Editing Financial economics (section)

===Departures from normality===
{{See also|Capital asset pricing model#Problems|Black–Scholes model#Criticism and comments}}
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{| class="wikitable floatright" | width="250"
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|{{smalldiv|
:<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"], priceonomics.com</ref>}}
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[[Image:Ivsrf.gif|thumb|right|Implied volatility surface. The Z-axis represents implied volatility in percent, and X and Y axes represent the [[Greeks (finance)#Delta|option delta]], and the days to maturity.]]

As discussed, the assumptions that market prices follow a [[random walk]] and that asset returns are normally distributed are fundamental. Empirical evidence, however, suggests that these assumptions may not hold, and that in practice, traders, analysts [[Financial risk management#Banking|and risk managers]] frequently modify the "standard models" (see [[kurtosis risk]], [[skewness risk]], [[long tail]], [[model risk]]). 
In fact, [[Benoit Mandelbrot]] had discovered already in the 1960s<ref>{{cite journal |title=The Variation of Certain Speculative Prices |first= Benoit |last=Mandelbrot|author-link1=Benoit Mandelbrot |journal=[[The Journal of Business]] |volume=36|issue=Oct|year=1963|pages=394–419|doi= 10.1086/294632 |url =http://web.williams.edu/Mathematics/sjmiller/public_html/341Fa09/econ/Mandelbroit_VariationCertainSpeculativePrices.pdf}}</ref> 
that changes in financial prices do not follow a [[normal distribution]], the basis for much option pricing theory, although this observation was slow to find its way into mainstream financial economics.
<ref name="Taleb_Mandelbrot">{{cite web |url=http://www.fooledbyrandomness.com/fortune.pdf |title=How the Finance Gurus Get Risk All Wrong |access-date=2010-06-15 |url-status=dead |archive-url=https://web.archive.org/web/20101207045925/http://www.fooledbyrandomness.com/fortune.pdf |archive-date=2010-12-07|author = Nassim Taleb and Benoit Mandelbrot}}</ref>

[[Financial models with long-tailed distributions and volatility clustering]] have been introduced to overcome problems with the realism of the above "classical" financial models; while [[Jump diffusion#In economics and finance|jump diffusion models]] allow for (option) pricing incorporating [[jump process|"jumps"]] in the [[spot price]].<ref name="holes">{{cite journal |title=How to use the holes in Black–Scholes |first= Fischer |last=Black|author-link1=Fischer Black |journal=[[Journal of Applied Corporate Finance]] |volume=1|issue=Jan|year=1989|pages=67–73|doi=10.1111/j.1745-6622.1989.tb00175.x}}</ref> 
Risk managers, similarly, complement (or substitute) the standard [[value at risk]] models with [[Historical simulation (finance)|historical simulations]], [[Mixture model#A financial model|mixture models]], [[principal component analysis]], [[extreme value theory]], as well as models for [[volatility clustering]].<ref>See for example III.A.3, in Carol Alexander, ed. (January 2005). ''The Professional Risk Managers' Handbook''. PRMIA Publications. {{ISBN|978-0976609704}}</ref> 
For further discussion see {{section link|Fat-tailed distribution|Applications in economics}}, and {{section link|Value at risk|Criticism}}. Portfolio managers, likewise, have modified their optimization criteria and algorithms; see {{slink|#Portfolio theory}} above.

Closely related is the [[volatility smile]], where, as above, [[implied volatility]] – the volatility corresponding to the BSM price – is observed to ''differ'' as a function of [[strike price]] (i.e. [[moneyness]]), true only if the price-change distribution is non-normal, unlike that assumed by BSM (i.e. <math>N(d_1)</math> and <math>N(d_2)</math> above). The term structure of volatility describes how (implied) volatility differs for related options with different maturities. An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure. These empirical phenomena negate the assumption of constant volatility – and [[log-normal]]ity – upon which Black–Scholes is built.<ref name="Haug Taleb"/><ref name="holes"/>  
Within institutions, the function of Black–Scholes is now, largely, to ''communicate'' prices via implied volatilities, much like bond prices are communicated via [[yield to maturity|YTM]]; see {{section link|Black–Scholes model|The volatility smile}}.

