Editing Financial economics (section)

==Resultant models==
Applying the above economic concepts, we may then derive various [[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.
* [[Option (finance)|Options]]: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.
* [[Perfect information|Information]]: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with [[Future value|future monetary value]] (FMV).
Applying this framework, with the above concepts, leads to the required models. This derivation begins with the assumption of "no uncertainty" and is then expanded to incorporate the other considerations.<ref name="Fama and Miller"/> (This division sometimes denoted "[[deterministic]]" and "random",<ref name="Luenberger"/> or "[[stochastic]]".)

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

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

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

For both stocks and bonds, "under certainty, with the focus on cash flows from securities over time," valuation based on a [[yield curve|term structure of interest rates]] is in fact consistent with arbitrage-free pricing.<ref>See footnote 3 under Rubinstein (2005). "The Fundamental Theorem (Part I)", refenced below.</ref>
Indeed, a corollary of [[#Arbitrage-free_pricing_and_equilibrium|the above]] is that "[[Rational pricing#The law of one price|the law of one price]] implies the existence of a discount factor";<ref>§ 4.1 "Law of one price and existence of a discount factor" in Cochrane (2005).</ref>
correspondingly, as formulated, {{Nowrap|<math display=inline>\sum_{s}\pi_{s} = 1/r</math>}}.

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

===Uncertainty===
[[Image:markowitz frontier.jpg|thumb|right|Efficient Frontier. The hyperbola is sometimes referred to as the 'Markowitz Bullet', and its upward sloped portion is the efficient frontier if no risk-free asset is available. With a risk-free asset, the straight line is the efficient frontier. The graphic displays the CAL, [[Capital allocation line]], formed when the risky asset is a single-asset rather than the market, in which case the line is the CML.]]
[[Image:CML-plot.png|thumb|right|The [[Capital market line]] is the tangent line drawn from the point of the risk-free asset to the [[feasible region]] for risky assets. The tangency point M represents the [[market portfolio]]. The CML results from the combination of the market portfolio and the risk-free asset (the point L). Addition of leverage (the point R) creates levered portfolios that are also on the CML.]]
{| class="wikitable floatright" | width="250"
|- style="text-align:center;"
|{{smalldiv|1=The capital asset pricing model (CAPM):
:<math>E(R_i) = R_f + \beta_{i}(E(R_m) - R_f)</math>}}
{{small|The [[required rate of return|expected return]] used when discounting cashflows on an asset <math>i</math>, is the risk-free rate plus the [[Capital asset pricing model#Formula|market premium]] multiplied by [[Beta (finance)|beta]] {{Nowrap|(<math>\rho_{i,m} \frac {\sigma_{i}}{\sigma_{m}}</math>)}}, the asset's correlated volatility relative to the overall market <math>m</math>.}}
|}
[[Image:SML-chart.png|thumb|right|[[Security market line]]: the representation of the CAPM displaying the expected rate of return of an individual security as a function of its systematic, non-diversifiable risk.]]
For [[Decision theory#Choice under uncertainty|"choice under uncertainty"]] the twin assumptions of rationality and [[Financial market efficiency|market efficiency]], as more closely defined, lead to [[modern portfolio theory]] (MPT) with its [[capital asset pricing model]] (CAPM) – an ''equilibrium-based'' result – and to the [[Black–Scholes model|Black–Scholes–Merton theory]] (BSM; often, simply Black–Scholes) for [[Valuation of options|option pricing]] – an ''arbitrage-free'' result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitrage-free with respect to the more fundamental, equilibrium determined, securities prices; see {{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"]. ''[[Financial Analysts Journal]]'', September/October 1965, Vol. 21, No. 5: 55–59.</ref> 
Given the ability to profit from [[privacy|private information]], self-interested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. ''efficient'', prices: the [[efficient-market hypothesis]], or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See [[earnings response coefficient]].)
The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information  are identical to the ''best guess of the future'': the assumption of [[rational expectations]]. 
The EMH does allow that when faced with new information, some investors may overreact and some may underreact, 
<ref name="affirmative case">[[Mark Rubinstein]] (2001). [https://escholarship.org/content/qt22q318mh/qt22q318mh_noSplash_0a514e78eb5c8e16a9ead63a4f4a628e.pdf?t=krnd5a "Rational Markets: Yes or No? The Affirmative Case"]. ''[[Financial Analysts Journal]]'', May - Jun., 2001, Vol. 57, No. 3: 15-29</ref>
but what is required, however, is that investors' reactions follow a [[normal distribution]] – so that the net effect on market prices cannot be reliably exploited <ref name="affirmative case"/> to make an abnormal profit.
In the competitive limit, then, market prices will reflect all available information and prices can only move in response to news:<ref name="Shiller"/> the [[random walk hypothesis]]. 
This news, of course, could be "good" or "bad", minor or, less common, major; and these moves are then, correspondingly, normally distributed; with the price therefore following a log-normal distribution. 
{{NoteTag|The EMH was presented by [[Eugene Fama]] in a 1970 [[review paper]],<ref>Fama, Eugene (1970). [https://www.jstor.org/stable/2325486 "Efficient Capital Markets: A Review of Theory and Empirical Work"]. ''[[Journal of Finance]]''. Vol. 25, No. 2.</ref> consolidating previous works re random walks in stock prices: [[Jules Regnault]] (1863); [[Louis Bachelier]] (1900); [[Maurice Kendall]] (1953); [[Paul Cootner]] (1964); and [[Paul Samuelson]] (1965), among others.}}

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

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

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

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