This year we plan to hold three London/Oxford/Warwick financial mathematics workshops to bring together researchers at our different institutions and to establish new research networks.

The first London/Oxford/Warwick Financial Mathematics Workshop will take place online on Tuesday 11 and Wednesday 12 January 2022. The planned face-to-face event has been cancelled due to the increasing spread of coronavirus. The second event will take place on Monday 4 and Tuesday 5 April at the University of Warwick and we hope will be held face-to-face. Details of the Oxford events will be confirmed in the new year.

Registration

The event is open to all, but please register in advance by clicking 'Register for this event' so that we can send you the meeting link.

Speakers

• Chris Kenyon
• Christoph Czichowsky
• Miryana Grigorova
• Leandro Sanchez-Betancourt
• Zhezhun Chen
• Chris Salvi
• Nazem Khan
• Wolfgang Stockinger
• Alex Pannier
• Holly Brannelly

Announcements

Next London/Oxford/Warwick event

The next event will take place in Warwick on 4 and 5 April. You can find the website for the next event by following this link.

We are taking submissions of proposed talks for the next London/Oxford/Warwick workshop. If you would like to give a talk you should submit your title and abstract using this link by 17:00 on Wed 26 Jan. We particularly encourage Postdocs and PhD students to submit proposed talks.

We plan to open registration for the Warwick event next week.

FinMathMentor Mentoring Scheme

The SIAM Activity Group on Financial Mathematics and Engineering and the Bachelier Finance Society have launched a new mentoring scheme for early career financial mathematicians and engineers from underrepresented groups. You can find out more here.

Agenda

Tuesday 11 January

10:30 Welcome

10:35-11:30, Chris Kenyon, Sustainability framework for financial products: Carbon Equivalence Principle (CEP)

Sustainability is a key point for financial markets and we use equivalent carbon impact as a useful simplification to capture its key axis. We propose the Carbon Equivalence Principle (CEP) for financial products: i.e. that the carbon effect of a financial product be included as a linked term sheet compatible with existing bank systems. The CEP implies that the carbon impact of investment follows the money. Making carbon impacts consistent with existing bank systems enables direct alignment of financial product use, management to net zero, and sustainability, improving on non-compatible disclosure proposals. The CEP enables pricing and risk management by including potential future costs and benefits. It is motivated by bank self-interest, and so aligns financial markets with sustainability without need for any additional consideration. We demonstrate one application of the CEP by calculating the potential liability cost due to carbon emissions for the construction and running of power plants using examples from the US Energy Administration. The annuity spread required for financing the carbon emission liability can be interpreted both as cost, and as a change to the credit rating of the projects. We observe that many non-renewable technologies are already non-viable for new-builds in the USA under all carbon price scenarios provided by the Network for Greening the Financial System (NGFS) and most – but not all – renewable technologies are viable. Other applications of the CEP beyond project finance include equities, crypto-currencies, commodities, and derivatives.

11:30-11:40 Break

11:40-12:25, Cris Salvi, Neural Stochastic Partial Differential Equations.

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling dynamical systems evolving under the influence of randomness. We introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs, SDEs, RDEs -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving at an arbitrary resolution, both in space and in time. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can be used to learn solution operators of SPDEs (a.k.a. Ito maps) depending simultaneously on the initial condition and a realization of the driving noise. By transferring some of its operations to the spectral domain, we show how a Neural SPDE can be evaluated either calling an ODE solver or solving a fixed point problem, inheriting in both cases memory-efficient backpropagation capabilities for training provided by existing adjoint-based or implicit-differentiation-based methods. Experiments on various semilinear SPDEs (including stochastic Navier-Stokes) demonstrate how our model is capable of learning complex spatiotemporal dynamics with better accuracy and using only a modest amount of training data compared to all alternative models, and its evaluation is up to 3 orders of magnitude faster than traditional solvers.

12:25-13:30 Break

13:30-14:15, Miryana Grigorova, Superhedging of options in non-linear incomplete market models

We will study the superhedging price (and superhedging strategies) of European and American options in non-linear incomplete market models with default, with a particular focus on the case of the American options which is more involved. We will provide a dual representation of the seller’s (superhedging) price for the American option in terms of a mixed stochastic control/stopping problem with non-linear expectations/evaluations, and in terms of non-linear Reflected BSDEs with constraints. If time permits, we will also present a duality result for the buyer’s price in terms of a stochastic game of control and stopping with non-linear expectations/evaluations. The talk will be based on joint works with Marie Claire Quenez and Agnès Sulem.

14:15-15:00, Nazem Khan, Sensitivity to large losses and ρ-arbitrage for convex risk measures

We study mean-risk portfolio selection in a one-period financial market, where risk is quantified by a star-shaped risk measure ρ. 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 ρ-arbitrage. This leads to a new class of risk measures that are suitable for portfolio selection. We show that ρ belongs to this class if and only if ρ is real-valued and the smallest positively homogeneous risk measure dominating ρ is the worst-case risk measure. We then specialise to the case that ρ is convex and admits a dual representation. We derive necessary and sufficient dual characterisations of (strong) ρ-arbitrage as well as the property that ρ 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.

15:00-15:15 Break

15:15-16:00, Alex Pannier, Rough multi-factor volatility models for SPX and VIX options

After a short review of VIX and rough volatility models, we provide explicit small-time formulae for the at-the-money implied volatility, skew and curvature in a large class of models. Our general setup encompasses both European options on a stock and VIX options, thereby providing new insights on their joint calibration. This framework also allows to consider rough volatility models and their multi-factor versions; in particular we develop a detailed analysis of the two-factor rough Bergomi model. The tools used are essentially based on Malliavin calculus for Gaussian processes. This is a joint work with A. Jacquier and A. Muguruza.

