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{{Short description|Academic discipline concerned with the exchange of money}}
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'''Financial economics''' is the branch of [[economics]] characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on ''both sides'' of a trade".
<ref name="stanford1">[[William F. Sharpe]], [http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm "Financial Economics"] {{Webarchive|url=https://web.archive.org/web/20040604105441/http://www.stanford.edu/~wfsharpe/mia/int/mia_int2.htm |date=2004-06-04 }}, in {{cite web |url=https://web.stanford.edu/~wfsharpe/mia/MIA.HTM |title=''Macro-Investment Analysis'' |publisher=Stanford University (manuscript) |access-date=2009-08-06 |archive-url=https://web.archive.org/web/20140714034144/https://web.stanford.edu/~wfsharpe/mia/mia.htm |archive-date=2014-07-14 |url-status=live }} (Own emphasis.)</ref> 
Its concern is thus the interrelation of financial variables, such as [[share price]]s, [[interest rate]]s and [[exchange rate]]s, as opposed to those concerning the [[real economy]]. 
It has two main areas of focus:<ref name="Miller">[[Merton H. Miller]], (1999). The History of Finance: An Eyewitness Account, ''Journal of Portfolio Management''. Summer 1999.</ref> [[asset pricing]] and [[corporate finance]]; the first being the perspective of providers of [[Financial capital|capital]], i.e. investors, and the second of users of capital.
It thus provides the theoretical underpinning for much of [[finance]].

The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".<ref>[[Robert C. Merton]] {{cite web |url=http://nobelprize.org/nobel_prizes/economics/laureates/1997/merton-lecture.pdf |title=Nobel Lecture |access-date=2009-08-06 |archive-url=https://web.archive.org/web/20090319202149/http://nobelprize.org/nobel_prizes/economics/laureates/1997/merton-lecture.pdf |archive-date=2009-03-19 |url-status=live }}</ref><ref name="Fama and Miller">See Fama and Miller (1972), ''The Theory of Finance'', in Bibliography.</ref> It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant [[economic model|economic]] and [[financial model]]s and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. 
It thus also includes a formal study of the [[financial market]]s themselves, especially [[market microstructure]] and [[financial regulation|market regulation]].
It is built on the foundations of [[microeconomics]] and [[decision theory]].

[[Financial econometrics]] is the branch of financial economics that uses [[Econometrics|econometric]] techniques to parameterise the relationships identified.
[[Mathematical finance]] is related in that it will derive and extend the mathematical or numerical models suggested by financial economics.
Whereas financial economics has a primarily microeconomic focus, [[monetary economics]] is primarily [[macroeconomic]] in nature.

==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, [[Problem of points#Pascal and Fermat|in discussing]] the "[[Problem of points]]".}}

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=  Journal of 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 [[fundamental theorem of asset pricing]] formalizes this. A market with "[[physical measure|physical]]" probabilities, denoted by "P": 
is (i) considered arbitrage-free if, and only if, there exists at least one risk neutral probability measure "Q" that is [[Equivalence (measure theory)|
equivalent]] to P; 
and (ii) is considered complete, if Q is unique ''and'' there is a risk-free bond which may act as the [[numeraire]], or pricing reference.
<ref>Walter Schachermayer (2004). [https://www.mat.univie.ac.at/~schachermayer/pubs/preprnts/prpr0118a.pdf  "The Notion of Arbitrage and Free Lunch in Mathematical Finance"].</ref>

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 {{section link|#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 {{section link|#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.

Although mathematical finance separates into [[Mathematical finance#History: Q versus P|two analytic regimes]], these - per the fundamental theorem - are at base consistent.
Thus, risk and portfolio management generally use P, physical (or actual or actuarial) probabilities, while derivatives pricing uses Q, risk-neutral  (or arbitrage-pricing) probabilities. (In specific applications the lower case symbols p and q are used, as in the above equations.) But here, [[quantitative analyst|"quants"]], in fact, construct the latter from the former (as outlined); and this as a modelling "convenience", allowing them to operate ''as if'' risk-neutral and to discount expected payoffs at the risk-free rate.

===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]].)

==Resultant models==
Applying the above economic concepts, we may then derive various [[economic model|economic-]] and financial models and principles. As outlined, the two areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information",<ref name="stanford1"/><ref name="Farmer_Geanakoplos"/> as will be seen below.
* Time: money now is traded for money in the future.
* Uncertainty (or risk): The amount of money to be transferred in the future is uncertain.
* [[Option (finance)|Options]]: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.
* [[Perfect information|Information]]: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with [[Future value|future monetary value]] (FMV).
Applying this framework, with the above concepts, leads to the required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations.<ref name="Fama and Miller"/> (This division sometimes denoted "[[deterministic]]" and "random",<ref name="Luenberger"/> or "[[stochastic]]".)

===Certainty===
{| class="wikitable floatright" | width="250"
|- style="text-align:left;"
|{{smalldiv|
:<math> \sum_{t=1}^n\frac{C}{(1+i)^t} + \frac{F}{(1+i)^n} </math>
{{small|[[Bond valuation#Present value approach|Bond valuation formula]] where Coupons and Face value are discounted at the appropriate rate, "i": typically reflecting a spread over the risk free rate [[Bond valuation#Relative price approach|as a function of credit risk]]; often quoted as a "[[yield to maturity]]". See body for discussion re the relationship with the above pricing formulae.}} 
|}
{| class="wikitable floatright" | width="250"
|- style="text-align:left;"
|{{smalldiv|
:<math>
\sum_{t=1}^n \frac{FCFF_t}{(1+WACC_{t})^t} +
 \frac{\left[\frac{FCFF_{n+1}}{(WACC_{n+1}-g_{n+1})}\right]}{(1+WACC_{n})^n}
</math>
}}
{{small|[[Valuation using discounted cash flows#Basic formula for firm valuation using DCF model|DCF valuation formula]], where the [[business valuation|value of the firm]], is its forecasted [[free cash flow]]s discounted to the present using the [[weighted average cost of capital]], i.e. [[cost of equity]] and [[cost of debt]], with the former (often) derived using the below CAPM. The result here is the [[net present value]] of the investment opportunity.
For [[stock valuation|share valuation]] investors use the related [[dividend discount model]]. }}
|}
[[Image:MM2.png|thumb|right|Modigliani–Miller Proposition II with risky debt. Even if [[leverage (finance)|leverage]] ([[Debt to equity ratio|D/E]]) increases, the [[weighted average cost of capital|WACC]] (k0) stays constant.]]
The starting point here is "Investment under certainty", and usually framed in the context of a corporation.
The [[Fisher separation theorem]], asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders. 
Related is the [[Modigliani–Miller theorem]], which shows that, under certain conditions, the value of a firm is unaffected by [[capital structure|how that firm is financed]], and depends neither on its [[dividend policy]] nor [[Corporate finance#Related considerations|its decision]] to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as a benchmark<ref name="Varian"/> for evaluating the effects of factors outside the model that do affect value.
{{NoteTag|The theorem of [[Franco Modigliani]] and [[Merton Miller]] is often called the "capital structure irrelevance principle"; it is presented in two key papers 
of 1958,<ref name="MM1">{{cite journal | last = Modigliani | first = F. |author2=Miller, M. | year = 1958 | title = The Cost of Capital, Corporation Finance and the Theory of Investment | journal = American Economic Review | volume = 48 | issue = 3 | pages = 261–297 | jstor = 1809766 }}</ref>
and 1963.<ref name="MM2">{{cite journal | last = Modigliani | first = F. |author2=Miller, M. | year = 1963 | title = Corporate income taxes and the cost of capital: a correction | journal = American Economic Review | volume = 53 | issue = 3 | pages = 433–443 | jstor = 1809167 }}</ref>}}

The mechanism for determining (corporate) value is provided by<ref name="New School">{{cite web |url=http://cepa.newschool.edu/het/schools/finance.htm |title=Finance Theory|author=[[The New School]]|accessdate=2006-06-28 |url-status=dead |archiveurl=https://web.archive.org/web/20060702212228/http://cepa.newschool.edu/het/schools/finance.htm |archivedate=2006-07-02 }}</ref><ref name="Rubinstein_2">{{cite web|url=http://www.in-the-money.com/artandpap/I%20Present%20Value.doc |title=Great Moments in Financial Economics: I. Present Value|accessdate=2007-06-28 |url-status=dead |archiveurl=https://web.archive.org/web/20070713043745/http://www.in-the-money.com/artandpap/I%20Present%20Value.doc |archivedate=2007-07-13|date=2002|author=[[Mark Rubinstein]] }}</ref> 
[[John Burr Williams]]' ''[[The Theory of Investment Value]]'', which proposes that the value of an asset should be calculated using "evaluation by the rule of present worth". Thus, for a common stock, the [[Intrinsic value (finance)#Equity|"intrinsic"]], long-term worth is the present value of its future net cashflows, in the form of [[dividend]]s; in [[Corporate finance#Investment and project valuation|the corporate context]], "[[free cash flow]]" as aside.  What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. [[Net present value]] (NPV), the result of this [[Valuation using discounted cash flows|discounted cashflow valuation]], is the direct extension of these ideas typically applied to Corporate Finance decisioning. For other results, as well as specific models developed here, see the list of "Equity valuation" topics under {{section link|Outline of finance|Discounted cash flow valuation}}.
{{NoteTag|[[John Burr Williams]] published his "Theory" in 1938; NPV was recommended to corporate managers by [[Joel Dean (economist)|Joel Dean]] in 1951.}}

[[Bond valuation]], in that cashflows ([[Coupon (finance)|coupons]] and return of principal, or "[[Face value]]") are deterministic, may proceed in the same fashion.<ref name="Luenberger">See Luenberger's ''Investment Science'', under Bibliography.</ref> An immediate extension, [[Bond valuation#Arbitrage-free pricing approach|Arbitrage-free bond pricing]], discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding [[zero rate]], and of equivalent credit worthiness – as opposed to an overall rate.
In many treatments bond valuation precedes [[equity valuation]], under which cashflows (dividends) are not "known" ''per se''. Williams and onward allow for forecasting as to these – based on [[Dividend payout ratio|historic ratios]] or published [[dividend policy]] – and cashflows are then treated as essentially deterministic; see below under {{section link|#Corporate finance theory}}.

For both stocks and bonds, "under certainty, with the focus on cash flows from securities over time," valuation based on a [[yield curve|term structure of interest rates]] is in fact consistent with arbitrage-free pricing.<ref>See footnote 3 under Rubinstein (2005). "The Fundamental Theorem (Part I)", refenced below.</ref>
Indeed, a corollary of [[#Arbitrage-free_pricing_and_equilibrium|the above]] is that "[[Rational pricing#The law of one price|the law of one price]] implies the existence of a discount factor".<ref>§ 4.1 "Law of one price and existence of a discount factor" in Cochrane (2005).</ref>
Corresponding to these: {{Nowrap|<math display=inline>\sum_{s}\pi_{s} = 1/r</math>}}; and equivalently, for the above "numeraire bond", {{Nowrap|<math display=inline>B_0=\sum_{s}\pi_{s}</math>}}. In practice, the latter is interchangeable with the risk-free [[yield curve]]: this ensures consistency between the theoretical no-arbitrage framework and applied valuation, providing a common benchmark for discounting future cash flows.

Whereas these "certainty" results are all commonly employed under corporate finance, uncertainty is the focus of "asset pricing models" as follows. [[Irving Fisher#Interest and capital|Fisher's formulation]] of the theory here - developing [[Intertemporal choice#Fisher's model of intertemporal consumption|an intertemporal equilibrium model]] - underpins also<ref name="New School"/> the below applications to uncertainty; 
{{NoteTag|In fact, "Fisher (1930, [The Theory of Interest]) is the seminal work for most of the financial theory of investments during the twentieth century… Fisher develops the first formal equilibrium model of an economy with both intertemporal exchange and production. In so doing, at one swoop, he not only derives present value calculations as a natural economic outcome in calculating wealth, he also justifies the maximization of present value as the goal of production and derives determinants of the interest rates that are used to calculate present value."<ref name="Rubinstein2"/>{{rp|55}}}}
see<ref>Gonçalo L. Fonseca (N.D.). [https://web.archive.org/web/20080429203224/http://cepa.newschool.edu/het/essays/capital/fisherinvest.htm Irving Fisher's Theory of Investment]. ''History of Economic Thought'' series, [[The New School]].</ref> for the development.

