Editing Finance (section)

===Financial mathematics===
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:<math>\begin{align}
  C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\
     d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
     d_2 &= d_1 - \sigma\sqrt{T - t} \\
\end{align}</math>
[[Black–Scholes model#Black–Scholes formula|The Black–Scholes formula]] for the value of a [[call option]]. Although lately its use is [[Financial economics#Departures from normality|considered naive]], it has underpinned the development of derivatives-theory, and financial mathematics more generally, since its introduction in 1973.<ref>[https://priceonomics.com/the-history-of-the-black-scholes-formula/ "The History of the Black-Scholes Formula"] {{Webarchive|url=https://web.archive.org/web/20211126125626/https://priceonomics.com/the-history-of-the-black-scholes-formula/ |date=2021-11-26 }}, priceonomics.com</ref>}}
|}
[[File:OAS valuation tree (es).png|thumb|[[Lattice model (finance)|"Trees"]] are widely applied in mathematical finance; here used in calculating an [[option adjusted spread|OAS]]. Other common pricing-methods are [[Monte Carlo methods in finance#Overview|simulation]] and [[Finite difference methods for option pricing|PDEs]]. These are used for settings beyond [[Black–Scholes model#Fundamental hypotheses|those envisaged]] by Black-Scholes.
[[Valuation of options#Post crisis|Post crisis]], even in those settings, banks use [[local volatility|local]] and [[stochastic volatility]] models to incorporate the [[volatility surface]], while the [[xVA]] adjustments accommodate [[counterparty credit risk|counterparty]] and capital considerations.]]
{{Main|Mathematical finance}}
{{see also|Quantitative analysis (finance)|Financial modeling#Quantitative finance}}
Financial mathematics<ref name="SIAM2">[https://www.siam.org/research-areas/detail/financial-mathematics-and-engineering# Research Area: Financial Mathematics and Engineering] {{Webarchive|url=https://web.archive.org/web/20220516114958/https://www.siam.org/research-areas/detail/financial-mathematics-and-engineering |date=2022-05-16 }}, Society for Industrial and Applied Mathematics</ref> is the field of [[applied mathematics]] concerned with [[financial market]]s;
[[Louis Bachelier#The doctoral thesis|Louis Bachelier's doctoral thesis]], defended in 1900, is considered to be the first scholarly work in this area. The field is largely focused on the [[Outline of finance#Derivatives pricing|modeling of derivatives]]—with much emphasis on [[Interest rate derivative|interest rate-]] and [[Credit derivative|credit risk modeling]]—while other important areas include [[actuarial science|insurance mathematics]] and [[Outline of finance#Mathematical techniques|quantitative portfolio management]]. Relatedly, the techniques developed [[contingent claim analysis|are applied]] to pricing and hedging a wide range of [[Asset-backed security|asset-backed]], [[Government bond|government]], and [[Capital structure|corporate]]-securities.

As [[#Quantitative finance|above]], in terms of practice, the field is referred to as quantitative finance and / or mathematical finance, and comprises primarily the three areas discussed.
The [[Outline of finance#Mathematical tools|main mathematical tools]] and techniques are, correspondingly:
* for derivatives,<ref name="Mastro">For a survey, see [https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf "Financial Models"] {{Webarchive|url=https://web.archive.org/web/20211113134700/https://catalogimages.wiley.com/images/db/pdf/9781118487716.excerpt.pdf |date=2021-11-13 }}, from Michael Mastro (2013). ''Financial Derivative and Energy Market Valuation'', John Wiley & Sons. {{ISBN| 978-1-118-48771-6}}.</ref> [[Itô calculus|Itô's stochastic calculus]], [[Monte Carlo methods in finance|simulation]], and [[partial differential equation]]s; see aside boxed discussion re the prototypical [[Black-Scholes model]] and [[Valuation of options#Pricing models|the various numeric techniques]] now applied
* for risk management,<ref name="DeMeo">See generally, Roy E. DeMeo (N.D.) [https://mathfinance.charlotte.edu/sites/mathfinance.charlotte.edu/files/media/An%20Introduction%20to%20Value%20At%20Risk%20New.pdf Quantitative Risk Management: VaR and Others] {{Webarchive|url=https://web.archive.org/web/20211112082350/https://mathfinance.charlotte.edu/sites/mathfinance.charlotte.edu/files/media/An%20Introduction%20to%20Value%20At%20Risk%20New.pdf |date=2021-11-12 }}</ref> [[value at risk]], [[stress test (financial)|stress testing]] and [[PnL Explained#Sensitivities method|"sensitivities" analysis]] (applying the "greeks"); the underlying mathematics comprises [[Mixture model#A financial model|mixture models]], [[Principal component analysis#Quantitative finance|PCA]], [[volatility clustering]] and [[Copula (probability theory)#Quantitative finance|copulas]].<ref>See for example III.A.3, in Carol Alexander, ed. (2005). ''The Professional Risk Managers' Handbook''. PRMIA Publications. {{ISBN|978-0-9766097-0-4}}</ref>
* in both of these areas, and particularly for portfolio problems, quants employ [[Outline of finance#Mathematical techniques|sophisticated optimization techniques]]

Mathematically, these separate into [[Mathematical finance#History: Q versus P|two analytic branches]]: derivatives pricing uses [[Risk-neutral measure|risk-neutral probability]] (or [[rational pricing|arbitrage-pricing]] probability), denoted by "Q"; while risk and portfolio management generally use physical (or actual or actuarial) probability, denoted by "P". These are interrelated through the above "[[Fundamental theorem of asset pricing]]".

The subject has a close relationship with financial economics, which, as outlined, is concerned with much of the underlying theory that is involved in financial mathematics:  generally, financial mathematics will derive and extend the [[mathematical model]]s suggested. [[Computational finance]] is the branch of (applied) [[computer science]] that deals with problems of practical interest in finance, and especially<ref name="SIAM2"/> emphasizes the [[numerical methods]] applied here.