This paper introduces the notion of option pricing in the context of financial markets. The discrete time, one-period binomial model is explored and generalized to the multi-period binomial model. The multi-period model is then redeveloped using the sophisticated tools of martingale theory. The paper concludes with a brief extension of the results to continuous time, giving a heuristic derivati...

For the pricing of options on equity shares, the Black-Scholes equation has become an indispensable tool for agents on the financial market. Under the assumption that the value of the underlying share evolves in time according to a stochastic differential equation and some further assumptions on the financial market, the equation can be derived by an application of Itô’s calculus. It represents...

We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed. Introduction The Black-Scholes-Merton model for option prices is an ...

Option valuation has been a challenging issue of financial engineering and optimization for a long time. The increasing complexity of market conditions requires utilization of advanced models that, commonly, do not lead to closed-form solutions. Development of novel numerical procedures, which prove to be efficient within various option valuation problems, is therefore worthwhile. Notwithstan...

Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter q. A generalized form of the Black-Scholes (BS) partial differential equation and some closed-form solutions are obtained. The standard BS equation (q=1) which is used by economists to calculate option prices requires multiple values o...

Differentiation matrices provide a compact and unified formulation for a variety of differential equation discretisation and timestepping algorithms. This paper illustrates their use for solving three differential equations of finance: the classic Black-Scholes equation (linear initial-boundary value problem), an American option pricing problem (linear complementarity problem), and an optimal m...

In this paper we consider a backward parabolic partial differential equation, called the Black-Scholes equation, governing American and European option pricing. We present a numerical method combining the Crank-Nicolson method in the time discretization with a hybrid finite difference scheme on a piecewise uniform mesh in the spatial discretization. The difference scheme is stable for the arbit...

In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for fractional diffusion equation. We also show stability convergence analysis fully discrete scheme a Crank–Nicolson time. A Black–Scholes equation is proposed tested numerically. Some numerical experiments are carried out to confirm a...

We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coeffic...

We apply path integration techniques to obtain option pricing with stochastic volatility using a generalized Black-Scholes equation known as the Merton and Garman equation. We numerically simulate the option prices using the technique of path integration. Using market data, we determine the parameters of the model. It is found that the market chooses a special class of models for which a more e...