===Uncertainty===
[[Image:markowitz frontier.jpg|thumb|right|Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and its upward sloped portion is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier. The graphic displays the CAL, [[Capital allocation line]], formed when the risky asset is a single-asset rather than the market, in which case the line is the CML.]]
[[Image:CML-plot.png|thumb|right|The [[Capital market line]] is the tangent line drawn from the point of the risk-free asset to the [[feasible region]] for risky assets. The tangency point M represents the [[market portfolio]]. The CML results from the combination of the market portfolio and the risk-free asset (the point L). Addition of leverage (the point R) creates levered portfolios that are also on the CML.]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|{{smalldiv|1=The capital asset pricing model (CAPM):
:<math>E(R_i) = R_f + \beta_{i}(E(R_m) - R_f)</math>}}
{{small|The [[required rate of return|expected return]] used when discounting cashflows on an asset <math>i</math>, is the risk-free rate plus the [[Capital asset pricing model#Formula|market premium]] multiplied by [[Beta (finance)|beta]] {{Nowrap|(<math>\rho_{i,m} \frac {\sigma_{i}}{\sigma_{m}}</math>)}}, the asset's correlated volatility relative to the overall market <math>m</math>.}}
|}
[[Image:SML-chart.png|thumb|right|[[Security market line]]: the representation of the CAPM displaying the expected rate of return of an individual security as a function of its systematic, non-diversifiable risk.]]
For [[Decision theory#Choice under uncertainty|"choice under uncertainty"]] the twin assumptions of rationality and [[Financial market efficiency|market efficiency]], as more closely defined, lead to [[modern portfolio theory]] (MPT) with its [[capital asset pricing model]] (CAPM) – an ''equilibrium-based'' result – and to the [[Black–Scholes model|Black–Scholes–Merton theory]] (BSM; often, simply Black–Scholes) for [[Valuation of options|option pricing]] – an ''arbitrage-free'' result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see {{section link|Asset pricing|Interrelationship}}.

Briefly, and intuitively – and consistent with {{section link|#Arbitrage-free pricing and equilibrium}} above – the relationship between rationality and efficiency is as follows.<ref>For a more formal treatment, see, for example: Eugene F. Fama. (1965). [http://www.cfapubs.org/toc/faj/1965/21/5 "Random Walks in Stock Market Prices"]. ''[[Financial Analysts Journal]]'', September/October 1965, Vol. 21, No. 5: 55–59.</ref> 
Given the ability to profit from [[privacy|private information]], self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. ''efficient'', prices: the [[efficient-market hypothesis]], or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See [[earnings response coefficient]].)
The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information  are identical to the ''best guess of the future'': the assumption of [[rational expectations]]. 
The EMH does allow that when faced with new information, some investors may overreact and some may underreact,<ref name="affirmative case">[[Mark Rubinstein]] (2001). [https://escholarship.org/content/qt22q318mh/qt22q318mh_noSplash_0a514e78eb5c8e16a9ead63a4f4a628e.pdf?t=krnd5a "Rational Markets: Yes or No? The Affirmative Case"]. ''[[Financial Analysts Journal]]'', May - Jun., 2001, Vol. 57, No. 3: 15-29</ref>
but what is required, however, is that investors' reactions follow a [[normal distribution]] – so that the net effect on market prices cannot be reliably exploited<ref name="affirmative case"/> to make an abnormal profit.
In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news:<ref name="Shiller"/> the [[random walk hypothesis]]. 
This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution. 
{{NoteTag|The EMH was presented by [[Eugene Fama]] in a 1970 [[review paper]],<ref>Fama, Eugene (1970). [https://www.jstor.org/stable/2325486 "Efficient Capital Markets: A Review of Theory and Empirical Work"]. ''[[Journal of Finance]]''. Vol. 25, No. 2.</ref> consolidating previous works re random walks in stock prices: [[Jules Regnault]] (1863); [[Louis Bachelier]] (1900); [[Maurice Kendall]] (1953); [[Paul Cootner]] (1964); and [[Paul Samuelson]] (1965), among others.}}

Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow;<ref name="affirmative case"/> correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium.
Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends,<ref name="Cochrane & Culp">Christopher L. Culp and [[John H. Cochrane]]. (2003). "[http://faculty.chicagobooth.edu/john.cochrane/research/Papers/cochrane-culp%20asset%20pricing.pdf "Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management"] {{Webarchive|url=https://web.archive.org/web/20160304190225/http://faculty.chicagobooth.edu/john.cochrane/research/Papers/cochrane-culp%20asset%20pricing.pdf |date=2016-03-04 }}, in ''Modern Risk Management: A History''. Peter Field, ed. London: Risk Books, 2003. {{ISBN|1904339050}}</ref><ref name="Shiller">{{cite journal|last= Shiller|first= Robert J.|author-link= Robert J. Shiller|date= 2003|title= From Efficient Markets Theory to Behavioral Finance|journal= [[Journal of Economic Perspectives]]|volume= 17|issue= 1 (Winter 2003)|pages= 83–104|url= http://www.econ.yale.edu/~shiller/pubs/p1055.pdf|doi= 10.1257/089533003321164967|archive-url= https://web.archive.org/web/20150412081613/http://www.econ.yale.edu/~shiller/pubs/p1055.pdf|archive-date= 2015-04-12|url-status= live|doi-access= free}}</ref><ref name="Farmer_Geanakoplos"/> 
as based on currently available information.
What is required though, is a theory for determining the appropriate discount rate, i.e. "required return", given this uncertainty: this is provided by the MPT and its CAPM. Relatedly, rationality – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM.

In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are.
{{NoteTag|The efficient frontier was introduced by [[Harry Markowitz]] in 1952. The CAPM was derived by [[Jack L. Treynor|Jack Treynor]] (1961, 1962), [[William F. Sharpe]] (1964), [[John Lintner]] (1965), and [[Jan Mossin]] (1966) independently. Already in 1940, [[Bruno de Finetti]]<ref>de Finetti, B. (1940): ''Il problema dei “Pieni”''. Giornale dell’ Istituto Italiano degli Attuari 11, 1–88; translation (Barone, L. (2006)): "The problem of full-risk insurances". Chapter I. The risk within a single accounting period. ''[[Journal of Investment Management]]'' 4(3), 19–43.</ref> had described the mean-variance method, in the context of [[reinsurance]].}}
This result will be independent of the investor's level of risk aversion and assumed [[utility function]], thus providing a readily determined discount rate for corporate finance decision makers [[#Certainty|as above]],<ref name="Jensen&Smith">[[Michael C. Jensen|Jensen, Michael C.]] and Smith, Clifford W., "The Theory of Corporate Finance: A Historical Overview". In: ''The Modern Theory of Corporate Finance'', New York: McGraw-Hill Inc., pp. 2–20, 1984.</ref> and for other investors.
The argument [[Modern portfolio theory#Mathematical model|proceeds as follows]]:<ref name="Bollerslev">See, e.g., [[Tim Bollerslev]] (2019). [http://public.econ.duke.edu/~boller/Econ.471-571.F19/Lec3_471-571_F19.pdf "Risk and Return in Equilibrium: The Capital Asset Pricing Model (CAPM)"]</ref> 
If one can construct an [[efficient frontier]] – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the [[Risk-free interest rate|risk-free asset]] and the "[[market portfolio]]" (the [[Mutual fund separation theorem]]), with the combinations here plotting as the [[capital market line]], or CML. 
Then, given this CML, the required return on a risky security will be independent of the investor's [[utility function]], and solely determined by its [[covariance]] ("beta") with aggregate, i.e. market, risk. 
This is because investors here can then maximize utility through leverage as opposed to stock selection; see [[Separation property (finance)]], {{section link|Markowitz model|Choosing the best portfolio}} and CML diagram aside. 
As can be seen in the formula aside, this result is consistent with [[#Arbitrage-free pricing and equilibrium|the preceding]], equaling the riskless return plus an adjustment for risk.<ref name="Cochrane & Culp"/> 
A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive [[Outline of finance#Asset pricing models|other equilibrium-pricing models]].

[[Image:Stockpricesimulation.jpg|thumb|right|Simulated geometric Brownian motions with parameters from market data]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|{{smalldiv|[[Black–Scholes equation|The Black–Scholes equation:]]
:<math>\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} = rV</math>
[[Black–Scholes equation#Financial interpretation of the Black–Scholes PDE|Interpretation:]] by arbitrage arguments, the instantaneous impact of time <math>t</math> and change in [[spot price]] <math>s</math> on an option price <math>V</math> will (must) realize as growth at <math>r</math>, the risk free rate, when the option is correctly [[Rational pricing#Delta hedging|"manufactured"]].}}
|}
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|{{smalldiv|[[Black–Scholes model#Black–Scholes formula|The Black–Scholes formula]] for the value of a call option:
:<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#Interpretation|Interpretation]]: The value of a call is the [[risk free rate]]d present value of its expected [[in the money]] value - i.e. a specific formulation of the fundamental valuation result. <math>N(d_2)</math> is [[Normal distribution#Standard normal distribution|the standard normal probability]] that the call will be exercised; <math>N(d_1)S</math> is the present value of the expected asset price at expiration, [[Conditional probability|given that]] the asset price at expiration is above the exercise price.}}
|}
Black–Scholes provides a mathematical model of a financial market containing [[Derivative (finance)|derivative]] instruments, and the resultant formula for the price of [[option style|European-styled options]].
{{NoteTag|"BSM" – two seminal 1973 papers by [[Fischer Black]] and [[Myron Scholes]],<ref name="BlackScholes_paper">{{cite journal|title=The Pricing of Options and Corporate Liabilities|last=Black|first=Fischer|author2=Myron Scholes|journal=Journal of Political Economy|year=1973|volume=81|issue=3|pages=637–654|doi=10.1086/260062|s2cid=154552078}} [https://www.jstor.org/stable/1831029]</ref> and [[Robert C. Merton]]<ref name="Merton_paper">{{cite journal|title=Theory of Rational Option Pricing|last=Merton|first=Robert C.|journal=Bell Journal of Economics and Management Science|year=1973|volume=4|issue=1|pages=141–183|doi=10.2307/3003143|jstor=3003143|url=http://dml.cz/bitstream/handle/10338.dmlcz/135817/Kybernetika_43-2007-6_6.pdf|hdl=1721.1/49331|hdl-access=free}} [https://www.jstor.org/stable/3003143]</ref> – is consistent with "previous versions of the formula" of [[Louis Bachelier]] (1900) and [[Edward O. Thorp]] (1967);<ref name="Haug Taleb">Haug, E. G; [[Nassim Nicholas Taleb|Taleb, N. N.]] (2008). [https://polymer.bu.edu/hes/rp-haug08.pdf "Why We Have Never Used the Black–Scholes–Merton Option Pricing Formula"]. ''Wilmott Magazine''.</ref> although these were more "actuarial" in flavor, and had not established risk-neutral discounting.<ref name="Derman"/>  [[Case Sprenkle]] (1961)<ref>{{cite journal|title=Warrant prices as indicators of expectations and preferences|last=Sprenkle|first=Case M.|journal=Yale Economic Essays |year=1961|volume=1|issue=2|pages=179–231}}</ref> had published a formula for the price of a call-option which, with adjustments, satisfied the BSM partial differential equation.<ref>[https://mathshistory.st-andrews.ac.uk/Biographies/Black_Fischer/ Black, Fischer] [[MacTutor History of Mathematics Archive]]</ref>
James Boness (1964),<ref>A. James Boness (1964). [https://www.jstor.org/stable/1828962 "Elements of a Theory of Stock-Option Value"]. ''[[Journal of Political Economy]]''. Vol. 72, No. 2.</ref>
in fact, derived a formula identical to BSM, though through a different argument.<ref name="Haug Taleb"/>
[[Vinzenz Bronzin]] (1908) produced very early results, also.}}
The model is expressed as the Black–Scholes equation, a [[partial differential equation]] describing the changing price of the option over time; it is derived assuming log-normal, [[geometric Brownian motion]] (see [[Brownian model of financial markets]]).
The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing (<math>V</math>, the value, or price, of the option, grows at <math>r</math>, the risk-free rate).<ref name="Rubinstein"/><ref name="Cochrane & Culp"/>
This hedge, in turn, implies that there is only one right price – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is the [[Partial differential equation#Analytical solutions|solution]] to the equation.)
Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with {{section link|#Arbitrage-free pricing and equilibrium}} above. Relatedly, therefore, the pricing formula [[Black–Scholes model#Derivations|may also be derived]] directly via risk neutral expectation.
[[Itô's lemma]] provides [[Itô's lemma#Black–Scholes formula|the underlying mathematics]], and, with [[Itô calculus]] more generally, remains fundamental in quantitative finance.
{{NoteTag|[[Kiyosi Itô]] published his Lemma in 1944. [[Paul Samuelson]]<ref>{{cite journal | author = Samuelson Paul | author-link = Paul Samuelson | year = 1965 | title = A Rational Theory of Warrant Pricing | url = http://www.dse.unisalento.it/c/document_library/get_file?folderId=1344637&name=DLFE-157230.pdf | journal = Industrial Management Review | volume = 6 | page = 2 | access-date = 2017-02-28 | archive-url = https://web.archive.org/web/20170301092720/http://www.dse.unisalento.it/c/document_library/get_file?folderId=1344637&name=DLFE-157230.pdf | archive-date = 2017-03-01 | url-status = live }}</ref> introduced this area of mathematics into finance in 1965;
Robert Merton promoted continuous [[stochastic calculus]] and continuous-time processes from 1969.<ref>Merton, Robert C. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case." The Review of Economics and Statistics 51 (August 1969): 247-257.</ref> }}

