Financial derivatives – and options in particular – form an important component of financial instruments. Considerable negative criticism has been directed at options and derivatives in light of the 2008 economic crisis, some of it being justified and some being off the mark. As long as there are assets and liabilities, there will be a need to protect the future value of assets, as well as of finding ways for maximizing returns on assets. Derivatives play a central role in achieving these twin objectives.

Given the uncertainties of the financial markets, there is a strong demand from banks, financial organizations, and investors for predicting the future behavior of securities. Derivative instruments, and options in particular, are a response to this need of the market and are widely traded in the financial markets.

Options and other derivatives of underlying financial securities have contributed significantly to the explosion of the capital markets and their general principles are discussed. There are three broad categories of derivatives, namely forwards, futures, and options. Option theory is developed for equities using a path integral formulation of white noise.

A series expansion of an option's price is defined for a generic case, in powers of the underlying security's volatility. The volatility expansion is of great generality and will turn out to be crucial in developing approximation schemes for a variety of interest rates and coupon bond options.