust CLT and Bayraktar-Munk’s robust limit theorem for alphastable distribution. To prove the convergence, we develop a novel weak convergence approach based on the notion of tightness and weak compactness on a sublinear expectation space. We further prove a new type of Levy-Khintchine representation formula to characterise the limit nonlinear Levy process. To establish the convergence rate, we use and extend techniques introduced by Krylov and Barles-Jakobsen for the monotone schemes for viscosity solutions. Based on a series of joint works with Mingshang Hu, Shuo Huang, Lianzi Jiang and Shige Peng.

**1530– Coffee Break**

**1600– Patrick Chang (Oxford)**

Algorithmic Game Theory: The Learning Equations

We provide a framework to characterize the interaction of state-dependent learning algorithms as a system of deterministic ODEs in a dynamic game where the joint actions affect the transition of the state process. Our methodology covers general learning algorithms including temporal difference algorithms with complicated information structures when players receive perfect, private, and public signals. We demonstrate our methodology in a repeated 2×2 Prisoner’s dilemma with perfect monitoring. We show that learning algorithms can learn to tacitly collude through a reward-punishment mechanism.

**1630– Eghbal Rahimikia (Manchester)**

NLP for Financial Forecasting (from FinText to GPT3)

This talk focuses on two projects I have done in collaboration with the Oxford–Man Institute of Quantitative Finance (during my internship there with Stefan Zohren) and one upcoming project for developing a financial GPT model in partnership with GraphCore. I’ll review FinText (https://www.rahimikia.com/fintext), the first financial word embedding we developed, and its use cases in volatility and return forecasting. I also cover how we can use more advanced NLP models (like GPT3, GPT-J, etc.) for financial forecasting (upcoming paper).

**1700– Yifan Jiang (Oxford)**

Sensitivity of robust optimization over an adapted Wasserstein ambiguity set

In this talk, we consider the sensitivity of distributionally robust optimization problem under an adapted Wasserstein perturbation. We extend the classical results in a static setting to the dynamic multi-period setting. Under mild conditions, we give an explicit expression for the first order approximation to the value function. An optimization problem with a cost of weak type will also be discussed.

**1730– Collaboration time**

**1900– Dinner (New College dining hall, by invitation)**

**Tuesday 6th September **

**0930– Coffee and discussion**

**1030– Ales Cerny (Bayes Business School)**

Simplified Stochastic Calculus via Semimartingale Representations

We introduce a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. The resulting calculus makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The new calculus is very effective when it comes to computing drifts and expected values that possibly involve a change of measure. Such drift calculations yield, for example, partial integro-differential equations, Hamilton–Jacobi–Bellman equations, Feynman–Kac formulae, or exponential moments needed in numerous applications. We provide several illustrations of the new technique, among them (1) a novel result on the Margrabe option to exchange one defaultable asset for another; and (2) a novel technique for computing the distribution of a signed stochastic exponential with application in mean-variance analysis. The talk is based on a series of joint works with Johannes Ruf; see ssrn.com/abstract=3500384,ssrn.com/abstract=3633638, ssrn.com/abstract=3633622, and arxiv.org/abs/1909.03020.

**1130– Coffee Break**

**1200– Puru Gupta (Warwick)**

Portfolio Choice in Dynamic Thin Markets: Merton meets Cournot

We consider an augmented version of Merton’s canonical portfolio choice problem, wherein trading by individual investors influences the price of the underlying traded financial asset. In a market with two such large investors, this gives rise to strategic interaction among investors, in which investors decide their trading rates independently and simultaneously at each instant in the spirit of dynamic Cournot competition, which we model and analyze as a non-zero sum singular stochastic differential game. We establish an equivalence result for an investor’s best response problem, which is a singular stochastic optimal control problem, and an auxiliary classical optimal control problem by proving that the valuefunctions for the two problems are equal. The equivalence result is obtained by exploiting the invariance of the value functions with respect to an integral flow associated with the drift coefficient of the original control problem. Under mild regularity conditions, we also show that the optimal trajectories of the two control problems coincide, which permits analytical characterization of investors’ best-response mappings. Finally, in the special case when stock price volatility is constant, we show that the unique Nash equilibrium is deterministic.

**1230– Faycal Drissi (Oxford/Paris I)**

Decentralised Finance and Automated Market Making: Optimal Liquidity Provision

Automated Market Makers (AMMs) are a new prototype of trading venues which are revolutionising the way market participants interact. At present, the majority of AMMs are Constant Function Market Makers (CFMMs) where a deterministic trading function determines how markets are cleared. A distinctive characteristic of CFMMs is that execution costs for liquidity takers, and revenue for liquidity providers, are given by a closed-form function of price, liquidity, and transaction size. This gives rise to a new class of trading problems. We focus on Constant Product Market Makers with Concentrated Liquidity and show how to optimally provide liquidity. We use Uniswap v3 data to study liquidity dynamics and to motivate the models. In particular, we describe how the wealth from liquidity provision decomposes into a fee component and an asset component. Finally, we perform consecutive runs of in-sample estimation of model parameters and out-of-sample liquidity provision to showcase the performance of the strategies.

**1300– Lunch**

**1430– Henry Chiu (Imperial)**

A model-free approach to continuous-time finance

We present a non-probabilistic, pathwise approach to continuous-time finance based on causal functional calculus. We introduce a definition of self-financing, free from any integration concept and show that the value of a self-financing portfolio is a pathwise integral (every self-financing strategy is a gradient) and that generic domain of functional calculus is inherently arbitrage-free. We then consider the problem of hedging a path dependent payoff across a generic set of scenarios. We apply the transition principle of Isaacs in optimal control and obtain a verification theorem for the solution, which is characterised by a (fully non-linear) path-dependent equation. For the Asian option, we obtain explicit solution.

**1430– Saad Labyad (Oxford)**

Gradient-based estimation of linear Hawkes processes with general kernels

Linear multivariate Hawkes processes (MHP) are a fundamental class of point processes with self excitation. When estimating parameters for these processes, a difficulty is that the two main error functionals, the log-likelihood and the least squares error (LSE), as well as the evaluation of their gradients, have a quadratic complexity in the number of observed events. In practice, this prohibits the use of exact gradient-based algorithms for parameter estimation. We construct an adaptive stratified sampling estimator of the gradient of the LSE. This results in a fast parametric estimation method for MHP with general kernels, applicable to large datasets, which compares favourably with existing

methods.

**1530– Coffee Break**

**1600– Vicky Henderson (Warwick)**

Regret in Trading Decisions: A Model and Empirical Study

In this talk we will present a simple model of asset sales under dynamic regret. We will use this to motivate our empirical study of trading decisions using a large discount brokerage dataset. Our focus is to test how regret induced by not selling a stock at its maximum price shapes the propensity to sell. We undertake a number of descriptive analyses and more formal analysis via proportional hazard modelling.

**1700– Close**