As implied by the Fundamental Theorem, [[Asset pricing#Interrelationship|the two major results are consistent]].<!-- ; then, as is to be expected, "classical" financial economics is thus unified. --> 
Here, the Black-Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM.<ref name="Chance1">Don M. Chance (2008). [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN03-01.pdf "Option Prices and Expected Returns"] {{Webarchive|url=https://web.archive.org/web/20150923195335/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN03-01.pdf |date=2015-09-23 }}</ref><ref name="Derman">Emanuel Derman, [http://www.emanuelderman.com/media/Scientific_Approach_to_Finance.pdf ''A Scientific Approach to CAPM and Options Valuation''] {{Webarchive|url=https://web.archive.org/web/20160330002200/http://www.emanuelderman.com/media/Scientific_Approach_to_Finance.pdf |date=2016-03-30 }}</ref> 
The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. 
Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the above fundamental equations – further explaining, and if required demonstrating, this consistency.<ref name="Rubinstein">[[Mark Rubinstein|Rubinstein, Mark]]. (2005). "Great Moments in Financial Economics: IV. The Fundamental Theorem (Part I)", ''Journal of Investment Management'', Vol. 3, No. 4, Fourth Quarter 2005; <br> ~ (2006). Part II, Vol. 4, No. 1, First Quarter 2006. (See under "External links".)</ref> 
Here, the CAPM is derived<ref name="Backus"/> by linking <math>Y</math>, risk aversion, to overall market return, and setting the return on security <math>j</math> as <math>X_j/Price_j</math>; see {{section link|Stochastic discount factor|Properties}}.
The Black–Scholes formula is found, [[Binomial distribution#Normal approximation|in the limit]],<ref>Gregory Gundersen (2023).  [https://gregorygundersen.com/blog/2023/06/03/hsia-proof-black-scholes/ Proof the Binomial Model Converges to Black–Scholes]</ref> by attaching a [[binomial probability]]<ref name="Varian"/> to each of numerous possible [[spot price|spot-prices]] (i.e. states) and then rearranging for the terms corresponding to <math>N(d_1)</math> and <math>N(d_2)</math>, per the boxed description; see {{section link|Binomial options pricing model|Relationship with Black–Scholes}}.

==Extensions==
More recent work further generalizes and extends these models.  As regards [[asset pricing]], developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing.  As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models.

===Portfolio theory===
{{Further|Post-modern portfolio theory|Mathematical finance#Risk and portfolio management: the P world}}
{{see also|Financial risk management#Investment management}}
<!-- [[File:Asset Allocation.pdf|thumb|right|250px|[[Modern portfolio theory]] suggests a diversified portfolio of [[shares]] and other [[asset classes]] (such as debt in [[corporate bonds]], [[treasury bond]]s, or [[money market funds]]) will realise more predictable returns if there is prudent market regulation.]] -->
[[Image:Pareto Efficient Frontier for the Markowitz Portfolio selection problem..png|thumb|right|200px|Plot of two criteria when maximizing return and minimizing risk in financial portfolios (Pareto-optimal points in red)]]
[[File:Four_Correlations.svg|thumb|right|alt=Examples of bivariate copulæ used in finance.|Examples of bivariate copulæ used in finance.]]

Much of the development here relates to required return, i.e. pricing, extending the basic CAPM. [[Multiple factor models|Multi-factor models]] such as the [[Fama–French three-factor model]] and the [[Carhart four-factor model]], propose factors other than market return as relevant in pricing. The [[intertemporal CAPM]] and [[Consumption-based capital asset pricing model|consumption-based CAPM]] similarly extend the model. With [[intertemporal portfolio choice]], the investor now repeatedly optimizes her portfolio; while the inclusion of [[Consumption (economics)|consumption (in the economic sense)]] then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return.

Whereas the above extend the CAPM, the [[single-index model]] is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization.

The [[arbitrage pricing theory]] (APT)  similarly differs as regards its assumptions. APT "gives up the notion that there is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns."<ref>''The Arbitrage Pricing Theory,'' Chapter VI in Goetzmann, under External links.</ref> It returns the required (expected) return of a financial asset as a linear function of various macro-economic factors, and assumes that arbitrage should bring incorrectly priced assets back into line.{{NoteTag|The single-index model was developed by William Sharpe in 1963.<ref>{{cite journal |author=Sharpe, William F. |s2cid=55778045|title=A Simplified Model for Portfolio Analysis|journal=Management Science|year=1963 |volume=9|issue=2 |pages=277–93 |doi=10.1287/mnsc.9.2.277}}</ref>
APT was developed by [[Stephen Ross (economist)|Stephen Ross]] in 1976.<ref>{{Cite journal|last=Ross|first=Stephen A|date=1976-12-01|title=The arbitrage theory of capital asset pricing|journal=Journal of Economic Theory|volume=13|issue=3|pages=341–360|doi=10.1016/0022-0531(76)90046-6|issn=0022-0531}}
</ref>}}

The linear factor [[Multivariate linear regression|model ''structure'']] of the APT is used as the basis for many of the commercial risk and [[Financial software#Fund management|fund management systems]] employed by asset managers. Here,<ref name="Practitioner's Guide">[https://www.cfainstitute.org/-/media/documents/book/rf-publication/1994/rf-v1994-n4-4445-pdf.ashx "A Practitioner's Guide to Factor Models"]. [[CFA Institute]] Research Foundation</ref><ref name="Kempthorne">P. Kempthorne (2013). [https://ocw.mit.edu/courses/18-s096-topics-in-mathematics-with-applications-in-finance-fall-2013/3c8e688538b350a73a8c1cd46056fc5a_MIT18_S096F13_lecnote15.pdf "Factor Models"]. [[MIT OpenCourseWare]]</ref> 
managers apply various of the abovementioned multi-factor models - often 
[[Goldman Sachs asset management factor model|bespoke extensions]] - such that their portfolio has the desired exposure ("tilt") to macroeconomic, market and / or fundamental [[Risk factor (finance)|risk factors]];<ref name="Sharpe">[[William F. Sharpe]] (1999). [https://web.stanford.edu/~wfsharpe/mia/fac/mia_fac3.htm#Factor-based%20Portfolio%20Expected%20Returns%20and%20Risks "Factor-based Expected Returns, Risks and Correlations"]</ref> respectively: [[global macro|Macro-]], [[factor investing|Factor-]], and [[style investing|Style Funds]].
Research here, and application (e.g. "[[Smart_beta#Approach|Smart Beta]]"), is ongoing, and over the years "hundreds of factors attempt to explain the cross-section of expected returns"; 
<ref>{{Cite journal|last1=Harvey|first1=Campbell R.|last2=Liu|first2=Yan|last3=Zhu|first3=Caroline|date=2015|title=…and the Cross-Section of Expected Returns|language=en|location=Rochester, NY|ssrn=2249314|journal=[[The Review of Financial Studies]]}}</ref>
with this "factor zoo", 
<ref>Guanhao Fen; Stefano Giglio; Dacheng Xiu (2020).
[https://www.aqr.com/About-Us/AQR-Insight-Award/2018/Taming-the-Factor-Zoo "Taming the Factor Zoo"]. ''[[Journal of Finance]]''.</ref>
there is a risk of [[data mining]], and researchers have proposed various criteria for [[Statistical_significance#Significance_thresholds_in_specific_fields|establishing significance]].

At the same time, "classic" [[Modern portfolio theory#Efficient frontier with no risk-free asset|mean-variance optimization]] — i.e. building an [[efficient portfolio]] as described [[#Uncertainty|above]] — is still widely used by [[asset allocation|Asset Allocation Funds]].<ref>{{cite journal | url=https://www.pm-research.com/content/iijpormgmt/47/5/24 | title=Mean–Variance Optimization for Asset Allocation| author1=Jang Ho Kim |author2=Yongjae Lee |author3=Woo Chang Kim|author4= Frank J. Fabozzi|year=2021|journal = Journal of Portfolio Management|
volume =47| issue = 5|pages = 24–40| doi=10.3905/jpm.2021.1.219|url-access=subscription}}</ref> Here, given [[Modern portfolio theory#Criticisms|issues noted with this approach]], the application is typically in combination with other techniques.
The [[Black–Litterman model]]<ref>Black F. and Litterman R. (1991). [http://www.iijournals.com/doi/abs/10.3905/jfi.1991.408013 "Asset Allocation Combining Investor Views with Market Equilibrium"]. ''[[Journal of Fixed Income]]''. September 1991, Vol. 1, No. 2: pp.&nbsp;7-18</ref> is often employed.
Black–Litterman  departs from the original [[Markowitz model]] approach: it starts with an equilibrium assumption, as for the latter, but this is then modified to take into account the "views" (i.e., the specific opinions about asset returns) of the investor in question to arrive at a bespoke <ref>Guangliang He and Robert Litterman (1999). [https://people.duke.edu/~charvey/Teaching/BA453_2004/GS_The_intuition_behind.pdf "The Intuition Behind Black-Litterman Model Portfolios"]. [[Goldman Sachs]] Quantitative Resources Group</ref> asset allocation. 
A further modification often used <ref>James Chen (2021). [https://www.investopedia.com/terms/r/risk-parity.asp "Risk Parity"]. Investopedia</ref> relates to the subsequent optimization: under [[Tail risk parity|(Tail)]] [[risk parity]], the focus is on the allocation of risk, rather than the allocation of capital.

