In this paper, the meshless local Petrov-Galerkin (MLPG) method is applied for solving a generalized Black-Scholes equation in financial problems. This equation is a PDE governing the price evolution of a European call or a European put under the Black-Scholes model. The θ-weighted method and MLPG are used for discretizing the governing equation in time variable and option pricing, respectively...

The aim of this paper is to present a stochastic model that accounts for the effects of a long-memory in volatility on option pricing. The starting point is the stochastic Black-Scholes equation involving volatility with long-range dependence. We consider the option price as a sum of classical Black-Scholes price and random deviation describing the risk from the random volatility. By using the ...

In this paper, Laplace homotopy perturbation method, which is combined form of the Laplace transform and the homotopy perturbation method, is employed to obtain a quick and accurate solution to the fractional Black Scholes equation with boundary condition for a European option pricing problem. The Black-Scholes formula is used as a model for valuing European or American call and put options on ...

Standard derivative pricing theory is based on the assumption of agents acting as price takers on the market for the underlying asset. We relax this hypothesis and study if and how a large agent whose trades move prices can replicate the payoff of a derivative security. Our analysis extends prior work of Jarrow to economies with continuous security trading. We characterize the solution to the h...

Since the 2007/2008 financial crisis, total value adjustment (XVA) should be included when pricing derivatives. In present paper, derivative values of European and American options have been priced where we take into account counterparty risk. Whereas considering risk already under Black-Scholes dynamics in [2], here novel contribution is introduction stochastic volatility resulting a Heston ty...

There are many references showing that a classical solution to the Black–Scholes equation is a stochastic solution. However, it is the converse of this theorem which is most relevant in applications and the converse is also more mathematically interesting. In the present article we establish such a converse. We find a Feynman–Kac type theorem showing that the stochastic representation yields a ...

When we studied discrete-time models we used martingale pricing to derive the Black-Scholes formula for European options. It was clear, however, that we could also have used a replicating strategy argument to derive the formula. In this part of the course, we will use the replicating strategy argument in continuous time to derive the Black-Scholes partial differential equation. We will use this...

European options can be priced using the analytical solution of the Black-Scholes-Merton differential equation with the appropriate boundary conditions. A different approach and the one commonly used in situations where no analytical solution is available is the Monte Carlo Simulation. We present the results of Monte Carlo simulations for pricing European options and we compare with the analyti...