Analytical and numerical results for American style of perpetual put options through transformation into nonlinear stationary Black-Scholes equations

We analyze and calculate the early exercise boundary for a class of stationary generalized Black-Scholes equations in which the volatility function depends on the second derivative of the option price itself. A motivation for studying the nonlinear Black Scholes equation with a nonlinear volatility arises from option pricing models including, e.g., non-zero transaction costs, investors preferences, feedback and illiquid markets effects and risk from unprotected portfolio. We present a method how to transform the problem of American style of perpetual put options into a solution of an ordinary differential equation and implicit equation for the free boundary position. We finally present results of numerical approximation of the early exercise boundary, option price and their dependence on model parameters

Mathematical models in finance

In this paper we illustrate the interplay between Mathematics and Finance, pointing out the relevance of stochastic calculus and mathematical modelling in some important aspects of modern finance. We present two types of mathematical models: the binomial asset pricing model and continuous-time models. We point out some sensitive points of research.info:eu-repo/semantics/publishedVersio

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

Spatial approximation of nondivergent type parabolic PDEs with unbounded coefficients related to finance

We study the spatial discretisation of the Cauchy problem for a multidimensional linear parabolic PDE of second order, with nondivergent operator and unbounded time- and space-dependent coefficients. The equation free termand the initial data are also allowed to grow. Under a nondegeneracy assumption, we consider the PDE solvability in the framework of the variational approach and approximate in space the PDE problem’s generalised solution, with the use of finite-difference methods.Therate of convergence is estimated.info:eu-repo/semantics/publishedVersio

A fully nonlinear problem arising in financial modelling

We state existence and localisation results for a fully nonlinear boundary value problem using the upper and lower solutions method. With this study we aim to contribute to a better understanding of some analytical features of a problem arising in financial modelling related to the introduction of transaction costs in the classical Black-Scholes model. Our result concerns stationary solutions which become interesting in finance when the time does not play a relevant role such as, for instance, in perpetual options.info:eu-repo/semantics/publishedVersio

Loss given default : a backtesting exercise

Mestrado em Mathematical FinanceAfter January 2018, the new accounting standard IFRS 9 Financial Instruments was mandatory practice for all Financial Institutions. Introducing the new impairment model, which focus on expected credit losses (ECL) instead of incurred losses established previous in IAS 39 Measurement and Recognition. According to the new standard, the risk parameters involved in the computation of the ECL are required to be periodically revised. The Loss Given Default (LGD) is a risk input which represents the loss in case of a financial instrument defaults. Hence, the aim of the present report is to validate the risk input through a back testing exercise, considering statistical tests.info:eu-repo/semantics/publishedVersio

note on the numerical approximation of parabolic equations in Holder spaces

We consider the initial-boundary value problem for a multidimensional linear parabolic PDE of second order. This problem is solvable in Holder spaces. The solution is numerically approximated, using finite differences, and the rate of convergence of the time-space finite difference scheme is estimated. Both explicit and implicit discrete operators are given.info:eu-repo/semantics/publishedVersio

The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation

In this paper we study the existence of solutions of the nonlinear fourth-order equation [1] under the asymmetric nonlinear boundary conditions [2], [3] where g is a strictly monotonous function that may have some “one-sided” discontinuities and f and h exhibit some singularities.info:eu-repo/semantics/publishedVersio