Abstract
Dividend paying European stock options are modeled using a time-fractional Black–Scholes (tfBS) partial differential equation (PDE). The underlying fractional stochastic dynamics explored in this work are appropriate for capturing market fluctuations in which random fractional white noise has the potential to accurately estimate European put option premiums while providing a good numerical convergence. The aim of this paper is two fold: firstly, to construct a time-fractional (tfBS) PDE for pricing European options on continuous dividend paying stocks, and, secondly, to propose an implicit finite difference method for solving the constructed tfBS PDE. Through rigorous mathematical analysis it is established that the implicit finite difference scheme is unconditionally stable. To support these theoretical observations, two numerical examples are presented under the proposed fractional framework. Results indicate that the tfBS and its proposed numerical method are very effective mathematical tools for pricing European options.
Highlights
Since the discovery of the most celebrated Black–Scholes–Merton asset pricing formula in the early 1970s, the application of Black–Scholes (BS) partial differential equations (PDEs) in valuation of derivative instruments has become very popular
Ballerster et al [4] derived a robust numerical method for pricing vanilla options with discrete dividend payments. Another notable setback of the classical BS approach and many of the revised versions discussed above is that their PDEs involve integer order derivatives
In [20] a novel semianalytical technique for solving a Schrödinger-type option pricing time-fractional model governed by a controlled Brownian motion was proposed
Summary
Since the discovery of the most celebrated Black–Scholes–Merton asset pricing formula in the early 1970s, the application of Black–Scholes (BS) partial differential equations (PDEs) in valuation of derivative instruments has become very popular. It is a financial contract that gives the holder the right, but not obligation, to buy or sell a specified quantity of an underlying asset, e.g., a stock, at a fixed price, called the strike price, before or on the expiration date. Since it is a right and not an obligation, the holder of the contract can decide not to exercise the option and let it expire worthless
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