arXiv subjects

John Armstrong

Publications and source records attributed to John Armstrong.

At least 19 recordsLinked to original sources

arXiv2026

An ergodic theorem for multi-period mutual insurance

Suppose there are $N$ heterogeneous agents in a market with idiosyncratic risks but no uninsurable systematic risk factors. These agents may agree arbitrary financial contracts with one another, subject to the condition that contracts are self-enforcing under coalitions of agents in a common state. We show that, under mild conditions, this uniquely determines the limiting utility of every agent as $N$ tends to infinity. The result is an ergodic theorem: as the population grows, the number of degrees of freedom in the problem collapses, so that agents in the same state are treated identically in the limit. We exhibit an explicit, practically realisable mechanism achieving this limit using only short-dated contracts. The model can be applied either to an economy of heterogeneous agents pooling idiosyncratic risk through self-enforcing contracts or to the design of optimal insurance products such as pensions.

q-fin.MF↗

arXiv2026

Gamma Hedging without Rough Paths

We show how the robustness of gamma hedging can be understood without using rough-path theory. Instead, we use the concepts of $p^{th}$ variation along a partition sequence and Taylor's theorem directly, rather than defining an integral and proving a version of It\^o's lemma. The same approach allows classical results on delta-hedging to be proved without defining an integral and without the need to define the concept of self-financing in continuous time. We show that the approach can also be applied to barrier options and Asian options

math.PR↗

arXiv2025

Machine-learning a family of solutions to an optimal pension investment problem

We use a neural network to identify the optimal solutions to a family of pension investment problems, where the parameters determining an investor's risk and consumption preferences are given as inputs to the neural network in addition to economic variables. Training a single network across such a family fails without modification. Our main contribution is a scaling of the loss function that resolves this, together with a proof that the resulting algorithm converges. We use this to develop a practical tool for exploring how pension outcomes vary with preference parameters. We use a Black-Scholes economic model so that we may validate the accuracy of the network using a classical and provably convergent numerical method developed using the duality approach.

q-fin.CP↗

arXiv2025

Study on P-Type Doping of Mid-Wave and Long-Wave Infrared Mercury Cadmium Telluride

We present in depth study of p-type doping concentration of mid-wave infrared (MWIR) and long-wave infrared (LWIR) mercury cadmium Telluride (HgCdTe) thin films. Annealing time was changed under specific conditions to achieve a stable copper (Cu) doping concentration for HgCdTe thin films. Both MWIR and LWIR HgCdTe material were grown by molecular beam epitaxy (MBE), where different trends were observed between LWIR and MWIR HgCdTe thin films by increasing anneal time. We also report the impact of different thickness (4 micron, 6 micron and 9 micron) along with annealing time on doping level of LWIR HgCdTe thin films.

cond-mat.mtrl-sci↗

arXiv2025

A comparison of the effectiveness of alternative DC and CDC designs in a UK market

We use three stochastic models to evaluate the effectiveness of a number of possible pension designs which have been proposed for use in the UK. We consider individual DC schemes followed by full annuitisation and a flex-and-fix strategy which combines drawdown with gradual annuitisation. We compare these approaches with collective designs including: a flat-accrual shared-indexation CDC scheme that is similar to the Royal Mail Collective Pension Plan; a dynamic-accrual shared-indexation CDC scheme modelled on the approach considered in the DWP consultation on multi-employer CDC; and an alternative collective design based on a tontine structure. In our comparisons, we tune each strategy to give optimal performance given the stochastic model and a choice of representative risk preferences. We find the collective design based on a tontine structure consistently achieves the best performance in terms of member utility. We discuss the importance of leverage in the optimal investment strategies.

q-fin.PM↗

arXiv2024

Intergenerational cross-subsidies in UK Collective Defined Contribution (CDC) funds

We evaluate the performance and level of intergenerational cross-subsidy in flat-accrual and dynamic-accrual collective defined contribution (CDC) schemes which have been designed to be compatible with UK legislation. In the flat-accrual scheme, all members accrue the benefits at the same rate irrespective of age. This captures the most significant feature of the Royal Mail Collective Pension Plan, which is currently the only UK CDC scheme. The dynamic-accrual schemes seeks to reduce intergenerational cross-subsidies by varying the rate of benefit-accrual in accordance to the age of members and the current funding level. We find that these CDC schemes can often be successful in smoothing pension outcomes post-retirement while outperforming a defined contribution scheme followed by annuity purchase at the point of retirement. However, this out-performance is not guaranteed in a flat-accrual scheme and there is little smoothing of projected pension outcomes before retirement. There are significant intergenerational cross-subsidies in the flat-accrual scheme. These qualitatively mirror the cross-subsidies seen in existing defined benefit schemes, but we find the magnitude of the cross-subsidies is much larger in flat accrual CDC schemes. The dynamic-accrual scheme design is intended to reduce such cross-subsidies, but we find they still arise due to the approximate pricing methodology used to determine the benefits accrued by each contribution. Although the cross-subsidies tend to cancel out over time, in any given year they can be large. Thus, the benefits accrued by contributions should be calculated rigorously to reduce cross-subsidies.