Other developments re [[portfolio optimization]] include the following. 
Where factors additional to volatility are considered (kurtosis, skew...) then [[multiple-criteria decision analysis]] can be applied; here deriving a [[Pareto efficient]] portfolio. The [[universal portfolio algorithm]] identifies the "growth optimal portfolio" [[Kelly_criterion#Application_to_the_stock_market|per the Kelly criterion]], applying [[information theory]] to asset selection. [[Behavioral portfolio theory]] recognizes that investors have varied aims and create an investment portfolio that meets a broad range of goals.  Copulas have [[Copula (probability theory)#Quantitative finance|lately been applied here]]; recently this is the case also [[List of genetic algorithm applications#Finance and Economics|for genetic algorithms]] and <ref name="de Prado"/> [[Machine learning#Applications|machine learning, more generally]]; see [[#Financial_markets|below]].<ref name="Bagnara">Bagnara, Matteo (2021). "Asset Pricing and Machine Learning: A Critical Review". {{SSRN|3950568}}</ref>

===Derivative pricing===
{{Further|Mathematical finance#Derivatives pricing: the Q world}}
{{see also|Quantitative analyst#History}}
[[File:Arbre Binomial Options Reelles.png|thumb|right| Binomial Lattice with [[Binomial options pricing model#STEP 1: Create the binomial price tree|CRR formulae]] ]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
| {{smalldiv|1=PDE for a zero-coupon bond:
:<math>\frac{1}{2}\sigma(r)^{2}\frac{\partial^2 P}{\partial r^2}+[a(r)+\sigma(r)+\varphi(r,t)]\frac{\partial P}{\partial r}+\frac{\partial P}{\partial t} = rP</math>}}
{{small|[[Bond valuation#Stochastic calculus approach|Interpretation:]] Analogous to Black–Scholes,<ref>For a derivation see, for example, [https://www.math.fsu.edu/~dmandel/Primers/Understanding%20Market%20Price%20of%20Risk.pdf "Understanding Market Price of Risk"] (David Mandel, [[Florida State University]], 2015)</ref>
arbitrage arguments describe the instantaneous change in the bond price <math>P</math> for changes in the (risk-free) short rate <math>r</math>; the analyst selects the specific [[short-rate model]] to be employed.
}}
|}
[[Image:volatility smile.svg|thumb|right|Stylized volatility smile: showing the (implied) volatility by strike-price, for which the [[Black–Scholes formula]] returns market prices.]]
In pricing derivatives, the [[binomial options pricing model]] provides a discretized version of Black–Scholes, useful for the valuation of [[American option|American styled options]]. Discretized models of this type are built – at least implicitly – using state-prices ([[#State prices|as above]]); relatedly, a large number of researchers [[contingent claim analysis|have used options]] to extract state-prices for a variety of other applications in financial economics.<ref name="Rubinstein"/><ref name="Chance1"/><ref name="Chance2">Don M. Chance (2008). [http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf "Option Prices and State Prices"] {{Webarchive|url=https://web.archive.org/web/20120209215717/http://www.bus.lsu.edu/academics/finance/faculty/dchance/Instructional/TN97-13.pdf |date=2012-02-09 }}</ref> For [[Option style#Non-vanilla path-dependent "exotic" options|path dependent derivatives]], [[Monte Carlo methods for option pricing]] are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various [[Option (finance)#Model implementation|other numeric techniques]] have also been developed. The theoretical framework too has been extended such that [[martingale pricing]] is now the standard approach.
{{NoteTag|
The binomial model was first proposed by [[William F. Sharpe|William Sharpe]] in the 1978 edition of ''Investments'' ({{ISBN|013504605X}}), 
and in 1979 formalized by [[John Carrington Cox|Cox]], [[Stephen Ross (economist)|Ross]] and [[Mark Rubinstein|Rubinstein]]<ref>{{Cite journal |last1=Cox |first1=J. C. |authorlink1=John Carrington Cox |last2=Ross |first2=S. A. |authorlink2=Stephen Ross (economist)|last3=Rubinstein |first3=M. |authorlink3=Mark Rubinstein|doi=10.1016/0304-405X(79)90015-1 |title=Option pricing: A simplified approach |journal=[[Journal of Financial Economics]]|volume=7 |issue=3 |page=229 |year=1979|citeseerx=10.1.1.379.7582 }}</ref> 
and by [[Richard Rendleman|Rendleman]] and Bartter.<ref>Richard J. Rendleman, Jr. and Brit J. Bartter. 1979. "Two-State Option Pricing". ''[[Journal of Finance]]'' 24: 1093-1110.  {{doi|10.2307/2327237}}</ref>
[[Finite difference methods for option pricing]] were due to [[Eduardo Schwartz]] in 1977.<ref name="Schwartz">{{cite journal |last=Schwartz  |first= E.|date=January 1977|title= The Valuation of Warrants: Implementing a New Approach|journal= [[Journal of Financial Economics]]|volume= 4|pages= 79–94 |url= http://ideas.repec.org/a/eee/jfinec/v4y1977i1p79-93.html|doi=10.1016/0304-405X(77)90037-X }}</ref> 
[[Monte Carlo methods for option pricing]] were originated by [[Phelim Boyle]] in 1977;<ref>{{cite journal|last1=Boyle |first1=Phelim P. |url=http://ideas.repec.org/a/eee/jfinec/v4y1977i3p323-338.html |access-date=June 28, 2012 |title=Options: A Monte Carlo Approach |journal=Journal of Financial Economics |volume=4 |number=3 |year=1977 |pages=323–338 |doi=10.1016/0304-405x(77)90005-8|url-access=subscription }}</ref>
In 1996, methods were developed for [[American option|American]]<ref>{{Cite journal|title=Valuation of the early-exercise price for options using simulations and nonparametric regression |last=Carriere|first=Jacques|date=1996|journal=Insurance: Mathematics and Economics |doi=10.1016/S0167-6687(96)00004-2|volume=19|pages=19–30}}</ref>
and [[Asian option]]s.<ref>{{cite journal|last1=Broadie |first1=M. |first2=P. |last2=Glasserman |url=http://www.columbia.edu/~mnb2/broadie/Assets/bg_ms_1996.pdf |access-date=June 28, 2012 |title=Estimating Security Price Derivatives Using Simulation |journal=Management Science |volume=42 |issue=2 |year=1996 |pages=269–285 |doi=10.1287/mnsc.42.2.269|citeseerx=10.1.1.196.1128 }}</ref>
}}

Drawing on these techniques, models for various other underlyings and applications have also been developed, all based on the same logic (using "[[contingent claim analysis]]"). [[Real options valuation]] allows that option holders can influence the option's underlying; models for [[Employee stock option#Valuation|employee stock option valuation]] explicitly assume non-rationality on the part of option holders; [[Credit derivative]]s allow that payment obligations or delivery requirements might not be honored. [[Exotic derivative]]s are now routinely valued. Multi-asset underlyers are handled via simulation or [[Copula (probability theory)#Quantitative finance|copula based analysis]].

Similarly, the various [[short-rate model]]s allow for an extension of these techniques to [[Fixed income#Derivatives|fixed income-]] and [[interest rate derivative]]s. (The [[Vasicek model|Vasicek]] and [[Cox–Ingersoll–Ross model|CIR]] models are equilibrium-based, while [[Ho–Lee model|Ho–Lee]] and subsequent models are based on arbitrage-free pricing.) The more general [[Heath–Jarrow–Morton framework|HJM Framework]] describes the dynamics of the full [[forward rate|forward-rate]] curve – as opposed to working with short rates – and is then more widely applied. The valuation of the underlying instrument – additional to its derivatives – is relatedly extended, particularly for [[Hybrid security|hybrid securities]], where credit risk is combined with uncertainty re future rates;  see {{section link|Bond valuation|Stochastic calculus approach}} and {{section link|Lattice model (finance)|Hybrid securities}}. 
{{NoteTag|
[[Oldrich Vasicek]] developed his pioneering short-rate model in 1977.<ref>{{cite journal |last=Vasicek |first=O. |date=1977 |title=An equilibrium characterization of the term structure |journal=[[Journal of Financial Economics]] |volume=5 |issue=2 |pages=177–188 |doi=10.1016/0304-405X(77)90016-2 |citeseerx=10.1.1.164.447 }}</ref> 
The HJM framework originates from the work of [[David Heath (probabilist)|David Heath]], [[Robert A. Jarrow]], and Andrew Morton in 1987.<ref>David Heath, Robert A. Jarrow, and Andrew Morton (1987). [https://www.jstor.org/stable/2951677 ''Bond pricing and the term structure of interest rates: a new methodology'']  – working paper, Cornell University</ref>
}}

Following the [[Black Monday (1987)|Crash of 1987]], equity options traded in American markets began to exhibit what is known as a "[[volatility smile]]";  that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus [[implied volatility|implied volatilities]], than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here – as well as implications re the standard theory – are discussed [[#Departures from normality|in the next section]].

After the [[2008 financial crisis]], a further development:<ref name="Youmbi">Didier Kouokap Youmbi (2017). "[https://ssrn.com/abstract=2511585  Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach]". [[Bank of England]] – [[Prudential Regulation Authority (United Kingdom)|Prudential Regulation Authority]]</ref> as outlined, ([[Over-the-counter (finance)|over the counter]]) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. 
Addressing this, therefore, issues such as [[counterparty credit risk]], funding costs and costs of capital are now additionally considered when pricing,<ref>[http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf "Post-Crisis Pricing of Swaps using xVAs"] {{Webarchive|url=https://web.archive.org/web/20160917015231/http://pure.au.dk/portal-asb-student/files/96440392/Master_Thesis_Pure.pdf |date=2016-09-17 }}, Christian Kjølhede & Anders Bech, Master thesis, [[Aarhus University]]</ref> and a [[credit valuation adjustment]], or CVA – and potentially other ''valuation adjustments'', collectively [[xVA]] – is generally added to the risk-neutral derivative value.
The standard economic arguments can be extended to incorporate these various adjustments.<ref name="Hull_White_2">John C. Hull and Alan White (2014). [https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2212953 Collateral and Credit Issues in Derivatives Pricing]. Rotman School of Management Working Paper No. 2212953</ref>

A related, and perhaps more fundamental change, is that discounting is now on the [[Overnight index swap|Overnight Index Swap]] (OIS) curve, as opposed to [[LIBOR]] as used previously.<ref name="Youmbi"/> This is because post-crisis, the [[overnight rate]]  is considered a better proxy for the "risk-free rate".<ref>{{cite journal |title=LIBOR vs. OIS: The Derivatives Discounting Dilemma |first1=John |last1=Hull |first2=Alan |last2=White |journal=[[Journal of Investment Management]] |volume=11 |issue=3 |year=2013 |pages=14–27}}</ref> (Also, practically, the interest paid on cash [[collateral (finance)|collateral]] is usually the overnight rate; OIS discounting is then, sometimes, referred to as "[[Credit Support Annex|CSA]] discounting".) [[Swap (finance)#Valuation|Swap pricing]] – and, therefore, [[yield curve]] construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "[[multi-curve framework]]" where "forecast curves" are constructed for each floating-leg [[Libor#Maturities|LIBOR tenor]], with discounting on the ''common'' OIS curve.

===Corporate finance theory===
{{see also|Outline of corporate finance #Theory}}
[[Image:Manual decision tree.jpg|right|thumb|Project valuation via decision tree.]]
Mirroring the [[#Certainty|above]] developments, corporate finance valuations and decisioning no longer need assume "certainty".
{{NoteTag|
Simulation was first applied to (corporate) finance by [[David B. Hertz]] in 1964. 
<ref>David Bendel Hertz (1964). "Risk analysis in capital investment". ''[[Harvard Business Review]]''.  Vol. 42.1964, 1, p. 95-106</ref>
Decision trees, a standard [[operations research]] tool, were applied to corporate finance also in the 1960s.
<ref>See for example: {{cite journal |title= Decision Trees for Decision Making |first= John F. |url= https://hbr.org/1964/07/decision-trees-for-decision-making |last= Magee |journal= [[Harvard Business Review]] |volume= July 1964 |year= 1964 |pages= 795–816 |access-date= 2017-08-16 |archive-url= https://web.archive.org/web/20170816192517/https://hbr.org/1964/07/decision-trees-for-decision-making |archive-date= 2017-08-16 |url-status= live }}</ref>
Real options in corporate finance were first discussed by [[Stewart Myers]] in 1977.
<ref>Stewart Myers (1977). “Determinants of Corporate Borrowing.” ''[[Journal of Financial Economics]]''.</ref>
}}
[[Monte Carlo methods in finance]] allow financial analysts to construct "[[stochastic]]" or [[probabilistic]] corporate finance models, as opposed to the traditional static and [[deterministic]] models;<ref name="Damodaran_Risk"/> see {{section link|Corporate finance|Quantifying uncertainty}}. 
Relatedly, [[real options|Real Options theory]] allows for owner – i.e. managerial – actions that impact underlying value: by incorporating option pricing logic, these actions are then applied to a distribution of future outcomes, changing with time, which then determine the "project's" valuation today.<ref name="Damodaran"/> 
More traditionally, [[decision tree]]s – which are complementary – have been used to evaluate projects, by incorporating in the valuation (all) [[Event (probability theory)|possible events]] (or states) and consequent [[Decision making#Decision making in business and management|management decisions]];<ref>{{cite journal |title=Valuing Risky Projects: Option Pricing Theory and Decision Analysis |first1=James E. |last1=Smith |first2=Robert F. |last2=Nau |url=https://faculty.fuqua.duke.edu/~jes9/bio/Valuing_Risky_Projects.pdf |journal=Management Science |volume=41 |issue=5 |year=1995 |pages=795–816 |doi=10.1287/mnsc.41.5.795 |access-date=2017-08-17 |archive-url=https://web.archive.org/web/20100612170613/http://faculty.fuqua.duke.edu/%7Ejes9/bio/Valuing_Risky_Projects.pdf |archive-date=2010-06-12 |url-status=live }}</ref><ref name="Damodaran_Risk">[[Aswath Damodaran]] (2007). [http://www.stern.nyu.edu/~adamodar/pdfiles/papers/probabilistic.pdf "Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations"]. In ''Strategic Risk Taking: A Framework for Risk Management''. Prentice Hall. {{ISBN|0137043775}}</ref> the correct discount rate here reflecting each decision-point's "non-diversifiable risk looking forward."<ref name="Damodaran_Risk"/>

