Finance:Stochastic finance

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Short description: Branch of mathematical finance based on stochastic processes

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 “based on stochastic methods,” spanning both theory and applications at the interface of probability and finance.[1][2]

History

Louis Bachelier’s 1900 thesis in the Annales scientifiques de l’École Normale Supérieure modelled price changes with Brownian motion and anticipated later diffusion-based approaches.[3] A modern synthesis emerged with the Black–Scholes article in 1973, which connected dynamic hedging to a pricing partial differential equation and a closed-form solution.[4] 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.[5]

Mathematical foundations

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 Feynman–Kac.[6][7][8]

Representative models

A small set of models has shaped practice and pedagogy.

Models often cited in research and teaching
Model/framework Purpose in the literature Canonical source
Black–Scholes diffusion Baseline diffusion for dynamic hedging and valuation, yielding a pricing PDE and closed-form solutions for European options. Black, Fischer; Scholes, Myron (May–June 1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy (University of Chicago Press) 81 (3): 637–654. doi:10.1086/260062. https://www.journals.uchicago.edu/doi/10.1086/260062. Retrieved 13 September 2025. 
Stochastic volatility (Heston) Variance follows its own diffusion, producing volatility smiles/skews while retaining analytic tractability through characteristic functions. Heston, Steven L. (April 1993). "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". The Review of Financial Studies (Oxford University Press) 6 (2): 327–343. doi:10.1093/rfs/6.2.327. https://academic.oup.com/rfs/article/6/2/327/1574747. Retrieved 13 September 2025. 
Jump and Lévy models Discontinuities and heavy tails are modelled with jump-diffusions or pure-jump processes, improving fit to returns and option smiles. Cont, Rama (2003). Financial Modelling with Jump Processes. New York: Chapman & Hall/CRC. ISBN 1584884134. https://www.routledge.com/Financial-Modelling-with-Jump-Processes/Cont-Tankov/p/book/9781584884132. Retrieved 13 September 2025. 
Heath–Jarrow–Morton (HJM) rates Forward-rate dynamics are specified under no-arbitrage for the entire curve, unifying interest-rate derivative valuation. Heath, David; Jarrow, Robert; Morton, Andrew (January 1992). "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation". Econometrica (Wiley) 60 (1): 77–105. doi:10.2307/2951677. https://www.jstor.org/stable/2951677. Retrieved 13 September 2025. 

Themes and results

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.[5] Continuous-time portfolio choice and stochastic control supply consumption–investment results and verification through Hamilton–Jacobi–Bellman equations, while optimal stopping handles American-style exercise and related free-boundary problems.[7]

Journals and texts

Research commonly appears in Finance and Stochastics and Mathematical Finance; widely used books include Karatzas & Shreve’s graduate text on Brownian motion and stochastic calculus, Shreve’s two-volume sequence on continuous-time models, Baxter & Rennie’s introduction to derivative pricing, and Cont & Tankov’s monograph on jump processes.[1][2][6][9][8][10]

See also

References

  1. ↑ 1.0 1.1 "Aims and scope". Springer Nature. 2025. https://link.springer.com/journal/780/aims-and-scope. 
  2. ↑ 2.0 2.1 "Overview — Aims and scope". Wiley. 2025. doi:10.1111/(ISSN)1467-9965. https://onlinelibrary.wiley.com/page/journal/14679965/homepage/productinformation.html. 
  3. ↑ Bachelier, Louis (1900). "Théorie de la spéculation". Annales scientifiques de l'École Normale Supérieure (Gauthier-Villars) 17: 21–86. doi:10.24033/asens.476. https://www.numdam.org/item/ASENS_1900_3_17__21_0/. Retrieved 13 September 2025. 
  4. ↑ Black, Fischer; Scholes, Myron (May–June 1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy (University of Chicago Press) 81 (3): 637–654. doi:10.1086/260062. https://www.journals.uchicago.edu/doi/10.1086/260062. Retrieved 13 September 2025. 
  5. ↑ 5.0 5.1 Delbaen, Freddy; Schachermayer, Walter (September 1994). "A general version of the fundamental theorem of asset pricing". Mathematische Annalen (Springer) 300 (3): 463–520. doi:10.1007/BF01450498. https://link.springer.com/article/10.1007/BF01450498. Retrieved 13 September 2025. 
  6. ↑ 6.0 6.1 Karatzas, Ioannis (1991). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics. 113. New York: Springer. doi:10.1007/978-1-4612-0949-2. ISBN 9781461209492. https://link.springer.com/book/10.1007/978-1-4612-0949-2. Retrieved 13 September 2025. 
  7. ↑ 7.0 7.1 Shreve, Steven E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer. ISBN 0387401016. https://link.springer.com/book/9780387401010. Retrieved 13 September 2025. 
  8. ↑ 8.0 8.1 Baxter, Martin (1996). Financial Calculus: An Introduction to Derivative Pricing. Cambridge: Cambridge University Press. ISBN 0521552893. https://www.cambridge.org/core/books/financial-calculus/36E8CB4E4314F115464E4F95603376D8. Retrieved 13 September 2025. 
  9. ↑ Shreve, Steven E. (2004). Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer. ISBN 0387401016. https://link.springer.com/series/6863. Retrieved 13 September 2025. 
  10. ↑ Cont, Rama (2003). Financial Modelling with Jump Processes. New York: Chapman & Hall/CRC. ISBN 1584884134. https://www.routledge.com/Financial-Modelling-with-Jump-Processes/Cont-Tankov/p/book/9781584884132. Retrieved 13 September 2025. 



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