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Financial economics
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## Underlying economics{| class"wikitable floatright" | width"250"

style="text-align:center;"|Fundamental valuation result
Price_{j} =sum_{s}(p_{s}Y_{s}X_{sj})/r}}
{{small|=sum_{s}p_{s}X_{sj}Z_{s} = E[X_{s}Z_{s}] }} {{small|=sum_{s}(q_{s}X_{sj})/r}} {{small|=sum_{s}pi_{s} X_{sj}}}
{{small|Four equivalent formulations, where:}}
{{small|j is the asset or security}} {{small|s are the various states}} {{small|r is the risk-free return}} {{small|X_{sj} dollar payoffs in each state}} {{small|p_{s} a subjective, personal probability assigned to the state; sum_{s}p_{s}=1}} {{small|Y_{s} risk aversion factors by state, normalized s.t. sum_{s}q_{s}=1}} {{small|Zequiv Y/r the stochastic discount factor}} {{small|q_{s}equiv p_{s}Y_{s}, risk neutral probabilities}} {{small|pi_{s}=q_{s}/r state prices; sum_{s}pi_{s} = 1/r}}
As above, the discipline essentially explores how
rational investors would apply decision theory to the problem of investment. The subject is thus built on the foundations of microeconomics and decision theory, and derives several key results for the application of decision making under uncertainty to the financial markets. The underlying economic logic distills to a â€fundamental valuation resultâ€, as aside, which is developed in the following sections.

### Present value, expectation and utility

Underlying all of financial economics are the concepts of present value and expectation.Calculating their present value 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 therefore the starting point for financial decision making. (Its history is correspondingly early: Richard Witt discusses compound interest in depth already in 1613, in his book "Arithmeticall Questions";C. Lewin (1970). An early book on compound interest {{Webarchive|url=https://web.archive.org/web/20161221163926weblink |date=2016-12-21 }}, Institute and Faculty of Actuaries further developed by Johan de Witt and Edmond Halley.)An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, X_{s} and p_{s} respectively. (These ideas originate with Blaise Pascal and Pierre de Fermat.)This decision method, however, fails to consider risk aversion ("as any student of finance knows"). In other words, since individuals receive greater 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, Y_{s}. See Indifference price. (Some investors may in fact be risk seeking as opposed to 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 states that, if certain axioms are satisfied, the subjective value associated with a gamble by an individual is that individual{{'}}s 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; see also Ellsberg paradox. (The development here is originally due to Daniel Bernoulli, and later formalized by John von Neumann and Oskar Morgenstern.)">

### Arbitrage-free pricing and equilibrium{| class"wikitable floatright" | width"250"

style="text-align:center;"|JEL classification codes
JEL classification codes>Journal of Economic Literature classification codes, Financial Economics is one of the 19 primary classifications, at JEL: G. It follows monetary economics and International economics>International Economics and precedes public economics. For detailed subclassifications see {{sectionlink>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. 863â€“64. The below have links to entry abstracts of The New Palgrave Online for each primary or secondary JEL category (10 or fewer per page, similar to Google searches): Tertiary category entries can also be searched.For example,weblink {{Webarchive|url=https://web.archive.org/web/20130529074942weblink |date=2013-05-29 }}.
The concepts of
arbitrage-free, "rational", pricing and equilibrium are then coupled with the above to derive "classical"See Rubinstein (2006), under "Bibliography". (or "neo-classical") financial economics.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.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 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. 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, shows that where markets are as described â€”and are additionally (implicitly and correspondingly) completeâ€”one may then make financial decisions by constructing a 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, s, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction), essentially solving simultaneously for n (risk-neutral) probabilities, q_{s}, given n prices. The formal derivation will proceed by arbitrage arguments.Freddy Delbaen and Walter Schachermayer. (2004). "What is... a Free Lunch?" {{Webarchive|url=https://web.archive.org/web/20160304061252weblink |date=2016-03-04 }} (pdf). Notices of the AMS 51 (5): 526â€“528 For a simplified example see {{sectionlink|Rational pricing|Risk neutral valuation}}, where the economy has only two possible statesâ€”up and downâ€”and where q_{up} and q_{down} (=1-q_{up}) are the two corresponding (i.e. implied) probabilities, and in turn, the derived distribution, or "measure".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", i.e. a security-specific risk premium, compensating for the extent to which its cashflows are unpredictable. All pricing models are then essentially variants of this, given specific assumptions and/or conditions. 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 derivative instrument its forecasted cashflows in the up- and down-states, X_{up} and X_{down}, are multiplied through by q_{up} and q_{down}, and are then discounted at the risk-free interest rate; the third equation above. In pricing a â€œfundamentalâ€, 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 Y and r combined. In general, this may be derived by the CAPM (or extensions) as will be seen under #Uncertainty. The difference is explained as follows: By arbitrage arguments, the value of the derivative will (must) grow at the risk free rate; in the case of an option, this is achieved by â€œconstructingâ€ the instrument as a combination of the underlying and a risk free â€œbondâ€; see {{sectionlink|Rational pricing|Delta hedging}} (and #Uncertainty below). Where the underlying is itself being priced, such construction is of course not possible, the instrument being "fundamental".

