The evaluation of derivatives of double barrier options of the Bessel processes by methods of spectral analysis

This article studies the derivatives pricing using a method of spectral analysis, a theory of singular and regular perturbations. Using a risk-neutral assessment, the authors obtain the Cauchy problem, which allows to calculate the approximate price of derivative assets and their volatility based on the diffusion equation with fast and slow variables of nonlocal volatility, and they obtain a model with multidimensional stochastic volatility. Applying a spectral theory of self-adjoint operators in Hilbert space and a theory of singular and regular perturbations, an analytic formula for approximate asset prices is established, which is described by the CEV model with stochastic volatility dependent on l-fast variables and r-slowly variables, l ≥ 1, r ≥ 1, l ∈ N, r ∈ N and a local variable. Applying the Sturm-Liouville theory, Fredholm’s alternatives, as well as the analysis of singular and regular perturbations at different time scales, the authors obtained explicit formulas for derivatives price approximations. To obtain explicit formulae, it is necessary to solve 2l Poisson equations.

The main aim of the paper is to measure hedging efficiency using the Short Put Ladder strategy formed by barrier options in the equity market. The researchers hedge full protection against price’s drop, combining the European down and knock-in put options with the lowest exercise price and vanilla or barrier put options with the higher exercise prices. The authors chose the analyzed alternatives according to the requirement of the zero-cost strategy. The aim of the investigated hedging variants is to secure the minimum constant selling price for the underlying asset’s price drop. Theoretical results of this approach were applied in the equity market, i.e., SPDR S&P 500 ETF. The authors analyzed and compared all hedging variants to each other, however, only the selected techniques were presented in the paper. The findings reveal that the barrier options used for managing the equity risk produce significant reductions of that risk. The right combination of options with the strike prices and the barrier levels wisely selected plays a significant role in risk elimination. Finally, according to the findings, the recommendations for potential investors are introduced.

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.