New ADI scheme speeds up option pricing in stochastic models.
problem Efficiently pricing options in models with stochastic volatility.
method High-order ADI scheme combining spatial and time discretizations.
result Fourth-order accuracy in space and second-order in time.
Efficiently prices options in stochastic volatility models using sparse grids.
problem Option pricing in stochastic volatility models.
method Sparse grid high-order ADI scheme for second-order in time and fourth-order in space.
result Computational efficiency gains confirmed by numerical experiments.
In this paper a simple, effective adaptation of Alternating Direction Implicit (ADI) time discretization schemes is proposed for the numerical pricing of American-style options under the Heston model via a partial differential complementarity problem. The stability and convergence of the new methods are extensively inv…
In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider t…
In this paper the unconditional stability of four well-known ADI schemes is analyzed in the application to time-dependent multidimensional diffusion equations with mixed derivative terms. Necessary and sufficient conditions on the parameter theta of each scheme are obtained that take into account the actual size of the…
A new FV-ADI method calibrates SLV models efficiently.
problem Calibrating SLV models to their underlying local volatility models.
method Finite volume - Alternating Direction Implicit (ADI) approach for solving 1D and 2D forward Kolmogorov equations.
result The proposed method efficiently calibrates SLV models without requiring PDE transformations and conserves numerical mass.
This research extends Leland's work to multi-asset options with variable transaction costs.
problem Pricing multi-asset options with variable transaction costs.
method Generalized Leland's condition, proved existence of viscosity solution using Perron method, developed numerical ADI scheme.
result Existence of a viscosity solution for the fully nonlinear initial value problem.
The paper develops and tests operator splitting schemes for American options in a complex model.
problem Efficient numerical solution of American options under a two-asset Merton jump-diffusion model.
method Adaptation of IMEX and ADI operator splitting schemes to solve the two-dimensional PIDCP.
result Investigates and compares the convergence and performance of eight operator splitting methods.
A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parame…
The paper efficiently solves a complex option valuation equation for two assets.
problem Valuation of European options under a two-asset Kou jump-diffusion model.
method Extends an efficient algorithm for a one-dimensional integral to a two-dimensional one, using operator splitting schemes for time discretization.
result The method achieves optimal computational cost and stable convergence for various operator splitting schemes.
Efficient PDE method calibrates local volatility with stochastic interest rates.
problem Calibrating local volatility models with stochastic interest rates is time-consuming.
method Developed a PDE approach using ADI method to solve the forward equation.
result Effective and sufficient information for calibration and pricing is provided.
Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generaliz…
Stochastic volatility (SV) and local stochastic volatility (LSV) processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes. Many issues remain, thou…
The paper compares different models for GLWB pricing and hedging.
problem Valuing GLWB products in various models and interest rate scenarios.
method Hybrid tree-finite difference, Monte Carlo, and finite difference schemes.
result Numerical methods determine no-arbitrage fees and Greeks for GLWB contracts.
Extends return extrapolation to nonlinear, asymmetric functions under stochastic volatility.
problem Behavioral anomalies in portfolio choice under stochastic volatility.
method Smooth, nonlinear, asymmetric extrapolation function; CRRA investor; Heston stochastic volatility; Hamilton-Jacobi-Bellman equation; Numerical solutions (finite-difference ADI, deep learning-driven iterative).
result Saturation acts as an endogenous correction mechanism, reducing welfare loss.
We extend return extrapolation to incorporate asymmetry and saturation, finding that asymmetric nonlinear extrapolation leads to lower welfare loss.
problem Optimal portfolio choice under stochastic volatility
method Smooth, nonlinear extrapolation function with sentiment and variance hedging
result Lower welfare loss with asymmetric nonlinear extrapolation
The study proposes methods to price and hedge GMWB using stochastic models.
problem Valuing and hedging GMWB in the presence of stochastic interest rates and volatility.
method Hybrid tree-finite difference, Hybrid Monte Carlo, ADI finite difference, Standard Monte Carlo methods.
result Demonstrates sensitivity of no-arbitrage fee to various assumptions.