### State prices

With the above relationship established, the further specialized Arrowâ€“Debreu model may be derived. This important 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 analysis here is often undertaken assuming a representative agent.The Arrowâ€“Debreu model applies to economies with maximally complete markets, 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 Arrowâ€“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 pi_{s} of this particular state of the world.In the above example, the state prices, pi_{up}, pi_{down}would equate to the present values of \$q_{up} and \$q_{down}: 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 [pi_{up}Ã—X_{up} + pi_{down}Ã—X_{down}]; the second 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 martingale pricing and the related risk-neutral measure. See also Stochastic discount factor.State prices find immediate application as a conceptual tool ("contingent claim analysis"); but can also be applied to valuation problems.See de Matos, as well as Bossaerts and Ã˜degaard, under bibliography. Given the pricing mechanism described, one can decompose the derivative value â€” true in fact for "every security" â€” as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices. 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. (Breeden and Litzenberger's work in 1978JOURNAL, Prices of State-Contingent Claims Implicit in Option Prices, Douglas T., Breeden, Robert H., Litzenberger, Robert Litzenberger, Journal of Business, 51, 4, 1978, 621â€“651, 2352653, 10.1086/296025, established the use of state prices in financial economics.)

## Resultant models

missing image!
- MM2.png -
Modiglianiâ€“Miller Proposition II with risky debt. As leverage (D/E) increases, the WACC (k0) stays constant.
missing image!
- markowitz frontier.jpg -
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.
missing image!
- CML-plot.png -
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;"
The capital asset pricing model (CAPM):
E(R_i) = R_f + beta_{i}(E(R_m) - R_f){edih}
{{small|The expected return used when discounting cashflows on an asset i, is the risk-free rate plus the market premium multiplied by beta (rho_{i,m} frac {sigma_{i}}{sigma_{m}}), the asset's correlated volatility relative to the overall market m.}}
missing image!
- SML-chart.png -
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.
missing image!
- Stockpricesimulation.jpg -
Simulated geometric Brownian motions with parameters from market data.
{| class="wikitable floatright" | width="250" style="text-align:center;"
(Blackâ€“Scholes equation|The Blackâ€“Scholes equation:)
frac{partial V}{partial t} + frac{1}{2}sigma^2 S^2 frac{partial^2 V}{partial S^2} + rSfrac{partial V}{partial S} = rV
(Blackâ€“Scholes_equation#Financial_interpretation|Interpretation:) by arbitrage arguments, the instantaneous impact of time t and changes in spot price s on an option price V will (must) realize as growth at r, the risk free rate, when the option is correctly hedged.}}
{| class="wikitable floatright" | width="250" style="text-align:center;"
The Blackâ€“Scholes formula for the value of a call option:
begin{align}
C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)}
d_1 &= frac{1}{sigmasqrt{T - t{edih}left[lnleft(frac{S}{K}right) + left(r + frac{sigma^2}{2}right)(T - t)right]
d_2 &= d_1 - sigmasqrt{T - t}
end{align}Interpretation: The value of a call is the risk free rated present value of its expected in the money value. N(d_2) is the probability that the call will be exercised; N(d_1)S is the present value of the expected asset price at expiration, given that the asset price at expiration is above the exercise price. (A specific formulation of the fundamental valuation result.)}}
Applying the above economic concepts, we may then derive various 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", 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.
• Options: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.
• Information: knowledge of the future can reduce, or possibly eliminate, the uncertainty associated with 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. (This division sometimes denoted "deterministic" and "random", or "stochastic".)

