Front matter

summary:The Equadiff is a series of biannual conferences on mathematical analysis, numerical approximation and applications of differential equations. Proceedings of Equadiff 2017 Conference contain peer-reviewed contributions of participants of the conference. The proceedings cover a wide range of topics presented by plenary, minisymposia and contributed talks speakers. The scope of papers ranges from ordinary differential equations, differential inclusions and dynamical systems towards qualitative and numerical analysis of partial differential equations, stochastic PDEs and their applications

Pricing american call option by the Black-Scholes equation with a nonlinear volatility function

In this paper we analyze a nonlinear Black-Scholes equation for pricing American style call option in which the volatility may depend on the underlying asset price and the Gamma of the option. We study the generalized Black-Scholes equation by means of transformation of the free boundary problem (variational inequalities) into the so-called Gamma equation for the new variable H = S@2SV . Moreover, we reformulate our new problem with PSOR method and construct an effective numerical scheme for discretization of the Gamma equation. Finally,we solve numerically our nonlinear complementarity problem applying PSOR method.info:eu-repo/semantics/publishedVersio

Pricing perpetual put options by the Black–Scholes Equation with a nonlinear volatility function

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself.We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.info:eu-repo/semantics/publishedVersio

Pricing perpetual put options by the Black-Scholes equation with a nonlinear volatility function

We investigate qualitative and quantitative behavior of a solution to the problem of pricing American style of perpetual put options. We assume the option price is a solution to a stationary generalized Black-Scholes equation in which the volatility may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.info:eu-repo/semantics/publishedVersio

Comparison of analytical approximation formula and Newton's method for solving a class of nonlinear Black-Scholes parabolic equations

Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE-based option pricing models can be described by solutions to the generalized Black–Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. In this paper, different linearization techniques such as Newton’s method and the analytic asymptotic approximation formula are adopted and compared for a wide class of nonlinear Black–Scholes equations including, in particular, the market illiquidity model and the risk-adjusted pricing model. Accuracy and time complexity of both numerical methods are compared. Furthermore, market quotes data was used to calibrate model parameters

The C1^1 stability of slow manifolds for a system of singularly perturbed evolution equations

summary:In this paper we investigate the singular limiting behavior of slow invariant manifolds for a system of singularly perturbed evolution equations in Banach spaces. The aim is to prove the C$^{1}$ stability of invariant manifolds with respect to small values of the singular parameter

Limiting behavior of global attractors for singularly perturbed beam equations with strong damping

summary:The limiting behavior of global attractors \Cal A_\varepsilon for singularly perturbed beam equations $$\varepsilon^2 \frac{\partial^2u}{\partial t^2}+ \varepsilon\delta \frac{\partial u}{\partial t}+A \frac{\partial u}{\partial t}+\alpha Au+g(\|u\|_{1/4}^2)A^{1/2}u=0$$ is investigated. It is shown that for any neighborhood \Cal U of \Cal A_0 the set \Cal A_\varepsilon is included in \Cal U for $\varepsilon$ small