Analysis of an uncertain volatility model

This paper explores the application of Machine Learning techniques for pricing high-dimensional options within the framework of the Uncertain Volatility Model (UVM). The UVM is a robust framework that accounts for the inherent unpredictability of market volatility by setting upper and lower bounds on volatility and the correlation among underlying assets. By leveraging historical data and extreme values of estimated volatilities and correlations, the model establishes a confidence interval for future volatility and correlations, thus providing a more realistic approach to option pricing. By integrating advanced Machine Learning algorithms, we aim to enhance the accuracy and efficiency of option pricing under the UVM, especially when the option price depends on a large number of variables, such as in basket or path-dependent options. In this paper, we consider two approaches based on Machine Learning. The first one, termed GTU, evolves backward in time, dynamically selecting at each time step the most expensive volatility and correlation for each market state. Specifically, it identifies the particular values of volatility and correlation that maximize the expected option value at the next time step, and therefore, an optimization problem must be solved. This is achieved through the use of Gaussian Process regression, the computation of expectations via a single step of a multidimensional tree and the Sequential Quadratic Programming optimization algorithm. The second approach, referred to as NNU, leverages neural networks and frames pricing in the UVM as a control problem. Specifically, we train a neural network to determine the most adverse volatility and correlation for each simulated market state, generated via random simulations. The option price is then obtained through Monte Carlo simulations, which are performed using the values for the uncertain parameters provided by the neural network. The numerical results demonstrate that the proposed approaches can significantly improve the precision of option pricing and risk management strategies compared with methods already in the literature, particularly in high-dimensional contexts.

This paper aims to study the asymptotic behavior of double barrier American-style put option prices under an uncertain volatility model, which degenerates to a single point. We give an approximation of the double barrier American-style option prices with a small volatility interval, expressed by the Black–Scholes–Barenblatt equation. Then, we propose a novel representation for the early exercise boundary of American-style double barrier options in terms of the optimal stopping boundary of a single barrier contract.

The option pricing model with constant volatility in illiquid markets has been expanded by introducing two uncertain volatility models in this paper volatility. To conquer some insufficient existed in some literature, for example the time step should be small enough to satisfy the stability, the implicit difference equation has been established and numerical solution of the modified model with uncertain volatility has been discussed. Numerical results show that the method is nice and the accurate results can be gained with less computation.

The uncertain volatility model has long ago attracted the attention of practitioners as it provides worst-case pricing scenario for the sell-side. The valuation of a financial derivative based on this model requires solving a fully non-linear PDE. One can rely on finite difference schemes only when the number of variables (that is, underlyings and path-dependent variables) is small - in practice no more than three. In all other cases, numerical valuation seems out of reach. In this paper, we outline two accurate, easy-to-implement Monte-Carlo-like methods which hardly depend on dimensionality. The first method requires a parameterization of the optimal covariance matrix and consists in a series of backward low-dimensional optimizations. The second method relies heavily on a recently established connection between second-order backward stochastic differential equations and non-linear second-order parabolic PDEs. Both methods are illustrated by numerical experiments.

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case.

In this article we propose a $\alpha$-hypergeometric model with uncertain volatility (UV) where we derive a worst-case scenario for option pricing. The approach is based on the connexion between a certain class of nonlinear partial differential equations of HJB-type (G-HJB equations), that govern the nonlinear expectation of the UV model and that provide an alternative to the difficult model calibration problem of UV models, and second-order backward stochastic differential equations (2BSDEs). Using asymptotic analysis for the G-HJB equation and the equivalent 2BSDE representation, we derive a limit model that provides an accurate description of the worst-case price scenario in cases when the bounds of the UV model are slowly varying. The analytical results are tested by numerical simulations using a deep learning based approximation of the underlying 2BSDE.

We discuss robust scheduling models under uncertain environments described by scenario approach. Using the insights revealed by the analysis of traditional uncertain scheduling models involving the conflicting and balancing twofold relevance, which are the motivation of pursuing better performance and the conservatism of resisting risk, we establish a kind of new robust scheduling model. The optimization objective combines expected performance and robustness measure with a balance factor. A risk-resisting robustness measure is defined based on the concept of bad-scenario set, in which the number of bad scenarios can be adjusted by a standard performance. Thus, a set of robust scheduling models is established as the balance factor or the standard performance varies. A series of theorems reveal the relationship among the set of new models proposed in this paper and traditional uncertain scheduling models. And the condition of effectiveness of robustness for the set of new models is proposed as a theorem. Furthermore, an extensive experiment was conducted for job-shop scheduling problems with uncertain processing time. The computational results provide evidence that the set of new models is more comprehensive and more integrated in terms of pursuing better statistic performance and resisting the risk of performance deterioration. Thus, the new model can realize better balance between expected performance and risk-resisting robustness, as comparied against existing uncertain scheduling models.