### Certainty

The starting point here is â€œInvestment under certainty". The Fisher separation theorem, asserts that the objective of a corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is the Modiglianiâ€“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.The mechanism for determining (corporate) value is provided by The Theory of Investment Value (John Burr Williams, 1938), 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, long-term worth is the present value of its future net cashflows, in the form of dividends. 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 (introduced by Joel Dean in 1951). For other results, as well as specific models developed here, see the list of "Equity valuation" topics under {{sectionlink|Outline of finance|Discounted cash flow valuation}}.Bond valuation, in that cashflows (coupons and return of principal) are deterministic, may proceed in the same fashion.See Luenberger's Investment Science, under Bibliography. An immediate extension, Arbitrage-free bond pricing, discounts each cashflow at the market derived rate â€” i.e. at each coupon's corresponding zero-rate â€” as opposed to an overall rate. Note that 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 â€” based on historic ratios or published policy â€” and cashflows are then treated as essentially deterministic; see below under #Corporate finance theory.These "certainty" results are all commonly employed under corporate finance; uncertainty is the focus of "asset pricing models", as follows.

### Uncertainty

JOURNAL, Theory of Rational Option Pricing, Merton, Robert C., Bell Journal of Economics and Management Science, 1973, 4, 1, 141â€“183, 10.2307/3003143, 3003143, weblink

## Extensions

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

### Portfolio theory

missing image!
- Pareto Efficient Frontier for the Markowitz Portfolio selection problem..png -
Plot of two criteria when maximizing return and minimizing risk in financial portfolios (Pareto-optimal points in red)
(File:Four Correlations.png|thumb|right|alt=Examples of bivariate copulÃ¦ used in finance.|Examples of bivariate copulÃ¦ 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 Famaâ€“French three-factor model and the Carhart four-factor model, propose factors other than market return as relevant in pricing. The intertemporal CAPM and consumption-based CAPM similarly extend the model. With intertemporal portfolio choice, the investor now repeatedly optimizes her portfolio; while the inclusion of consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return.Whereas the above extend the CAPM, the single-index model is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization. The arbitrage pricing theory (APT; Stephen Ross, 1976) 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."The Arbitrage Pricing Theory, Chapter VI in Goetzmann, under External links. 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.As regards portfolio optimization, the Blackâ€“Litterman model departs from the original Markowitz model - i.e. of constructing portfolios via an efficient frontier. Blackâ€“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 asset allocation. Where factors additional to volatility are considered (kurtosis, skew...) then multiple-criteria decision analysis can be applied; here deriving a Pareto efficient portfolio. The universal portfolio algorithm (Thomas M. Cover, 1991) applies machine learning to asset selection, learning adaptively from historical data. Behavioral portfolio theory recognizes that investors have varied aims and create an investment portfolio that meets a broad range of goals. Copulas have lately been applied here; recently this is the case also for genetic algorithms. See {{sectionlink|Portfolio optimization|Improving portfolio optimization}} for other techniques and / or objectives.

