FROM ITO AND STRATONOVICH TO BLACK-SCHOLES

‎Options are financial instruments designed to protect investors from the stock market randomness‎. ‎In 1973 Black‎, ‎Scholes and Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation‎. ‎Herein‎, ‎We have reviewed Black-Scholes theory using Ito calculus‎, ‎which is standard to mathematical finance‎. ‎Moreover‎, ‎the Black-Scholes equation obtained using Stratonovich calculus is the same as the one obtained by means of the Ito calculus‎. ‎In fact‎, ‎this is the result we expected in advance because Ito and Stratonovich conventions are just different rules of calculus‎. ‎The option pricing method obtains the so-called Black-Scholes equation which is a partial differential equation of the same kind as the diffusion equation‎. ‎In fact‎, ‎it was this similarity that led Black and Scholes to obtain their option price formula as the solution of the diffusion equation with the initial and boundary conditions given by the option contract terms‎.