Mathematical and Computational Finance Seminar

We study optimal investment and consumption in an incomplete stochastic factor model for a power utility investor on the infinite horizon. When the state space of the stochastic factor is finite, we give a complete characterisation of the well-posedness of the problem and provide an efficient numerical algorithm for computing the value function. When the state space is a (possibly infinite) open interval and the stochastic factor is represented by an Ito diffusion, we develop a general theory of sub- and supersolutions for second-order ordinary differential equations on open domains without boundary values to prove existence of the solution to the Hamilton-Jacobi-Bellman (HJB) equation along with explicit bounds for the solution. By characterising the asymptotic behaviour of the solution, we are also able to provide rigorous verification arguments for various models, including the Heston model. Finally, we link the discrete and continuous setting and show that that the value function in the diffusion setting can be approximated very efficiently through a fast discretisation scheme.

Conventional data feeds from exchanges, even L3 feeds, generally only tell one what happened: accepted submissions of maker and taker orders,  cancellations, and the evolution of the order book and the best bid and ask prices. However, by analyzing a dataset derived from the blockchain of the highly liquid cryptocurrency exchange Hyperliquid, we are able to see all messages (4.5 bn in our one-month sample), including rejections. Unexpectedly, almost 60% of message traffic is generated by submission and subsequent rejection of a single order type: post-only limit orders sent to the 'wrong' (aggressive) side of the book, for example a buy limit order at a price at or above the best ask. Such orders are automatically rejected on arrival except in the (rare) case that the price moves up while the order is in transit. Nearly 30% of message traffic relates to cancellations, leaving a small fraction for all other messages.

I shall describe this order flow in detail, then address the question of why message traffic is dominated by rejected submissions which, by their nature, do not influence the order book in any way at all, and are invisible to all traders except the submitter. We propose that the reason lies in a market-making strategy whose aim is to gain queue priority immediately after any price change, and I shall show how the evidence supports this hypothesis. I shall also discuss the risk/return characteristics of the strategy, and finally discuss its pivotal role in replenishing liquidity following a price move.

Joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.

In the past years, deep learning algorithms have been applied to numerous classical problems from mathematical finance. In particular, deep learning has been employed to numerically solve high-dimensional derivatives pricing and hedging tasks. Theoretical foundations of deep learning for these tasks, however, are far less developed. In this talk, we start by revisiting deep hedging and introduce a recently developed adversarial training approach for making it more robust. We then present our recent results on theoretical foundations for approximating option prices, solutions to jump-diffusion PDEs and optimal stopping problems using (random) neural networks, allowing to obtain more explicit convergence guarantees. We address neural network expressivity, highlight challenges in analysing optimization errors and show the potential of random neural networks for mitigating these difficulties.

TBA

TBA

TBA