Application of the Kusuoka approximation to pricing barrier options

Abstract

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.

Computation of first-order Greeks for barrier options using chain rules for Wiener path integrals

“…Since barrier options are widely used financial products, computation of their prices and Greeks is an important issue in mathematical finance. Previous works have treated computation of barrier options [1][2][3][4][5] and their Greeks [6] under the framework of specific payoff functions (e.g., European-or Lookback-type functions) with constant trigger levels. This paper proposes a new method for the computation of the Greeks for barrier options under the general framework.…”

Ninomiya-Victoir scheme : Multilevel Monte Carlo estimators and discretization of the involved Ordinary Differential Equations

“…It is not clear at all how to directly address the influence of integration schemes for ODEs on the order of weak convergence and Ninomiya and Ninomiya rather look for sufficient conditions ensuring that the strong error and therefore the weak error generated by these schemes converge with order two. In [13] p437 (see also Remark 2.2 p173 [12]), they claim that this is achieved when using a Runge-Kutta scheme with order five (resp. order two) for the ODEs associated with the Brownian vector fields σ j , j ∈ {1, .…”