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{{Short description|Academic discipline concerned with the exchange of money}}
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'''Financial economics''' is the branch of [[Finance:Economics|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">[[Biography:William F. Sharpe|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 }}</ref> 
Its concern is thus the interrelation of financial variables, such as share prices, [[Finance:Interest rate|interest rate]]s and exchange rates, as opposed to those concerning the [[Finance:Real economy|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> [[Finance:Asset pricing|asset pricing]] and [[Finance:Corporate finance|corporate finance]]; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.
It thus provides the theoretical underpinning for much of [[Finance:Finance|finance]].

The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".<ref>[[Biography:Robert C. Merton|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 [[Finance:Economic model|economic]] and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. 
It thus also includes a formal study of the financial markets themselves, especially [[Finance:Market microstructure|market microstructure]] and [[Finance:Financial regulation|market regulation]].
It is built on the foundations of [[Finance:Microeconomics|microeconomics]] and [[Decision theory|decision theory]].

[[Finance:Financial econometrics|Financial econometrics]] is the branch of financial economics that uses [[Econometrics|econometric]] techniques to parameterise the relationships identified.
[[Finance:Mathematical finance|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, [[Finance:Monetary economics|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> \sum_{s}p_{s}=1</math>}}
:{{small|<math>Y_{s}</math> risk aversion factors by state, normalized s.t. <math>\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>\sum_{s}\pi_{s} = 1/r</math>}}
|}
Financial economics studies how [[Homo economicus|rational investors]] would apply [[Decision theory|decision theory]] to [[Social:Investment management|investment management]]. The subject is thus built on the foundations of [[Finance:Microeconomics|microeconomics]] and derives several key results for the application of decision making under uncertainty to the [[Finance:Financial market|financial market]]s. The underlying economic logic yields the [[Fundamental theorem of asset pricing|fundamental theorem of asset pricing]], which gives the conditions for [[Arbitrage|arbitrage]]-free asset pricing.<ref name="Rubinstein"/><ref name="Cochrane & Culp"/>
The aside formulae result directly.

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

Calculating their present value <math>X_{sj}/r</math> allows the decision maker to aggregate the cashflows (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 the starting point for financial decision making. {{NoteTag|Its history is correspondingly early: [[Biography:Fibonacci|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 [[Biography:Johan de Witt|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 [[Biography:Edmond Halley|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>}}

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 [[Biography:Blaise Pascal|Blaise Pascal]] and [[Biography:Pierre de Fermat|Pierre de Fermat]] in 1654.}}

This decision method, however, fails to consider [[Risk aversion|risk aversion]] ("as any student of finance knows"<ref name="Rubinstein"/>). 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 to therefore "adjust" the weight assigned to the various outcomes ("states") correspondingly, <math>Y_{s}</math>. See [[Finance:Indifference price|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 characterized as the maximization of expected utility. More formally, the resulting [[Expected utility hypothesis|expected utility hypothesis]] states that, if certain axioms are satisfied, the [[Finance: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 arise 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 [[Biography:Daniel Bernoulli|Daniel Bernoulli]] in 1738, which was later formalized by [[Biography:John von Neumann|John von Neumann]] and [[Biography:Oskar Morgenstern|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 [[Finance:Monetary economics|Monetary]] and International Economics and precedes [[Finance:Public economics|Public Economics]]. For detailed subclassifications see {{section link|JEL classification codes|G. Financial Economics}}.

''[[The New Palgrave Dictionary of Economics]]'' (2008, 2nd ed.) also uses the JEL codes to classify its entries in v. 8, Subject Index, including Financial Economics at pp.&nbsp;863–64. The below have links to entry [[Abstract (summary)|abstracts]] of The New Palgrave [https://web.archive.org/web/20130210185054/http://www.dictionaryofeconomics.com/dictionary Online] for each primary or secondary JEL category (10 or fewer per page, similar to [[Company:Google|Google]] searches):
:[http://www.dictionaryofeconomics.com/search_results?,q=&field=content&edition=all&topicid=G JEL: G] – Financial Economics
:[http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G0 JEL: G0] – General
:[http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G1 JEL: G1] – [[Finance:Financial market|General Financial Markets]]
:[http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G2 JEL: G2] – [[Finance:Financial institution|Financial institution]]s and [[Finance:Financial services|Services]]
:[http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G3 JEL: G3] – [[Finance:Corporate finance|Corporate finance]] and [[Finance:Corporate governance|Governance]]

Tertiary category entries can also be searched.<ref>For example, http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00 {{Webarchive|url=https://web.archive.org/web/20130529074942/http://www.dictionaryofeconomics.com/search_results?q=&field=content&edition=all&topicid=G00 |date=2013-05-29 }}.</ref>
|}
The concepts of [[Arbitrage|arbitrage]]-free, "rational", pricing and equilibrium are then coupled with the above to derive "classical"<ref name="Rubinstein2">See Rubinstein (2006), under "Bibliography".</ref>  (or [[Finance:Neoclassical economics|"neo-classical"]]<ref name="Derman"/>) financial economics.

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

[[Finance:Economic equilibrium|Economic equilibrium]] is, in general, 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 do not allow for 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 are therefore not in equilibrium.<ref name="Delbaen_Schachermayer"/> An arbitrage equilibrium is thus a precondition for a general economic equilibrium.

The immediate, and formal, extension of this idea, the [[Fundamental theorem of asset pricing|fundamental theorem of asset pricing]], shows that where markets are as described – and are additionally (implicitly and correspondingly) [[Finance:Complete market|complete]] – one may then make financial decisions by constructing a [[Finance:Risk-neutral measure|risk neutral probability measure]] corresponding to the market.

"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 [[Finance: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> (=<math>1-q_{up}</math>) are the two corresponding probabilities, and in turn, the derived distribution, or [[Probability measure|"measure"]].