The paper calibrates SLV models to LV models using an adjoint method.
problem Calibrating stochastic local volatility models to their underlying local volatility models.
method An adjoint semidiscretization of the forward Kolmogorov equation to solve for the leverage function.
result The method ensures that the fair values of European-style options in SLV and LV models match.
The study examines pricing American options with both exogenous and endogenous transaction costs.
problem Pricing American options with transaction costs and liquidity risks.
method Modeling liquidity risks as a mean-reverting process and transaction costs as proportional to trading amount. Two nonlinear PDEs are used to characterize option values. Numerical solution via ADI method and model calibration using maximum likelihood estimation.
result The model incorporating liquidity risks significantly outperforms the Leland model.
In this paper, using blow-up analysis, we prove a quantization result for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary. Similar results for Euclidean space were obtained by Adimurthi-Struwe \cite{Adi-Stru}, Druet \cite{Druet}, Lamm-Robert-Struwe \cite{L-R-S}, Mart…
Paper proposes a new numerical scheme for solving BSDEs.
problem Solving backward stochastic differential equations (BSDEs).
method Uses Lagrange interpolation to approximate derivatives and changes sample point distributions for different stability and convergence.
result Guarantees convergence of the scheme under certain conditions on sample point distributions.
New high-order scheme reduces BSDE truncation errors.
problem Numerical solution of backward stochastic differential equations (BSDEs).
method Proposes a new θ-scheme with careful θ selection for every subinterval. result Error estimates and verification of scheme order.
New numerical method for quantile hedging in imperfect markets.
problem Quantile hedging in non-linear markets with imperfections.
method Piecewise Constant Policy Timestepping (PCPT) coupled with monotone finite difference approximation.
result Convergence of the proposed numerical scheme proved using BSDE arguments.
In the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. For getting the static hedging formula, the underlying process needs to have…
A new scheme for FBSDEs simplifies computation without Monte Carlo.
problem Numerical solution for decoupled FBSDEs with reduced complexity.
method Recursive marginal quantization for fully quantization-based scheme.
result Effective numerical procedure for financial applications.
New method preserves positivity in financial model simulations.
problem Preserving positivity in financial model simulations.
method Combining semi discrete technique with split step method.
result Explicit and positivity preserving numerical scheme for Ait-Sahalia model.
We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero …
New numerical methods for evolving curves on curved spaces.
problem Evolve curves on Riemannian manifolds efficiently and accurately.
method Variational approximations and numerical schemes for curvature flow, curve diffusion, and elastic flow.
result Effective numerical schemes for geometric evolution equations on Riemannian manifolds.
A new numerical scheme approximates nonlinear filtering densities for noisy and partial measurements.
problem Approximating nonlinear filtering densities for noisy and partial measurements.
method Deep splitting scheme applied to the Fokker--Planck equation followed by Bayes' formula.
result Convergence rate established for the numerical scheme under parabolic Hörmander condition.
New high-order compact scheme improves basket option pricing accuracy.
problem Improving accuracy in pricing European Put options on a basket of assets.
method Developed a second-order accurate in time and fourth-order accurate in space high-order compact scheme.
result Standard second-order schemes are significantly outperformed by the new scheme.
We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of genera…
Paper adapts numerical scheme for parallel transport on diffeomorphism manifolds.
problem Efficient computation of parallel transport on high-dimensional manifold-valued data.
method Adapts a numerical scheme for parallel transport to finite-dimensional manifolds of diffeomorphisms.
result Qualitative and quantitative analysis of scheme's behavior on high-dimensional manifolds.
Novel IMEX scheme solves financial PDEs with mixed derivatives.
problem Numerical approximations for financial PDEs with mixed derivatives.
method Second order finite volume IMEX Runge-Kutta scheme.
result Achieves true second order convergence with non-regular initial conditions.