Uncertain Volatility Model: A Monte-Carlo Approach

This work presents the development and physical analysis of a sweat transport model that couples the fluctuations in air and vapour concentrations, and temperature, in a one-dimensional porous clothing assembly. The clothing is exposed to inherent time-varying conditions due to variations in the body temperature and ambient conditions. These fluctuations are governed by a coupled system of nonlinear relaxation–transport–diffusion PDEs of Petrovskii parabolic type. A condition for the well-posedness of the resulting system of equations is derived. It is shown that the energy of the diffusion part of the system is exponentially decreasing. The boundedness and stability of the system of equations is thus confirmed. The variational formulation of the system is derived, and the existence and uniqueness of a weak solution is demonstrated analytically. This system is shown to conserve positivity. The difficulty of obtaining an analytical solution due to the complexity of the problem, urges for a numerical approach. A comparison of three cases is made using the Crank–Nicolson finite difference method (FDM). Numerical experiments show the existence of singular coefficient matrices at the site of phase change. Furthermore, the steady-state profiles of temperature, air and vapour concentrations influence the attenuation of fluctuations. Numerical results verify the analytical findings of this work.

SummaryWe derive instability criteria for Lur'e systems with sector‐bounded nonlinearities and uncertain external signals. First, we define absolute instability of an equilibrium and derive an absolute instability condition for a fixed equilibrium point in terms of a linear matrix inequality, which is analogous to the well‐known circle stability criterion. Then, the condition is extended to a parametric absolute instability condition, which is applicable to the instability test of a Lur'e system with an equilibrium point whose location is affected by uncertain nonlinearities and uncertain external signals. Finally, we show that the proposed analysis method is useful through the oscillation analysis of an uncertain genetic network model. Copyright © 2014 John Wiley & Sons, Ltd.

Chemical model reduction under uncertainty

The paper is devoted to performance robustness analysis of uncertain state-space models with linear parametric uncertainties and output feedback. The purpose is to find bounds on uncertain parameters within which the system H/sub 2/ norm performance index is kept below a prespecified value. Through matrix nonsingularity analysis, it is shown that the bounds can be computed from structured singular values of certain composite matrices. To fully utilise the structural information of uncertainties, the authors also develop a method for reducing matrix sizes involved in the computation of structured singular value.

Torsional stick-slip vibrations decrease the performance, reliability and fail-safety of drilling systems used for the exploration and harvesting of oil, gas, minerals and geo-thermal energy. Current industrial controllers regularly fail to eliminate stick-slip vibrations, especially when multiple torsional flexibility modes in the drill-string dynamics play a role in the onset of stick-slip vibrations. This chapter presents the experimental validation of novel robust output-feedback controllers designed to eliminate stick-slip vibrations in the presence of multiple dominant torsional flexibility modes. For this purpose, a representative experimental test setup is designed, using a model of a real-life drilling rig as a basis. The model of the dynamics of the experimental setup can be cast in Lur’e-type form with set-valued nonlinearities representing an (uncertain) model for the complex bit-rock interaction and the interaction between the drill-string and the borehole. The proposed controller design strategy is based on skewed-\(\mu \)-DK-iteration and aims at optimizing the robustness with respect to uncertainty in the non-smooth bit-rock interaction. Moreover, a closed-loop stability analysis for the non-smooth drill-string model is provided. Experimental results confirm that stick-slip vibrations are indeed eliminated using the designed controller in realistic drilling scenarios in which state-of-practice controllers have failed to achieve the same.

Relative gain array (RGA) analysis has been widely used in process control to identify promising control structures and to characterize the degree of process interactions between controlled and manipulated variables. However, the influence of process model uncertainty on RGA analysis has received little attention. Analytical expressions for RGA uncertainty bounds are derived for 2 × 2 control problems and for general, n × n control problems. Both worst‐case bounds and statistical uncertainty bounds are derived. Several simulation examples illustrate the new results. The RGA uncertainty bounds provide useful information concerning model accuracy requirements and the robustness of decentralized control systems.

This work extends to third-order previously published work on developing the adjoint sensitivity and uncertainty analysis of the numerical model of a polyethylene-reflected plutonium (acronym: PERP) OECD/NEA reactor physics benchmark. The PERP benchmark comprises 21,976 imprecisely known (uncertain) model parameters. Previous works have used the adjoint sensitivity analysis methodology to compute exactly and efficiently all of the 21,976 first-order and (21,976)2 second-order sensitivities of the PERP benchmark’s leakage response to all of the benchmark’s uncertain parameters, showing that the largest and most consequential 1st- and 2nd-order response sensitivities are with respect to the total microscopic cross sections. These results have motivated extending the previous adjoint-based derivations to third-order, leading to the derivation, in this work, of the exact mathematical expressions of the (180)3 third-order sensitivities of the PERP leakage response with respect to these total microscopic cross sections. The formulas derived in this work are valid not only for the PERP benchmark but can also be used for computing the 3rd-order sensitivities of the leakage response of any nuclear system involving fissionable material and internal or external neutron sources. Subsequent works will use the adjoint-based mathematical expressions obtained in this work to compute exactly and efficiently the numerical values of these (180)3 third-order sensitivities (which turned out to be very large and consequential) and use them for a third-order uncertainty analysis of the PERP benchmark’s leakage response.