arXiv subjects

John Armstrong

Publications and source records attributed to John Armstrong.

At least 37 records · Page 2Linked to original sources

arXiv2020

Curved Schemes for SDEs on Manifolds

Given a stochastic differential equation (SDE) in $\mathbb{R}^n$ whose solution is constrained to lie in some manifold $M \subset \mathbb{R}^n$, we propose a class of numerical schemes for the SDE whose iterates remain close to $M$ to high order. Our schemes are geometrically invariant, and can be chosen to give perfect solutions for any SDE which is diffeomorphic to $n$-dimensional Brownian motion. Unlike projection-based methods, our schemes may be implemented without explicit knowledge of M. Our approach does not require simulating any iterated Itô interals beyond those needed to implement the Euler--Maryuama scheme. We prove that the schemes converge under a standard set of assumptions, and illustrate their practical advantages by considering a stochastic version of the Kepler problem.

math.NA↗

arXiv2020

Classifying Financial Markets up to Isomorphism

Two markets should be considered isomorphic if they are financially indistinguishable. We define a notion of isomorphism for financial markets in both discrete and continuous time. We then seek to identify the distinct isomorphism classes, that is to classify markets. We classify complete one-period markets. We define an invariant of continuous time complete markets which we call the absolute market price of risk. This invariant plays a role analogous to the curvature in Riemannian geometry. We classify markets when the absolute market price of risk is deterministic. We show that, in general, markets with non-trivial automorphism groups admit mutual fund theorems. We prove a number of such theorems.

q-fin.MF↗

arXiv2020

Non-Geometric Rough Paths on Manifolds

We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold TM, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.

math.CA↗

arXiv2020

Option pricing models without probability: a rough paths approach

We describe the pricing and hedging of financial options without the use of probability using rough paths. By encoding the volatility of assets in an enhancement of the price trajectory, we give a pathwise presentation of the replication of European options. The continuity properties of rough-paths allow us to generalise the so-called fundamental theorem of derivative trading, showing that a small misspecification of the model will yield only a small excess profit or loss of the replication strategy. Our hedging strategy is an enhanced version of classical delta hedging where we use volatility swaps to hedge the second order terms arising in rough-path integrals, resulting in improved robustness.

q-fin.MF↗

arXiv2020

Stochastic modeling of assets and liabilities with mortality risk

This paper describes a general approach for stochastic modeling of assets returns and liability cash-flows of a typical pensions insurer. On the asset side, we model the investment returns on equities and various classes of fixed-income instruments including short- and long-maturity fixed-rate bonds as well as index-linked and corporate bonds. On the liability side, the risks are driven by future mortality developments as well as price and wage inflation. All the risk factors are modeled as a multivariate stochastic process that captures the dynamics and the dependencies across different risk factors. The model is easy to interpret and to calibrate to both historical data and to forecasts or expert views concerning the future. The simple structure of the model allows for efficient computations. The construction of a million scenarios takes only a few minutes on a personal computer. The approach is illustrated with an asset-liability analysis of a defined benefit pension fund.

q-fin.RM↗

arXiv2020

Collectivised Post-Retirement Investment

We quantify the benefit of collectivised investment funds, in which the assets of members who die are shared among the survivors. For our model, with realistic parameter choices, an annuity or individual fund requires approximately 20\% more initial capital to provide as good an outcome as a collectivised investment fund. We demonstrate the importance of the new concept of pension adequacy in defining investor preferences and determining optimal fund management. We show how to manage heterogeneous funds of investors with diverse needs. Our framework can be applied to existing pension products, such as Collective Defined Contribution schemes.

q-fin.PM↗

arXiv2020

Asymptotically Optimal Management of Heterogeneous Collectivised Investment Funds

A collectivised fund is a proposed form of pension investment, in which all investors agree that any funds associated with deceased members should be split among survivors. For this to be a viable financial product, it is necessary to know how to manage the fund even when it is heterogeneous: that is when different investors have different preferences, wealth and mortality. There is no obvious way to define a single objective for a heterogeneous fund, so this is not an optimal control problem. In lieu of an objective function, we take an axiomatic approach. Subject to our axioms on the management of the fund, we find an upper bound on the utility that can be achieved for each investor, assuming a complete markets and the absence of systematic longevity risk. We give a strategy for the management of such heterogeneous funds which achieves this bound asymptotically as the number of investors tends to infinity.

q-fin.PM↗

arXiv2019

Collectivised Pension Investment with Exponential Kihlstrom--Mirman Preferences

In a collectivised pension fund, investors agree that any money remaining in the fund when they die can be shared among the survivors. We give a numerical algorithm to compute the optimal investment-consumption strategy for an infinite collective of identical investors with exponential Kihlstrom--Mirman preferences, investing in the Black--Scholes market in continuous time but consuming in discrete time. Our algorithm can also be applied to an individual investor. We derive an analytic formula for the optimal consumption in the special case of an individual who chooses not to invest in the financial markets. We prove that our problem formulation for a fund with an infinite number of members is a good approximation to a fund with a large, but finite number of members.