Compact scheme solves fractional Black-Scholes on non-uniform grids.
problem Solving time-fractional Black-Scholes equation on non-uniform grids.
method Three-point compact finite difference scheme on graded meshes.
result Fourth-order accuracy in space for special meshes.
Study on numerical analysis for corporate bonds using a unified 2 factor model.
problem Develop a numerical method to solve a unified 2 factor model for corporate bonds with fixed discrete coupons.
method Used explicit finite difference scheme to analyze stability and compute bond prices.
result Found conditions for the explicit finite difference scheme to be stable and computed bond prices, credit spread, and duration.
New probabilistic scheme combines deep learning with Runge-Kutta methods for solving PDEs.
problem Solving high-dimensional semi-linear parabolic PDEs efficiently.
method Probabilistic scheme using deep learning and Runge-Kutta methods.
result Crank-Nicolson schemes are efficient in terms of precision, computational cost, and numerical implementation.
We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows us to numerically solve stochastic control problems with controlled volatility,…
Novel numerical scheme for G-heat equation with uncertainty.
problem Efficiently quantify G-expectation for financial products.
method Proposes a novel numerical scheme for the two-dimensional G-heat equation.
result The scheme is monotonic, stable, and convergent, showing high efficiency.
New schemes for SDEs on manifolds keep solutions close to the manifold.
problem Solving SDEs constrained to manifolds in high accuracy.
method Geometrically invariant numerical schemes that remain close to the manifold.
result The schemes converge under standard assumptions and outperform existing methods.
ISALT uses inference to simulate SDEs with large time-steps, improving efficiency.
problem Efficiently simulating ergodic SDEs with large time-steps.
method Inference-based schemes adaptive to large time-steps (ISALT) from data.
result ISALT achieves significant time reduction and optimal accuracy.
Study validates numerical method for singular FBSDEs convergence.
problem Solving singular FBSDEs and associated PDEs.
method Particles approximation for transport operator and tree approximation for diffusion operator.
result Convergence rate of numerical method proved under reasonable conditions.
We develop high-order approximations for the Heston model.
problem Modeling the Heston model with high accuracy and efficiency.
method Combining approximation schemes on different random grids to achieve any order of convergence.
result Achieve any order of convergence for the Heston model.
High-order compact schemes improve option pricing accuracy for stochastic volatility models.
problem Improving option pricing accuracy for stochastic volatility models with non-uniform grids.
method Fourth-order accurate compact schemes applied to option pricing PDEs for stochastic volatility models on non-uniform grids.
result Fourth-order accuracy achieved for non-zero correlation, outperforming standard schemes.
The paper compares numerical schemes for nonholonomic systems using retraction maps.
problem Optimal control of nonholonomic systems with numerical approximations.
method Retraction maps used as seed for geometric integrators of Hamilton equations.
result Performance comparison of symplectic and non-symplectic integrators.
Efficient simulation scheme for rough Heston model reduces computational cost.
problem Accurate and efficient simulation of the rough Heston model for option pricing.
method Weak simulation scheme based on Markovian approximations of the rough Heston process.
result The new scheme exhibits second order weak convergence with linear computational cost.
Develops a numerical scheme for solving path-dependent FBSDEs and PDEs.
problem Solving path-dependent FBSDEs and PDEs numerically.
method Picard iteration method for FBSDEs, concentration inequality for estimator, supervised learning with neural networks for PDEs.
result Proves convergence and rate of convergence for the Picard iteration method.
Valuing FF contracts in time-dependent models
problem Valuing American options and Flexible Forwards contracts
method Recursive Riccati solution and Volterra equation
result FF contracts priced faster than traditional methods
The latter author, together with collaborators, proposed a numerical scheme to calculate the price of barrier options. The scheme is based on a symmetrization of diffusion process. The present paper aims to give a mathematical credit to the use of the numerical scheme for Heston or SABR type stochastic volatility model…