A fixed strike Asian option and comments on its numerical solution

Abstract

A boundary value formulation of an Asian option is solved with a wide range of standard textbook explicit finite difference methods including also artificial diffusion methods. We investigate the dependence of the numerical methods on the various degeneracies and approximations in the boundary value formulation. It is concluded that numerical solutions are generally oscillatory, that a simple artificial diffusion approach does not resolve this problem, but that by selecting the appropriate methods, oscillations can be avoided.

Numerical pricing of geometric asian options with barriers

“…We have chosen two particular FD methods in the remarkable amount of possibilities, considering ease and practicability in implementation, but in the work of Hugger, there is a deep analysis of a large number of them pointing out a generalized slowness in their performance.…”

“…Equation is proven to be hypoelliptic (see other works), property that guarantees smooth solution, but it is strongly degenerate at S = 0 because the logarithmic coefficient of is not defined and in A ‐variable because the term lacks. These degenerations may cause oscillations in FD numerical solutions and the adopted repair strategies in general deteriorate accuracy (see the work of Hugger).…”

“…The method here is applied to Asian barrier options evaluated with geometric mean that, although not common among practitioners, give some information also about the evaluation of Asian barrier options with arithmetic mean. The obtained results are then compared with numerical results obtained by two FD methods among the great variety that we can find in literature (resumed, eg, in the work of Hugger). The application of FD methods is not straightforward since explicit boundary conditions have to be derived from financial conditions in such a way to ensure wellposedness of the starting differential problem and to avoid oscillations.…”

“…Anyway, FD methods result to be very slow, and computing power and time is necessary to reduce the error (as concluded also in the work of Hugger in the case of arithmetic Asian options ); therefore, SABO seems to be a numerical method that overcomes this issue. The method is developed in this paper for continuously sampled geometric Asian option with no restriction w.r.t.…”

An Alternating-Direction Implicit Difference Scheme for Pricing Asian Options

“…Zvan et al [5] apply a flux-limiting method from computational fluid dynamics to tackle the problem of spurious oscillations that arise in Asian options. Hugger [6] proposes an artificial viscosity numerical method for Asian options to avoid the oscillations. Večeř [7] presents that the pricing techniques of an option on a traded account could be applied to price the Asian option, and that the price of the Asian option is characterized by a simple one-dimensional PDE.…”