The formal derivation will proceed by arbitrage arguments.<ref name="Rubinstein"/><ref name="Delbaen_Schachermayer">Freddy Delbaen and Walter Schachermayer. (2004). [https://www.ams.org/notices/200405/what-is.pdf "What is... a Free Lunch?"] {{Webarchive|url=https://web.archive.org/web/20160304061252/http://www.ams.org/notices/200405/what-is.pdf |date=2016-03-04 }} (pdf). Notices of the AMS 51 (5): 526–528</ref> 
The analysis here is often undertaken assuming a ''[[Finance:Representative agent|representative agent]]'', 
<ref name="Farmer_Geanakoplos"/> 
essentially treating all market-participants, "[[Philosophy:Agent (economics)|agents]]", as identical (or, at least, that they [[Finance: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 the problems are then mathematically tractable.

With this measure in place, the expected, i.e. required, return of any security (or portfolio) will then equal the riskless return, plus an "adjustment for risk",<ref name="Rubinstein"/> i.e. a security-specific [[Risk premium|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 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.

Thus, continuing the example, in pricing a [[Finance:Derivative|derivative instrument]] its forecasted cashflows in the 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 [[Finance: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. In general, this premium may be derived by the [[Finance:Capital asset pricing model|CAPM]] (or extensions) as will be seen under {{slink|#Uncertainty}}.

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

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

===State prices===
With the above relationship established, the further specialized [[Finance:Arrow–Debreu model|Arrow–Debreu model]] may be derived. 
{{NoteTag|State prices originate with [[Biography:Kenneth Arrow|Kenneth Arrow]] and [[Biography:Gérard Debreu|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>
[[Biography:Lionel W. McKenzie|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> 
Breeden and Litzenberger's work in 1978<ref>{{cite journal |title=Prices of State-Contingent Claims Implicit in Option Prices |first1=Douglas T. |last1=Breeden |first2=Robert H. |last2=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 [[Finance:Complete market|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; 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 [<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 integrating over the state price "density". These concepts are extended to [[Finance:Martingale pricing|martingale pricing]] and the related [[Finance:Risk-neutral measure|risk-neutral measure]].

State prices find immediate application as a conceptual tool ("[[Finance:Contingent claim analysis|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 = [[Finance:Annual Review of Financial Economics|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|stochastic discount factor]] - 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: [[Biography:David K. Backus|David K. Backus]] (2015). [http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf Fundamentals of Asset Pricing], Stern NYU</ref>
the third equation above.
Essentially, this factor divides expected 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 [[Finance:Marginal rate of substitution|marginal rate of substitution]]".

==Resultant models==
{| 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|[[Finance:Valuation using discounted cash flows#Basic formula for firm valuation using DCF model|DCF valuation formula]], where the [[Finance:Business valuation|value of the firm]], is its forecasted [[Finance:Free cash flow|free cash flow]]s discounted to the present using the [[Finance:Weighted average cost of capital|weighted average cost of capital]].
For [[Finance:Stock valuation|share valuation]] investors use the related [[Finance:Dividend discount model|dividend discount model]].}}
|}
[[Image:MM2.png|thumb|right|Modigliani–Miller Proposition II with risky debt. Even if [[Finance:Leverage|leverage]] (D/E) increases, the WACC (k0) stays constant.]]
[[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, [[Finance:Capital allocation line|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|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 [[Finance:Required rate of return|expected return]] used when discounting cashflows on an asset <math>i</math>, is the risk-free rate plus the [[Finance:Capital asset pricing model#Formula|market premium]] multiplied by [[Finance:Beta|beta]] (<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|[[Finance:Security market line|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.]]
[[Image:Stockpricesimulation.jpg|thumb|right|Simulated geometric Brownian motions with parameters from market data]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|{{smalldiv|[[Finance: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>
[[Finance: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 [[Finance: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 rated 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 the 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.}}
|}
Applying the above economic concepts, we may then derive various [[Finance:Economic model|economic-]] and financial models and principles. As above, the two usual 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.
* [[Finance:Option|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 [[Finance: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|stochastic]]".)

===Certainty===
The starting point here is "Investment under certainty", and usually framed in the context of a corporation.
The [[Finance:Fisher separation theorem|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 [[Finance:Modigliani–Miller theorem|Modigliani–Miller theorem]], which shows that, under certain conditions, the value of a firm is unaffected by how that firm is financed, and depends neither on its dividend policy nor its decision to raise capital by issuing stock or selling debt. The proof here proceeds using arbitrage arguments, and acts as a benchmark for evaluating the effects of factors outside the model that do affect value.
{{NoteTag|The theorem of [[Biography:Franco Modigliani|Franco Modigliani]] and [[Biography:Merton Miller|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=[[Organization:The New School|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' ''[[Finance:The Theory of Investment Value|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 [[Finance:Intrinsic value#Equity|"intrinsic"]], long-term worth is the present value of its future net cashflows, in the form of [[Finance:Dividend|dividend]]s. 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. [[Finance:Net present value|Net present value]] (NPV) 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 in 1951.}}

[[Finance:Bond valuation|Bond valuation]], in that cashflows ([[Finance:Coupon|coupons]] and return of principal) are deterministic, may proceed in the same fashion.<ref name="Luenberger">See Luenberger's ''Investment Science'', under Bibliography.</ref> An immediate extension, [[Finance: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 – 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 historic ratios or published policy – and cashflows are then treated as essentially deterministic; see below under {{slink|#Corporate finance theory}}.

These "certainty" results are all commonly employed under corporate finance; uncertainty is the focus of "asset pricing models", as follows. [[Biography:Irving Fisher#Interest and capital|Fisher's formulation]] of the theory here - developing [[Philosophy: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, [[Organization:The New School|The New School]].</ref> for the development.

