新奇選擇權不斷推陳出新，而它的評價也愈趨複雜，隨著電腦的進步，愈能運算更複雜的選擇權，但電腦的進步有限，所以仍有很多學者致力於路徑相依選擇權評價之研究。本文針對Kim, Chang, and Byun (2003)所評價的算術平均重設型選擇權，提出不同的演算架構，由於Kim, Chang, and Byun (2003)的評價模型是對狀態變數 (state variables)取對數後在等距做分割，隨然可以逼近答案，但是需要很多的運算空間，但有些點可藉有線性關係而求得，因此造成了浪費運算空間與時間。而我依據Dai, Wang, and Wei (2008)的方法衍生出二維的線性誤差來判斷是否需插入狀態變數之代表值 (representative values)。此方法是以遞迴的方式來判斷每對鄰近的平均股價跟履約價的線性誤差，而此方法不同於取對數等份切割(logarithmically equally-spaced)，除了減少非線性的誤差，也減少了運算的空間。由數值結果得知，此方法成功的減少內插誤差並且改善收斂速度。

There are more new exotic options, and the pricing problems associated with these options are complicated. Because of the technology progress of computers, we can price more kinds of options. However, there are still pricing problems that are not solved perfectly, especially for path-dependent options with more than one state variable. For instance, Kim, Chang, and Byun (2003) have an idea of pricing Arithmetic Average Reset Options. In this paper, we offer a different algorithm to improve their pricing algorithm. Since Kim, Chang, and Byun (2003) use logarithmically equally-spaced placement rule, they do not place representative average prices and strike prices optimally and thus waste both the memory space and the computational time. However, some information for this option can be found by linear relationship, so the method wastes space.
Extending the adaptive placement method in Dai, Wang, and Wei (2008).I develop the two-dimension adaptive placement method to pricing arithmetic average reset options. The notion of this method is to design a recursive algorithm to limit the error of the bilinear interpolation between each pair of adjacent representative average prices and strike prices. In addition to reducing the nonlinear error, less space is required than Kim, Chang, and Byun (2003). From the results of numerical experiments, this method successfully decreases the interpolation error and improves the convergence rate.

Chen, W.Y. , and S. Zhang. “The Analytics of Reset Options.” Journal of Derivatives, 8 (2000), pp. 59-71.
Cho, H.Y., and H.Y. Lee. “A Lattice Model for Pricing Geometric and Arithmetic Average Options.” Journal of Financial Engineering, 6 (1997), pp. 179-191.
Cox, J., S. Ross, and M. Rubinstein. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7 (1979), pp.229-263.
Dai, T.S., J.Y. Wang, and H.S. Wei. “Adaptive Placement Method on Pricing Arithmetic Average Options.” Review of Derivatives Research (2008), pp. 83-118.
Hull, J., and A. White. “ Efficient Procedures for Valuing European and American Path-Dependent Options.” Journal of Derivatives, 1 (1993), pp. 21-31.
Kemna, A.G.Z., and A.C.F. Vorst. ”A Pricing Method Based on Average Asset Values.” Journal of Banking and Finance (1990), pp. 113-129.
Kim, I.J. , G.H. Chang, and S.J. Byun. “Valuation of Arithmetic Average Reset Options.” Journal of Derivatives (2003), pp. 70-80.
Kwok, Y.K., and K.W. Lau. “Pricing Algorithms for Options with Exotic Path-Dependence.” Journal of Derivatives, 9 (2001), pp. 28-38.
Liao, S.L., and C.W. Wang. “Pricing Arithmetic Average Reset Options with Control Variates.” Journal of Derivatives, 10 (2002), pp. 59-74.
Liao, S.L., and C.W. Wang.”The Valuation of Reset Options with Multiple Strike Resets and Resets Dates.” Journal of Futures Markets, 23 (2003), pp.87-107.