q-fin.GN↗

arXiv2024

Optimal mutual insurance against systematic longevity risk

We mathematically demonstrate how and what it means for two collective pension funds to mutually insure one another against systematic longevity risk. The key equation that facilitates the exchange of insurance is a market clearing condition. This enables an insurance market to be established even if the two funds face the same mortality risk, so long as they have different risk preferences. Provided the preferences of the two funds are not too dissimilar, insurance provides little benefit, implying the base scheme is effectively optimal. When preferences vary significantly, insurance can be beneficial.

q-fin.MF↗

arXiv2024

Abundant Superintegrable Systems and Hessian Structures

We show that a large class of non-degenerate second-order (maximally) superintegrable systems gives rise to Hessian structures, which admit natural (Hessian) coordinates adapted to the superintegrable system. In particular, abundant superintegrable systems on Riemannian manifolds of constant sectional curvature fall into this class. We explicitly compute the natural Hessian coordinates for examples of non-degenerate second-order superintegrable systems in dimensions two and three.

nlin.SI↗

arXiv2024

Deep Gamma Hedging

We train neural networks to learn optimal replication strategies for an option when two replicating instruments are available, namely the underlying and a hedging option. If the price of the hedging option matches that of the Black--Scholes model then we find the network will successfully learn the Black-Scholes gamma hedging strategy, even if the dynamics of the underlying do not match the Black--Scholes model, so long as we choose a loss function that rewards coping with model uncertainty. Our results suggest that the reason gamma hedging is used in practice is to account for model uncertainty rather than to reduce the impact of transaction costs.

q-fin.CP↗

arXiv2024

Optimal post-retirement investment under longevity risk in collective funds

We study the optimal investment problem for a homogeneous collective of $n$ individuals investing in a Black-Scholes model subject to longevity risk with Epstein--Zin preferences. %and with preferences given by power utility. We compute analytic formulae for the optimal investment strategy, consumption is in discrete-time and there is no systematic longevity risk. We develop a stylised model of systematic longevity risk in continuous time which allows us to also obtain an analytic solution to the optimal investment problem in this case. We numerically solve the same problem using a continuous-time version of the Cairns--Blake--Dowd model. We apply our results to estimate the potential benefits of pooling longevity risk over purchasing an insurance product such as an annuity, and to estimate the benefits of optimal longevity risk pooling in a small heterogeneous fund.

q-fin.MF↗

arXiv2024

Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach

Predicting the conditional evolution of Volterra processes with stochastic volatility is a crucial challenge in mathematical finance. While deep neural network models offer promise in approximating the conditional law of such processes, their effectiveness is hindered by the curse of dimensionality caused by the infinite dimensionality and non-smooth nature of these problems. To address this, we propose a two-step solution. Firstly, we develop a stable dimension reduction technique, projecting the law of a reasonably broad class of Volterra process onto a low-dimensional statistical manifold of non-positive sectional curvature. Next, we introduce a sequentially deep learning model tailored to the manifold's geometry, which we show can approximate the projected conditional law of the Volterra process. Our model leverages an auxiliary hypernetwork to dynamically update its internal parameters, allowing it to encode non-stationary dynamics of the Volterra process, and it can be interpreted as a gating mechanism in a mixture of expert models where each expert is specialized at a specific point in time. Our hypernetwork further allows us to achieve approximation rates that would seemingly only be possible with very large networks.

math.NA↗

arXiv2023

The decolonisation of mathematics

We describe a mainstream "universalist" approach to the understanding of mathematics. We then conduct a systematic (but not exhaustive) review of the academic literature on the decolonisation of mathematics and identify how this challenges the universalist view. We examine evidence of whether the experience of mathematics in the UK is systemically racist, examining both the decolonial arguments and the empirical evidence. We find that there may be some benefit in teaching the history of mathematics, but that this should be weighed against the opportunity cost. We find some prima-facie evidence of discrimination in the descriptive statistics on the representation of ethnic minorities in academic roles in UK higher education.