===Uncertainty===
For [[Decision theory#Choice under uncertainty|"choice under uncertainty"]] the twin assumptions of rationality and [[Finance:Financial market efficiency|market efficiency]], as more closely defined, lead to [[Finance:Modern portfolio theory|modern portfolio theory]] (MPT) with its [[Finance:Capital asset pricing model|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 [[Finance: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 {{slink|Asset pricing|Interrelationship}}.

Briefly, and intuitively – and consistent with {{slink|#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]. ''[[Finance:Financial Analysts Journal|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 [[Finance:Efficient-market hypothesis|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 [[Finance:Earnings response coefficient|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, but what is required, however, is that investors' reactions follow a [[Normal distribution|normal distribution]] – so that the net effect on market prices cannot be reliably exploited 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|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 [[Biography:Eugene Fama|Eugene Fama]] in a 1970 review paper,<ref>Fama, Eugene (1970). "Efficient Capital Markets: A Review of Theory and Empirical Work". Journal of Finance.</ref> consolidating previous works re random walks in stock prices: Jules Regnault (1863); [[Biography:Louis Bachelier|Louis Bachelier]] (1900); [[Biography:Maurice Kendall|Maurice Kendall]] (1953); Paul Cootner (1964); and [[Biography:Paul Samuelson|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; 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 [[Biography:John H. Cochrane|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.|date= 2003|title= From Efficient Markets Theory to Behavioral Finance|journal= [[Finance:Journal of Economic Perspectives|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 [[Biography:Harry Markowitz|Harry Markowitz]] in 1952. The CAPM was derived by Jack Treynor (1961, 1962), [[Biography:William F. Sharpe|William F. Sharpe]] (1964), John Lintner (1965), and Jan Mossin (1966) independently.}}
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 as above,<ref name="Jensen&Smith">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 [[Finance:Markowitz model|proceeds as follows]]: 
If one can construct an [[Finance:Efficient frontier|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 [[Finance:Risk-free interest rate|risk-free asset]] and the "[[Finance:Market portfolio|market portfolio]]" (the [[Finance:Mutual fund separation theorem|Mutual fund separation theorem]]), with the combinations here plotting as the [[Finance:Capital market line|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|covariance]] ("beta") with aggregate, i.e. market, risk. 
This is because investors here can then maximize utility through leverage as opposed to pricing; see [[Finance:Separation property|Separation property]], {{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 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 other equilibrium-pricing models.

Black–Scholes provides a mathematical model of a financial market containing [[Finance:Derivative|derivative]] instruments, and the resultant formula for the price of [[Finance:Option style|European-styled options]].
{{NoteTag|"BSM" – two seminal 1973 papers by Fischer Black and [[Biography:Myron Scholes|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 [[Biography:Robert C. Merton|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 [[Biography:Louis Bachelier|Louis Bachelier]] (1900) and Edward O. Thorp (1967);<ref name="Haug Taleb">Haug, E. G. and [[Biography:Nassim Nicholas Taleb|Taleb, N. N.]] (2008). [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.581.884&rep=rep1&type=pdf Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula], ''Wilmott Magazine'' January 2008</ref> although these were more "actuarial" in flavor, and had not established risk-neutral discounting.<ref name="Derman"/> Vinzenz Bronzin (1908) produced very early results, also.}}
The model is expressed as the Black–Scholes equation, a [[Partial differential equation|partial differential equation]] describing the changing price of the option over time; it is derived assuming log-normal, [[Geometric Brownian motion|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 {{slink|#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|[[Biography:Kiyosi Itô|Kiyosi Itô]] published his Lemma in 1944. [[Biography:Paul Samuelson|Paul Samuelson]]<ref>{{cite journal | author = Samuelson Paul |  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|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 mentioned, it can be shown that the two models 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 fundamental result above – further explaining, and if required demonstrating, this unity.<ref name="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; ~ (2006). Part II, Vol. 4, No. 1, First Quarter 2006. See under "External links".</ref> 
Here, the CAPM is derived 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, in the limit, by attaching a binomial probability to each of numerous possible spot-prices (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 [[Finance:Asset pricing|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===
[[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.]]
The majority of developments here relate to required return, i.e. pricing, extending the basic CAPM. Multi-factor models such as the [[Finance:Fama–French three-factor model|Fama–French three-factor model]] and the [[Finance:Carhart four-factor model|Carhart four-factor model]], propose factors other than market return as relevant in pricing. The [[Finance:Intertemporal CAPM|intertemporal CAPM]] and [[Finance:Consumption-based capital asset pricing model|consumption-based CAPM]] similarly extend the model. With [[Finance:Intertemporal portfolio choice|intertemporal portfolio choice]], the investor now repeatedly optimizes her portfolio; while the inclusion of [[Finance: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 [[Finance:Single-index model|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 [[Finance:Arbitrage pricing theory|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 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|language=en|volume=13|issue=3|pages=341–360|doi=10.1016/0022-0531(76)90046-6|issn=0022-0531}}
</ref>
The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers.}}

As regards [[Finance:Portfolio optimization|portfolio optimization]], the [[Finance:Black–Litterman model|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> 
departs from the original [[Finance:Markowitz model|Markowitz model]] – i.e. of constructing portfolios via an [[Finance:Efficient frontier|efficient frontier]]. Black–Litterman instead starts with an equilibrium assumption, and 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"]. [[Company:Goldman Sachs|Goldman Sachs]] Quantitative Resources Group</ref> asset allocation. Where factors additional to volatility are considered (kurtosis, skew...) then [[Multiple-criteria decision analysis|multiple-criteria decision analysis]] can be applied; here deriving a Pareto efficient portfolio. The [[Universal portfolio algorithm|universal portfolio algorithm]] applies [[Machine learning|machine learning]] to asset selection, learning adaptively from historical data. [[Finance:Behavioral portfolio theory|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 [[Machine learning#Applications|Machine learning, more generally]].  
[[Finance:Tail risk parity|(Tail)]] [[Risk parity|risk parity]] focuses on allocation of risk, rather than allocation of capital.
{{NoteTag|The universal portfolio algorithm  was published by [[Biography:Thomas M. Cover|Thomas M. Cover]] in 1991. The Black–Litterman model was developed in 1990 at [[Company:Goldman Sachs|Goldman Sachs]] by Fischer Black and Robert Litterman, and published in 1991.}}
See {{section link|Portfolio optimization|Improving portfolio optimization}} for other techniques and objectives, and {{slink|Financial risk management|Investment management}} for discussion.