In consequence traders ([[Financial risk management#Banking|and risk managers]]) now, instead, use "smile-consistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. non-quoted, strike/maturity combinations, or of non-European derivatives, and generally for hedging purposes.  
The two main approaches are [[local volatility]] and [[stochastic volatility]]. The first returns the volatility which is "local" to each spot-time point of the [[Finite difference methods for option pricing|finite difference-]] or [[Monte Carlo methods for option pricing|simulation-based valuation]]; i.e. as opposed to implied volatility, which holds overall.  In this way calculated prices – and numeric structures – are market-consistent in an arbitrage-free sense.  The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first [[Stochastic volatility#Calibration and estimation|calibrated to observed prices]], and are then applied to the valuation or hedging in question; the most common are [[Heston model|Heston]], [[SABR volatility model|SABR]] and [[Constant elasticity of variance model|CEV]]. This approach addresses certain problems identified with hedging under local volatility.<ref>{{cite journal |title=  Managing smile risk |first= Patrick |last=Hagan |display-authors=etal|journal=[[Wilmott Magazine]] |issue=Sep|year=2002 |pages=84–108}}</ref>

Related to local volatility are the [[Lattice model (finance)|lattice]]-based [[Implied binomial tree|implied-binomial]] and [[Implied trinomial tree|-trinomial trees]] – essentially a discretization of the approach – which are similarly, but less commonly,<ref name="Figlewski"/> used for pricing; these are built on state-prices recovered from the surface. [[Edgeworth binomial tree]]s allow for a specified (i.e. non-Gaussian) [[Skewness|skew]] and [[kurtosis]] in the spot price; priced here, options with differing strikes will return differing implied volatilities, and the tree can be calibrated to the smile as required.<ref>See for example Pg 217 of: Jackson, Mary; Mike Staunton (2001). ''Advanced modelling in finance using Excel and VBA''. New Jersey: Wiley. {{ISBN|0-471-49922-6}}.</ref> 
Similarly purposed (and derived) [[Closed-form expression|closed-form models]] were also developed.
<ref>These include: [[Robert A. Jarrow|Jarrow]] and Rudd (1982); Corrado and Su (1996); Brown and Robinson (2002); [[David K. Backus|Backus]], Foresi, and Wu (2004). 
See, e.g.: E. Jurczenko, B. Maillet, and B. Negrea (2002). [http://eprints.lse.ac.uk/24950/1/dp430.pdf "Revisited multi-moment approximate option pricing models: a general comparison (Part 1)"]. Working paper, [[London School of Economics and Political Science]].</ref>

As discussed, additional to assuming log-normality in returns, "classical" BSM-type models also (implicitly) assume the existence of a credit-risk-free environment, where one can perfectly replicate cashflows so as to fully hedge, and then discount at "the" risk-free-rate.  
And therefore, post crisis, the various x-value adjustments must be employed, effectively correcting the risk-neutral value for [[counterparty credit risk|counterparty-]] and [[XVA#Valuation adjustments|funding-related]] risk.
These xVA are ''additional'' to any smile or surface effect: with the surface built on price data for fully-collateralized positions, there is therefore no "[[double counting (accounting)|double counting]]" of credit risk (etc.) when appending xVA. (Were this not the case, then each counterparty would have its own surface...)

As mentioned at top, mathematical finance (and particularly [[financial engineering]]) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light.  Recognizing this, critics of financial economics - especially vocal since the [[2008 financial crisis]] - suggest that instead, the theory needs revisiting almost entirely:
{{NoteTag|
This quote, from banker and author [[James Rickards]], is representative. 
Prominent and earlier criticism<ref name="Taleb_Mandelbrot"/> is from [[Benoit Mandelbrot]], [[Emanuel Derman]], [[Paul Wilmott]], [[Nassim Taleb]], [[Financial engineering#Criticisms|and others]].
Well known popularizations include Taleb's ''[[Fooled by Randomness]]'' and [[The Black Swan: The Impact of the Highly Improbable|''The Black Swan'']], Mandelbrot's [[Benoit Mandelbrot#Bibliography|''The Misbehavior of Markets'']], and Derman's [[Emanuel Derman#Models.Behaving.Badly|''Models.Behaving.Badly'']] and, with Wimott, the ''[[Financial Modelers' Manifesto]]''.
}}
{{Blockquote|The  current  system,  based  on  the  idea  that  risk  is  distributed  in  the  shape  of  a  bell  curve,  is  flawed...  The  problem  is  [that economists and practitioners]  never  abandon  the  bell  curve.  They  are  like  medieval  astronomers  who believe the sun revolves around the earth and are [[Geocentric model#Ptolemaic system|furiously tweaking their geo-centric  math]]  in  the  face  of  contrary  evidence.  They  will  never  get  this  right; [[Copernican Revolution|they need their Copernicus]].<ref>[https://www.govinfo.gov/content/pkg/CHRG-111hhrg51925/pdf/CHRG-111hhrg51925.pdf ''The Risks of  Financial  Modeling: VAR  and  the  Economic  Meltdown''], Hearing before the [[United States House Science Subcommittee on Investigations and Oversight|Subcommittee  on  Investigations  and  Oversight]], [[United States House Committee on Science, Space, and Technology|Committee on Science and Technology]], [[United States House of Representatives|House of Representatives]], One  Hundred  Eleventh  Congress, first  session, September 10, 2009</ref>}}