Abstract. The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the early exercise boundary for a linear as well as a class of nonline...

This paper uses risk-adjusted lognormal probabilities to derive the BlackScholes formula and explain the factors N(d1) and N(d2). It also shows how the one-period and multi-period binomial option pricing formulas can be restated so that they involve analogues of N(d1) and N(d2) which have the same interpretation as in the Black-Scholes model. Cet article utilise les probabilités lognormaux corr...

This paper assumes that the underlying asset prices are lognormally distributed, and derives necessary and su cient conditions for the valuation of options using a Black-Scholes type methodology. It is shown that the price of a futures-style, marked-to-market option is given by Black's formula if the pricing kernel is lognormally distributed. Assuming that this condition is ful lled, it is then...

This paper deals with the numerical solution of Black–Scholes option pricing partial differential equations by means of semidiscretization technique. For the linear case a fourth-order discretization with respect to the underlying asset variable allows a highly accurate approximation of the solution. For the nonlinear case of interest modeling option pricing with transaction costs, semidiscreti...

The security dynamics described by the Black-Scholes equation with price-dependent variance can be approximated as a damped discrete-time hopping process on a recombining binomial tree. In a previous working paper, such a nonuniform tree was explicitly constructed in terms of the continuous-time variance. The present note outlines how the previous procedure could be extended to multifactor Blac...

A better understanding of the empirical dynamics of Black-Scholes implied volatility surface has long been of considerable interest to both practitioners and academics. Basing on some findings about the ad hoc Black-Scholes valuation approach suggested in Dumas, Flemming and Whaley (1998), this essay studies the empirical performance of various volatility function forms that characterize the re...

– We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset fluctuations: “fat tails” and correlations in the scale of fluctuations. These aspects, although not included in the pricing models, are very precisely reflected in t...

In the original Black-Scholes model, the risk is quantified by a constant volatility parameter. It has been proposed by many authors that the volatilities should be modeled by a stochastic process to obtain a more realistic model. The volatility that corresponds to actual market data for option prices in Black-Scholes model is called the implied volatility. This volatility is in general depende...

This is the first of two papers in which we consider a stock with price process defined by a stochastic differential equation driven by a process Y (·) different from Brownian motion. The adoption of such a colored noise input is motivated by an analysis of real market data. The process Y (·) is defined by a continuous-time AR(∞)-type equation and may have either short or long memory. We show t...

This article presents a new option pricing principle that is more useful than the no-arbitrage principle, especially for incomplete markets. The focus here is on ideas behind mathematics— why the new theory is warranted, and how common sense dictates its construction. I. Complete and Incomplete Markets The real financial world is too complicated for anyone to understand it completely. Thus a si...