===Derivative pricing===
[[File:Arbre Binomial Options Reelles.png|thumb|right| Binomial Lattice with 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|[[Finance: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, [[Organization:Florida State University|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 [[Finance:Short-rate model|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|binomial options pricing model]] provides a discretized version of Black–Scholes, useful for the valuation of American styled options. Discretized models of this type are built – at least implicitly – using state-prices (as above); relatedly, a large number of researchers 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 [[Finance: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 [[Finance:Option#Model implementation|other numeric techniques]] have also been developed. The theoretical framework too has been extended such that [[Finance:Martingale pricing|martingale pricing]] is now the standard approach.
{{NoteTag|
The binomial model was first proposed by [[Biography:William F. Sharpe|William Sharpe]] in the 1978 edition of ''Investments'' ({{ISBN|013504605X}}), 
and in 1979 formalized by Cox, Ross and Rubinstein 
<ref>{{Cite journal |last1=Cox |first1=J. C. |last2=Ross |first2=S. A. |last3=Rubinstein |first3=M. |authorlink3=Mark Rubinstein|doi=10.1016/0304-405X(79)90015-1 |title=Option pricing: A simplified approach |journal=[[Finance:Journal of Financial Economics|Journal of Financial Economics]]|volume=7 |issue=3 |page=229 |year=1979|citeseerx=10.1.1.379.7582 }}</ref> 
and by 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 [[Biography:Eduardo Schwartz|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= [[Finance:Journal of Financial Economics|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 [[Biography:Phelim Boyle|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}}</ref>
In 1996, methods were developed for 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 [[Finance:Asian option|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 "[[Finance:Contingent claim analysis|contingent claim analysis]]"). [[Finance:Real options valuation|Real options valuation]] allows that option holders can influence the option's underlying; models for [[Finance:Employee stock option#Valuation|employee stock option valuation]] explicitly assume non-rationality on the part of option holders; [[Finance:Credit derivative|Credit derivative]]s allow that payment obligations or delivery requirements might not be honored. [[Finance:Exotic derivative|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 [[Finance:Short-rate model|short-rate model]]s allow for an extension of these techniques to [[Finance:Fixed income#Derivatives|fixed income-]] and [[Finance:Interest rate derivative|interest rate derivative]]s. (The [[Finance:Vasicek model|Vasicek]] and [[Cox–Ingersoll–Ross model|CIR]] models are equilibrium-based, while [[Finance:Ho–Lee model|Ho–Lee]] and subsequent models are based on arbitrage-free pricing.) The more general [[Finance:Heath–Jarrow–Morton framework|HJM Framework]] describes the dynamics of the full [[Finance: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 [[Finance: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=[[Finance:Journal of Financial Economics|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 [[Biography: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 [[Finance:Black Monday (1987)|Crash of 1987]], equity options traded in American markets began to exhibit what is known as a "[[Finance:Volatility smile|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 [[Finance: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 in the next section.

After the [[Finance:Financial crisis of 2007–2008|financial crisis of 2007–2008]], 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 – [[Finance:Prudential Regulation Authority (United Kingdom)|Prudential Regulation Authority]]</ref> ([[Finance:Over-the-counter|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|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, [[Organization:Aarhus University|Aarhus University]]</ref> and a [[Credit valuation adjustment|credit valuation adjustment]], or CVA – and potentially other ''valuation adjustments'', collectively [[XVA|xVA]] – is generally added to the risk-neutral derivative value.
The economic arguments underlying derivatives valuation 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 (OIS) curve, as opposed to LIBOR as used previously.<ref name="Youmbi"/> This is because post-crisis, the [[Finance:Overnight rate|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=[[Finance:Journal of Investment Management|Journal of Investment Management]] |volume=11 |issue=3 |year=2013 |pages=14–27}}</ref> (Also, practically, the interest paid on cash [[Finance:Collateral|collateral]] is usually the overnight rate; OIS discounting is then, sometimes, referred to as "CSA discounting".) [[Finance:Swap#Valuation|Swap pricing]] – and, therefore, [[Finance:Yield curve|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 "[[Finance:Multi-curve framework|multi-curve framework]]" where "forecast curves" are constructed for each floating-leg [[Finance:Libor#Maturities|LIBOR tenor]], with discounting on the ''common'' OIS curve.

===Corporate finance theory===
[[Image:Manual decision tree.jpg|right|thumb|Project valuation via decision tree.]]
Corporate finance theory has also been extended: mirroring the above developments, asset-valuation and decisioning no longer need assume "certainty". 
[[Finance:Monte Carlo methods in finance|Monte Carlo methods in finance]] allow financial analysts to construct "[[Stochastic|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 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"/> 
{{NoteTag|Simulation was first applied to (corporate) finance by [[Biography:David B. Hertz|David B. Hertz]] in 1964; Real options in corporate finance were first discussed by Stewart Myers in 1977.}}

More traditionally, [[Decision tree|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 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"/> {{NoteTag|This technique predates the use of real options in corporate finance;<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> it is borrowed from [[Operations research|operations research]], and is not a "financial economics development" ''per se''.}}

Related to this, is the treatment of forecasted cashflows in equity valuation. In many cases, following Williams 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 [[Finance:Financial modeling#Accounting|Financial modeling § Accounting]]. 
In more modern treatments, then, it is the ''expected'' cashflows (in the [[Expected value|mathematical sense]]: {{small|<math>\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 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> 
[[Finance:Clean surplus accounting|Clean surplus accounting]] and the related [[Finance:Residual income valuation|residual income valuation]] provide a model that returns price as a function of earnings, expected returns, and change in [[Finance:Book value|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|Irrelevance of dividend policy}}.

