Editing Financial economics (section)

===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]].)