math.HO↗

arXiv2023

Gamma Hedging and Rough Paths

We apply rough-path theory to study the discrete-time gamma-hedging strategy. We show that if a trader knows that the market price of a set of European options will be given by a diffusive pricing model, then the discrete-time gamma-hedging strategy will enable them to replicate other European options so long as the underlying pricing signal is sufficiently regular. This is a sure result and does not require that the underlying pricing signal has a quadratic variation corresponding to a probabilisitic pricing model. We show how to generalise this result to exotic derivatives when the gamma is defined to be the Gubinelli derivative of the delta by deriving rough-path versions of the Clark--Ocone formula which hold surely. We illustrate our theory by proving that if the stock price process is sufficiently regular, as is the implied volatility process of a European derivative with maturity $T$ and smooth payoff $f(S_T)$ satisfying $f^{\prime \prime}>0$, one can replicate with certainty any European derivative with smooth payoff and maturity $T$.

q-fin.MF↗

arXiv2022

Projections of SDEs onto Submanifolds

In [ABF19] the authors define three projections of Rd-valued stochastic differential equations (SDEs) onto submanifolds: the Stratonovich, Ito-vector and Ito-jet projections. In this paper, after a brief survey of SDEs on manifolds, we begin by giving these projections a natural, coordinate-free description, each in terms of a specific representation of manifold-valued SDEs. We proceed by deriving formulae for the three projections in ambient $\mathbb R^d$-coordinates. We use these to show that the Ito-vector and Ito-jet projections satisfy respectively a weak and mean-square optimality criterion for small t: this is achieved by solving constrained optimisation problems. These results confirm, but do not rely on the approach taken in [ABF19], which is formulated in terms of weak and strong Ito-Taylor expansions. In the final section we exhibit examples showing how the three projections can differ, and explore alternative notions of optimality.

math.PR↗

arXiv2022

Itô Stochastic differentials

We give an infinitesimal meaning to the symbol $dX_t$ for a continuous semimartingale $X$ at an instant in time $t$. We define a vector space structure on the space of differentials at time $t$ and deduce key properties consistent with the classical Itô integration theory. In particular, we link our notion of a differential with Itô integration via a stochastic version of the Fundamental Theorem of Calculus. Our differentials obey a version of the chain rule, which is a local version of Itô's lemma. We apply our results to financial mathematics to give a theory of portfolios at an instant in time.

math.PR↗

arXiv2020

Anomalous Recurrence Properties of Markov Chains on Manifolds of Negative Curvature

We present a recurrence-transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that there exist Markov chains on a large class of such manifolds which are both recurrent and have zero average drift at every point. We give an explicit example of such a chain on hyperbolic space of arbitrary dimension, and also on a stochastically incomplete manifold. We also prove that such recurrent chains cannot be uniformly elliptic, in contrast with the Euclidean case.

math.PR↗

arXiv2020

The importance of dynamic risk constraints for limited liability operators

Previous literature shows that prevalent risk measures such as Value at Risk or Expected Shortfall are ineffective to curb excessive risk-taking by a tail-risk-seeking trader with S-shaped utility function in the context of portfolio optimisation. However, these conclusions hold only when the constraints are static in the sense that the risk measure is just applied to the terminal portfolio value. In this paper, we consider a portfolio optimisation problem featuring S-shaped utility and a dynamic risk constraint which is imposed throughout the entire trading horizon. Provided that the risk control policy is sufficiently strict relative to the asset performance, the trader's portfolio strategies and the resulting maximal expected utility can be effectively constrained by a dynamic risk measure. Finally, we argue that dynamic risk constraints might still be ineffective if the trader has access to a derivatives market.

q-fin.PM↗

arXiv2020

The ineffectiveness of coherent risk measures

We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term $ρ$-arbitrage for a risk measure $ρ$. We show how to determine analytically whether such $ρ$-arbitrage portfolios exist in complete markets and in the Markowitz model. We also consider realistic numerical examples of incomplete markets and determine whether expected shortfall constraints are ineffective in these markets. We find that the answer depends heavily upon the probability model selected by the risk manager but that it is certainly possible for expected shortfall constraints to be ineffective in realistic markets. Since value at risk constraints are weaker than expected shortfall constraints, our results can be applied to value at risk. By contrast, we show that reasonable expected utility constraints are effective in any arbitrage-free market.

q-fin.RM↗