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Term 3

Date: 15th June 2022

Cosimo Andrea Munari (University of Zurich): Fundamental theorem of asset pricing with acceptable risk in markets with frictions

We study the range of prices at which a rational agent should contemplate transacting a financial contract outside a given securities market. Trading is subject to nonproportional transaction costs and portfolio constraints and full replication by way of market instruments is not always possible. Rationality is defined in terms of consistency with market prices and acceptable risk thresholds. We obtain a direct and a dual description of market-consistent prices with acceptable risk. The dual characterization requires an appropriate extension of the classical Fundamental Theorem of Asset Pricing where the role of arbitrage opportunities is played by good deals, i.e., costless investment opportunities with acceptable risk-reward tradeoff. In particular, we highlight the importance of scalable good deals, i.e., investment opportunities that are good deals regardless of their volume.

Date: 9th June 2022

Urvashi Behary Paray (University of Warwick): Convexity Corrections via a Markov-functional approach

Markov-functional models (MFMs) constitute a form of interest rate model. They are attractive to practitioners working in banks and hedge funds because of their efficiency and ability to calibrate to a variety of liquid market instruments. They are used to price and hedge exotic interest-rate derivative instruments whose prices are not available in the market and so must be priced in house. The aim during the talk is two-fold: Firstly, we will explore the setup of a (two-factor) Markov-functional approach to price Constant Maturity swaps (CMS); CMS are an interest rate swap whereby the interest in one leg is reset periodically with reference to a market swap rate, usually with a long-term maturity. When pricing a CMS, some form of adjustments/corrections, referred to as convexity corrections is required. Convexity corrections can be loosely defined as the adjustment term that has to be introduced when taking an expectation of an interest rate with respect to a measure under which it is not a martingale. The model in itself is computationally intensive but provides a rich mathematical framework through which we can explore convexity corrections. We then use the Markov-functional model as a benchmark model to construct an efficient and easy-to-implement (one-factor or two-factor) model that is able to capture the most relevant market information when pricing convexity-related products.

Date: 26th May 2022

Harto Saarinen (University of Turku): Optimal control problems of one-dimensional diffusions with random intervention times

We study ergodic and discounted bounded variation control of a regular one-dimensional diffusion process. The agent is allowed to control the diffusion only at jump times of an observable, independent Poisson process. Under relatively weak set of assumptions we solve these problems in closed form. We discuss the connection of these problems with the usual singular control problems. The results are illustrated with geometric Brownian motion and Ornstein-Uhlenbeck process.

Term 2

Date: 4th February 2022

Ruiqi Liu (University of Warwick): The Optimal Control of Inventory and Production for a Hybrid Energy Producer

We study a continuous-time, infinite horizon optimal energy storage and production problem. The primary source of production is modelled as an uncontrolled one-dimensional diffusion process with general dynamics. By controlling the secondary source of production and total energy output, which are both bounded variation processes, we aim to optimize the storage level under a general running reward function and maximize the profit generated from the production. Through associating the control problem with Dykin's game, the optimal control is closely related to two free boundaries, and we show that one can be directly computed and the other is characterized via an integral equation. After establishing the smooth-pasting principle on these boundaries, a viscosity approach is used to prove the smoothness of the value function, which leads to the verification of the proposed optimal control.

Date: 25th February 2022

Huy Chau (University of Manchester): Super-replication with transaction costs under model uncertainty for continuous processes

We formulate a superhedging theorem in the presence of transaction costs and model uncertainty. Asset prices are assumed continuous and uncertainty is modelled in a parametric setting. Our proof relies on a new topological framework in which no Krein-Smulian type theorem is available. This is joint with Masaaki Fukasawa (Osaka University) and Miklos Rasonyi (Alfred Renyi Institute of Mathematics)

Date: 18th March 2022

Mikko Pakkanen (Imperial College London): Rough volatility – Re-examining empirical evidence using the generalised method of moments

In late 2014, Jim Gatheral, Thibault Jaisson and Mathieu Rosenbaum released the first preprint of their ground-breaking paper "Volatility is rough", arguing that financial market volatility should be modelled by stochastic processes with rough trajectories, such as a fractional Brownian motion with Hurst index below 0.5.

