We consider the problem of finding optimal exercise policies for American options, both under constant and stochastic volatility settings. Rather than work with the usual equations that characterize the price exclusively, we derive and use boundary evolution equations that characterize the evolution of the optimal exercise boundary. Using these boundary evolution equations we show how one can c...

nonstandard finite difference schemes for the black-scholes partial differential equation preserving the positivity property are proposed. computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the black-scholes equation. unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

Nonstandard finite difference schemes for the Black-Scholes partial differential equation preserving the positivity property are proposed. Computationally simple schemes are derived by using a nonlocal approximation in the reaction term of the Black-Scholes equation. Unlike the standard methods, the solutions of new proposed schemes are positive and free of the spurious oscillations.

• The total value adjustment (XVA) under the Heston model is studied for European and American options . Linear nonlinear partial differential equations have been deduced modelling XVA. Several numerical methods suitable boundary conditions are proposed in order to solve obtained PDEs A comparison with Black-Scholes made results. Expected exposures models also computed. Since 2007/2008 financia...

An alternative approach is proposed for constructing a strongly continuous semigroup based on the classical method of successive approximations, or Picard iterations, together with generating functions. application to Black–Scholes integro-differential operator which arises in pricing European options under jump-diffusion dynamics provided. The expressed as Mellin convolution time-inhomogeneous...

We consider some well-known partial differential equations that arise in Financial Mathematics, namely the Black–Scholes–Merton, Longstaff, Vasicek, Cox–Ingersoll–Ross and Heath equations. Our central aim is to discover any underlying connections taking into account Lie remarkability property of heat equation. For a few these there known connection with equation through coordinate transformatio...