Abstract
We show that the performances of the finite difference method for double barrier option pricing can be strongly enhanced by applying both a repeated Richardson extrapolation technique and a mesh optimization procedure. In particular, first we construct a space mesh that is uniform and aligned with the discontinuity points of the solution being sought. This is accomplished by means of a suitable transformation of coordinates, which involves some parameters that are implicitly defined and whose existence and uniqueness is theoretically established. Then, a finite difference scheme employing repeated Richardson extrapolation in both space and time is developed. The overall approach exhibits high efficacy: barrier option prices can be computed with accuracy close to the machine precision in less than one second. The numerical simulations also reveal that the improvement over existing methods is due to the combination of the mesh optimization and the repeated Richardson extrapolation.
Highlights
The most common approach for pricing double barrier options is based on solving a partial differential equation of Black–Scholes type
We show that the performances of the finite difference method for double barrier option pricing can be strongly enhanced by applying both a repeated Richardson extrapolation technique and a mesh optimization procedure
We show that the performances of the finite difference method for double barrier option pricing can be considerably improved by applying a suitable mesh optimization and a Richardson extrapolation technique
Summary
The most common approach for pricing double barrier options is based on solving a partial differential equation of Black–Scholes type (see Wilmott 1998) Such an equation does not have a closed-form solution and requires numerical approximation. We show that the performances of the finite difference method for double barrier option pricing can be considerably improved by applying a suitable mesh optimization and a Richardson extrapolation technique. An aligned and uniform mesh is straightforward to obtain only if the strike price and the barriers are the terms of some arithmetic progression This is the case that is usually considered when proposing finite difference schemes for double barrier option pricing, see, e.g., Duffy (2004), Wade et al (2007), Ndogmo and Ntwiga (2011), and Milev and Tagliani (2013). Our attention is focused on the case of discretely monitored barriers, but continuously monitored barriers could be considered as well; in Sect. 3 the procedure for constructing an optimal mesh is developed and theoretically analyzed; in Sect. 4 it is shown how to compute the optimal mesh by means of suitable bisection/Newton algorithms whose convergence is a-priory theoretically guaranteed; in Sect. 5 the finite difference scheme is briefly sketched; in Sect. 6 some numerical results are presented and discussed; in Sect. 7 some conclusions are drawn
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