Unit:
Κατεύθυνση Εφαρμοσμένα ΜαθηματικάLibrary of the School of Science
Author:
Αλευρομάγειρος Αριστείδης
Supervisors info:
Ιώαννης Στρατής Καθηγητής (επιβλέπων), Γεράσιμος Μπαρμπάτης Αναπλ. Καθηγητής, Αθανάσιος Γιαννακόπουλός Καθηγητής
Original Title:
Αντίστροφα προβλήματα σε παραβολικές εξισώσεις και εφαρμογές στα Χρηματοοικονομικά
Translated title:
Ιnverse problems in parabolic equations and applications to finance
Summary:
The MSc thesis describes the solution of the Black-Scholes partial differential
equation which is a fundamental tool for calculating the prices of options. In
the first chapter an analysis of martingale theory is provided whereas in the
second chapter an introduction to option theory is given. As far as the third
chapter is concerned, an analysis of the heat equation (parabolic partial
differential equation) is quoted in order to explain in the chapter four how
the Black-Scholes equation is related to heat equation through some
transformations and which is its solution. In chapter four it is also mentioned
how the Black-Scholes equation is derived by using the risk neutral measure.
Finally in the last chapter, the solution of the inverse problem of finance is
stated which is based on a proof made by the mathematician Bruno Dupire.
Specifically, if for fixed values of the price of a stock and the time we are
able to know all the prices of options for all the strike prices and maturities
then we can find a unique local volatility function. This result gives us
information about the volatility in the future.
Keywords:
Black-Scholes Equation, The inverse problem in finance, Risk Neutral Measure, Options, Volatility
