The technique of dynamic hedging, combined with the application of Ito calculus and the absence of arbitrage hypothesis, provides a methodology for the valuation of financial derivatives by models of partial differential equations of Black-Scholes type. This document is intended to summarize in a simple way the concepts and techniques used in this methodology up to get the prices of the more tr...

Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving a nonlinear Black-Scholes equation.

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.

The state price density of a basket, even under uncorrelated Black–Scholes dynamics, does not allow for a closed from density. (This may be rephrased as statement on the sum of lognormals and is especially annoying for such are used most frequently in Financial and Actuarial Mathematics.) In this note we discuss short time and small volatility expansions, respectively. The method works for gene...

A generalized option-pricing formula is found, based on a nonGaussian stock price model. The dynamics of the underlying stock are assumed to follow a stochastic process with anomalous nonlinear diffusion, phenomenologically modelled as a statistical feedback process within the framework of the generalized thermostatistics proposed by Tsallis. A generalized form of the Black-Scholes differential...

This work presents a theoretical analysis for the Black–Scholes equation. Given a terminal condition, the analytical solution of the Black–Scholes equation is obtained by using the Adomian approximate decomposition technique. The mathematical technique employed in this work also has significance in studying some other problems in finance theory.

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.

Expanding the ideas of the author’s paper ’Nonexpansive maps and option pricing theory’ (Kibernetica, 34:6, 1998, 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so ...

Using the dynamic programming principle in optimal stopping theory, we derive a semilinear Black and Scholes type partial differential equation set in a fixed domain for the value of an American (call/put) option. The nonlinearity in the semilinear Black and Scholes equation depends discontinuously on the American option value, so that standard theory for partial differential equation does not ...