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...

We survey some new progress on the pricing models driven by fractional Brownian motion or mixed fractional Brownian motion. In particular, we give results on arbitrage opportunities, hedging, and option pricing in these models. We summarize some recent results on fractional Black & Scholes pricing model with transaction costs. We end the paper by giving some approximation results and indicating...

The field of mathematical finance has gained significant attention since Black and Scholes (1973) published their Nobel Prize work in 1973. Using some simplifying economic assumptions, they derived a linear partial differential equation (PDE) of convection–diffusion type which can be applied to the pricing of options. The solution of the linear PDE can be obtained analytically. In this paper we...

Option trading forms part of our financial markets. A traded option gives to its owner the right to buy (call option) or to sell (put option) a fixed quantity of assets of a specified stock at a fixed price (exercise or strike price). There are two major types of traded options. One is the American option that can be exercised at any time prior to its expiry date, and the other option, which ca...

Stochastic differential equations and the Black-Scholes PDE. We derived the BlackScholes formula by using arbitrage (risk-neutral) valuation in a discrete-time, binomial tree setting, then passing to a continuum limit. This section explores an alternative, continuoustime approach via the Ito calculus and the Black-Scholes differential equation. This material is very standard; I like Wilmott-How...

Abstract This paper aims to estimate the parameters of time-fractional Black-Scholes (TFBS) partial differential equation with Caputo fractional derivative by using real option prices S &amp;P 500 index options. First, numerical solution is obtained developing a high-order scheme order ( $$3-\alpha $$ 3 <mml:m...

The Black-Scholes equation is a hallmark of mathematical finance, and any study of this growing field would be incomplete without having seen and understood the logic behind this equation. The initial focus of this paper will be to explore the arguments leading to the equation and the financial background necessary to understand the arguments. The problem of estimating the only parameter which ...

In this paper we apply the innovative Laplace transformation method introduced by Sheen, Sloan, and Thomée (IMA J. Numer. Anal., 2003) to solve the Black-Scholes equation. The algorithm is of arbitrary high convergence rate and naturally parallelizable. It is shown that the method is very efficient for calculating various option prices. Existence and uniqueness properties of the Laplace transfo...