The Role of the Volatility in the Option Market

We review some general aspects about the Black�Scholes equation, which is used for predicting the fair price of an option inside the stock market. Our analysis includes the symmetry properties of the equation and its solutions. We use the Hamiltonian formulation for this purpose. Taking into account that the volatility inside the Black�Scholes equation is a parameter, we then introduce the Merton�Garman equation, where the volatility is stochastic, and then it can be perceived as a field. We then show how the Black�Scholes equation and the Merton�Garman one are locally equivalent by imposing a gauge symmetry under changes in the prices over the Black�Scholes equation. This demonstrates that the stochastic volatility emerges naturally from symmetry arguments. Finally, we analyze the role of the volatility on the decisions taken by the holders of the options when they use the solution of the Black�Scholes equation as a tool for making investment decisions.