"Corporate finance" as a discipline more generally, per Fisher above, relates to the long term objective of maximizing the [[Finance:Enterprise value|value of the firm]] - and its [[Finance:Total shareholder return|return to shareholders]] - and thus also incorporates the areas of [[Finance:Capital structure|capital structure]] and [[Finance:Dividend policy|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) [[Finance:Corporate finance#Capitalization structure|optimization re capitalization structure]], and theories here as to corporate choices and behavior: [[Finance:Capital structure substitution theory|Capital structure substitution theory]], [[Finance:Pecking order theory|Pecking order theory]], [[Finance:Market timing hypothesis|Market timing hypothesis]], [[Finance:Trade-off theory of capital structure|Trade-off theory]];
(ii) [[Finance:Corporate finance#Dividend policy|considerations and analysis re dividend policy]], additional to - and sometimes contrasting with - Modigliani-Miller, include: 
the [[Finance:Dividend policy#Walter's model|Walter model]], Lintner model, and  [[Finance:Dividend policy#Residuals theory of dividends|Residuals theory]], as well as discussion re the observed [[Finance:Clientele effect|clientele effect]] and [[Finance:Dividend puzzle|dividend puzzle]].

As described, the typical application of real options is to [[Finance:Capital budgeting|capital budgeting]] type problems. 
However, here, they are 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 related agency problems.
<ref name="Damodaran">{{cite journal|last=Damodaran|first=Aswath|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, as above, which is then applied to the analysis. For example, [[Finance:Convertible bond|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===
The discipline, as outlined, also includes a formal study of [[Finance:Financial market|financial market]]s. Of interest especially are market regulation and [[Finance:Market microstructure|market microstructure]], and their relationship to [[Finance:Financial market efficiency|price efficiency]].

[[Finance:Regulatory economics|Regulatory economics]] studies, in general, the economics of regulation. In the context of finance, it will address the impact of [[Finance:Financial regulation|financial regulation]] on the functioning of markets and the efficiency of prices, while also weighing the corresponding increases in market confidence and [[Finance:Financial stability|financial stability]].
Research here considers how, and to what extent, regulations relating to disclosure ([[Finance:Earnings guidance|earnings guidance]], [[Finance:Annual report|annual report]]s), [[Insider trading|insider trading]], and [[Finance:Short#Regulations|short-selling]] will impact price efficiency, the [[Finance:Cost of equity|cost of equity]], and [[Finance:Market liquidity|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-, [[Finance:Fisher market|Fisher-]], and 
Arrow-Debreu markets as prototypes), and "analyzes how specific trading mechanisms affect the price formation process",<ref>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 [[Finance: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 [[Finance:Equity premium puzzle|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) [[Finance:Frictionless market|market frictions]] and other imperfections;
see also [[Market design|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 [[Finance:Agent-based computational economics#Example: finance|examine any impact]] due to a change in structure or policy - or [[Finance:Artificial economics#Method|to make inferences]] re market dynamics - by testing these in an artificial financial market, or AFM.
{{NoteTag|The Benchmark here is the pioneering AFM of the [[Organization:Santa Fe Institute|Santa Fe Institute]]  developed in the early 1990s. See <ref name="LeBaron2"/> for discussion of other early models.}}
This approach, essentially [[Computer experiment|simulated]] trade between numerous [[Philosophy:Agent (economics)|agents]], "typically uses [[Artificial intelligence|artificial intelligence]] technologies [often genetic algorithms and [[Artificial neural network|neural nets]]] to represent the 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'' [[Organization:Erasmus Research Institute of Management|ERIM]] Report Series</ref>

These [[Finance: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 A'', 1, 20.</ref> with participants modifying their trading strategies having learned over time, and "are able to describe macro features [i.e. [[Finance:Stylized fact|stylized fact]]s] [[Philosophy:Emergence#Economics|emerging]] from a soup of individual interacting strategies".<ref name="LeBaron2"/>
Agent-based models depart further from the classical approach — the [[Finance:Representative agent|representative agent]], as outlined — in that they introduce [[Finance:Heterogeneity in economics|heterogeneity]] into the environment (thereby addressing, also, the [[Finance:Aggregation problem|aggregation problem]]).

==Challenges and criticism==
<!-- |Financial Modelers' Manifesto|Unreasonable ineffectiveness of mathematics#Economics and finance|Physics envy -->
As above, there is a very close link between (i) the [[Random walk hypothesis|random walk hypothesis]], with the associated belief that price changes should follow a [[Normal distribution|normal distribution]], on the one hand, and (ii) 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===
<!-- 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 [[Finance:Call option|call option]]. Although lately its use is [[Finance: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 option delta, and the days to maturity.]]

