Matematiksel finanstaki stokastik diferensiyel denklemlerin davranışsal sınıflandırması

We study the behavior of solutions for stochastic differential equations such as Heston stochastic volatility model, Merton-Black Scholes model and Merton's Jump Diffusion model. We examine the numerical solutions using Euler Maruyama, Milstein and stochastic Runge-Kutta methods when we analyze Heston stochastic volatility model 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). 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. While we explore strengths and limitations of Heston stochastic volatility model, Merton-Black Scholes model and Merton's Jump Diffusion model based on behavior of their numerical solutions, we suggest some model improvements in the light of the applications. Moreover, we introduce 3-dimensional matrix norms as generalizations of the matrix norms and prove the related lemmas, Duran and İzgi 2015, by using the applicable numerical linear algebra and analysis arguments. Furthermore, we define moving matrix for 2D and 3D matrices. Afterwards, we define market impression matrix norm as an application to the 3-dimensional matrix norms using moving matrices, Duran and İzgi, 2015. We can benefit from it to quantify market impression approximately by means of the numerical solutions for the stochastic differential equations. We analyze the simulation results for various parameters such that we perform high peak and fat-tail analysis for the impact of Heston, Merton-Black Scholes and Merton's jump diffusion models parameters' on the simulations of the extreme situations by using the first four standardized moments and extreme value tools such as quantile quantile (QQ), mean excess (ME) and Hill plots to examine the fat-tailness of the distributions. We also illustrate high peak and fat-tail analysis for BIST-100 index. On the other hand, we investigate 3D dynamics of the average logarithmic stock return, interest rate and speed of mean reversion variables, together. In addition, we believe that polarization and the transitions between polarizations and comovements are important part of extreme situation picture. Therefore, we investigate comovement and polarization of interest rates and daily returns of BIST-100 index in order to understand the corresponding behavioral dynamics. Heston stochastic volatility model predicts that the average logarithmic stock return increases as interest rate rises. Actually, we observe that there are also sufficiently large time intervals where interest rates were decreased and stock prices increased gradually in US stock markets and Borsa Istanbul, unlike the Heston stochastic volatility model suggests. Moreover, we analyze and compare the behavior of solutions for Merton-Black Scholes model and Merton's Jump Diffusion model. Especially, we focus on analyses of logarithmic stock price distributions obtained for these models using impression matrix norm and extreme value theory perspective. We achieved to present jump parameters' effects onto the behavior of solutions and also logarithmic stock price distributions using jump-adapted approximation method. Finally, we present price fluctuations for the Merton's jump diffusion and the Merton-Black Scholes models using impression matrix norm which reflects the effects of the jump parameters explicitly.