Pricing theory for derivative securities is a highly technical topic in finance; its foundations rest on trading practices and its theory relies on advanced methods from stochastic calculus and numerical analysis. This chapter summarizes the main concepts while presenting the essential theory and basic mathematical tools for which the modeling and pricing of financial derivatives can be achieved.

Financialassetsaresubdividedintoseveralclasses,somebeingquitebasicwhileothersare structured as complex contracts referring to more elementary assets. Examples of elementary asset classes include stocks, which are ownership rights to a corporate entity; bonds, which are promises by one party to make cash payments to another in the future; commodities, which are assets, such as wheat, metals, and oil that can be consumed; and real estate assets, which have a convenience yield deriving from their use. A more general example of an asset is that of a contractual contingent claim associated with the obligation of one party to enter a stream of more elementary financial transactions, such as cash payments or deliveries of shares, with another party at future dates. The value of an individual transaction is called a pay-off or payout. Mathematically, a pay-off can be modeled by means of a payoff function in terms of the prices of other, more elementary assets. There are numerous examples of contingent claims. Insurance policies, for instance, are structured as contracts that envision a payment by the insurer to the insured in case a specific event happens, such as a car accident or an illness, and whose pay-off is typically linked to the damage suffered by the insured party. Derivative assets are claims that distinguish themselves by the property that the payoff function is expressed in terms of the price of an underlying asset. In finance jargon, one often refers to underlying assets simply as underlyings. To some extent, there is an overlap between insurance policies and derivative assets, except the nomenclature differs because the first are marketed by insurance companies while the latter are traded by banks.

Financialassetsaresubdividedintoseveralclasses,somebeingquitebasicwhileothersare structured as complex contracts referring to more elementary assets. Examples of elementary asset classes include stocks, which are ownership rights to a corporate entity; bonds, which are promises by one party to make cash payments to another in the future; commodities, which are assets, such as wheat, metals, and oil that can be consumed; and real estate assets, which have a convenience yield deriving from their use. A more general example of an asset is that of a contractual contingent claim associated with the obligation of one party to enter a stream of more elementary financial transactions, such as cash payments or deliveries of shares, with another party at future dates. The value of an individual transaction is called a pay-off or payout. Mathematically, a pay-off can be modeled by means of a payoff function in terms of the prices of other, more elementary assets. There are numerous examples of contingent claims. Insurance policies, for instance, are structured as contracts that envision a payment by the insurer to the insured in case a specific event happens, such as a car accident or an illness, and whose pay-off is typically linked to the damage suffered by the insured party. Derivative assets are claims that distinguish themselves by the property that the payoff function is expressed in terms of the price of an underlying asset. In finance jargon, one often refers to underlying assets simply as underlyings. To some extent, there is an overlap between insurance policies and derivative assets, except the nomenclature differs because the first are marketed by insurance companies while the latter are traded by banks.