As discussed, the assumptions that market prices follow a [[Random walk|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 [[Finance:Financial risk management#Banking|and risk managers]] frequently modify the "standard models" (see [[Kurtosis risk]], [[Skewness risk]], [[Long tail]], [[Model risk]]). 
In fact, [[Biography:Benoit Mandelbrot|Benoit Mandelbrot]] had discovered already in the 1960s 
<ref>{{cite journal |title=The Variation of Certain Speculative Prices |first= Benoit |last=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|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|journal=[[Finance:Journal of Applied Corporate Finance|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|value at risk]] models with [[Finance:Historical simulation|historical simulations]], [[Mixture model#A financial model|mixture models]], [[Principal component analysis|principal component analysis]], [[Extreme value theory|extreme value theory]], as well as models for [[Finance:Volatility clustering|volatility clustering]].<ref>See for example III.A.3, in Carol Alexander, ed. (January 2005). ''The Professional Risk Managers' Handbook''. PRMIA Publications. {{ISBN|978-0976609704}}</ref> 
For further discussion see {{section link|Fat-tailed distribution|Applications in economics}}, and {{section link|Value at risk|Criticism}}.
Portfolio managers, likewise, have modified their optimization criteria and algorithms; see {{slink|#Portfolio theory}} above.

Closely related is the [[Finance:Volatility smile|volatility smile]], where, as above, [[Finance:Implied volatility|implied volatility]] – the volatility corresponding to the BSM price – is observed to ''differ'' as a function of [[Finance:Strike price|strike price]] (i.e. [[Finance:Moneyness|moneyness]]), true only if the price-change distribution is non-normal, unlike that assumed by BSM. 
The term structure of volatility describes how (implied) volatility differs for related options with different maturities. 
An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure. 
These empirical phenomena negate the assumption of constant volatility – and log-normality – 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 [[Finance:Yield to maturity|YTM]]; see {{section link|Black–Scholes model|The volatility smile}}.

In consequence traders ([[Finance:Financial risk management#Banking|and risk managers]]) now, instead, use "smile-consistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. non-quoted, strike/maturity combinations, or of non-European derivatives, and generally for hedging purposes.  
The two main approaches are [[Finance:Local volatility|local volatility]] and [[Finance:Stochastic volatility|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 [[Finance:Stochastic volatility#Calibration and estimation|calibrated to observed prices]], and are then applied to the valuation or hedging in question; the most common are [[Finance:Heston model|Heston]], [[Finance:SABR volatility model|SABR]] and [[Finance: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=0|journal=Wilmott Magazine |issue=Sep|year=2002 |pages=84–108}}</ref>

Related to local volatility are the [[Finance:Lattice model|lattice]]-based [[Finance:Implied binomial tree|implied-binomial]] and [[Finance: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. [[Finance:Edgeworth binomial tree|Edgeworth binomial tree]]s allow for a specified (i.e. non-Gaussian) [[Skewness|skew]] and [[Kurtosis|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: Jarrow and Rudd (1982); Corrado and Su (1996); Brown and Robinson (2002); [[Biography:David K. Backus|Backus]], Foresi, and Wu (2004). 
See: Emmanuel Jurczenko, Bertrand Maillet, and Bogdan 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. This is valid as the surface is built on price data relating to fully collateralized positions, and there is therefore no "[[Finance: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 [[Finance:Financial engineering|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, James Rickards, amongst other critics <ref name="Taleb_Mandelbrot"/> of financial economics, suggests that, instead, the theory needs revisiting almost entirely: 
:"The  current  system,  based  on  the  idea  that  risk  is  distributed  in  the  shape  of  a  bell  curve,  is  flawed...  The  problem  is  [that economists and practitioners]  never  abandon  the  bell  curve.  They  are  like  medieval  astronomers  who believe the sun revolves around the earth and are [[Geocentric model#Ptolemaic system|furiously tweaking their geo-centric  math]]  in  the  face  of  contrary  evidence.  They  will  never  get  this  right; [[Astronomy: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 Subcommittee  on  Investigations  and  Oversight, Committee  on  Science  and  Technology, House of Representatives, One  Hundred  Eleventh  Congress, first  session, September 10, 2009</ref>

===Departures from rationality===
{|class="wikitable floatright" | width="200"
|- style="font-size:75%"
|-align="center"
|colspan="1" | [[Finance:Market anomaly|Market anomalies]] and economic puzzles
|-
| rowspan="2" |
* [[Finance:Calendar effect|Calendar effect]]
** [[Finance:January effect|January effect]]
** [[Finance:Sell in May|Sell in May]]
** Mark Twain effect
** [[Finance:Santa Claus rally|Santa Claus rally]]
* Closed-end fund puzzle
* [[Finance:Dividend puzzle|Dividend puzzle]]
* [[Finance:Equity home bias puzzle|Equity home bias puzzle]]
* [[Finance:Equity premium puzzle|Equity premium puzzle]]
* [[Finance:Forward premium anomaly|Forward premium anomaly]]
* [[Finance:Low-volatility anomaly|Low-volatility anomaly]]
* [[Finance:Momentum|Momentum anomaly]]
* [[Finance:Neglected firm effect|Neglected firm effect]]
* [[Finance:Post-earnings-announcement drift|Post-earnings-announcement drift]]
* [[Finance:Real exchange-rate puzzles|Real exchange-rate puzzles]]
|-
|}
As seen, a common assumption is that financial decision makers act rationally; see [[Homo economicus]]. Recently, however, researchers in [[Finance:Experimental economics|experimental economics]] and [[Finance:Experimental finance|experimental finance]] have challenged this assumption [[Empirical evidence|empirically]]. These assumptions are also challenged [[Theory|theoretically]], by behavioral finance, a discipline primarily concerned with the limits to rationality of economic agents.
{{NoteTag|
An early anecdotal treatment is Benjamin Graham's "[[Finance:Mr. Market|Mr. Market]]", discussed in his ''[[Finance:The Intelligent Investor|The Intelligent Investor]]'' in 1949. 
See also [[Biography:John Maynard Keynes|John Maynard Keynes]]' 1936 discussion of [[Finance:Animal spirits (Keynes)|"Animal spirits"]], and the related [[Keynesian beauty contest]], in his ''General Theory'', Ch. 12.
''Extraordinary Popular Delusions and the Madness of Crowds'' is a study of crowd psychology by Scottish journalist 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>

