Editing Stochastic process (section)

=== Applications in Finance ===

==== Black-Scholes Model ====
One of the most famous applications of stochastic processes in finance is the '''[[Black-Scholes model]]''' for option pricing. Developed by [[Fischer Black]], [[Myron Scholes]], and [[Robert Solow]], this model uses '''[[Geometric Brownian motion]]''', a specific type of stochastic process, to describe the dynamics of asset prices.<ref>{{Cite journal |last1=Black |first1=Fischer |last2=Scholes |first2=Myron |date=1973 |title=The Pricing of Options and Corporate Liabilities |url=https://www.jstor.org/stable/1831029 |journal=Journal of Political Economy |volume=81 |issue=3 |pages=637–654 |doi=10.1086/260062 |jstor=1831029 |issn=0022-3808}}</ref><ref>{{Citation |last=Merton |first=Robert C. |title=Theory of rational option pricing |date=July 2005 |work=Theory of Valuation |pages=229–288 |url=http://www.worldscientific.com/doi/abs/10.1142/9789812701022_0008 |access-date=2024-09-30 |edition=2 |publisher=WORLD SCIENTIFIC |language=en |doi=10.1142/9789812701022_0008 |isbn=978-981-256-374-3|hdl=1721.1/49331 |hdl-access=free }}</ref> The model assumes that the price of a stock follows a continuous-time stochastic process and provides a closed-form solution for pricing European-style options. The Black-Scholes formula has had a profound impact on financial markets, forming the basis for much of modern options trading.

The key assumption of the Black-Scholes model is that the price of a financial asset, such as a stock, follows a '''[[log-normal distribution]]''', with its continuous returns following a normal distribution. Although the model has limitations, such as the assumption of constant volatility, it remains widely used due to its simplicity and practical relevance.

==== Stochastic Volatility Models ====
Another significant application of stochastic processes in finance is in '''[[Stochastic volatility|stochastic volatility models]]''', which aim to capture the time-varying nature of market volatility. The '''[[Heston model]]'''<ref>{{Cite journal |last=Heston |first=Steven L. |date=1993 |title=A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options |url=https://www.jstor.org/stable/2962057 |journal=The Review of Financial Studies |volume=6 |issue=2 |pages=327–343 |doi=10.1093/rfs/6.2.327 |jstor=2962057 |issn=0893-9454}}</ref> is a popular example, allowing for the volatility of asset prices to follow its own stochastic process. Unlike the Black-Scholes model, which assumes constant volatility, stochastic volatility models provide a more flexible framework for modeling market dynamics, particularly during periods of high uncertainty or market stress.