Adaptive placement method on pricing arithmetic average options

Since there is no analytic solution for arithmetic average options until present, developing an efficient numerical algorithm becomes a promising alternative. One of the most famous numerical algorithms is introduced by Hull and White (J Deriv 1:21–31, 1993). Motivated by the common idea of reducing the nonlinearity error in the adaptive mesh model in Figlewski and Gao (J Financ Econ 53:313–351, 1999) and the adaptive quadrature method, we propose an adaptive placement method to replace the logarithmically equally-spaced placement rule in the Hull and White’s model by placing more representative average prices in the highly nonlinear area of the option value as the function of the arithmetic average stock price. The basic idea of this method is to design a recursive algorithm to limit the error of the linear interpolation between each pair of adjacent representative average prices. Numerical experiments verify the superior performance of this method for reducing the interpolation error and hence improving the convergence rate. To show that the adaptive placement method can improve any numerical algorithm with the techniques of augmented state variables and the piece-wise linear interpolation approximation, we also demonstrate how to integrate the adaptive placement method into the GARCH option pricing algorithm in Ritchken and Trevor (J Finance 54:377–402, 1999). Similarly great improvement T.-S. Dai Department of Information and Finance Management, National Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected] J.-Y. Wang (B) Department of International Business, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd, Taipei 106, Taiwan e-mail: [email protected] H.-S. Wei Department of Finance, National Central University, Taoyuan County, Taiwan e-mail: [email protected]