Finding the derivative price using the Vasicek model with multidimensional stochastic volatility

Abstract

Methods of calculating the approximate price of options using instruments of spectral analysis, singular and regular wave theory in the context of influence of fast and slow acting factors are developed. By combining methods from the spectral theory of singular and regular disturbances, one can approximate the price of derivative financial instruments as a schedule of its own functions. The article uses the theory of spectral analysis and the singular and regular theory of perturbations, which are applied to the short-term interest rates described by the Vasicek model with multidimensional stochastic volatility. The approximate price of derivatives and their profitability are calculated. Applying the Sturm-Liouville theory, the Fredholm alternative, and the analysis of singular and regular disturbances in different time scales, explicit formulas were obtained for the approximation of bond prices and yields based on the development of their own functions and eigenvalues of self-adjoint operators using boundary value problems for singular and regular perturbations. The theorem for estimating the accuracy of derivatives price approximation is established. Such a technique, in contrast to existing ones, makes it possible to study the stock market dynamics and to monitor the financial flows in the market. This greatly facilitates the statistical evaluation of their parameters in the process of monitoring the derivatives pricing and the study of volatility behavior for the profitability analysis and taking strategic management decisions on the stock market transactions.