We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe – theoretically and by typical data – that the prices coincide very well. We illustrate Louis Bachelier’s efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we explain – by simple methods from chaos expansion – why Bachelier’s model yields good short-time approx...

In the context of a Black-Scholes economy and with a no-arbitrage argument, we derive arbitrarily accurate lower and upper bounds for the value of European options on a stock paying a discrete dividend. Setting the option price error below the smallest monetary unity, both bounds coincide, and we obtain the exact value of the option.

Barrier options were first priced by Merton in 1973 using partial differential equation. In this work, we present a closed form formula for pricing European barrier option with a moving barrier that increases with time to expiration. We adopted a three-step approach which include; justifying that barrier options satisfy the Black-Scholes partial differential equation under certain conditions, p...

The analogue of Black–Scholes formula for vanilla call option price in conditions of (B, S)-securities market with delayed response is derived. A special case of continuous-time version of GARCH is considered. The results are compared with the results of Black and Scholes. © 2006 IMACS. Published by Elsevier B.V. All rights reserved.

We survey recent results on the behavior of the Black-Scholes implied volatility at extreme strikes. There are simple and universal formulae that give quantitative links between tail behavior and moment explosions of the underlying on one hand, and growth of the famous volatility smile on the other hand. Some original results are included as well.

Due to the impressive amount of new data provided by the RXTE satellite in the past decade, our knowledge of the phenomenology of accretion onto black holes has increased considerably. In particular, it has been possible to schematize the outburst evolution of transient systems on the basis of their spectral and timing properties, and link them to the ejection of relativistic jets as observed i...

This article studies the behavior of an index It which is assumed to be a tradable security, to satisfy the BSM model dIt/It = μdt + σdWt, and to be efficient in the following sense: we do not expect a prespecified trading strategy whose value is almost surely always nonnegative to outperform the index greatly. The efficiency of the index imposes severe restrictions on its appreciation rate; in...

Given that the options market is now very large and significant part of the trade of financial instruments, the evaluation of pricing of these derivatives becomes very important for regulators as well as market participants. The value of an option can be estimated by using a variety of quantitative techniques based on the concept of risk neutral pricing. In the famous Black-Scholes pricing form...

We study the conditions for 2-dimensional dilaton gravity models to have dynamical formation of black holes and construct all such models. Furthermore we present a parametric representation of the general solutions of the black holes.