Editing Geometric Brownian motion (section)

==Extensions==
In an attempt to make GBM more realistic as a model for stock prices, also in relation to the [[volatility smile]] problem, one can drop the assumption that the volatility (<math>\sigma</math>) is constant. If we assume that the volatility is a [[Deterministic system|deterministic]] function of the stock price and time, this is called a [[local volatility]] model. A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.<ref name="musielarutkowski"/><ref>Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI https://doi.org/10.1007/3-540-30591-2</ref><ref>{{cite journal | title = Lognormal-mixture dynamics and calibration to market volatility smiles | first1 = Damiano | last1 = Brigo | author-link1 = Damiano Brigo | first2 = Fabio | last2 = Mercurio | author-link2 = Fabio Mercurio | pages = 427–446 | journal = International Journal of Theoretical and Applied Finance | volume = 5 | year = 2002 | issue = 4 | doi = 10.1142/S0219024902001511 }}</ref><ref>Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, {{ISSN|1469-7688}}</ref> If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a [[stochastic volatility]] model, see for example the [[Heston model]].<ref>{{cite journal | title = A closed-form solution for options with stochastic volatility with applications to bond and currency options | first = Steven L. | last = Heston | author-link1 = Steven L. Heston | pages = 327–343 | journal = Review of Financial Studies | volume = 6 | issue = 2  | year = 1993 | jstor = 2962057 | doi = 10.1093/rfs/6.2.327| s2cid = 16091300 }}</ref>