### Derivative pricing

File:Arbre Binomial Options Reelles.png|thumb|right| Binomial Lattice with (Binomial options pricing model#STEP 1: Create the binomial price tree|CRR formulae) ]]{{See also|Mathematical finance#Derivatives pricing: the Q world}}{| class="wikitable floatright" | width="250" style="text-align:center;"
PDE for a zero-coupon bond:
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{edih}
{{small|(Bond_valuation#Stochastic_calculus_approach |Interpretation:) Analogous to Black-Scholes, arbitrage arguments describe the instantaneous change in the bond price P for changes in the (risk-free) short rate r; the analyst selects the specific short-rate model to be employed.}}
missing image!
- volatility smile.svg|thumb|right|Stylized volatility smile: showing the (implied) volatility by strike-price, where the Black-Scholes formulaBlack-Scholes formulaAs regards derivative pricing, the binomial options pricing model provides a discretized version of Blackâ€“Scholes, useful for the valuation of American styled options. Discretized models of this type are builtâ€”at least implicitlyâ€”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.Don M. Chance (2008). "Option Prices and State Prices" {{Webarchive|url=https://web.archive.org/web/20120209215717weblink |date=2012-02-09 }} For 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 other numeric techniques have also been developed. The theoretical framework too has been extended such that martingale pricing is now the standard approach.Drawing on these techniques, models for various other underlyings and applications have also been developed, all based off the same logic (using "contingent claim analysis"). Real options valuation allows that option holders can influence the option's underlying; models for employee stock option valuation explicitly assume non-rationality on the part of option holders; Credit derivatives allow that payment obligations and / or delivery requirements might not be honored. Exotic derivatives are now routinely valued. Multi-asset underlyers are handled via simulation or copula based analysis.Similarly, beginning with the Vasicek model (Oldrich Vasicek, 1977), various short rate models, as well as the HJM and BGM forward rate-based techniques, allow for an extension of these techniques to fixed income- and interest rate derivatives. (The Vasicek and CIR models are equilibrium-based, while Hoâ€“Lee and subsequent models are based on arbitrage-free pricing.) Bond valuation is relatedly extended: the Stochastic calculus approach, employing these methods, allows for rates that are "random" (while returning a price that is arbitrage free, as above); lattice models for hybrid securities allow for non-deterministic cashflows (and stochastic rates).Following the Crash of 1987, equity options traded in American markets began to exhibit what is known as a "volatility smile"; that is, for a given expiration, options whose strike price differs substantially from the underlying asset's price command higher prices, and thus implied 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 â€” as well as implications re the standard theory â€” are discussed in the next section.Post the financial crisis of 2008, a further development. As above, (OTC) derivative pricing has 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 â€” called into question during the crisis. Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered,"Post-Crisis Pricing of Swaps using xVAs" {{Webarchive|url=https://web.archive.org/web/20160917015231weblink |date=2016-09-17 }}, Christian KjÃ¸lhede & Anders Bech, Master thesis, Aarhus University and a Credit Valuation Adjustment, or CVAâ€”and potentially other valuation adjustments, collectively xVAâ€”is generally added to the risk-neutral derivative value.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. This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate".JOURNAL, LIBOR vs. OIS: The Derivatives Discounting Dilemma, John, Hull, Alan, White, Journal of Investment Management, 11, 3, 2013, 14â€“27, (Also, practically, the interest paid on cash collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "CSA discounting".) Swap pricing - and, therefore, curve construction - is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "multi-curve" framework where "forecast curves" are constructed for each floating-leg LIBOR tenor, with discounting on the common OIS curve; see {{sectionlink|Interest rate swap|Valuation and pricing}}.

### Corporate finance theory

Manual decision tree.jpg
-
Corporate finance theory has also been extended: mirroring the above developments, asset-valuation and decisioning no longer need assume "certainty". Monte Carlo methods in finance allow financial analysts to construct "stochastic" or probabilistic corporate finance models, as opposed to the traditional static and deterministic models; see {{sectionlink|Corporate finance|Quantifying uncertainty}}. Relatedly, Real Options theory allows for ownerâ€”i.e. managerialâ€”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. (Simulation was first applied to (corporate) finance by David B. Hertz in 1964; Real options in corporate finance were first discussed by Stewart Myers in 1977.)More traditionally, decision treesâ€”which are complementaryâ€”have been used to evaluate projects, by incorporating in the valuation (all) possible events (or states) and consequent management decisions;JOURNAL, Valuing Risky Projects: Option Pricing Theory and Decision Analysis, James E., Smith, Robert F., Nau,weblink Management Science, 41, 5, 1995, 795â€“816, 10.1287/mnsc.41.5.795, 2017-08-17,weblink" title="web.archive.org/web/20100612170613weblink">weblink 2010-06-12, no, Aswath Damodaran (2007). "Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations". In Strategic Risk Taking: A Framework for Risk Management. Prentice Hall. {{ISBN|0137043775}} the correct discount rate here reflecting each point's "non-diversifiable risk looking forward." (This technique predates the use of real options in corporate finance;See for example: JOURNAL, Decision Trees for Decision Making, John F.,weblink Magee, Harvard Business Review, July 1964, 1964, 795â€“816, 2017-08-16,weblink 2017-08-16, no, it is borrowed from 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,JOURNAL, Kritzman, Mark, An Interview with Nobel Laureate Harry M. Markowitz
, Financial Analysts Journal, 73, 4, 2017, 16â€“21, 10.2469/faj.v73.n4.3, as opposed to a more correct state-by-state treatment under uncertainty; see comments under Financial modeling Â§ Accounting.

