Mathematical Modeling with Multidisciplinary Applications

CHAPTER 9

LINEAR AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS IN FINANCIAL ENGINEERING

L. A. BOUKAS¹, K. I. VASILEIADIS², S. Z. XANTHOPOULOS², A. N. YANNACOPOULOS³

¹ Department of Information and Communication Systems Engineering, University of the Aegean, Greece

²Laboratory for Financial and Actuarial Mathematics, Department of Statistics and Actuarial − Financial Mathematics, University of the Aegean, Greece

³ Department of Statistics, Athens University of Economics and Business, Greece

9.1 FINANCIAL DERIVATIVES

Financial markets are mechanisms that facilitate the trading (buying and selling) of various assets, as, for example, financial securities (e.g., stocks and bonds), commodities (e.g., precious metals or agricultural products), etc. Among the various types of financial markets (e.g., capital markets, money markets, commodities markets, etc.), the market for derivative securities has seen an enormous growth during the last 30 years. One could say that the main role of the derivatives market is to facilitate the transfer of the financial risk that is inherent in other assets.

Generally speaking, a derivative security (or derivative contract or simply derivative) is a contract, the value of which depends on a reference value of some other underlying asset (s).

Usually, the underlying asset, to which a derivative refers, is a tradable financial security or a commodity, and the reference value is the price of this underlying asset. However, the reference value could ...