Related to this, is the treatment of forecasted cashflows in [[equity valuation]]. In many cases, following Williams [[#Certainty|above]], the average (or most likely) cash-flows were discounted,<ref name="Markowitz_interview">{{cite journal |last= Kritzman |first= Mark |title= An Interview with Nobel Laureate Harry M. Markowitz
 |journal=Financial Analysts Journal  |volume=73 |issue= 4|year= 2017|pages= 16–21|doi=10.2469/faj.v73.n4.3|s2cid= 158093964 }}</ref> as opposed to a theoretically correct state-by-state treatment under uncertainty; see comments under [[Financial modeling#Accounting|Financial modeling § Accounting]]. 
In more modern treatments, then, it is the ''expected'' cashflows (in the [[Expected value|mathematical sense]]: <math display=inline>\sum_{s}p_{s}X_{sj}</math>) combined into an overall value per forecast period which are discounted.<ref name="Kruschwitz and Löffler"/><ref name="welch">[http://book.ivo-welch.info/read/chap13.pdf "Capital Budgeting Applications and Pitfalls"] {{Webarchive|url=https://web.archive.org/web/20170815234404/http://book.ivo-welch.info/read/chap13.pdf |date=2017-08-15 }}. Ch 13 in [[Ivo Welch]] (2017). ''Corporate Finance'': 4th Edition</ref><ref>George Chacko and Carolyn Evans (2014). ''Valuation: Methods and Models in Applied Corporate Finance''. FT Press. {{ISBN|0132905221}}</ref><ref name="Damodaran_Risk"/>
And using the CAPM – or extensions – the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows<ref name="Damodaran_Risk"/>
(essentially, <math>Y</math> and <math>r</math> combined).

Other developments here include<ref>See Jensen and Smith under "External links", as well as Rubinstein under "Bibliography".</ref> [[agency theory]], which analyses the difficulties in motivating corporate management (the "agent"; in a different sense to the above) to act in the best interests of shareholders (the "principal"), rather than in their own interests; here emphasizing [[Corporate finance#Corporate governance|the issues interrelated]] with capital structure.<ref>{{cite journal|title=Theory of the firm: Managerial behavior, agency costs and ownership structure|last1=Jensen|first1=Michael |last2=Meckling|first2=William|journal=Journal of Financial Economics |year=1976|volume=3|issue=4|pages=305–360|doi=10.1016/0304-405X(76)90026-X|doi-access=free}}</ref> 
[[Clean surplus accounting]] and the related [[residual income valuation]] provide a model that returns price as a function of earnings, expected returns, and change in [[book value]], as opposed to dividends. This approach, to some extent, arises due to the implicit contradiction of seeing value as a function of dividends, while also holding that dividend policy cannot influence value per Modigliani and Miller's "[[Irrelevance principle]]"; see {{section link|Dividend policy|Relevance of dividend policy}}.

"Corporate finance" as a discipline more generally, building on Fisher [[#Certainty|above]], relates to the long term objective of maximizing the [[Enterprise value|value of the firm]] - and its [[Total shareholder return|return to shareholders]] - and thus also incorporates the areas of [[capital structure]] and [[dividend policy]].<ref>[http://pages.stern.nyu.edu/~adamodar/New_Home_Page/AppldCF/other/Image2.gif Corporate Finance: First Principles], from [[Aswath Damodaran]] (2022). ''Applied Corporate Finance: A User's Manual''. Wiley. {{ISBN|978-1118808931}}</ref>
Extensions of the theory here then also consider these latter, as follows:
(i) [[Corporate finance#Capitalization structure|optimization re capitalization structure]], and theories here as to corporate choices and behavior: [[Capital structure substitution theory]], [[Pecking order theory]], [[Market timing hypothesis]], [[Trade-off theory of capital structure|Trade-off theory]];
(ii) [[Corporate finance#Dividend policy|considerations and analysis re dividend policy]], additional to - and sometimes contrasting with - Modigliani-Miller, include: 
the [[Dividend policy#Walter's model|Walter model]], [[John Lintner#Lintner's dividend policy model|Lintner model]], [[Dividend policy#Residuals theory of dividends|Residuals theory]] and [[Dividend policy#Dividend signaling hypothesis|signaling hypothesis]], as well as discussion re the observed [[clientele effect]] and [[dividend puzzle]].

As described, the typical application of real options is to [[capital budgeting]] type problems. 
However, here, they are [[Corporate finance#Corporate governance|also applied]] to problems of capital structure and dividend policy, and to the related design of corporate securities;<ref name="Garbade">Kenneth D. Garbade (2001). ''Pricing Corporate Securities as Contingent Claims.'' [[MIT Press]]. {{ISBN|9780262072236}}</ref>  
and since stockholder and bondholders have different objective functions, in the analysis of the [[Corporate finance#Corporate governance|related agency problems]].<ref name="Damodaran">{{cite journal|last=Damodaran|first=Aswath|author-link=Aswath Damodaran|title=The Promise and Peril of Real Options|journal=NYU Working Paper|issue=S-DRP-05-02|year=2005|url=http://stern.nyu.edu/~adamodar/pdfiles/papers/realopt.pdf|access-date=2016-12-14|archive-url=https://web.archive.org/web/20010613082802/http://www.stern.nyu.edu/~adamodar/pdfiles/papers/realopt.pdf|archive-date=2001-06-13|url-status=live}}</ref> 
In all of these cases, state-prices can provide the market-implied information relating to the corporate, [[#State prices|as above]], which is then applied to the analysis. For example, [[convertible bond]]s can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.<ref name="corp fin state prices"/><ref name="Kruschwitz and Löffler">See Kruschwitz and Löffler under Bibliography.</ref>

===Financial markets===
[[File:Trend scanning spans method.svg|thumb|[[$SPY|SPY]] daily closing prices (2018-2025)  with corresponding [[machine learning]] produced <ref name="de Prado">Marcos M. López de Prado (2010). [https://www.cambridge.org/core/books/machine-learning-for-asset-managers/6D9211305EA2E425D33A9F38D0AE3545 ''Machine Learning for Asset Managers'']. [[Cambridge University Press]]. {{ISBN|9781108883658}}</ref> [[Trend_following|"trend-scanning"]] [[Supervised learning|labels]] and related statistics, used in subsequent [[statistical classification|classification-]] and [[Outline_of_machine_learning#Regression_analysis|regression problems]].]]

The discipline, as outlined, also includes a formal study of [[financial market]]s. Of interest especially are market regulation and [[market microstructure]], and their relationship to [[Financial market efficiency|price efficiency]].

[[Regulatory economics]] studies, in general, the economics of regulation. In the context of finance, it will address the impact of [[financial regulation]] on the functioning of markets and the efficiency of prices, while also weighing the corresponding increases in market confidence and [[financial stability]].
Research here considers how, and to what extent, regulations relating to disclosure ([[earnings guidance]], [[annual report]]s), [[insider trading]], and [[Short (finance)#Regulations|short-selling]] will impact price efficiency, the [[cost of equity]], and [[market liquidity]].<ref>See for example: 
Hazem Daouk, Charles M.C. Lee, David Ng. (2006). [https://www.sciencedirect.com/science/article/abs/pii/S0929119905000507 "Capital Market Governance: How Do Security Laws Affect Market Performance?"]. ''Journal of Corporate Finance'', Volume 12, Issue 3;
Emilios Avgouleas (2010). [https://core.ac.uk/download/pdf/6631008.pdf "The Regulation of Short Sales and its Reform"] [https://dice.ifo.de/ DICE Report], Vol. 8, Iss. 1. 
</ref>

Market microstructure is concerned with the details of how exchange occurs in markets 
(with [[Walrasian auction|Walrasian-]], [[Matching market|matching-]], [[Fisher market|Fisher-]], and 
[[Arrow–Debreu model#Intuitive description of the Arrow–Debreu model|Arrow-Debreu markets]] as prototypes), 
and "analyzes how specific trading mechanisms affect the [[price formation]] process",<ref>[[Maureen O'Hara (financial economist)|O'Hara, Maureen]], Market Microstructure Theory, Blackwell, Oxford, 1995, {{ISBN|1-55786-443-8}}, p.1.</ref> examining the ways in which the processes of a market affect determinants of [[transaction costs]], prices, quotes, volume, and trading behavior.
It has been used, for example, in providing explanations for [[Real exchange-rate puzzles|long-standing exchange rate puzzles]],<ref>King, Michael, Osler, Carol and Rime, Dagfinn (2013). [https://www.sciencedirect.com/science/article/abs/pii/S0261560613000594 "The market microstructure approach to foreign exchange: Looking back and looking forward"], ''Journal of International Money and Finance''. Volume 38, November 2013, Pages 95-119</ref> and for the [[equity premium puzzle]].<ref>Randi Næs, Johannes Skjeltorp (2006). [https://www.norges-bank.no/globalassets/upload/english/publications/economic-bulletin/2006-03/naes.pdf "Is the market microstructure of stock markets important?"]. [[Norges Bank]] Economic Bulletin 3/06 (Vol. 77)</ref> 
In contrast to the above classical approach, models here explicitly allow for (testing the impact of) [[Frictionless market|market frictions]] and other [[Perfect market|imperfections]];
see also [[market design]].

For both regulation<ref>See, e.g., Westerhoff, Frank H. (2008). [https://ideas.repec.org/a/jns/jbstat/v228y2008i2-3p195-227.html "The Use of Agent-Based Financial Market Models to Test the Effectiveness of Regulatory Policies"], ''Journal of Economics and Statistics''</ref> and microstructure,<ref>See, e.g., Mizuta, Takanobu (2019). [https://arxiv.org/pdf/1906.06000.pdf "An agent-based model for designing a financial market that works well"]. 2020 IEEE Symposium Series on Computational Intelligence (SSCI).</ref> and generally,<ref name="LeBaron"/> [[Agent-based model#In economics and social sciences|agent-based models]] can be developed<ref name="ERIM"/> to [[Agent-based computational economics#Example: finance|examine any impact]] due to a change in structure or policy - or [[Artificial economics#Method|to make inferences]] re market dynamics - [[Computer experiment|by testing these]] in an artificial financial market, or AFM.
{{NoteTag|The Benchmark here is the pioneering AFM of the [[Santa Fe Institute]]  developed in the early 1990s. See<ref name="LeBaron2"/> for discussion of other early models.}}
This approach, essentially [[Discrete-event simulation|simulated]] trade between numerous [[Agent (economics)|agents]], "typically uses [[artificial intelligence]] technologies [often [[genetic algorithms]] and [[Artificial neural network|neural nets]]] to represent the [[adaptive market hypothesis|adaptive behaviour]] of market participants".<ref name="ERIM">Katalin Boer, Arie De Bruin, Uzay Kaymak (2005). [https://core.ac.uk/download/pdf/18517225.pdf "On the Design of Artificial Stock Markets"].  ''Research In Management'' [[Erasmus Research Institute of Management|ERIM]] Report Series</ref>

These [[Microfoundations|'bottom-up' models]] "start from first principals of agent behavior",<ref name="LeBaron2">LeBaron, B. (2002). [https://www2.econ.iastate.edu/tesfatsi/blake.sfisum.pdf "Building the Santa Fe artificial stock market"]. ''[[Physica (journal)|Physica A]]'', 1, 20.</ref> with participants modifying their trading strategies having learned over time, and "are able to describe macro features [i.e. [[stylized fact]]s] [[Emergence#Economics|emerging]] from a soup of individual interacting strategies".<ref name="LeBaron2"/>
Agent-based models depart further from the classical approach — the [[representative agent]], as outlined — in that they introduce [[Heterogeneity in economics|heterogeneity]] into the environment (thereby addressing, also, the [[aggregation problem]]).

