We give a study result to analyze a rather different, semi-analytical numerical\nalgorithms based on splitting-step methods with their applications to mathematical\nfinance. As certain subsistent numerical schemes may fail due to producing\nnegative values for financial variables which require non-negativity preserving.\nThese algorithms which we are analyzing preserve not only the\nnon-negativity, but also the character of boundaries (natural, reflecting, absorbing,\netc.). The derivatives of the CIR process and the Heston model are\nbeing extensively studied. Beyond plain vanilla European options, we creatively\napply our splitting-step methods to a path-dependent option valuation.\nWe compare our algorithms to a class of numerical schemes based on Euler\ndiscretization which are prevalent currently. The comparisons are given with\nrespect to both accuracy and computational time for the European call option\nunder the CIR model whereas with respect to convergence rate for the\npath-dependent option under the CIR model and the European call option\nunder the Heston model.
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