Conclusion

Since the introduction of Brownian motion into financial market modelling by Bachelier and the seminal paper by Black and Scholes, the prodigious development of stochastic calculus has led to the emergence of stochastic models, notably with local volatility but also with stochastic volatility, whose source of randomness is generated by Brownian motions supposed to model the uncertainty observed on financial markets. These models are very popular and are actually used by banking industry practitioners, although they are built on idealized assumptions that are not necessarily verified. Indeed, the absence of arbitrage opportunity is questioned by some practitioners who can observe price mismatches from one market to another, and the very definition of arbitrage opportunity is sometimes called into question, see