More recent research <ref name="Bagnara"/> focuses on the potential impact of [[machine learning]] on market functioning and efficiency.
As these methods become more prevalent in financial markets, economists would expect greater [[information acquisition]] and improved price efficiency.<ref name="Barbopoulos">Barbopoulos, Leonidas G. ''et al''. (2023) "Market Efficiency When Machines Access Information".  [[NYU Stern School of Business]]. {{SSRN|3783221}}</ref> 
In fact, an apparent rejection of market efficiency (see [[#Departures_from_rationality|below]]) might simply represent "the unsurprising consequence of investors not having precise knowledge of the parameters of a data-generating process that involves thousands of predictor variables".<ref>Ian W.R. Martin, and Stefan Nagel (2022).  [https://www.sciencedirect.com/science/article/pii/S0304405X21004566 "Market efficiency in the age of big data"]. [[Journal of Financial Economics]]. Volume 145, Issue 1, July 2022, Pages 154-177</ref>
At the same time, it is acknowledged that a potential downside of these methods, in this context, is their lack of [[Mechanistic interpretability|interpretability]] "which translates into difficulties in attaching economic meaning to the results found."<ref name="Bagnara"/>

==Challenges and criticism==
{{see also|Neoclassical economics#Criticisms |Financial mathematics#Criticism|Financial engineering#Criticisms}}
<!-- |Financial Modelers' Manifesto|Unreasonable ineffectiveness of mathematics#Economics and finance|Physics envy -->
As above, there is a very close link between:
the [[random walk hypothesis]], with the associated belief that price changes should follow a [[normal distribution]], on the one hand; 
and market efficiency and [[rational expectations]], on the other.
Wide departures from these are commonly observed, and there are thus, respectively, two main sets of challenges.

===Departures from normality===
{{See also|Capital asset pricing model#Problems|Black–Scholes model#Criticism and comments}}
<!-- replicating above...
{| class="wikitable floatright" | width="250"
|- style="text-align:left;"
|{{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>}}
|} -->
[[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.]]

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#Investment 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 {{section link|#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 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#Investment 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 [[Geocentrism#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>}}

===Departures from rationality===
{{See also|Efficient-market hypothesis#Criticism|Rational expectations#Criticism}}
{|class="wikitable floatright" | width="200"
|- style="font-size:75%"
|-align="center"
|colspan="1" | [[Market anomaly|Market anomalies]] and [[economic puzzle]]s
|-
| rowspan="2" |
* [[Calendar effect]]
** [[January effect]]
** [[Sell in May]]
** [[Mark Twain effect]]
** [[Santa Claus rally]]
* [[Closed-end fund puzzle]]
* [[Dividend puzzle]]
* [[Equity home bias puzzle]]
* [[Equity premium puzzle]]
* [[Excess volatility puzzle]]
* [[Forward premium anomaly]]
* [[Low-volatility anomaly]]
* [[Momentum (finance)|Momentum anomaly]]
* [[Neglected firm effect]]
* [[Post-earnings-announcement drift]]
* [[Real exchange-rate puzzles]]
|-
|}
As seen, a common assumption is that financial decision makers act rationally; see [[Homo economicus]]. Recently, however, researchers in [[experimental economics]] and [[experimental finance]] have challenged this assumption [[Empirical evidence|empirically]]. These assumptions are also challenged [[Theory|theoretically]], by [[behavioral finance]], 
<ref>Bruce Stangle, [[Burton Malkiel]] and [[Sendhil Mullainathan]] (2005). [https://sendhil.org/wp-content/uploads/2019/08/Other-Article-11.pdf "Market Efficiency versus Behavioral Finance"]. ''[[Journal of Applied Corporate Finance]]''. Volume 17 Number 3.</ref>
a discipline primarily concerned with the limits to rationality of economic agents.
{{NoteTag|
An early anecdotal treatment is [[Benjamin Graham]]'s "[[Mr. Market]]", discussed in his ''[[The Intelligent Investor]]'' in 1949. 
See also [[John Maynard Keynes]]' 1936 discussion of [[Animal spirits (Keynes)|"Animal spirits"]], and the related [[Keynesian beauty contest]], in his [[The General Theory of Employment, Interest and Money#Chapter 12: Animal spirits|''General Theory'', Ch. 12.]]
''[[Extraordinary Popular Delusions and the Madness of Crowds]]'' is a study of [[crowd psychology]] by Scottish journalist [[Charles Mackay (author)|Charles Mackay]], first published in 1841, with Volume I discussing [[economic bubbles]].
}}
For related criticisms re corporate finance theory vs its practice see:.<ref>{{cite SSRN|author = Pablo Fernandez|title = Common Sense and Illogical Models: Finance and Financial Economics|year=2019|SSRN =2906887}}</ref>

Various [[Market anomaly|market anomalies]] have been documented in parallel. These comprise price "distortions", e.g. [[size premium]]s, or return "predictability" <ref name="Rossi">Matteo Rossi (2016). [https://www.inderscienceonline.com/doi/epdf/10.1504/IJMFA.2015.074905 "The efficient market hypothesis and calendar anomalies: a literature review"]
''International Journal of Managerial and Financial Accounting'' Vol. 7, No. 3-4</ref> as exemplified by the various [[calendar effect]]s, and are anomalous in the sense that <ref name="Rossi"/> they cannot be explained by traditional economic theories, (apparently) contradicting the [[efficient-market hypothesis]].  Related to these are various of the [[economic puzzle]]s, similarly contradicting the theory. The [[equity premium puzzle]], as one example, arises in that the difference between the observed returns on stocks as compared to government bonds is consistently higher than the [[risk premium]] rational equity investors should demand, an "[[abnormal return]]". For further context see [[Random walk hypothesis#A non-random walk hypothesis|Random walk hypothesis § A non-random walk hypothesis]], and sidebar for specific instances.

More generally, and, again, particularly following the [[2008 financial crisis]], financial economics (and [[mathematical finance]]) has been subjected to deeper criticism.
Notable here is [[Nassim Taleb]], whose critique overlaps the above, but extends<ref>See, e.g., this [[opinion piece]]: [https://today.thefinancialexpress.com.bd/print/the-pseudo-science-hurting-markets "The pseudo-science hurting markets"] ([[Financial Times]], November 2007).</ref> 
also to the institutional<ref name="Taleb_crisis"/><ref name="Taleb_Goldstein_Spitznagel">Nassim N. Taleb, [[Daniel G. Goldstein]], and [[Mark Spitznagel|Mark W. Spitznagel]] (2009). [https://hbr.org/2009/10/the-six-mistakes-executives-make-in-risk-management "The Six Mistakes Executives Make in Risk Management"], ''[[Harvard Business Review]]''</ref>
aspects of finance - [[Ivory tower#Academic usage|including academic]].<ref>[[Nassim Taleb]] (2009). [http://www.fooledbyrandomness.com/Triana-fwd.pdf "History Written By The Losers"], Foreword to Pablo Triana's ''Lecturing Birds How to Fly'' {{ISBN|978-0470406755}}</ref><ref name="Haug Taleb"/>
His [[Black swan theory]] posits that although [[List of stock market crashes and bear markets|events of large magnitude and consequence]] play a major role in finance, since these are (statistically) unexpected, [[Fooled by Randomness#Thesis|they are "ignored"]] by economists and traders  - who often [[Skin_in_the_Game_(book)#The_centrality_of_negative_incentives|have no skin in the game]].  
Thus, although a "[[Taleb distribution]]" - which normally provides a payoff of small positive returns, while carrying a small but significant risk of catastrophic losses - more realistically describes markets than current models, the latter continue to be preferred; and even with [[risk manager|professionals here]] acknowledging that it only "generally works" or only "works on average".<ref>Nassim Taleb (1997). [https://www.fooledbyrandomness.com/jorion.html "Against Value-at-Risk: Nassim Taleb Replies to Philippe Jorion"]. fooledbyrandomness.com</ref>

Here,<ref name="Taleb_crisis">Nassim Taleb (2011). [https://www.fooledbyrandomness.com/crisis.pdf “Why Did the Crisis of 2008 Happen?”]</ref> [[financial crises]] have been a topic of interest<ref>From ''[[The New Palgrave Dictionary of Economics]]'', Online Editions, 2011, 2012, with abstract links:<br />&nbsp;&nbsp; • [http://www.dictionaryofeconomics.com/article?id=pde2012_F000330&edition=1 "regulatory responses to the financial crisis: an interim assessment"] {{Webarchive|url=https://web.archive.org/web/20130529101109/http://www.dictionaryofeconomics.com/article?id=pde2012_F000330&edition=1 |date=2013-05-29 }} by [[Howard Davies (economist)|Howard Davies]]<br />&nbsp;&nbsp; • [http://www.dictionaryofeconomics.com/article?id=pde2011_C000621&edition= "Credit Crunch Chronology: April 2007–September 2009"] {{Webarchive|url=https://web.archive.org/web/20130529092712/http://www.dictionaryofeconomics.com/article?id=pde2011_C000621&edition= |date=2013-05-29 }} by The Statesman's Yearbook team<br />&nbsp;&nbsp; • [http://www.dictionaryofeconomics.com/article?id=pde2011_M000430&edition=current&q= "Minsky crisis"] {{Webarchive|url=https://web.archive.org/web/20130529172102/http://www.dictionaryofeconomics.com/article?id=pde2011_M000430&edition=current&q= |date=2013-05-29 }} by [[L. Randall Wray]]<br />&nbsp;&nbsp; • [http://www.dictionaryofeconomics.com/article?id=pde2011_E000326&edition=current&q= "euro zone crisis 2010"] {{Webarchive|url=https://web.archive.org/web/20130529092726/http://www.dictionaryofeconomics.com/article?id=pde2011_E000326&edition=current&q= |date=2013-05-29 }} by [[Daniel Gros]] and Cinzia Alcidi.<br />&nbsp;&nbsp; • [[Carmen M. Reinhart]] and [[Kenneth S. Rogoff]], 2009. ''This Time Is Different: Eight Centuries of Financial Folly'', Princeton. [http://press.princeton.edu/titles/8973.html Description] {{Webarchive|url=https://web.archive.org/web/20130118213207/http://press.princeton.edu/titles/8973.html |date=2013-01-18 }}, ch. 1 ("Varieties of Crises and their Dates". pp. [http://press.princeton.edu/chapters/s8973.pdf 3-20)] {{Webarchive|url=https://web.archive.org/web/20120925065855/http://press.princeton.edu/chapters/s8973.pdf |date=2012-09-25 }}, and chapter-preview [https://books.google.com/books?id=ak5fLB24ircC&pg=PR7gbs_atb links.]</ref> 
and, in particular, [[2008 financial crisis#Prediction by economists|the failure]]<ref name="Taleb_Goldstein_Spitznagel"/> of (financial) economists - as well as<ref name="Taleb_crisis"/> [[2008 financial crisis#Incorrect pricing of risk|bankers]] and [[Government policies and the subprime mortgage crisis|regulators]] - to model and predict these.
See {{section link|Subprime crisis background information#Stages of the crisis}}.
The related problem of [[systemic risk]], has also received attention. Where companies hold securities in each other, then this interconnectedness may entail a "valuation chain" – and the performance of one company, or security, here will impact all, a phenomenon not easily modeled, regardless of whether the individual models are correct. See: [[Systemic risk#Inadequacy of classic valuation models|Systemic risk § Inadequacy of classic valuation models]]; [[Cascades in financial networks]]; [[Flight-to-quality]].