While Gatheral, Jaisson and Rosenbaum's empirical findings on the roughness of realised volatility have since been replicated across different asset classes and with thousands of assets, determining the roughness of realised volatility remains a delicate statistical problem. It is complicated by the fact that we can only observe volatility as a time integral (integrated variance) with measurement error (estimated by means of realised variance). Integration is a smoothing operation while measurement error increases the perceived roughness of the measurements, giving rise to two counteracting sources of bias whose net effect is unclear. In particular, critics have questioned to what degree roughness of volatility can be distinguished from measurement error.

In this talk, I will present a novel generalised method of moments (GMM) estimation technique for general log-normal volatility models, aiming to address this concern. The GMM estimator accommodates both the impact of integration and the presence of measurement error, the latter through a bias correction. After presenting asymptotic theory for the GMM estimator, I will demonstrate by Monte Carlo experiments that the bias correction is indispensable. Indeed, without it, non-rough volatility may be estimated as rough, but once it is incorporated, the bias problem is, for all practical purposes, resolved. Finally, by applying the GMM estimator to empirical realised volatility data on 29 stock market indices worldwide, I show that Gatheral, Jaisson and Rosenbaum's conclusion stands: realised volatility is indeed best described by a rough process.

Based on joint work with Anine Bolko, Kim Christensen and Bezirgen Veliyev.

Term 1

Date: 15th October 2021

Nazem Khan (University of Warwick): Sensitivity to large losses and rho-arbitrage for convex risk measures

In this talk, we revisit mean-risk portfolio selection in a one-period financial market, where risk is quantified by a superlinear risk measure rho. We introduce two new axioms: weak and strong sensitivity to large losses. We show that the first axiom is key to ensure the existence of optimal portfolios and the second one is key to ensure the absence of rho-arbitrage. This leads to a new class of risk measures that are suitable for portfolio selection. We show that rho belongs to this class if and only if rho is real-valued and the smallest positively homogeneous risk measure dominating rho is the worst-case risk measure. We then specialise to the case that rho is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) rho-arbitrage as well as the property that rho is suitable for portfolio selection. Finally, we introduce the new risk measure of Loss Sensitive Expected Shortfall, which is similar to and not more complicated to compute than Expected Shortfall but suitable for portfolio selection - which Expected Shortfall is not.

Date: 4th November 2021

Matteo Burzoni (Università degli Studi di Milano): Mean Field Games with absorption and a model of bank run

We consider a MFG problem obtained as the limit of N-particles systems with an absorbing region. Once a particle hits such a region, it leaves the game and the rest of the system continues to play with N-1 particles. We study existence of equilibria for the limiting problem in a framework with common noise and establish the existence of epsilon Nash equilibria for the N-particles problems. These results are applied to a novel model of bank run. This is a joint work with L. Campi.

Date: 26th November 2021

Johannes Muhle-Karbe (Imperial College London): Hedging with Market and Limit Orders

Trading with limit orders allows to earn rather than pay bid-ask spreads. However, limit orders do not guarantee immediate execution and, moreover, are exposed to adverse selection by counterparties with superior information. We study the resulting tradeoff between market and limit orders for option hedging in a tractable extension of Leland’s model. This serves as a stylised model for the management of “central risk books”, where risk management and trade implementation are centralised. (Joint work in progress with Kevin Webster (Citadel LLC) and Zexin Wang (Imperial).)

Date: 3rd December 2021

Roxana Dumitrescu (King's College London): Control and optimal stopping mean-field games: a linear programming approach

We develop the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via the controlled/stopped martingale approach, another relaxation method used in a few previous papers in the case when there is only control (joint work with M. Leutscher and P. Tankov)