Stochastic finance is a field of mathematical finance that models prices, interest rates and risk with stochastic processes, and applies probability, stochastic calculus and martingale techniques to valuation, hedging and risk measurement. Specialist journals frame the area as finance âÂÂbased on stochastic methods,â spanning both theory and applications at the interface of probability and finance.
Louis BachelierâÂÂs 1900 thesis in the Annales scientifiques de lâÂÂÃÂcole Normale Supérieure modelled price changes with Brownian motion and anticipated later diffusion-based approaches. A modern synthesis emerged with the BlackâÂÂScholes article in 1973, which connected dynamic hedging to a pricing partial differential equation and a closed-form solution. From the late 1980s, martingale and semimartingale methods supplied a measure-theoretic foundation, notably the fundamental theorem of asset pricing that links absence of arbitrage to the existence of an equivalent martingale measure.
Core tools come from continuous-time probability and stochastic analysis. Graduate texts present Brownian motion as a canonical model and develop Itô integration and stochastic differential equations; pricing is set on martingale grounds with changes of measure and links to parabolic PDEs via FeynmanâÂÂKac.
A small set of models has shaped practice and pedagogy.
Work in the field returns to a handful of ideas. The fundamental theorem of asset pricing states that absence of arbitrage corresponds to a probability measure under which discounted price processes are martingales; in complete markets this gives exact replication, while incompleteness leads to super-replication and risk-measure approaches. Continuous-time portfolio choice and stochastic control supply consumptionâÂÂinvestment results and verification through HamiltonâÂÂJacobiâÂÂBellman equations, while optimal stopping handles American-style exercise and related free-boundary problems.
Research commonly appears in Finance and Stochastics and Mathematical Finance; widely used books include Karatzas & ShreveâÂÂs graduate text on Brownian motion and stochastic calculus, ShreveâÂÂs two-volume sequence on continuous-time models, Baxter & RennieâÂÂs introduction to derivative pricing, and Cont & TankovâÂÂs monograph on jump processes.