Editing Financial economics (section)

===Arbitrage-free pricing and equilibrium===
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|JEL classification codes
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|In the [[JEL classification codes|Journal of Economic Literature classification codes]], Financial Economics is one of the 19 primary classifications, at JEL: G. It follows [[monetary economics|Monetary]] and [[International economics|International Economics]] and precedes [[public economics|Public Economics]]. ''[[The New Palgrave Dictionary of Economics]]'' also uses the JEL codes to classify its entries. The primary and secondary JEL categories are:
:JEL: G – [[Financial Economics]] ([https://web.archive.org/web/20130529054128/http://www.dictionaryofeconomics.com/search_results?,q=&field=content&edition=all&topicid=G archived link])
:JEL: G0 – General
:JEL: G1 – [[Financial market|General Financial Markets]]
:JEL: G2 – [[Financial institution]]s and [[Financial services|Services]]
:JEL: G3 – [[Corporate finance]] and [[Corporate governance|Governance]]
Each is further divided into its tertiary categories.
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The concepts of [[arbitrage]]-free, "rational", pricing and equilibrium are then coupled 
<ref name="Varian">{{cite journal |title=The Arbitrage Principle in Financial Economics|first1=Hal R. |last1=Varian |author-link=Hal Varian|journal=Economic Perspectives |volume=1 |issue=2 |year=1987 |pages=55–72 |doi=10.1257/jep.1.2.55 |jstor=1942981| url=https://pubs.aeaweb.org/doi/pdfplus/10.1257/jep.1.2.55}}</ref>
with the above to derive various of the "classical"<ref name="Rubinstein2">See Rubinstein (2006), under "Bibliography".</ref>  (or [[Neoclassical economics|"neo-classical"]]<ref name="Derman"/>) financial economics models.

[[Rational pricing]] is the assumption that asset prices (and hence asset pricing models) will reflect the [[Arbitrage-free|arbitrage-free price]] of the asset, as any deviation from this price will be [[Rational_pricing#Arbitrage_mechanics|arbitraged away]]: the [[Rational_pricing#The_law_of_one_price|"law of one price"]]. This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.

[[Economic equilibrium]] is a state in which economic forces such as supply and demand are balanced, and in the absence of external influences these equilibrium values of economic variables will not change. [[General equilibrium theory|General equilibrium]] deals with the behavior of supply, demand, and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium. (This is in contrast to partial equilibrium, which only analyzes single markets.)

The two concepts are linked as follows: where market prices are [[complete market|complete]] and do not allow profitable arbitrage, i.e. they comprise an arbitrage-free market, then these prices are also said to constitute an "arbitrage equilibrium". Intuitively, this may be seen by considering that where an arbitrage opportunity does exist, then prices can be expected to change, and they are therefore not in equilibrium.<ref name="Delbaen_Schachermayer"/> An arbitrage equilibrium is thus a precondition for a general economic equilibrium.

"Complete" here means that there is a price for every asset in every possible state of the world, <math>s</math>, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming [[Frictionless market|no friction]]): essentially [[System of linear equations|solving simultaneously]] for ''n'' (risk-neutral) probabilities, <math>q_{s}</math>, given ''n'' prices. For a simplified example see {{section link|Rational pricing|Risk neutral valuation}}, where the economy has only two possible states – up and down – and where <math>q_{up}</math> and <math>q_{down}</math> ({{Nowrap|{{=}}<math>1-q_{up}</math>}}) are the two corresponding probabilities, and in turn, the derived distribution, or [[probability measure|"measure"]].

The formal derivation will proceed by arbitrage arguments.<ref name="Rubinstein"/><ref name="Delbaen_Schachermayer">Freddy Delbaen and Walter Schachermayer. (2004). [https://www.ams.org/notices/200405/what-is.pdf "What is... a Free Lunch?"] {{Webarchive|url=https://web.archive.org/web/20160304061252/http://www.ams.org/notices/200405/what-is.pdf |date=2016-03-04 }} (pdf). Notices of the AMS 51 (5): 526–528</ref><ref name="Varian"/> The analysis here is often undertaken to assume a ''[[representative agent]]'',<ref name="Farmer_Geanakoplos"/> essentially treating all market participants, "[[agent (economics)|agents]]", as identical (or, at least, assuming that they [[Heterogeneity in economics#Economic models with heterogeneous agents|act in such a way that]] the sum of their choices is equivalent to the decision of one individual) with the effect that [[Unreasonable ineffectiveness of mathematics#Economics and finance|the problems are then]] mathematically tractable.

With this measure in place, the expected, [[Required return|i.e. required]], return of any security (or portfolio) will then equal the risk-free return, plus an "adjustment for risk",<ref name="Rubinstein"/> i.e. a security-specific [[risk premium]], compensating for the extent to which its cashflows are unpredictable. All pricing models are then essentially variants of this, given specific assumptions or conditions.<ref name="Rubinstein"/><ref name="Cochrane & Culp"/><ref name="Backus"/> This approach is consistent with [[#Present value, expectation and utility|the above]], but with the expectation based on "the market" (i.e. arbitrage-free, and, per the theorem, therefore in equilibrium) as opposed to individual preferences.

Continuing the example, in pricing a [[derivative (finance)|derivative instrument]], its forecasted cashflows in the abovementioned up- and down-states <math>X_{up}</math> and <math>X_{down}</math>, are multiplied through by <math>q_{up}</math> and <math>q_{down}</math>, and are then [[present value|discounted]] at the risk-free interest rate; per the second equation above.  In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with <math>Y</math> and <math>r</math> combined. This premium may be derived by the [[Capital asset pricing model|CAPM]] (or extensions) as will be seen under {{slink|#Uncertainty}}.

The difference is explained as follows: By construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by "manufacturing" the instrument as a combination of the [[underlying]] and a risk free "bond"; see {{section link|Rational pricing|Delta hedging}} (and {{slink|#Uncertainty}} below). Where the underlying is itself being priced, such "manufacturing" is of course not possible – the instrument being "fundamental", i.e. as opposed to "derivative" – and a premium is then required for risk.

(Correspondingly, mathematical finance separates into [[Mathematical finance#History: Q versus P|two analytic regimes]]:
risk and portfolio management (generally) use [[physical measure|physical-]] (or actual or actuarial) probability, denoted by "P"; while derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q".
In specific applications the lower case is used, as in the above equations.)