q-fin.PM↗

arXiv2019

Collectivised Pension Investment with Homogeneous Epstein-Zin Preferences

In a collectivised pension fund, investors agree that any money remaining in the fund when they die can be shared among the survivors. We compute analytically the optimal investment-consumption strategy for a fund of $n$ identical investors with homogeneous Epstein--Zin preferences, investing in the Black--Scholes market in continuous time but consuming in discrete time. Our result holds for arbitrary mortality distributions. We also compute the optimal strategy for an infinite fund of investors, and prove the convergence of the optimal strategy as $n\to \infty$. The proof of convergence shows that effective strategies for inhomogeneous funds can be obtained using the optimal strategies found in this paper for homogeneous funds, using the results of [2]. We find that a constant consumption strategy is suboptimal even for infinite collectives investing in markets where assets provide no return so long as investors are "satisfaction risk-averse." This suggests that annuities and defined benefit investments will always be suboptimal investments. We present numerical results examining the importance of the fund size, $n$, and the market parameters.

q-fin.PM↗

arXiv2019

VPLanet: The Virtual Planet Simulator

We describe a software package called VPLanet that simulates fundamental aspects of planetary system evolution over Gyr timescales, with a focus on investigating habitable worlds. In this initial release, eleven physics modules are included that model internal, atmospheric, rotational, orbital, stellar, and galactic processes. Many of these modules can be coupled simultaneously to simulate the evolution of terrestrial planets, gaseous planets, and stars. The code is validated by reproducing a selection of observations and past results. VPLanet is written in C and designed so that the user can choose the physics modules to apply to an individual object at runtime without recompiling, i.e., a single executable can simulate the diverse phenomena that are relevant to a wide range of planetary and stellar systems. This feature is enabled by matrices and vectors of function pointers that are dynamically allocated and populated based on user input. The speed and modularity of VPLanet enables large parameter sweeps and the versatility to add/remove physical phenomena to assess their importance. VPLanet is publicly available from a repository that contains extensive documentation, numerous examples, Python scripts for plotting and data management, and infrastructure for community input and future development.

astro-ph.EP↗

arXiv2018

The Markowitz Category

We give an algebraic definition of a Markowitz market and classify markets up to isomorphism. Given this classification, the theory of portfolio optimization in Markowitz markets without short selling constraints becomes trivial. Conversely, this classification shows that, up to isomorphism, there is little that can be said about a Markowitz market that is not already detected by the theory of portfolio optimization. In particular, if one seeks to develop a simplified low-dimensional model of a large financial market using mean--variance analysis alone, the resulting model can be at most two-dimensional.

q-fin.PM↗

arXiv2018

Exo-Milankovitch Cycles II: Climates of G-dwarf Planets in Dynamically Hot Systems

Using an energy balance model with ice sheets, we examine the climate response of an Earth-like planet orbiting a G dwarf star and experiencing large orbital and obliquity variations. We find that ice caps couple strongly to the orbital forcing, leading to extreme ice ages. In contrast with previous studies, we find that such exo-Milankovitch cycles tend to impair habitability by inducing snowball states within the habitable zone. The large amplitude changes in obliquity and eccentricity cause the ice edge, the lowest latitude extent of the ice caps, to become unstable and grow to the equator. We apply an analytical theory of the ice edge latitude to show that obliquity is the primary driver of the instability. The thermal inertia of the ice sheets and the spectral energy distribution of the G dwarf star increase the sensitivity of the model to triggering runaway glaciation. Finally, we apply a machine learning algorithm to demonstrate how this technique can be used to extend the power of climate models. This work illustrates the importance of orbital evolution for habitability in dynamically rich planetary systems. We emphasize that as potentially habitable planets are discovered around G dwarfs, we need to consider orbital dynamics.

astro-ph.EP↗

arXiv2018

Pricing index options by static hedging under finite liquidity

We develop a model for indifference pricing in derivatives markets where price quotes have bid-ask spreads and finite quantities. The model quantifies the dependence of the prices and hedging portfolios on an investor's beliefs, risk preferences and financial position as well as on the price quotes. Computational techniques of convex optimisation allow for fast computation of the hedging portfolios and prices as well as sensitivities with respect to various model parameters. We illustrate the techniques by pricing and hedging of exotic derivatives on S&P index using call and put options, forward contracts and cash as the hedging instruments. The optimized static hedges provide good approximations of the options payouts and the spreads between indifference selling and buying prices are quite narrow as compared with the spread between super- and subhedging prices.

