The Black-Scholes ( 1973) option pricing model is a universal standard among option valuation models. Despite its widespread popularity, however, the model has some known deficiencies in actual applications. For example, when calibrated to accurately price at-the-money options, the Black-Scholes model frequently misprices deep in-the-money and deep out-of-the-money options. Pricing biases assoc...

In this paper we develop a high-order adaptive finite difference space-discretization for the Black–Scholes (B–S) equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence is two, both for low-order and high-order standard finite difference (FD) schemes. To obtain a sixth-order scheme we use an extra grid in a limited spaceand time-do...

This paper compares the performance of Black-Scholes with an artificial neural network (ANN) in pricing European style call options on the FTSE 100 index. It is the first extensive study of the performance of ANNs in pricing UK options, and the first to allow for dividends in the closed-form model. For out-of themoney options, the ANN is clearly superior to Black-Scholes. For in-the-money optio...

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the...

We discuss the use of the WKB ansatz in a variety of parabolic problems involving a small parameter. We analyse the Stefan problem for small latent heat, the Black–Scholes problem for an American put option, and some nonlinear diffusion equations, in each case constructing an asymptotic solution by the use of ray methods.

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 ...

We present a review and some new results on form methods for generating holomorphic semigroups on Hilbert spaces. In particular, we explain how the notion of closability can be avoided. As examples we include the Stokes operator, the Black–Scholes equation, degenerate differential equations and the Dirichlet-to-Neumann operator. Mathematics Subject Classification (2000). Primary 47A07; Secondar...

We show that the problem of recovering the time-dependent parameters of an equation of Black-Scholes type can be formulated as an inverse Stieltjes moment problem. An application to the problem of implied volatility calculation in the case when the model parameters are time varying is provided and results of numerical simulations are presented.