## Challenges and criticism

{{see also|Financial mathematics#Criticism|Financial engineering#Criticisms|Financial Modelers' Manifesto|Unreasonable ineffectiveness of mathematics#Economics and finance|Physics envy}}As above, there is a very close link between (i) the random walk hypothesis, with the associated expectation that price changes should follow a normal distribution, on the one hand, and (ii) market efficiency and rational expectations, on the other. Note, however, that (wide) departures from these are commonly observed, and there are thus, respectively, two main sets of challenges.

### Departures from normality

missing image!
- Ivsrf.gif -
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.
"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 furiously tweaking their geo-centric math in the face of contrary evidence. They will never get this right; they need their Copernicus." 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
">

### Departures from rationality{|class"wikitable floatright" | width"200"

style="font-size:75%"
align="center"
Market anomalies and Economic puzzles
{{See also|Efficient-market hypothesis#Criticism|Rational expectations#Criticism}}As seen, a common assumption is that financial decision makers act rationally; see

{{Wikipedia books|Finance}}{{div col}}
{{div col end}}

{{Reflist|20em}}

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• BOOK, Frank J. Fabozzi, Edwin H. Neave and Guofu Zhou, Financial Economics, Wiley, 2011, 978-0470596203,
• BOOK, Christian Gollier, The Economics of Risk and Time (2nd Edition), MIT Press, 2004, 978-0-262-57224-8,
• BOOK, Thorsten Hens and Marc Oliver Rieger, Financial Economics: A Concise Introduction to Classical and Behavioral Finance, Springer Publishing, Springer, 2010, 978-3540361466,
• BOOK, Chi-fu Huang and Robert Litzenberger, Robert H. Litzenberger, Foundations for Financial Economics, Prentice Hall, 1998, 978-0135006535,
• BOOK, Jonathan E. Ingersoll, Theory of Financial Decision Making, Rowman & Littlefield, 1987, 978-0847673599,
• BOOK, Robert A. Jarrow, Finance theory, Prentice Hall, 1988, 978-0133148657,
• BOOK, Chris Jones, Financial Economics, Routledge, 2008, 978-0415375856,
• BOOK, Brian Kettell, Economics for Financial Markets, Butterworth-Heinemann, 2002, 978-0-7506-5384-8,
• BOOK, Yvan Lengwiler, Microfoundations of Financial Economics: An Introduction to General Equilibrium Asset Pricing, Princeton University Press, 2006, 978-0691126319,
• BOOK, Stephen F. LeRoy, Jan Werner, Principles of Financial Economics, Cambridge University Press, 2000, 978-0521586054,
• BOOK, Leonard C. MacLean, William T. Ziemba, Handbook of the Fundamentals of Financial Decision Making, World Scientific, 2013, 978-9814417341,
• BOOK, Antonio Mele (forthcoming), Financial Economics: Classics & Contemporary, MIT Press,
• BOOK, Frederic S. Mishkin, The Economics of Money, Banking, and Financial Markets (3rd Edition), Prentice Hall, 2012, 978-0132961974,
• BOOK, Harry Panjer, Harry H. Panjer, ed., Financial Economics with Applications, Actuarial Foundation, 1998, 978-0938959489,
• BOOK, Geoffrey Poitras, Pioneers of Financial Economics, Volume IEdward Elgar Publishing > location=