Consistent with, and complementary to these findings, various persistent [[Finance:Market anomaly|market anomalies]] have been documented, these being price or return distortions – e.g. [[Finance:Size premium|size premium]]s – which appear to contradict the [[Finance:Efficient-market hypothesis|efficient-market hypothesis]]; [[Finance:Calendar effect|calendar effect]]s are the best known group here. Related to these are various of the economic puzzles, concerning phenomena similarly contradicting the theory. The ''[[Finance:Equity premium puzzle|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|risk premium]] rational equity investors should demand, an "[[Finance:Abnormal return|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 particularly following the [[Finance:Financial crisis of 2007–2008|financial crisis of 2007–2008]], financial economics and [[Finance:Mathematical finance|mathematical finance]] have been subjected to deeper criticism; notable here is [[Biography:Nassim Nicholas Taleb|Nassim Nicholas Taleb]], who claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading; see [[Philosophy:Black swan theory|Black swan theory]], [[Taleb distribution]]. 
A topic of general interest has thus been financial crises,
<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<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 [[Biography:L. Randall Wray|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 the failure of (financial) economics to model (and predict) these.

A related problem is [[Finance:Systemic risk|systemic risk]]: 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: [[Finance:Systemic risk#Inadequacy of classic valuation models|Systemic risk § Inadequacy of classic valuation models]]; [[Finance:Cascades in financial networks|Cascades in financial networks]]; [[Finance:Flight-to-quality|Flight-to-quality]].

Areas of research 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> [[Finance:Noise trader|noise trading]], [[Finance:Market microstructure|market microstructure]] (as above), and Heterogeneous agent models. The latter is extended to [[Finance:Agent-based computational economics|agent-based computational models]], as mentioned; 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> price is treated as an emergent phenomenon, resulting from the interaction of the various market participants (agents). The [[Finance:Noisy market hypothesis|noisy market hypothesis]] argues that prices can be influenced by speculators and momentum traders, as well as by [[Insider trading|insiders]] and institutions that often buy and sell stocks for reasons unrelated to fundamental value; see [[Finance:Noise (economic)|Noise (economic)]]. The adaptive market hypothesis is an attempt to reconcile the efficient market hypothesis with behavioral economics, by applying the principles of [[Biology:Evolution|evolution]] to financial interactions. An [[Information cascade|information cascade]], alternatively, shows market participants engaging in the same acts as others ("[[Biology:Herd behavior|herd behavior]]"), despite contradictions with their private information. [[Copula (probability theory)#Quantitative finance|Copula-based modelling]] has similarly been applied. See also [[Biography:Hyman Minsky|Hyman Minsky]]'s [[Biography:Hyman Minsky#Minsky's financial instability-hypothesis|"financial instability hypothesis"]], as well as  George Soros' application of [[Social:Reflexivity (social theory)#In economics|"reflexivity"]].

On the obverse, however, various studies have shown that despite these departures from efficiency, asset prices do typically exhibit a random walk and that one cannot therefore consistently outperform market averages, i.e. attain "alpha".<ref>[[Biography:William F. Sharpe|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> The practical implication, therefore, is that passive investing (e.g. via low-cost [[Finance:Index fund|index fund]]s) should, on average, serve better than any other active strategy.<ref name=two>[[Biography:William F. Sharpe|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> {{NoteTag|Burton Malkiel's ''[[Finance:A Random Walk Down Wall Street|A Random Walk Down Wall Street]]'' – first published in 1973, and in its 13th edition as of 2023 – is a widely read popularization of these arguments.  See also John C. Bogle's ''[[Finance:Common Sense on Mutual Funds|Common Sense on Mutual Funds]]''; but compare Warren Buffett's ''[[Finance:The Superinvestors of Graham-and-Doddsville|The Superinvestors of Graham-and-Doddsville]]''.}}
Relatedly, institutionally inherent ''[[Finance:Limits to arbitrage|limits to arbitrage]]'' – as opposed to factors directly contradictory to the theory – are sometimes proposed as an explanation for these departures from efficiency.

