The Black-Scholes model is the standard approach used for pricing financial options. However, although being theoretically strong, option prices valued by the model often differ from the prices observed in the financial markets. This paper applies a hybrid neural network which preprocesses financial input data for improving the estimation of option market prices. This model is comprised of two ...

Classical explicit finite difference schemes are unsuitable for the solution of the famous Black-Scholes partial differential equation, since they impose severe restrictions on the time step. Furthermore, they may produce spurious oscillations in the solution. We propose a new scheme that is free of spurious oscillations and guarantees the positivity of the solution for arbitrary stepsizes. The...

in this paper two different methods are presented to approximate the solution of the fractional black-scholes equation for valuation of barrier option. also, the two schemes need less computational work in comparison with the traditional methods. in this work, we propose a new generalization of the two-dimensional differential transform method and decomposition method that will extend the appli...

Hedging a derivative security with non-risk-neutral number of shares leads to portfolio profit or loss. Unlike in the Black-Scholes world, the net present value of all future cash flows till maturity is no longer deterministic, and basis risk may be present at any time. The key object of our analysis is probability distribution of future P&L conditioned on the present value of the underlying. W...

In this paper, Heir-equations method is applied to investigate nonclassical symmetries and new solutions of the Black-Scholes equation. Nonlinear self-adjointness is proved and infinite number of conservation laws are computed by a new conservation laws theorem.

The price of an option can under some assumptions be determined by the solution of the Black–Scholes partial differential equation. Often options are issued on more than one asset. In this case it turns out that the option price is governed by the multi-dimensional version of the Black–Scholes equation. Options issued on a large number of underlying assets, such as index options, are of particu...

In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasi-linear parabolic partial (PDEs), which is related to FBSDEs from Pardoux-Peng theory. The rely on a process by minimizing path-wise difference between two discrete processes, are defined time discretization DNN r...

The celebrated optimal portfolio theory of R. C. Merton was successfully extended by the author to assets that do not obey Log-Normal price dynamics in [S. Stojanovic, Computational Financial Mathematics using Mathematica® : optimal trading in stocks and options, Birkhäuser, Boston, 2003]. Namely, a general one-factor model was solved, and applied in the case of appreciation-rate reversing mark...

The Black-Scholes (1973) option pricing model provides the foundation for the modern theory of options valuation. In actual applications, however, the model has certain well-known deficiencies. For example, when calibrated to accurately price at-the-money options the Black-Scholes (1973) model often misprices deep in-the-money and deep out-of-themoney options. This model-anomalous behavior give...