A master equation approach to the numerical solution of option pricing models is developed. The basic idea of the approach is to consider the Black–Scholes equation as the macroscopic equation of an underlying mesoscopic stochastic option price variable. The dynamics of the latter is constructed and formulated in terms of a master equation. The numerical efficiency of the approach is demonstrat...

In this paper we derive an effective equation for derivative pricing which accounts for the presence of virtual arbitrage opportunities and their elimination by the market. We model the arbitrage return by a stochastic process and find an equation for the average derivative price. This is an integro-differential equation which, in the absence of the virtual arbitrage or for an infinitely fast m...

We are concerned with a model for asset prices introduced by Koichiro Takaoka, which extends the well known Black-Scholes model. For the pricing of contingent claims, partial differential equation (PDE) is derived in a special case under the typical delta hedging strategy. We present an exact pricing formula by way of solving the equation.

This paper gives a connection between the theory of semigroups of operators (functional analysis) and mathematical finance through the Black-Scholes equation. Besides using this theory obtain the solution of this equation via the infinitesimal generator of a group of evolution. Mathematics Subject Classification: Primary 91G80; Secondary 47D06

The standard Black-Scholes theory of option pricing is extended to cope with underlying return fluctuations described by general probability distributions. A Langevin process and its related Fokker-Planck equation are devised to model the market stochastic dynamics, allowing us to write and formally solve the generalized Black-Scholes equation implied by dynamical hedging. A systematic expansio...

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...