13 research outputs found
Explicit Stabilised Gradient Descent for Faster Strongly Convex Optimisation
This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for
strongly convex optimisation problems. This new algorithm is based on explicit
stabilised integrators for stiff differential equations, a powerful class of
numerical schemes that avoid the severe step size restriction faced by standard
explicit integrators. For optimising quadratic and strongly convex functions,
this paper proves that RKCD nearly achieves the optimal convergence rate of the
conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the
condition number of the quadratic function worsens. It is established that this
optimal rate is obtained also for a partitioned variant of RKCD applied to
perturbations of quadratic functions. In addition, numerical experiments on
general strongly convex problems show that RKCD outperforms Nesterov's
accelerated gradient descent
On the acceleration of explicit finite difference methods for option pricing
Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black-Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.Numerical methods for option pricing, Black-Scholes model, Computational finance, Equity options, American options, Exotic options,