Areas of research [[Financial crisis#Theories|attempting to explain]] (or at least model) these phenomena, and crises, include<ref name="Farmer_Geanakoplos">{{cite journal | author = Farmer J. Doyne, Geanakoplos John | year = 2009 | title = The virtues and vices of equilibrium and the future of financial economics | url = https://campuspress.yale.edu/johngeanakoplos/files/2017/07/63.-The-Virtues-and-Vices-of-Equilbrium-and-the-Future-of-Financial-Economics-2009-26baz0x.pdf | journal = Complexity | volume = 14 | issue = 3 | pages = 11–38 | doi=10.1002/cplx.20261| arxiv = 0803.2996 | bibcode = 2009Cmplx..14c..11F | s2cid = 4506630 }}</ref> [[market microstructure]] and [[Heterogeneous agent model]]s, as above. The latter is extended to [[agent-based computational economics|agent-based computational models]]; here,<ref name="LeBaron">For a survey see: LeBaron, Blake (2006). [https://peeps.unet.brandeis.edu/~blebaron/wps/hbook.pdf "Agent-based Computational Finance"]. [https://www.sciencedirect.com/handbook/handbook-of-computational-economics ''Handbook of Computational Economics'']. Elsevier</ref> as mentioned, price is treated as an [[emergent phenomenon]], resulting from the interaction of the various market participants (agents). The [[noisy market hypothesis]] argues that prices can be influenced by speculators and [[momentum trader]]s, as well as by [[insider trading|insiders]] and institutions that often buy and sell stocks for reasons unrelated to [[fundamental value]]; see [[Noise (economic)]] and  [[Noise trader]]. The [[adaptive market hypothesis]] is an attempt to reconcile the efficient market hypothesis with behavioral economics, by applying the principles of [[evolution]] to financial interactions. An [[information cascade]], alternatively, shows market participants engaging in the same acts as others ("[[herd behavior]]"), despite contradictions with their private information. [[Copula (probability theory)#Quantitative finance|Copula-based modelling]] has similarly been applied. See also [[Hyman Minsky]]'s [[Hyman Minsky#Minsky's financial instability-hypothesis|"financial instability hypothesis"]], as well as  [[George Soros#Reflexivity, financial markets, and economic theory|George Soros' application]] of [[Reflexivity (social theory)#In economics|"reflexivity"]].
In the alternative, institutionally inherent [[limits to arbitrage]] - i.e. as opposed to factors directly contradictory to the theory - are sometimes referenced.

Note however, that despite the above inefficiencies, asset prices do ''effectively''<ref name="affirmative case"/> follow a random walk - i.e. (at least) in the sense that "changes in the stock market are unpredictable, lacking any pattern that can be used by an investor to beat the overall market".<ref>Albert Phung (2024). [https://www.investopedia.com/ask/answers/08/random-walk-theory.asp How Can Random Walk Theory Be Applied to Investing?], investopedia</ref>
Thus after [[Active management#Disadvantages of active management|fund costs]] - and given [[Passive management#Rationale for passive investing|other considerations]] -  it is difficult to consistently outperform market averages<ref>[[William F. Sharpe]] (1991). [http://www.stanford.edu/~wfsharpe/art/active/active.htm "The Arithmetic of Active Management"]  
{{Webarchive|url=https://web.archive.org/web/20131113071513/http://www.stanford.edu/~wfsharpe/art/active/active.htm |date=2013-11-13 }}. ''Financial Analysts Journal'' Vol. 47, No. 1, January/February</ref>  
and achieve [[Alpha (investment)|"alpha"]].
The practical implication<ref name="Prosaic">[[William F. Sharpe]] (2002). [http://www.stanford.edu/~wfsharpe/art/talks/indexed_investing.htm ''Indexed Investing: A Prosaic Way to Beat the Average Investor''] {{Webarchive|url=https://web.archive.org/web/20131114160728/http://www.stanford.edu/~wfsharpe/art/talks/indexed_investing.htm |date=2013-11-14 }}. 
Presentation: [[Monterey Institute of International Studies]]. Retrieved May 20, 2010.</ref> is that [[passive investing]], i.e. via low-cost [[index fund]]s, should, on average, serve better than [[Investment strategy#Strategies|any other]] [[active investing|active strategy]] - 
and, in fact, this practice is [[Index fund#Market size|now widely adopted]].
{{NoteTag|[[Burton Malkiel]]'s ''[[A Random Walk Down Wall Street]]'' – first published in 1973, and in its 13th edition as of 2024 – is a widely read popularization of these arguments.  See also [[John C. Bogle]]'s ''[[Common Sense on Mutual Funds]]''; but compare [[Warren Buffett]]'s ''[[The Superinvestors of Graham-and-Doddsville]]''.}}
Here, however, the following [[Passive management#Concerns about passive investing|concern is posited]]:
although in concept, it is "the research undertaken by active managers [that] keeps prices closer to value... [and] thus there is a fragile equilibrium in which some investors choose to index while the rest continue to search for mispriced securities";<ref name="Prosaic"/>
in practice, as more investors "pour money into index funds tracking the same stocks, valuations [[Index fund#Common market impact|for those companies]] become inflated",<ref>James Faris (2025). [https://www.businessinsider.com/stock-market-bubble-risk-investing-strategy-passive-investing-index-funds-2025-2 A troubling 'self-fulfilling prophecy' may be forming a market bubble], [[Business Insider]]</ref> potentially leading to [[economic bubble|asset bubbles]].
See [[Grossman-Stiglitz paradox]].

==See also==
{{div col}}
* [[:Category:Finance theories]]
* [[:Category:Financial models]]
* {{annotated link|Deutsche Bank Prize in Financial Economics}}
* {{section link|Finance|Financial theory}}
* {{annotated link|Fischer Black Prize}}
* [[Journal of Financial Economics]]
* [[List of financial economics articles]]
* {{annotated link|List of financial economists}}
* {{section link|List of unsolved problems in economics|Financial economics}}
* {{annotated link|Master of Financial Economics}}
* {{annotated link|Monetary economics}}
* {{annotated link|Outline of economics}}
* {{annotated link|Outline of corporate finance}}
* {{annotated link|Outline of finance}}

{{div col end}}

==Historical notes==
{{see also|Quantitative analyst#Seminal publications|List of quantitative analysts}}
{{NoteFoot}}

==References==
{{Reflist|20em}}

==Bibliography==
{{refbegin|30em}}
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* {{cite book | author= [[Thorsten Hens]] and Marc Oliver Rieger | title=Financial Economics: A Concise Introduction to Classical and Behavioral Finance| publisher= [[Springer Publishing|Springer]] | year=2010| isbn=978-3540361466}}
* {{cite book | author= [[Chi-fu Huang]] and [[Robert Litzenberger|Robert H. Litzenberger]] | title=Foundations for Financial Economics | publisher=Prentice Hall | year=1998| isbn=978-0135006535}}
* {{cite book | author= Jonathan E. Ingersoll | title=Theory of Financial Decision Making | url= https://archive.org/details/theoryoffinancia1987inge | url-access= registration | publisher=Rowman & Littlefield | year=1987| isbn=978-0847673599| author-link=Jonathan E. Ingersoll }}
* {{cite book | author= Robert A. Jarrow | title=Finance theory | publisher=Prentice Hall| year=1988| isbn=978-0133148657| author-link=Robert A. Jarrow }}
* {{cite book | author= Chris Jones | title=Financial Economics| publisher= [[Routledge]] | year=2008| isbn=978-0415375856 }}
* {{cite book | author= Brian Kettell | title= Economics for Financial Markets | publisher= [[Butterworth-Heinemann]] | year=2002| isbn=978-0-7506-5384-8}}
* {{cite book | author= Yvan Lengwiler | title=Microfoundations of Financial Economics: An Introduction to General Equilibrium Asset Pricing| publisher= Princeton University Press | year=2006| isbn=978-0691126319 }}
* {{cite book |author1=Stephen F. LeRoy |author2=Jan Werner | title=Principles of Financial Economics| publisher= Cambridge University Press | year=2000| isbn=978-0521586054}}
* {{cite book |author1=Leonard C. MacLean |author2=William T. Ziemba | title= Handbook of the Fundamentals of Financial Decision Making | publisher= World Scientific | year=2013| isbn=978-9814417341 }}
* {{cite book |author1=[[Antonio Mele]]|year=2022| title= Financial Economics | publisher= MIT Press|isbn=9780262046848}}
* {{cite book | author= Robert C. Merton | title=Continuous-Time Finance | publisher=[[Wiley-Blackwell|Blackwell]] | year=1992| isbn= 978-0631185086}}
* {{cite book | author= Frederic S. Mishkin | title=The Economics of Money, Banking, and Financial Markets| publisher= [[Prentice Hall]] | year=2012| isbn=978-0132961974| author-link=Frederic S. Mishkin| edition=3rd}}
* {{cite book |editor=[[Harry Panjer|Harry H. Panjer]] | title=Financial Economics with Applications | publisher= Actuarial Foundation | year=1998| isbn=978-0938959489}}
* {{cite book | editor= Geoffrey Poitras | title= Pioneers of Financial Economics
| publisher= [[Edward Elgar Publishing]] | year= 2007 }} Volume I {{ISBN|978-1845423810}}; Volume II {{ISBN|978-1845423827}}.
* {{cite book | editor=[[Richard Roll]]  | url=https://www.e-elgar.com/shop/gbp/book-series/economics-and-finance/the-international-library-of-critical-writings-in-financial-economics-series.html | title=The International Library of Critical Writings in Financial Economics | publisher=[[Edward Elgar Publishing]] | location=[[Cheltenham]] | year=2006 }}

'''Asset pricing'''
* {{cite book | author= Kerry E. Back | title=Asset Pricing and Portfolio Choice Theory| publisher= Oxford University Press | year=2010| isbn=978-0195380613 }}
* {{cite book | author= Tomas Björk | title= Arbitrage Theory in Continuous Time| publisher= Oxford University Press | year=2009| isbn=978-0199574742| edition= 3rd}}
* {{cite book | author= John H. Cochrane | title=Asset Pricing| publisher=[[Princeton University Press]] | year=2005| isbn=978-0691121376| author-link=John H. Cochrane}}
* {{cite book | author= Darrell Duffie | title=Dynamic Asset Pricing Theory| publisher= Princeton University Press | year=2001| isbn=978-0691090221 | author-link=Darrell Duffie| edition=3rd}}
* {{cite book |author=[[Edwin Elton|Edwin J. Elton]] |author2=[[Martin J. Gruber]] |author3=Stephen J. Brown |author4=[[William N. Goetzmann]] | title=Modern Portfolio Theory and Investment Analysis| publisher=[[John Wiley & Sons|Wiley]] | year=2014| isbn=978-1118469941| edition=9th}}
* {{cite book | author=Robert A. Haugen | title=Modern Investment Theory| publisher= Prentice Hall | year=2000| isbn=978-0130191700| author-link=Robert Haugen| edition=5th}}
* {{cite book | author= [[Mark S. Joshi]], Jane M. Paterson | title=Introduction to Mathematical Portfolio Theory| publisher= Cambridge University Press | year=2013| isbn=978-1107042315 }}
* {{cite book | author=Lutz Kruschwitz, Andreas Loeffler | title=Discounted Cash Flow: A Theory of the Valuation of Firms | publisher=Wiley | year=2005 | isbn=978-0470870440 | url=https://archive.org/details/discountedcashfl00krus }}
* {{cite book | author= David G. Luenberger | title=Investment Science| publisher=[[Oxford University Press]] | year=2013| isbn=978-0199740086| author-link=David Luenberger| edition=2nd}}
* {{cite book | author= Harry M. Markowitz | title=Portfolio Selection: Efficient Diversification of Investments| publisher= Wiley | year=1991| isbn=978-1557861085 | author-link=Harry M. Markowitz| edition=2nd|url=https://cowles.yale.edu/sites/default/files/2022-09/m16-all.pdf}}
* {{cite book | author= Frank Milne | title=Finance Theory and Asset Pricing | publisher= Oxford University Press | year=2003| isbn=978-0199261079| author-link=Frank Milne | edition=2nd }}
* {{cite book | author= George Pennacchi | title=Theory of Asset Pricing| publisher= Prentice Hall | year=2007| isbn=978-0321127204 | author-link=George Pennacchi}}
* {{cite book | author= Mark Rubinstein | title=A History of the Theory of Investments| publisher= Wiley| year=2006| isbn=978-0471770565| author-link=Mark Rubinstein}}
* {{cite book | author= William F. Sharpe | title=Portfolio Theory and Capital Markets: The Original Edition| publisher= [[McGraw-Hill]] | year=1999| isbn=978-0071353205| author-link=William F. Sharpe|url=https://archive.org/details/portfoliotheoryc0000shar/page/n5/mode/2up}}

