In 1966, Claude Berge proposed the following sorting problem. Given a string of n alternating white and black pegs, rearrange the pegs into a string consisting of ⌈n 2 ⌉ white pegs followed immediately by ⌊n 2 ⌋ black pegs (or vice versa) using only moves which take 2 adjacent pegs to 2 vacant adjacent holes. Berge’s original question was generalized by considering the same sorting problem usin...

I criticise the view that the relativity and equivalence principles are consequences of the small-scale structure of the metric in general relativity, by arguing that these principles also apply to systems with non-trivial self-gravitation and hence non-trivial spacetime curvature (such as black holes). I provide an alternative account, incorporating aspects of the criticised view, which allows...

The authors recently discovered some interesting relations between the Black-Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a …nite horizon. This is achieved in the present paper, where Yuri’s formula, as originally presented in Akahori, Imamura and Yano (2008), is also derived. We are most...

In [1], Borodin et al figured out a gap of [5], and gave a new proof with the similar technique. The purpose of this note is to fix the gap of [5] by slightly revising the definition of special faces, and adding a few lines of explanation in the proofs (new added text are all in black font).

We generalize the classical binomial approach of the model of Black and Scholes to a Markov binomial approach. This leads to a new formula for the cost of an option.

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.