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

{{div col end}}

==Historical notes==
{{NoteFoot}}

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

==Bibliography==
{{refbegin|30em}}
'''Financial economics'''
* {{cite book | author= Roy E. Bailey | title=The Economics of Financial Markets | publisher={{wipe|Cambridge University Press}} | year=2005| isbn=978-0521612807}}
* {{cite book | author= Marcelo Bianconi| title= Financial Economics, Risk and Information | publisher= [[Company:World Scientific|World Scientific]] | year=2013| isbn=978-9814355131 | edition= 2nd }}
* {{cite book | author= Zvi Bodie, [[Biography:Robert C. Merton|Robert C. Merton]] and David Cleeton | title=Financial Economics | publisher=Prentice Hall | year=2008| isbn=978-0131856158 | edition=2nd }}
* {{cite book | author= James Bradfield | title= Introduction to the Economics of Financial Markets | publisher= Oxford University Press | year=2007| isbn=978-0-19-531063-4 }}
* {{cite book |author1=Satya R. Chakravarty | title=An Outline of Financial Economics | publisher=Anthem Press | year=2014| isbn=978-1783083367 }}
* {{cite book | author= Jakša Cvitanić and Fernando Zapatero| title= Introduction to the Economics and Mathematics of Financial Markets | publisher=MIT Press| year=2004| isbn=978-0262033206}}
* {{cite book |editor=George M. Constantinides |editor2=Milton Harris |editor3=René M. Stulz | url=http://econpapers.repec.org/bookchap/eeefinchp/ | title=Handbook of the Economics of Finance| publisher=[[Company:Elsevier|Elsevier]] | year=2003| isbn=978-0444513632}}
* {{cite book |author1=Keith Cuthbertson |author2=Dirk Nitzsche | title=Quantitative Financial Economics: Stocks, Bonds and Foreign Exchange | publisher=Wiley | year=2004| isbn=978-0470091715 }}
* {{cite book | author= [[Biography:Jean-Pierre Danthine|Jean-Pierre Danthine]], John B. Donaldson | title=Intermediate Financial Theory| publisher= [[Company:Academic Press|Academic Press]] | year=2005| isbn=978-0123693808| edition=2nd}}
* {{cite book | author= Louis Eeckhoudt |author2=Christian Gollier, Harris Schlesinger | title=Economic and Financial Decisions Under Risk | publisher=Princeton University Press | year=2005| isbn=978-0-691-12215-1}}
* {{cite book |author1=Jürgen Eichberger |author2=Ian R. Harper | title=Financial Economics | publisher= Oxford University Press | year=1997| isbn=978-0198775409 }}
* {{cite book |author=Igor Evstigneev |author2=Thorsten Hens |author3=Klaus Reiner Schenk-Hoppé | title=Mathematical Financial Economics: A Basic Introduction  | publisher=Springer | year=2015| isbn=978-3319165707}}
* {{cite book | author= Frank J. Fabozzi, Edwin H. Neave and Guofu Zhou | title=Financial Economics | publisher= Wiley | year=2011| isbn=978-0470596203}}
* {{cite book | author= Eugene F. Fama and Merton H. Miller | title=The Theory of Finance | publisher=Holt, Rinehart and Winston | year=1972| isbn=0030867320|url=https://faculty.chicagobooth.edu/eugene.fama/research/Theory%20of%20Finance/The%20Theory%20of%20Finance%20Preface%20and%20Table%20of%20Contents.pdf}}
* {{cite book | author= Christian Gollier | title= The Economics of Risk and Time | publisher=MIT Press | year=2004| isbn=978-0-262-57224-8 | edition= 2nd }}
* {{cite book | author= Thorsten Hens and Marc Oliver Rieger | title=Financial Economics: A Concise Introduction to Classical and Behavioral Finance| publisher= [[Company:Springer Publishing|Springer]] | year=2010| isbn=978-3540361466}}
* {{cite book | author= Chi-fu Huang and 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| }}
* {{cite book | author= Robert A. Jarrow | title=Finance theory | publisher=Prentice Hall| year=1988| isbn=978-0133148657| }}
* {{cite book | author= Chris Jones | title=Financial Economics| publisher= [[Company:Routledge|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=[[Company: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|  edition=3rd}}
* {{cite book |editor=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| }}
* {{cite book | author= Darrell Duffie | title=Dynamic Asset Pricing Theory| publisher= Princeton University Press | year=2001| isbn=978-0691090221 |  edition=3rd}}
* {{cite book |author=Edwin J. Elton |author2=Martin J. Gruber |author3=Stephen J. Brown |author4=William N. Goetzmann | title=Modern Portfolio Theory and Investment Analysis| publisher={{wipe|John Wiley & Sons}} | 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|  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=[[Organization:Oxford University Press|Oxford University Press]] | year=2013| isbn=978-0199740086|  edition=2nd}}
* {{cite book | author= Harry M. Markowitz | title=Portfolio Selection: Efficient Diversification of Investments| publisher= Wiley | year=1991| isbn=978-1557861085 |  edition=2nd}}
* {{cite book | author= Frank Milne | title=Finance Theory and Asset Pricing | publisher= Oxford University Press | year=2003| isbn=978-0199261079|  edition=2nd }}
* {{cite book | author= George Pennacchi | title=Theory of Asset Pricing| publisher= Prentice Hall | year=2007| isbn=978-0321127204 | }}
* {{cite book | author= Mark Rubinstein | title=A History of the Theory of Investments| publisher= Wiley| year=2006| isbn=978-0471770565| }}
* {{cite book | author= William F. Sharpe | title=Portfolio Theory and Capital Markets: The Original Edition| publisher= McGraw-Hill | year=1999| isbn=978-0071353205| }}

'''Corporate finance'''
* {{cite book |author1=Jonathan Berk |author2=Peter DeMarzo | title=Corporate Finance | publisher= Pearson | year=2013| isbn=978-0132992473 |edition=3rd }}
* {{cite book | author= 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=Franklin Allen| title=Principles of Corporate Finance | publisher= Mcgraw-Hill | year=2013| isbn=978-0078034763|title-link=Principles of Corporate Finance |}}
*  {{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| }}
* {{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 |author2=Randolph Westerfield |author3=Jeffrey Jaffe | title=Corporate Finance| publisher= McGraw-Hill | year=2012| isbn=978-0078034770 | edition=10th}}
* {{cite book |editor=Joel M. Stern | title=The Revolution in Corporate Finance| publisher=[[Company:Wiley-Blackwell|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| }}
* {{cite book | author= Ivo Welch | title=Corporate Finance| year=2017| isbn=978-0-9840049-2-8|  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 | }}
* [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 [[Biography:Phelim Boyle|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,  [[Organization:The New School|The New School]]
{{refend}}
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{{refbegin|30em}}
'''Course material'''
* [http://pages.stern.nyu.edu/~dbackus/233/notes_econ_assetpricing.pdf Fundamentals of Asset Pricing], [[Biography:David K. Backus|David K. Backus]], 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]. [[Biography:William F. Sharpe|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, [[Organization:Yale University|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, [[Organization:University of Texas at Austin|University of Texas at Austin]]
* [http://jhqian.org/apt/apbook.pdf An Introduction to Asset Pricing Theory], Junhui Qian, Shanghai Jiao Tong University
{{refend}}

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{{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]
* [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)"], 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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{{Economics|state=autocollapse}}
{{Finance}}
{{Financial risk}}

[[Category:Financial economics| ]]
[[Category:Actuarial science]]

{{Sourceattribution|Financial economics}}