'''Corporate finance'''
* {{cite book |author1=Jonathan Berk |author2=Peter DeMarzo | title=Corporate Finance | publisher= [[Pearson Education|Pearson]] | year=2013| isbn=978-0132992473 |edition=3rd }}
* {{cite book | author= Peter Bossaerts |author-link=Peter Bossaerts|author2= Bernt Arne Ødegaard | title= Lectures on Corporate Finance| publisher= World Scientific | year=2006| isbn=978-981-256-899-1|edition= Second }}
* {{cite book | author= Richard Brealey |author2=Stewart Myers |author3-link=Franklin Allen |author3=Franklin Allen| title=Principles of Corporate Finance | publisher= Mcgraw-Hill | year=2013| isbn=978-0078034763|title-link=Principles of Corporate Finance |author-link=Richard Brealey |author2-link=Stewart Myers }}
*{{cite book |author1=CFA Institute | title=Corporate Finance: Economic Foundations and Financial Modeling | publisher= Wiley | year=2022| isbn=978-1119743767 |author1-link=CFA Institute |edition=3rd}}
*  {{cite book |author=Thomas E. Copeland |author2=J. Fred Weston |author3=Kuldeep Shastri | title=Financial Theory and Corporate Policy | publisher=Pearson| year=2004| isbn=978-0321127211 | edition=4th }}
*{{cite book  | title=Principles of Finance| year=2022| isbn=9781951693541 | author = Julie Dahlquist, Rainford Knight, Alan S. Adams | publisher=OpenStax, Rice University|url = https://open.umn.edu/opentextbooks/textbooks/principles-of-finance}}
* {{cite book| author=Aswath Damodaran| title=Corporate Finance: Theory and Practice| publisher=Wiley| year=1996| isbn=978-0471076803| url=https://archive.org/details/corporatefinance0000damo| author-link=Aswath Damodaran}}
* {{cite book | author= João Amaro de Matos | title=Theoretical Foundations of Corporate Finance | publisher= Princeton University Press | year=2001| isbn=9780691087948}}
*{{cite book |author1= C. Krishnamurti  |author2=S. R. Vishwanath | title=Advanced Corporate Finance | publisher= MediaMatics | year=2010| isbn=978-8120336117|url=https://www.phindia.com/Books/BookDetail/9788120336117/advanced-corporate-finance-vishwanath-krishnamurti }}
* {{cite book |author1=Joseph Ogden |author2=Frank C. Jen |author3=Philip F. O'Connor | title= Advanced Corporate Finance | publisher= Prentice Hall| year= 2002 | isbn= 978-0130915689}}
* {{cite book |author1=Pascal Quiry |author2=Yann Le Fur |author3=Antonio Salvi |author4=Maurizio Dallochio |author5=[[Pierre Vernimmen]] | title=Corporate Finance: Theory and Practice| publisher=Wiley| year=2011| isbn=978-1119975588|edition=3rd }}
* {{cite book |author=[[Stephen Ross (economist)|Stephen Ross]] |author2=Randolph Westerfield |author3=Jeffrey Jaffe | title=Corporate Finance| publisher= [[McGraw-Hill]] | year=2012| isbn=978-0078034770 | edition=10th}}
* {{cite book |editor=[[Joel Stern|Joel M. Stern]] | title=The Revolution in Corporate Finance| publisher=[[Wiley-Blackwell]]| year=2003| isbn=9781405107815| edition=4th}}
* {{cite book | author= Jean Tirole | title=The Theory of Corporate Finance| publisher= Princeton University Press | year=2006| isbn=978-0691125565| author-link=Jean Tirole}}
* {{cite book | author= Ivo Welch | title=Corporate Finance| year=2017| isbn=978-0-9840049-2-8| author-link=Ivo Welch| edition=4th}}
{{refend}}

==External links==
{|
|-
| valign="top" |
{{refbegin|30em}}
;Surveys
* {{cite journal | doi = 10.1111/j.1745-6622.2000.tb00050.x | volume=13 | issue=2 | year=2000 | journal=Journal of Applied Corporate Finance | pages=8–14 | author=Miller Merton H | title=The History of Finance: An Eyewitness Account | author-link = Merton H. Miller}}
* [https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/I%20Present%20Value.doc Great Moments in Financial Economics I], [https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/II%20Modigliani-Miller%20Theorem.doc II], [https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/III%20Short-Sales%20and%20Stock%20Prices.doc III], [https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/IV%20Fundamental%20Theorem%20-%20Part%20I.doc IVa], [https://web.archive.org/web/20070628225647/http://www.in-the-money.com/artandpap/IV%20Fundamental%20Theorem%20-%20Part%20II.doc IVb] ([[Internet Archive|archived]], 2007-06-27). [[Mark Rubinstein]]
* [https://web.archive.org/web/20030204203936/http://www.finance-and-physics.org/Library/Articles3/scienceandfinance/science.htm The Scientific Evolution of Finance] ([[Internet Archive|archived]], 2003-04-03). Don Chance and Pamela Peterson
* [https://web.archive.org/web/20180615164831/http://www.econ.boun.edu.tr/content/2017/spring/EC-42701/Lecture%20Note-Markowitz%20(1999)-04-05-2017.pdf The Early History of Portfolio Theory: 1600-1960], Harry M. Markowitz. ''Financial Analysts Journal'', Vol. 55, No. 4 (Jul. – Aug., 1999), pp.&nbsp;5–16
* [https://ssrn.com/abstract=244161 The Theory of Corporate Finance: A Historical Overview], [[Michael C. Jensen]] and Clifford W. Smith.
* [http://emanuelderman.com/a-stylized-history-of-quantitative-finance/ A Stylized History of Quantitative Finance], Emanuel Derman
* [https://web.archive.org/web/20120326213849/http://www.thederivativesbook.com/Chapters/10Chap.pdf Financial Engineering: Some Tools of the Trade] (discusses historical context of derivative pricing). Ch 10 in [[Phelim Boyle]] and Feidhlim Boyle (2001). "Derivatives: The Tools That Changed Finance". Risk Books (June 2001). {{ISBN|189933288X}}
* [https://www.inkling.com/read/principles-of-corporate-finance-brealey-10th/chapter-34/section-34-1 What We Do Know: The Seven Most Important Ideas in Finance]; [https://www.inkling.com/read/principles-of-corporate-finance-brealey-10th/chapter-34/section-34-2 What We Do Not Know: 10 Unsolved Problems in Finance], [[Richard A. Brealey]], [[Stewart Myers]] and [[Franklin Allen]].
* [https://dspace.mit.edu/handle/1721.1/48732 An Overview of Modern Financial Economics] ([[MIT]] [[Working paper]]). [[Chi-fu Huang]]
* [https://web.archive.org/web/20080429203224/http://cepa.newschool.edu/het/essays/capital/fisherinvest.htm Irving Fisher's Theory of Investment] Gonçalo L. Fonseca,  [[The New School]]
* [https://sites.bu.edu/perry/files/2019/04/Financial-Economics-Mehrling.pdf Financial Economics], [[Perry Mehrling]]. (From "Handbook of the History of Economic Analysis", 2016.)

{{refend}}
| valign="top" |
{{refbegin|30em}}
'''Course material'''
* [http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf Fundamentals of Asset Pricing], [[David K. Backus]], [[New York University Stern School of Business|NYU, Stern]]
* [http://antoniomele.org/financial_economics Financial Economics: Classics and Contemporary] {{Webarchive|url=https://web.archive.org/web/20190423203032/http://antoniomele.org/financial_economics/ |date=2019-04-23 }}, [[Antonio Mele]], [[Università della Svizzera Italiana]]
* [http://www.ulb.ac.be/cours/solvay/farber/PhD.htm Microfoundations of Financial Economics] André Farber, [[Solvay Business School]]
* [http://arquivo.pt/wayback/20160516071307/http%3A//viking.som.yale.edu/will/web_pages/will/finman540/classnotes/notes.html An introduction to investment theory], William Goetzmann, [[Yale School of Management]]
* [http://www.stanford.edu/~wfsharpe/mia/MIA.HTM Macro-Investment Analysis]. [[William F. Sharpe]], [[Stanford Graduate School of Business]]
* [http://ocw.mit.edu/courses/sloan-school-of-management/15-401-finance-theory-i-fall-2008/ Finance Theory] ([[MIT OpenCourseWare]]). [[Andrew Lo]], [[MIT]].
* [http://oyc.yale.edu/economics/econ-251 Financial Theory] ([[Open Yale Courses]]). [[John Geanakoplos]], [[Yale University]].
* {{Cite web |url=http://homepage.newschool.edu/~het/essays/capital/invest.htm |title=The Theory of Investment |access-date=March 3, 2014 |archive-url=https://web.archive.org/web/20120621123950/http://homepage.newschool.edu/~het/essays/capital/invest.htm |archive-date=June 21, 2012 |url-status=dead }}. G.L. Fonseca, [[New School for Social Research]]
* [https://www.ma.utexas.edu/rtgs/applied/school2009/rtg09_trans.pdf Introduction to Financial Economics]. Gordan Zitkovi, [[University of Texas at Austin]]
* [http://jhqian.org/apt/apbook.pdf An Introduction to Asset Pricing Theory], Junhui Qian, [[Shanghai Jiao Tong University]]
{{refend}}

| valign="top" |
{{refbegin|30em}}
'''Links and portals'''
* [http://www.aeaweb.org/jel/guide/jel.php?class=G JEL Classification Codes Guide]
* [http://rfe.org/showCat.php?cat_id=56 Financial Economics Links on AEA's RFE] {{Webarchive|url=https://web.archive.org/web/20171104233339/http://rfe.org/showCat.php?cat_id=56 |date=2017-11-04 }}
* [http://www.ssrn.com/en/index.cfm/fen/ SSRN Financial Economics Network]
* [http://www.economicsnetwork.ac.uk/subjects/Financial%20Economics Financial Economics listings on economicsnetwork.ac.uk]
* [https://fic.wharton.upenn.edu/policy-briefs/financial-economists-roundtable-fer/ Financial Economists Roundtable] {{Webarchive|url=https://web.archive.org/web/20171107025720/https://fic.wharton.upenn.edu/policy-briefs/financial-economists-roundtable-fer/ |date=2017-11-07 }}
* [http://www.nber.org/jel/G_index.html NBER Working Papers in Financial Economics]
* [https://web.archive.org/web/20140313133821/http://www.qfinance.com/information-sources?section=financial-economics Financial Economics Resources on QFINANCE] ([[Internet Archive|archived]] 2014-03-13)
* [https://web.archive.org/web/20160324132025/http://www.helsinki.fi/WebEc/webecg.html Financial Economics Links on WebEc] ([[Internet Archive|archived]] 2016-03-24)

'''Actuarial resources'''
* [http://www.soa.org/education/exam-req/edu-exam-mfe-detail.aspx "Models for Financial Economics (MFE)"] {{Webarchive|url=https://web.archive.org/web/20220118190929/https://www.soa.org/education/exam-req/edu-exam-mfe-detail.aspx |date=2022-01-18 }}, [[Society of Actuaries]]
* [http://www.actuaries.org.uk/students/pages/ct8-financial-economics "Financial Economics (CT8)"], [[Institute and Faculty of Actuaries]]
* [https://web.archive.org/web/20140326125149/http://66.216.104.121/files/sections/prime.pdf "A Primer In Financial Economics"], S. F. Whelan, D. C. Bowie and A. J. Hibbert. ''[[British Actuarial Journal]]'', Volume 8, Issue 1, April 2002, pp.&nbsp;27–65.
* [http://www.soa.org/files/sections/actuary-journal-final.pdf "Pension Actuary's Guide to Financial Economics"]. Gordon Enderle, Jeremy Gold, Gordon Latter and Michael Peskin. [[Society of Actuaries]] and [[American Academy of Actuaries]].
{{refend}}
|}

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