isbn= 978-1845423810, ; Volume II {{ISBN|978-1845423827}}.
Asset pricing
• BOOK, Kerry E. Back, Asset Pricing and Portfolio Choice Theory, Oxford University Press, 2010, 978-0195380613,
• BOOK, Tomas BjÃ¶rk, Arbitrage Theory in Continuous Time (3rd Edition), Oxford University Press, 2009, 978-0199574742,
• BOOK, John H. Cochrane, Asset Pricing, Princeton University Press, 2005, 978-0691121376,
• BOOK, Darrell Duffie, Dynamic Asset Pricing Theory (3rd Edition), Princeton University Press, 2001, 978-0691090221,
• BOOK, Edwin Elton, Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, William N. Goetzmann, Modern Portfolio Theory and Investment Analysis (9th Edition), John Wiley & Sons, Wiley, 2014, 978-1118469941,
• BOOK, Robert Haugen, Robert A. Haugen, Modern Investment Theory (5th Edition), Prentice Hall, 2000, 978-0130191700,
• BOOK, Mark S. Joshi, Jane M. Paterson, Introduction to Mathematical Portfolio Theory, Cambridge University Press, 2013, 978-1107042315,
• BOOK, Lutz Kruschwitz, Andreas Loeffler, Discounted Cash Flow: A Theory of the Valuation of Firms, Wiley, 2005, 978-0470870440,weblink
• BOOK, David Luenberger, David G. Luenberger, Investment Science (2nd Edition), Oxford University Press, 2013, 978-0199740086,
• BOOK, Harry M. Markowitz, Portfolio Selection: Efficient Diversification of Investments (2nd Edition), Wiley, 1991, 978-1557861085,
• BOOK, Frank Milne, Finance Theory and Asset Pricing (2nd Edition), Oxford University Press, 2003, 978-0199261079,
• BOOK, George Pennacchi, Theory of Asset Pricing, Prentice Hall, 2007, 978-0321127204,
• BOOK, Mark Rubinstein, A History of the Theory of Investments, Wiley, 2006, 978-0471770565,
• BOOK, William F. Sharpe, Portfolio Theory and Capital Markets: The Original Edition, McGraw-Hill, 1999, 978-0071353205,
Corporate finance
• BOOK, Jonathan Berk, Peter DeMarzo, Corporate Finance (3rd Edition), Pearson Education, Pearson, 2013, 978-0132992473,
• BOOK, Peter Bossaerts, Bernt Arne Ã˜degaard, Lectures on Corporate Finance (Second Edition), World Scientific, 2006, 978-981-256-899-1,
• BOOK, Richard Brealey, Stewart Myers, Franklin Allen, Principles of Corporate Finance, Mcgraw-Hill, 2013, 978-0078034763, Principles of Corporate Finance,
• BOOK, Aswath Damodaran, Corporate Finance: Theory and Practice, Wiley, 1996, 978-0471076803,weblink
• BOOK, JoÃ£o Amaro de Matos, Theoretical Foundations of Corporate Finance, Princeton University Press, 2001, 9780691087948,
• BOOK, Joseph Ogden, Frank C. Jen, Philip F. O'Connor, Advanced Corporate Finance, Prentice Hall, 2002, 978-0130915689,
• BOOK, Pascal Quiry, Yann Le Fur, Antonio Salvi, Maurizio Dallochio, Pierre Vernimmen, Corporate Finance: Theory and Practice (3rd Edition), Wiley, 2011, 978-1119975588,
• BOOK, Stephen Ross (economist), Stephen Ross, Randolph Westerfield, Jeffrey Jaffe, Corporate Finance (10th Edition), McGraw-Hill, 2012, 978-0078034770,
• BOOK, Joel Stern, Joel M. Stern, ed., The Revolution in Corporate Finance (4th Edition), Wiley-Blackwell, 2003, 9781405107815,
• BOOK, Jean Tirole, The Theory of Corporate Finance, Princeton University Press, 2006, 978-0691125565,
• BOOK, Ivo Welch, Corporate Finance (3rd Edition), 2014, 978-0-9840049-1-1,

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