This note outlines an approach to stress testing of covariance of financial time series, in the context of financial risk management. It discusses how the geodesic distance between covariance matrices implies a notion of plausibility of covariance stress tests. In this approach, correlation stress tests span a submanifold of constant determinant of the Fisher--Rao manifold of covariance matrices. A parsimonious geometrically invariant definition of arbitrarily large correlation stress tests is proposed, and a few examples are discussed.
Cristina Monforte-Royo, Blanca Goni-Fuste, Iris Crespo
et al.
Abstract Background Euthanasia has been incorporated into the health services of seven countries. The legalisation of these practices has important repercussions for the competences of nurses, and it raises questions about their role. When a patient with advanced disease expresses a wish to die, what is expected of nurses? What are the needs of these patients, and what kind of care plan do they require? What level of autonomy might nurses have when caring for these patients? The degree of autonomy that nurses might or should have when it comes to addressing such a wish and caring for these patients has yet to be defined. Recognising the wish to die as a nursing diagnosis would be an important step towards ensuring that these patients receive adequate nursing care. This study-protocol aims to define and validate the nursing diagnosis wish to die in patients with advanced disease, establishing its defining characteristics and related factors; to define nursing-specific interventions for this new diagnosis. Methods A prospective three-phase study will be carried out. Phase-A) Foundational knowledge: an umbrella review of systematic reviews will be conducted; Phase-B) Definition and validation of the diagnostic nomenclature, defining characteristics and related factors by means of an expert panel, a Delphi study and application of Fehring’s diagnostic content validation model; Phase-C) Definition of nursing-specific interventions for the new diagnosis. At least 200 academic and clinical nurses with expertise in the field of palliative care or primary health care will be recruited as participants across the three phases. Discussion The definition of the wish to die as a nursing diagnosis would promote greater recognition and autonomy for nurses in the care of patients who express such a wish, providing an opportunity to alleviate underlying suffering through nursing-specific interventions and drawing attention to the needs of patients with advanced disease. The new diagnosis would be an addition to nursing science and would provide a framework for providing care to people with advanced disease who express such a wish. Nurses would gain professional autonomy about identifying, exploring and responding clinically to such a wish.
AbstractThe National Institute for Health and Care Research (NIHR) Policy Research Unit in Behavioural Science (PRU-BS) was funded to inform government on the application of behavioural science in health and social care policy. What makes this unit different to other topic specific ones, was the wide range of its brief. Because of this, the PPI group would need to include a wide range of experience and expertise and be prepared to learn. We were a different type of public group for a different type of task. This paper deals with how we approached this. In this paper we outline how the PPI plan in the funding proposal for the PRU-BS was adapted to real world challenges. We describe the stages in the formation of the PPI Strategy Group and how a virtual platform was created to ensure good communication. We discuss our pragmatic approach of developing Terms of Reference and a PPI strategy document. Given the restrictions imposed by the Covid-19 pandemic we explain how we tackled PPI SG member induction sessions, meetings and training sessions. To illustrate how the group operates we provide an example of our involvement in a PRU-BS project. Central to our paper is the lessons we learned. We hope the challenges we met in forming the unique PPI SG, how these were overcome, and our recommendations will help others faced with a similar task.
What kind of implied volatility extrapolation is appropriate? Roger Lee proved that the Black-Scholes implied variance can not grow faster than linearly in log-moneyness. This paper investigates what happens in the Bachelier (or Normal) implied volatility world, making sure to cover the various aspects of vanilla option arbitrages.
In this paper we revisit Burnett (2021) \& Burnett and Williams (2021)'s notion of hedging valuation adjustment (HVA), originally intended to deal with dynamic hedging frictions such as transaction costs, in the direction of model risk. The corresponding HVA reconciles a global fair valuation model with the local models used by the different desks of the bank. Model risk and dynamic hedging frictions indeed deserve a reserve, but a risk-adjusted one, so not only an HVA, but also a contribution to the KVA of the bank. The orders of magnitude of the effects involved suggest that local models should not so much be managed via reserves, as excluded altogether.
Enrico Dall'Acqua, Riccardo Longoni, Andrea Pallavicini
In industrial applications it is quite common to use stochastic volatility models driven by semi-martingale Markov volatility processes. However, in order to fit exactly market volatilities, these models are usually extended by adding a local volatility term. Here, we consider the case of singular Volterra processes, and we extend them by adding a local-volatility term to their Markov lift by preserving the stylized results implied by these models on plain-vanilla options. In particular, we focus on the rough-Heston model, and we analyze the small time asymptotics of its implied local-volatility function in order to provide a proper extrapolation scheme to be used in calibration.
