# Adaptive Mesh Relocation-Refinement (AMrR) on Kim's Method for Options Pricing

Adaptive Mesh Relocation-Refinement (AMrR) on Kim's Method for Options Pricing

This Demonstration shows an adaptive mesh relocation-refinement (AMrR or sometimes AMR) strategy on Kim's method [1] for pricing American options, using the composite trapezoidal rule over a time mesh with four steps. A European financial option is an instrument that allows its holder the right to buy or sell an equity at a future maturity date for a fixed price called the "strike price." An American option allows its holder to exercise the contract at any time up to the maturity date, and because of this, it is worth more than the European option by an amount called the "early exercise premium." For the American call's holder, the early exercise becomes optimal when the underlying asset price exceeds a critical boundary , above which the intrinsic value of the option becomes greater than its holding value. According to Kim's method, the valuation of the American option derives from an integral expression of the early exercise premium as a function of the optimal exercise boundary plus the value of the European option.

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An adaptive mesh usually aims to control the size and the shape of each element so that for any particular solution, the overall error is controlled (Budd, Huang and Russel, 2009). There are numerous applications of mesh adaptivity, including computational fluid dynamics, groundwater flow, blow-up problems, chemotaxis systems, reaction-diffusion systems, the nonlinear Schrodinger equation, phase-change problems, shear layer calculations, gas dynamics, magneto-hydrodynamics, meteorological problems and others. In mathematical finance, adaptation strategies (e.g. [2, 3]) have been applied to optimize the mesh of space-time finite element approximations of the Black–Scholes model.

This Demonstration tests a similar AMrR strategy on Kim's method, which relies on a recursive process to approximate the critical boundary in conjunction with the composite trapezoidal rule. The r-adaptation strategy aims to minimize an adjoint functional that is assumed to be linearly related to the global error; practically, the r-adaptive algorithm maps the sensitivity of the solution to locally refined regions and generates weights to resize the time steps accordingly. This process is repeated until the sensitivity of the solution is distributed equally over the time steps. This AMrR algorithm is inspired by the methods in [4–6].

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The optimal mesh not only produces more accurate option price approximations compared with the uniform mesh, but also can generate tighter upper bounds for the theoretical option price, combined with the upper bound methods in [7, 8].

Use "display" to show one of these four different plots:

"option price convergence" shows the American option price convergence through the mesh refinement process. The table shows the option price approximation according to the uniform and the optimal mesh, respectively.

"mesh relocation" shows the adaptive mesh resizing through the refinement iterations. The horizontal dotted grid lines indicate the uniform mesh mapping. The table shows the temporal points at their optimal position.

"optimal exercise boundary" shows the approximation of the optimal exercise boundary obtained from the refined mesh, compared with the boundary obtained from the uniform mesh.

"absolute impacts" shows the distribution of the solution sensitivity obtained from the refined mesh, compared with the respective distribution obtained from the uniform mesh. The table shows the adjoint functional values, respectively.