Editing Financial economics (section)

==Underlying economics==
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|Fundamental valuation equation <ref name="Cochrane & Culp"/>
|-
|{{small|<math>Price_{j} =\sum_{s}(p_{s}Y_{s}X_{sj})/r</math>}}
::{{small|<math>=\sum_{s}(q_{s}X_{sj})/r</math>}}
::{{small|<math>=\sum_{s}p_{s}X_{sj}\tilde{m}_{s} = E[X_{s}\tilde{m}_{s}] </math>}}
::{{small|<math>=\sum_{s}\pi_{s} X_{sj}</math>}}
{{small|Four equivalent formulations,<ref name="Rubinstein"/> where:}}
:{{small|<math>j</math> is the asset or security}}
:{{small|<math>s</math> are the various states}}
:{{small|<math>r</math> is the risk-free return}}
:{{small|<math>X_{sj}</math> dollar payoffs in each state}}
:{{small|<math>p_{s}</math> a subjective, personal probability assigned to the state; <math display=inline> \sum_{s}p_{s}=1</math>}}
:{{small|<math>Y_{s}</math> risk aversion factors by state, normalized s.t. <math display=inline>\sum_{s}q_{s}=1</math>}}
:{{small|<math>q_{s}\equiv p_{s}Y_{s}</math>, risk neutral probabilities}}
:{{small|<math>\tilde{m}\equiv Y/r</math> the stochastic discount factor}}
:{{small|<math>\pi_{s}=q_{s}/r</math> state prices; <math display=inline>\sum_{s}\pi_{s} = 1/r</math>}}
|}
Financial economics studies how [[homo economicus|rational investors]] would apply [[decision theory]] to [[investment management]]. The subject is thus built on the foundations of [[microeconomics]] and derives several key results for the application of [[decision making]] under uncertainty to the [[financial market]]s. The underlying economic logic yields the [[fundamental theorem of asset pricing]], which gives the conditions for [[arbitrage]]-free asset pricing.<ref name="Rubinstein"/><ref name="Cochrane & Culp"/>
The various "fundamental" valuation formulae result directly.

===Present value, expectation and utility===
Underlying all of financial economics are the concepts of [[present value]] and [[Expected value|expectation]].<ref name="Rubinstein"/>

Calculating their present value, <math>X_{sj}/r</math> in the first formula, allows the decision maker to aggregate the [[cashflow]]s (or other returns) to be produced by the asset in the future to a single value at the date in question, and to thus more readily compare two opportunities; this concept is then the starting point for financial decision making.{{NoteTag|Its history is correspondingly early: [[Fibonacci]] developed the concept of present value already in 1202 in his ''[[Liber Abaci]]''. [[Compound interest]] was discussed in depth by [[Richard Witt]] in 1613, in his ''Arithmeticall Questions'',<ref>C. Lewin (1970). [https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf An early book on compound interest] {{Webarchive|url=https://web.archive.org/web/20161221163926/https://www.actuaries.org.uk/system/files/documents/pdf/0121-0132.pdf |date=2016-12-21 }}, Institute and Faculty of Actuaries</ref> and was further developed by [[Johan de Witt]] in 1671 <ref>James E. Ciecka. 2008. [https://fac.comtech.depaul.edu/jciecka/deWitt.pdf "The First Mathematically Correct Life Annuity"]. Journal of Legal Economics 15(1): pp. 59-63</ref> and by [[Edmond Halley]] in 1705.<ref>James E. Ciecka (2008). [https://fac.comtech.depaul.edu/jciecka/Halley.pdf "Edmond Halley’s Life Table and Its Uses"]. ''Journal of Legal Economics'' 15(1): pp. 65-74.</ref>}}
(Note that here, "<math>r</math>" represents a generic (or arbitrary) [[Discounted cash flow#Discount rate|discount rate]] applied to the cash flows, whereas in the valuation formulae, the [[risk-free rate]] is applied once these have been "adjusted" for their riskiness; see below.)

An immediate extension is to combine probabilities with present value, leading to the [[Expected value|expected value criterion]] which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, <math>X_{s}</math> and <math>p_{s}</math> respectively.{{NoteTag|These ideas originate with [[Blaise Pascal]] and [[Pierre de Fermat]] in 1654; see {{slink|Problem of points#Pascal and Fermat}}.}}

This decision method, however, fails to consider [[risk aversion]]. In other words, since individuals receive greater [[Utility#Applications|utility]] from an extra dollar when they are poor and less utility when comparatively rich, the approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly: <math>Y_{s}</math>. See [[indifference price]]. (Some investors may in fact be [[risk seeking]] as opposed to [[Risk aversion|risk averse]], but the same logic would apply.)

Choice under uncertainty here may then be defined as the maximization of [[expected utility]]. More formally, the resulting [[expected utility hypothesis]] states that, if certain axioms are satisfied, the [[Subjective theory of value|subjective]] value associated with a gamble by an individual is ''that individual''{{'}}s [[Expected value|statistical expectation]] of the valuations of the outcomes of that gamble.

