Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate neural SDE models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model, and will discuss some initial results using real data.

Based on joint work with Christoph Reisinger and Sheng Wang

Penrose's proposal of smooth conformal compactification is not only of geometric elegance, it also makes concrete predictions on physically measurable objects such as the "late-time tails" of gravitational waves.  At the same time, the physical motivation for a smooth null infinity remains itself unclear. In this talk, building on arguments due to Christodoulou, Damour and others, I will show that, in generic gravitational collapse, the "peeling property" of gravitational radiation is violated (so one cannot attach a smooth null infinity). Moreover, I will explain how this violation of peeling is in principle measurable in the form of leading-order deviations from the usual late-time tails of gravitational radiation.

This talk is based on https://arxiv.org/abs/2105.08079, https://arxiv.org/abs/2105.08084 and … .

It will be a hybrid seminar on both zoom and in-person in L5. 

4d N=2 SCFTs are extremely important structures. In the first minitalk we will introduce them, then we will show three areas of mathematics with which this area of physics interacts. The minitalks are independent. The talk will be hybrid, with teams link below.

The junior Geometry and Physics seminar aims to bring together people from both areas, giving talks which are interesting and understandable to both.

Website: https://sites.google.com/view/oxfordpandg/physics-and-geometry-seminar

Teams link: https://www.google.com/url?q=https%3A%2F%2Fteams.microsoft.com%2Fl%2Fme…

We introduce a new method for studying stochastic processes in random graphs controlled by degree information, involving combining enumeration techniques with an abstract second moment argument. We use it to constructively resolve a conjecture of Füredi from 1988: with high probability, the random graph G(n,1/2) admits a friendly bisection of its vertex set, i.e., a partition of its vertex set into two parts whose sizes differ by at most one in which n-o(n) vertices have at least as many neighbours in their own part as across. This work is joint with Asaf Ferber, Matthew Kwan, Bhargav Narayanan, and Mehtaab Sawhney.

We study a family of evolving directed random graphs that includes the directed preferential model and the directed uniform attachment model. The directed preferential model is of particular interest since it is known to produce scale-free graphs with regularly varying in-degree distribution. We start by describing the local weak limits for our family of random graphs in terms of randomly stopped continuous-time branching processes, and then use these limits to establish the asymptotic behavior of the corresponding PageRank distribution. We show that the limiting PageRank distribution decays as a power-law in both models, which is surprising for the uniform attachment model where the in-degree distribution has exponential tails. And even for the preferential attachment model, where the power-law hypothesis suggests that PageRank should follow a power-law, our result shows that the two tail indexes are different, with the PageRank distribution having a heavier tail than the in-degree distribution.

Suppose there are $n$ students in a class. But assume that not everybody is friends with everyone else, and there is a graph which determines the friendship structure. What is the chance that there are two friends in this class, both with birthdays on October 12? More generally, given a simple labelled graph $G_n$ on $n$ vertices, color each vertex with one of $c=c_n$ colors chosen uniformly at random, independent from other vertices. We study the question: what is the number of monochromatic edges of color 1?

As it turns out, the limiting distribution has three parts, the first and second of which are quadratic and linear functions of a homogeneous Poisson point process, and the third component is an independent Poisson. In fact, we show that any distribution limit must belong to the closure of this class of random variables. As an application, we characterize exactly when the limiting distribution is a Poisson random variable.

This talk is based on joint work with Bhaswar Bhattacharya and Somabha Mukherjee.

We will review how one can find families of 4d N=1 SCFTs starting from known 6d (1,0) SCFTs. 

Then we will discuss a relation between 6d RG-flows and 4d RG-flows, where the 4d RG-flow relates 4d N=1 models constructed from compactification of 6d (1,0) SCFTs related by the 6d RG-flow. We will show how we can utilize such a relation to find many "Lagrangians" for strongly coupled 4d models. Relating 6d SCFTs to 4d models as mentioned above will result in geometric reasoning behind some 4d phenomena such as dualities and symmetry enhancement.

Such a program generates a large database of known 4d N=1 SCFTs with many interrelations one can use in future efforts to construct 4d N=1 SCFTs from string theory directly.