We introduce the standard fourth order compact finite difference formulae. We show how these formulae apply in the special case of the heat equation. It is well known that the American option pricing problem may be formulated in terms of the Black Scholes partial differential equation (PDE) together with a free boundary condition. Standard methods allow this problem to be transformed into a mov...

in which CBS is the usual European call price of an option without dividends. The function φ(S0, S, td) is the log-normal no dividend density function. The integral representation given in (1) does not have an explicit solution. It can be approximated with a numerical integration method, which may be complicated due to the right side boundary at infinity. Another possibility is to solve the Bla...

An option pricing formula is obtained, based on a stochastic model with statistical feedback. The fluctuations evolve according to a Tsallis distribution which fits empirical data for stock returns. A generalized form of the Black-Scholes partial differential equation is derived, parametrized by the Tsallis entropic index q. A martingale representation is found, which allows us to use concepts ...

The objective of this paper is twofold. Firstly, to derive time-fractional evolution equation modeling the No-Arbitrage premium Asian option (with arithmetic and geometric averages ) contingent upon an underlying asset that satisfies fractional stochastic differential equation, in a setting when strike price fixed floating. Secondly, we have computed four versions put-call parities for options,...

A valuation model is presented for options on stocks for which BlackScholes arbitrage does not entirely eliminate risk. The price dynamics of a portfolio of options and the underlying security is quanti ed by requiring that the excess reward-to-risk ratio of the portfolio be identical to that of the underlying stock: excess return risk portfolio = excess return risk stock : The nonlinear evolut...

Abstract The nonlinear differential equation option pricing formula is invaluable in financial derivatives investment risk assessment. This article applies the theory of equations to deal with risks commodity and currency markets. Through this condition, we obtain fair price process contingent rights under classic Black-Scholes model optimal growth strategy. results show that measurement stable...

Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor’s preferences or illiquid markets, which may have an impact on the stock price, the volatility, the drift and the option pr...

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as “the curse of dimensionality”. This paper presents a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated as a control theory prob...