Modèle Black-Scholes | EPFL Graph Search

Le modèle de Black-Scholes est utilisé pour désigner deux concepts très proches :

le modèle Black-Scholes ou modèle Black-Scholes-Merton qui est un modèle mathématique du marché pour une action, dans lequel le prix de l'action est un processus stochastique en temps continu ; par opposition au « modèle Cox Ross-Rubinstein » qui suit un processus stochastique en temps discret (cf. les processus stochastiques sont des fonctions du temps aléatoires) ;
léquation aux dérivées partielles de Black-Scholes qui est l'équation satisfaite par le prix d'un dérivé d'un primitif.

Robert C. Merton a été le premier à publier un article développant l'aspect mathématique d'un modèle d'évaluation d'option en citant les travaux de Fischer Black et de Myron Scholes. Ceux-ci, publiés en 1973, se fondent sur les développements de théoriciens comme Louis Bachelier ou encore Paul Samuelson. L'apport fondamental du modèle de Black et Scholes est de mettre en rapport le prix implicite de l'option et les vari

Asian option pricing with orthogonal polynomials

Sander Félix M Willems

In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.

ROUTLEDGE JOURNALS, TAYLOR & FRANCIS LTD2019

Skew-Brownian motion and pricing European exchange options br

Puneet Pasricha

This article derives a closed-form pricing formula for European exchange options under a non-Gaussianframework for the underlying assets, intending to resolve mispricing associated with a geometric Brownianmotion. The dynamics of each of the two correlated underlying assets are assumed to be governed by theexponential of a skew-Brownian motion, which is specified as a sum of a standard Brownian motion and anindependent reflected Brownian motion. The proposed pricing formula does not incur additional computationalcosts than the standard Black-Scholes framework, which one can quickly recover as a particular case of theproposed framework. Finally, we present some numerical experiments followed by a valuable discussion onthe results

ELSEVIER SCIENCE INC2022

Pricing interest rate, dividend, and equity risk

Sander Félix M Willems

This thesis studies the valuation and hedging of financial derivatives, which is fundamental for trading and risk-management operations in financial institutions. The three chapters in this thesis deal with derivatives whose payoffs are linked to interest rates, equity prices, and dividend payments. The first chapter introduces a flexible framework based on polynomial jump-diffusions (PJD) to jointly price the term structures of dividends and interest rates. Prices for dividend futures, bonds, and the dividend paying stock are given in closed form. Option prices are approximated efficiently using a moment matching technique based on the principle of maximum entropy. An extensive calibration exercise shows that a parsimonious model specification has a good fit with Euribor interest rate swaps and swaptions, Euro Stoxx 50 index dividend futures and dividend options, and Euro Stoxx 50 index options. The second chapter revisits the problem of pricing a continuously sampled arithmetic Asian option in the classical Black-Scholes setting. An identity in law links the integrated stock price to a one-dimensional polynomial diffusion, a particular instance of the PJD encountered in the first chapter. The Asian option price is approximated by a series expansion based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however numerical experiments show that the bias can safely be ignored in practice. The last chapter presents a non-parametric method to construct a maximally smooth discount curve from observed market prices of linear interest rate products such as swaps, forward rate agreements, or coupon bonds. The discount curve is given in closed form and only requires basic linear algebra operations. The method is illustrated with several practical examples.

EPFL2019

Pricing interest rate, dividend, and equity risk

Sander Félix M Willems

EPFL2019