q-fin.PR↗

arXiv2018

Optimizing S-shaped utility and implications for risk management

We consider market players with tail-risk-seeking behaviour as exemplified by the S-shaped utility introduced by Kahneman and Tversky. We argue that risk measures such as value at risk (VaR) and expected shortfall (ES) are ineffective in constraining such players. We show that, in many standard market models, product design aimed at utility maximization is not constrained at all by VaR or ES bounds: the maximized utility corresponding to the optimal payoff is the same with or without ES constraints. By contrast we show that, in reasonable markets, risk management constraints based on a second more conventional concave utility function can reduce the maximum S-shaped utility that can be achieved by the investor, even if the constraining utility function is only rather modestly concave. It follows that product designs leading to unbounded S-shaped utilities will lead to unbounded negative expected constraining utilities when measured with such conventional utility functions. To prove these latter results we solve a general problem of optimizing an investor expected utility under risk management constraints where both investor and risk manager have conventional concave utility functions, but the investor has limited liability. We illustrate our results throughout with the example of the Black--Scholes option market. These results are particularly important given the historical role of VaR and that ES was endorsed by the Basel committee in 2012--2013.

q-fin.RM↗

arXiv2018

Coordinate-free Stochastic Differential Equations as Jets

We explain how Itô Stochastic Differential Equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can be used to derive graphical representations of Itô SDEs. We show how jets can be used to derive the differential operators associated with SDEs in a coordinate free manner. We relate jets to vector flows, giving a geometric interpretation of the Itô--Stratonovich transformation. We show how percentiles can be used to give an alternative coordinate free interpretation of the coefficients of one dimensional SDEs. We relate this to the jet approach. This allows us to interpret the coefficients of SDEs in terms of "fan diagrams". In particular the median of a SDE solution is associated to the drift of the SDE in Stratonovich form for small times.

math.PR↗

arXiv2017

Exo-Milankovitch Cycles I: Orbits and Rotation States

The obliquity of the Earth, which controls our seasons, varies by only ~2.5 degrees over ~40,000 years, and its eccentricity varies by only ~0.05 over 100,000 years. Nonetheless, these small variations influence Earth's ice ages. For exoplanets, however, variations can be significantly larger. Previous studies of the habitability of moonless Earth-like exoplanets have found that high obliquities, high eccentricities, and dynamical variations can extend the outer edge of the habitable zone by preventing runaway glaciation (snowball states). We expand upon these studies by exploring the orbital dynamics with a semi-analytic model that allows us to map broad regions of parameter space. We find that in general, the largest drivers of obliquity variations are secular spin-orbit resonances. We show how the obliquity varies in several test cases, including Kepler-62 f, across a wide range of orbital and spin parameters. These obliquity variations, alongside orbital variations, will have a dramatic impact on the climates of such planets.

astro-ph.EP↗

arXiv2017

Optimal approximation of SDEs on submanifolds: the Ito-vector and Ito-jet projections

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion of projecting a vector field onto a submanifold in order to derive approximations to ordinary differential equations, and improves the previous Stratonovich projection method by adding optimality analysis and results. Indeed, just as in the case of ordinary projection, our definitions of projection are based on optimality arguments and give in a well-defined sense "optimal" approximations to the original SDE in the mean-square sense. We also show that the Stratonovich projection satisfies an optimality criterion that is more ad hoc and less appealing than the criteria satisfied by the Ito projections we introduce. As an application we consider approximating the solution of the non-linear filtering problem with a Gaussian distribution and show how the newly introduced Ito projections lead to optimal approximations in the Gaussian family and briefly discuss the optimal approximation for more general families of distribution. We perform a numerical comparison of our optimally approximated filter with the classical Extended Kalman Filter to demonstrate the efficacy of the approach.

math.PR↗

arXiv2016

Small-time asymptotics for a general local-stochastic volatility model with a jump-to-default: curvature and the heat kernel expansion

We compute a sharp small-time estimate for implied volatility under a general uncorrelated local-stochastic volatility model. For this we use the Bellaiche \cite{Bel81} heat kernel expansion combined with Laplace's method to integrate over the volatility variable on a compact set, and (after a gauge transformation) we use the Davies \cite{Dav88} upper bound for the heat kernel on a manifold with bounded Ricci curvature to deal with the tail integrals. If the correlation $ρ< 0$, our approach still works if the drift of the volatility takes a specific functional form and there is no local volatility component, and our results include the SABR model for $β=1, ρ\le 0$. \bl{For uncorrelated stochastic volatility models, our results also include a SABR-type model with $β=1$ and an affine mean-reverting drift, and the exponential Ornstein-Uhlenbeck model.} We later augment the model with a single jump-to-default with intensity $\lm$, which produces qualitatively different behaviour for the short-maturity smile; in particular, for $ρ=0$, log-moneyness $x > 0$, the implied volatility increases by $\lm f(x) t +o(t) $ for some function $f(x)$ which blows up as $x \searrow 0$. Finally, we compare our result with the general asymptotic expansion in Lorig, Pagliarani \& Pascucci \cite{LPP15}, and we verify our results numerically for the SABR model using Monte Carlo simulation and the exact closed-form solution given in Antonov \& Spector \cite{AS12} for the case $ρ=0$.

q-fin.PR↗