The impetus for these ideas arises from various inconsistencies observed under the expected value framework, such as the [[St. Petersburg paradox]] and the [[Ellsberg paradox]].{{NoteTag|The development here is originally due to [[Daniel Bernoulli]] in 1738; it was later formalized by [[John von Neumann]] and [[Oskar Morgenstern]] in 1947.}}

===Arbitrage-free pricing and equilibrium===
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|JEL classification codes
|-
|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.
|}
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.)

===State prices===
With the above relationship established, the further specialized [[Arrow–Debreu model]] may be derived. 
{{NoteTag|State prices originate with [[Kenneth Arrow]] and [[Gérard Debreu]] in 1954.<ref>{{cite journal | last1 = Arrow | first1 = K. J. | last2 = Debreu | first2 = G. | year = 1954 | title = Existence of an equilibrium for a competitive economy | journal = Econometrica | volume = 22 | issue =3 | pages = 265–290 | doi = 10.2307/1907353 | jstor = 1907353 }}</ref>
[[Lionel W. McKenzie]] is also cited for his independent proof of equilibrium existence in 1954.<ref>{{cite journal |first=Lionel W. |last=McKenzie |title=On Equilibrium in Graham's Model of World Trade and Other Competitive Systems |journal=Econometrica |year=1954 |volume=22 |issue=2 |pages=147–161 |jstor=1907539 |doi=10.2307/1907539}}</ref> 
[[Douglas Breeden|Breeden]] and [[Robert Litzenberger|Litzenberger's]] work in 1978<ref>{{cite journal |title=Prices of State-Contingent Claims Implicit in Option Prices |first1=Douglas T. |last1=Breeden |author-link=Douglas Breeden|first2=Robert H. |last2=Litzenberger |author2-link=Robert Litzenberger |journal=[[Journal of Business]] |volume=51 |issue=4 |year=1978 |pages=621–651 |jstor=2352653 |doi=10.1086/296025|s2cid=153841737 }}</ref> established the use of state prices in financial economics.}}
This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. 
The Arrow–Debreu model applies to economies with maximally [[complete market]]s, in which there exists a market for every time period and forward prices for every commodity at all time periods.

A direct extension, then, is the concept of a [[state price]] security, also called an Arrow–Debreu security, a contract that agrees to pay one unit of a [[numeraire]] (a currency or a commodity) if a particular state occurs ("up" and "down" in the simplified example above) at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the ''state price'' <math>\pi_{s}</math> of this particular state of the world; the collection of these is also referred to as a "Risk Neutral Density".<ref name="Figlewski"/>

In the above example, the state prices, <math>\pi_{up}</math>,  <math>\pi_{down}</math>would equate to the present values of <math>$q_{up}</math> and <math>$q_{down}</math>: i.e. what one would pay today, respectively, for the up- and down-state securities; the [[state price vector]] is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be {{Nowrap|[<math>\pi_{up}</math>×<math>X_{up}</math> + <math>\pi_{down}</math>×<math>X_{down}</math>]}}: the fourth formula (see above regarding the absence of a risk premium here). For a [[continuous random variable]] indicating a continuum of possible states, the value is found by [[integration (mathematics)|integrating]] over the state price "density".

State prices find immediate application as a conceptual tool ("[[contingent claim analysis]]");<ref name="Rubinstein"/> but can also be applied to valuation problems.<ref name="corp fin state prices">See de Matos, as well as Bossaerts and Ødegaard, under bibliography.</ref> Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security"<ref name="Miller"/> – as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices.<ref name="Chance2"/><ref name="corp fin state prices"/> 
<ref name="Figlewski">{{cite journal | last1 =  Figlewski | first1 = Stephen | year = 2018 | title = Risk-Neutral Densities: A Review Annual Review of Financial Economics | journal = [[Annual Review of Financial Economics]] | volume = 10 | pages = 329–359| doi = 10.1146/annurev-financial-110217-022944 | ssrn = 3120028 | s2cid = 158075926 |url=https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3120028}}</ref> 
These recovered state-prices can then be used for valuation of other instruments with exposure to the underlyer, or for other decision making relating to the underlyer itself.

Using the related [[stochastic discount factor]] - SDF; also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor <math>\tilde{m}</math>, and then taking the expectation;<ref name="Backus">See: [[David K. Backus]] (2015). [http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf Fundamentals of Asset Pricing], Stern NYU</ref><ref>Lars Peter Hansen & Eric Renault (2020). [https://larspeterhansen.org/wp-content/uploads/2016/10/Pricing-Kernels-and-Stochastic-Discount-Factors.pdf "Pricing Kernels"] in: ''Encyclopedia of Quantitative Finance''. {{ISBN|0470057564}}</ref> the third equation above. Essentially, this factor divides expected [[Utility#Expected_utility|utility]] at the relevant future period - a function of the possible asset values realized under each state - by the utility due to today's wealth, and is then also referred to as "the intertemporal [[marginal rate of substitution]]".
Correspondingly, the SDF, <math>\tilde{m}_{s}</math>, may be thought of as the discounted value of Risk Aversion, <math>Y_{s}.</math>
(The latter may be inferred via the ratio of risk neutral- to physical-probabilities, <math>q_{s} / p_{s}.</math>
See [[Girsanov theorem]] and [[Radon-Nikodym derivative]].)