16:00-17:00 Informal discussion on Gather

Wednesday 12 January

09:00-10:00, Christoph Czichowsky, Rough volatility and portfolio optimisation under transaction costs

Rough volatility models have become quite popular recently, as they capture both the fractional scaling of the time series of the historic volatility (Gatheral et al. 2018) and the behaviour of the implied volatility surface (Fukasawa 2011, Bayer et al. 2016) remarkably well. In contrast to classical stochastic volatility models, the volatility process is neither a Markov process nor a semimartingale. Therefore, these models fall outside the scope of standard stochastic analysis and provide new mathematical challenges. In this talk, we illustrate the impact of rough volatility processes on portfolio optimisation under transaction costs connecting these two lines of research. The talk is based on joint work with Johannes Muhle-Karbe and Denis Schelling.

10:15-11:00, Wolfgang Stockinger, Numerical methods for high-dimensional mean-field control problems

In this talk, we discuss numerical methods for non-smooth mean-field control problems (MFC), which seek optimal control of McKean-Vlasov dynamics whose coefficients involve mean-field interactions both on the state and actions. At the core of our algorithm, we iteratively refine the approximate optimal feedback control by applying an accelerated proximal gradient method to the MFC problem, which does not require solving the first order optimality condition, arising from the stochastic maximum principle, analytically. Our method is a hybrid algorithm, i.e., Monte-Carlo and PDE based, which computes the optimal control in feedback form and solves the control problem along a flow of measures. Thereby, we improve the robustness of the numerically obtained feedback controls in terms of model perturbation compared to pure sampling-based algorithms. We also propose a version of our method based on a deep-neural network implementation, which is applicable for high-dimensional control problems, illustrated for the well-known Cucker-Smale model.

11:00-11:45, Holly Brannelly, A New Class of Quantile Processes for Risk Analysis and Valuation

This research focuses on the construction of a novel class of quantile processes, governing the stochastic dynamics of quantiles in continuous time. The marginals of such quantile processes are obtained by transforming the marginals of a cadlag process under a composite map consisting of a distribution and a quantile function, similar to rank transmutation maps. As a result, we obtain a one-step approach to constructing widely flexible classes of stochastic models, accommodating extensive ranges of higher-order moment behaviours (e.g., tail behaviours in the finite dimensional distributions, and asymmetry). Such features are directly parameterised in the composite map and are thus interpretable with respect to the driving process. It is shown that the quantile processes induce a distorted probability measure, and as such we propose a general, time-consistent, and dynamic risk valuation principle under the induced measures of quantile processes. The valuation principle allows for pricing in incomplete markets and thus has application in insurance pricing. Here, the distorted measures are considered `subjective’ and are constructed in such a way to account for external market characteristics and investor risk attitudes, leading to a parametric system of risk-sensitive probability measures, indexed by such factors.

11:45-13:00 Break

13:00-13:45, Leandro Sanchez-Betancourt, Internalise or Externalise: Brokers and Informed Trading

We study how a broker provides liquidity to an informed trader and to a noise trader. The broker decides how much of the flow she keeps in her books (i.e., internalisation) and how much she unwinds in an exchange (i.e., externalisation). We frame the interactions between the broker and traders as a Stackelberg game. The informed trader knows the stochastic process that drives the drift of the asset price. The order flow of the noise trader is uninformative and reverts to zero. We obtain the broker's internalisation and extarnalisation optimal strategy in closed form. We estimate model parameters from market data and show the performance of the broker, the noise trader, and the informed trader for a variety of scenarios. Lastly, we characterise the amount of transaction costs that the broker needs to charge to break even.

13:45-14:30, Zezhun Chen, EM Estimation for Bivariate Mixed Poisson INAR(1) Claim Count Regression Models with Correlated Random Effects

This work considers bivariate mixed Poisson INAR(1) regression models with correlated random effects for modelling correlations of different signs and magnitude among time series of different types of claim counts. This is the first time that the proposed family of INAR(1) models is used in a statistical or actuarial context. For expository purposes, the bivariate mixed Poisson INAR(1) claim count regression models with correlated Lognormal and Gamma random effects paired via a Gaussian copula are presented as competitive alternatives to the classical bivariate Negative Binomial INAR(1) claim count regression model which only allows for positive dependence between the time series of claim count responses. Our main achievement is that we develop novel alternative Expectation-Maximization type algorithms for maximum likelihood estimation of the parameters of the models which are demonstrated to perform satisfactory when the models are fitted to Local Government Property Insurance Fund data from the state of Wisconsin.

14:30-14:45 Closing Remarks/Announcements

Organising committee

• John Armstrong
• Sam Cohen
• Roxana Dumitrescu
• Martin Herdegen
• Gechun Liang

Scientific committee

• Umit Cetin
• Sam Cohen
• Martin Herdegen
• David Hobson
• Andrea Macrina
• Eyal Neuman
• Christoph Reisinger

Funding

We are grateful to the Heilbronn Institute of Mathematical Research for partial funding of our workshop series. This HIMR award was funded by the Additional Funding Programme for Mathematical Sciences, a £300 million investment over a five-year period from 2020/21 to 2024/25. This new funding is being delivered by EPSRC. The programme is providing a significant boost to the UK's world-leading mathematical sciences community, increasing support for this key discipline and increasing the pool of mathematicians able to participate in and deploy state-of-the-art research.