Using classical finite difference schemes often generates numerical drawbacks such as spurious oscillations in the solution of the famous Black–Scholes partial differential equation. We analyze the fully implicit scheme, frequently used numerical method in Finance, that in the presence of discontinuous payoff and low volatility arises spurious oscillations. We propose a modification of this sch...

Comparing with the linear Black–Scholes model, fractional option pricing models are constructed by taking account some more parameters like, for example, transaction cost, so that it becomes difficult to find exact analytical solution. In this paper, we analyze a nonlinear Black and Scholes solution using novel numerical method, based on mixture of efficient techniques. particular, combine (1) ...

Empirical studies show that the volatility may exhibit correlations that decay as a fractional power of the time offset. The paper presents a rigorous analysis for the case when the stationary stochastic volatility model is constructed in terms of a fractional Ornstein Uhlenbeck process to have such correlations. It is shown how the associated implied volatility has a term structure that is a f...

Keywords: Black–Scholes equation Completely monotonic function Finite difference scheme Laplace Transform M-Matrix Positivity-preserving Post–Widder formula a b s t r a c t In this paper we explore discrete monitored barrier options in the Black–Scholes framework. The discontinuity arising at each monitoring data requires a careful numerical method to avoid spurious oscillations when low volati...

The object of this study was to investigate some implications of the tenets of behavioral finance on the pricing of financial derivatives. In particular, based on the work by Wolff et al (2009) we have investigated how prospect theory (Kahneman and Tversky, 1979) can be intregrated into the Black and Scholes (1973) option pricing framework. We have then used the resulting " behavioral version "...

The present paper accomplishes a major step towards a reconciliation of two conflicting approaches in mathematical finance: on the one hand, the mainstream approach based on the notion of no arbitrage (Black, Merton & Scholes); and on the other hand, the consideration of non-semimartingale price processes, the archetype of which being fractional Brownian motion (Mandelbrot). Imposing (arbitrari...

The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black–Scholes equations with a nonlinear volatility which can be a function of 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 taking into account e.g. nontrivial tr...

This paper analyses the model of Black–Scholes option pricing from point view group theoretic approach. The study identified new independent variables that lead to transformation equation. Furthermore, corresponding determining equations were constructed and symmetries found. As a result, findings demonstrate integrability present an invariant solution for Ornstein–Uhlenbeck stochastic process.