Option Pricing Problems in Variational Formulation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F49777513%3A23510%2F17%3A43932863" target="_blank" >RIV/49777513:23510/17:43932863 - isvavai.cz</a>

Výsledek na webu

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DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

Option Pricing Problems in Variational Formulation

Popis výsledku v původním jazyce

This chapter deals with variational formulation of option pricing problems. Author start from the well-known case, the Black-Scholes model for a put option with strike price and maturity given, which assumes the underlying asset to follow a geometric Brownian motion. This problem provides a reasonable basic framework to follow basic steps of derivation of variational formulation of option pricing problem. In general, variational formulation consists of finding a continuous function defined on the time interval with the values in a properly defined functional space. Finite element method applied to option pricing problem in finance yields usually a system of ordinary differential equations if discretization process applies to space domain of underlying asset only. Pricing American options requires, due to the early exercise feature of such derivative contracts, the solution of optimal stopping problems for the price process. Unlike in the European case, the pricing function of an American option does not satisfy a partial differential equation, but a partial differential inequality, or a system of inequalities. Recasting such problem into a variational inequality problem is the next step, which is given in detail. Author mentions briefly the functional space which provides natural framework for weak formulation of American put option pricing problem. Both optimal exercise boundary and additive decomposition of American put option are discussed, as well. Finally, numerical solution of 2-D basket European put option pricing problem is discussed in detail. Author concerns with influence of various parameters upon the option price, with the correlation structure of underlying assets in particular. The details of FreeFem++ code are revealed, too.

Název v anglickém jazyce

Option Pricing Problems in Variational Formulation

Popis výsledku anglicky

This chapter deals with variational formulation of option pricing problems. Author start from the well-known case, the Black-Scholes model for a put option with strike price and maturity given, which assumes the underlying asset to follow a geometric Brownian motion. This problem provides a reasonable basic framework to follow basic steps of derivation of variational formulation of option pricing problem. In general, variational formulation consists of finding a continuous function defined on the time interval with the values in a properly defined functional space. Finite element method applied to option pricing problem in finance yields usually a system of ordinary differential equations if discretization process applies to space domain of underlying asset only. Pricing American options requires, due to the early exercise feature of such derivative contracts, the solution of optimal stopping problems for the price process. Unlike in the European case, the pricing function of an American option does not satisfy a partial differential equation, but a partial differential inequality, or a system of inequalities. Recasting such problem into a variational inequality problem is the next step, which is given in detail. Author mentions briefly the functional space which provides natural framework for weak formulation of American put option pricing problem. Both optimal exercise boundary and additive decomposition of American put option are discussed, as well. Finally, numerical solution of 2-D basket European put option pricing problem is discussed in detail. Author concerns with influence of various parameters upon the option price, with the correlation structure of underlying assets in particular. The details of FreeFem++ code are revealed, too.

Druh

C - Kapitola v odborné knize

CEP obor

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OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA15-20405S" target="_blank" >GA15-20405S: Modelování procesů na finančních trzích a predikce bankrotu firem aparátem reálných opcí</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název knihy nebo sborníku

Advanced Methods of Computational Finance

ISBN

978-80-245-2207-4

Počet stran výsledku

33

Strana od-do

77-109

Počet stran knihy

239

Název nakladatele

University of Economics, Prague, Oeconomica Publishing House

Místo vydání

Prague

Kód UT WoS kapitoly

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