Efficient Path Generation Method In Quasi-Monte Carlo And Weight Handling For Asian Option Pricing

Some financial problems such as option pricing and sensitivity calculation are essentially integral estimation problems.The quasi-Monte Carlo method(QMC)method is a very effective estimation method for these financial problems,and is more efficient than the Monte Carlo method.Due to the poor uniformity of the distribution of points in the high-dimensional QMC method,for some high-dimensional integral financial problems,such as the Asian option pricing problem,special Brownian motion path generation methods(PGMs)are often used to make the estimation results more depended on the points in the initial dimensions.The degree can be measured by effective dimension.These path generation methods include the Random Walk(RW)method,the Brownian Bridge(BB)method,and the Principal Component Analysis(PCA)method.In general,the PCA method has the most good results,which can significantly reduce the effective dimension,while the PCA method generation matrix is computationally complex.This paper proposes a new generation method called PCABB that combines the PCA and BB methods,which can achieve good results and low computational complexity.PCABB method is to first generate Brownian motion at a certain point in time by using the PCA method,and then use the BB method to generate Brownian motion at the remaining time points.PCABB method is weaker than the PCA method,but with proper settings,the PCABB method can be very close to the PCA method and has great advantages in computational complexity.On the other hand,for some weighted Asian options,PCA,BB and some other methods may not have a good effect.This paper proposes a PCA-based generation method,called WPCA method,which can significantly improve the effect of reducing the effective dimension.WPCA method is to multiply the covariance matrix of the Brownian motion by the weights and then calculate the eigenvalue decomposition of it,then calculate generating matrix.This method is better than the RW,BB,and PCA methods for some weighted Asian options mentioned in the paper.This method can be considered to be effective for most of the different weights.WPCA can also be combined with the PCABB method to reduce computational complexity.