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The main theme of this research is numerical verification of applicability of a higher-order approximation to pricing barrier options, which is a both mathematically and practically important path-dependent type problem in mathematical finance. The authors successfully extend two types of algorithms called NV and NN. Both algorithms are based on a higher-order approximation scheme called Kusuoka approximation and have been shown to attain second-order approximation of stochastic differential equations as long as applied to European-type problems. In extending the algorithms, the authors apply a function representing the probability of hitting a boundary. In the numerical experiments, these two algorithms are compared with the Euler–Maruyama scheme which is one of the most popular first-order approximation schemes. As a result, it is demonstrated that the speed of calculation of these two algorithms could be much higher than that of the Euler–Maruyama scheme extended by the same way. It is concluded that one of the keys to improvement of the results is the construction of the function for calculation of the probability of hitting a boundary.

Application of the Kusuoka approximation to pricing barrier options