We derive a series expansion by Hermite polynomials for the price of an arithmetic Asian option. This series requires the computation of moments and correlators of the underlying price process, but for a polynomial jump-diffusion, these are given in closed form, hence no numerical simulation is required to evaluate the series. This allows, for example, for the explicit computation of Greeks. The weight function defining the Hermite polynomials is a Gaussian density with scale $b$. We find that the rate of convergence for the series depends on $b$, for which we prove a lower bound to guarantee convergence. Numerical examples show that the series expansion is accurate but unstable for initial values of the underlying process far from zero, mainly due to rounding errors.
This paper studies the concept of instantaneous arbitrage in continuous time and its relation to the instantaneous CAPM. Absence of instantaneous arbitrage is equivalent to the existence of a trading strategy which satisfies the CAPM beta pricing relation in place of the market. Thus the difference between the arbitrage argument and the CAPM argument in Black and Scholes (1973) is this: the arbitrage argument assumes that there exists some portfolio satisfying the capm equation, whereas the CAPM argument assumes, in addition, that this portfolio is the market portfolio.
We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for a generic market dynamics given by a multidimensional Itô's process we specify and prove the equivalence between (NFLVR) and expected utility maximization. As a by-product we provide a geometric characterization of the (NUPBR) condition given by the zero curvature (ZC) condition. Finally, we extend the Black-Scholes PDE to markets allowing arbitrage.
In this paper, we introduce an analytical perturbative solution to the Merton Garman model. It is obtained by doing perturbation theory around the exact analytical solution of a model which possesses a two-dimensional Galilean symmetry. We compare our perturbative solution of the Merton Garman model to Monte Carlo simulations and find that our solutions performs surprisingly well for a wide range of parameters. We also show how to use symmetries to build option pricing models. Our results demonstrate that the concept of symmetry is important in mathematical finance.
We introduce a non-parametric method to recover physical probability distributions of asset returns based on their European option prices and some other sparse parametric information. Thus the main problem is similar to the one considered foir instance in the Recovery Theorem by Ross (2015), except that here we consider a non-dynamical setting. The recovery of the distribution is complete, instead of estimating merely a finite number of its parameters, such as implied volatility, skew or kurtosis. The technique is based on a reverse application of recently introduced Distribution Matching by the author and is related to the ideas in Distribution Pricing by Dybvig (1988) as well as comonotonicity.
We study American swaptions in the linear-rational (LR) term structure model introduced in [5]. The American swaption pricing problem boils down to an optimal stopping problem that is analytically tractable. It reduces to a free-boundary problem that we tackle by the local time-space calculus of [7]. We characterize the optimal stopping boundary as the unique solution to a nonlinear integral equation that can be readily solved numerically. We obtain the arbitrage-free price of the American swaption and the optimal exercise strategies in terms of swap rates for both fixed-rate payer and receiver swaps. Finally, we show that Bermudan swaptions can be efficiently priced as well.
An innovative extension of Geometric Brownian Motion model is developed by incorporating a weighting factor and a stochastic function modelled as a mixture of power and trigonometric functions. Simulations based on this Modified Brownian Motion Model with optimal weighting factors selected by goodness of fit tests, substantially outperform the basic Geometric Brownian Motion model in terms of fitting the returns distribution of historic data price indices. Furthermore we attempt to provide an interpretation of the additional stochastic term in relation to irrational behaviour in financial markets and outline the importance of this novel model.
We present a framework on how to hedge the interest rate sensitivity of liabilities discounted by an extrapolated yield curve. The framework is based on functional analysis in that we consider the extrapolated yield curve as a functional of an observed yield curve and use its Gâteaux variation to understand the sensitivity to any possible yield curve shift. We apply the framework to analyse the Smith-Wilson method of extrapolation that is proposed by the European Insurance and Occupational Pensions Authority (EIOPA) in the coming EU legislation Solvency II, and the method recently introduced, and currently prescribed, by the Swedish Financial Supervisory Authority.
Hedging in the presence of transaction costs leads to complex optimization problems. These problems typically lack closed-form solutions, and their implementation relies on numerical methods that provide hedging strategies for specific parameter values. In this paper we use a genetic programming algorithm to derive explicit formulas for near-optimal hedging strategies under nonlinear transaction costs. The strategies are valid over a large range of parameter values and require no information about the structure of the optimal hedging strategy.
Consider Least Squares Monte Carlo (LSM) algorithm, which is proposed by Longstaff and Schwartz (2001) for pricing American style securities. This algorithm is based on the projection of the value of continuation onto a certain set of basis functions via the least squares problem. We analyze the stability of the algorithm when the number of exercise dates increases and prove that, if the underlying process for the stock price is continuous, then the regression problem is ill-conditioned for small values of the time parameter.
We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta algorithm. The new formula turns out to be fast and accurate.