Application of the Heston stochastic volatility model for Borsa Istanbul using impression matrix norm

We study the behavior of solutions for stochastic differential equations such as the Heston stochastic volatility model. We examine the numerical solutions using Euler-Maruyama, Milstein and stochastic Runge-Kutta methods to investigate whether there is a role of the methods for different volatility cases or not, related to the impact of cumulative errors on this application. We perform simulations for different stock market conditions by using the large data set from Borsa Istanbul-100 (BIST-100) between 04.01.2007 and 31.12.2012. We use volatilities in terms of extreme values at the overlapping case when we examine initial and long term volatilities for the application of the Heston model. We also apply unit volatility based on extreme values to approximate volatilities in our analysis. We examine the advantages and limitations of the model. Moreover, we introduce 3-dimensional matrix norms. Furthermore, we define market impression matrix norm as an application to the 3-dimensional matrix norms. We can benefit from it to quantify market impression approximately by means of the numerical solutions for the stochastic differential equations. Finally, we analyze the simulation results for various parameters.