Journal of Computational Finance

Risk.net

Computing two-factor deltas using unstructured meshes

Amélie Bélanger, Bruce Simpson

ABSTRACT

The Black–Scholes value and risk-free portfolio for options that depend on two underlying assets are defined by the solution of the Black–Scholes partial differential equation, and its gradient. A finite-volume, or finiteelement, method using an unstructured mesh to discretize the underlying asset price region can be used to compute a piecewise linear (pwlinear) approximation for the valuation. A gradient recovery method applied to this computed valuation can be used to compute a pwlinear approximation to the Black–Scholes delta hedging parameters. In this paper, we discuss efficient unstructured mesh design for computing both of these contributions to the Black–Scholes risk-free portfolio function. Mesh designs are viewed as determining sequences of meshes with increasingly accurate computed valuations and deltas. We argue that meshes that provide quadratic convergence to zero of the errors in the deltas are more efficient than meshes with lower convergence rates for these errors. However, requiring quadratic convergence of the recovered gradient values of a function places strong restrictions on the mesh design. We present computations that demonstrate how meshes with suitably chosen uniform submeshes can produce quadratic convergence of the deltas in a weighted error measure.

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