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Financial economics
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{{morefootnotesdate=December 2018}}{{Economics sidebar}}Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade".William F. Sharpe, "Financial Economics" {{Webarchiveurl=https://web.archive.org/web/20040604105441weblink date=20040604 }}, in WEB,weblink MacroInvestment Analysis, Stanford University (manuscript), 20090806,weblink" title="web.archive.org/web/20140714034144weblink">weblink 20140714, no, Its concern is thus the interrelation of financial variables, such as prices, interest rates and shares, as opposed to those concerning the real economy. It has two main areas of focus:Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, Journal of Portfolio Management. Summer 1999. asset pricing and corporate finance; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".Robert C. Merton WEB,weblink Nobel Lecture, 20090806,weblink" title="web.archive.org/web/20090319202149weblink">weblink 20090319, no, It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions. It is built on the foundations of microeconomics and decision theory.Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise these relationships. Mathematical finance is related in that it will derive and extend the mathematical or numerical models suggested by financial economics. Note though that the emphasis there is mathematical consistency, as opposed to compatibility with economic theory. Financial economics has a primarily microeconomic focus, whereas monetary economics is primarily macroeconomic in nature.Financial economics is usually taught at the postgraduate level; see Master of Financial Economics. Recently, specialist undergraduate degrees are offered in the discipline.e.g.: Kent {{Webarchiveurl=https://web.archive.org/web/20140221212707weblink date=20140221 }}; City London {{Webarchiveurl=https://web.archive.org/web/20140223090217weblink date=20140223 }}; UC Riverside {{Webarchiveurl=https://web.archive.org/web/20140222044845weblink date=20140222 }}; Leicester {{Webarchiveurl=https://web.archive.org/web/20140222003101weblink date=20140222 }}; Toronto {{Webarchiveurl=https://web.archive.org/web/20140221102435weblink date=20140221 }}; UMBC {{Webarchiveurl=https://web.archive.org/web/20141230165120weblink date=20141230 }}This article provides an overview and survey of the field: for derivations and more technical discussion, see the specific articles linked."> the content below is remote from Wikipedia
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Underlying economics{ class"wikitable floatright"  width"250"
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}}}
{{smallj is the asset or security}}
{{smalls are the various states}}
{{smallr is the riskfree return}}
{{smallX_{sj} dollar payoffs in each state}}
{{smallp_{s} a subjective, personal probability assigned to the state; sum_{s}p_{s}=1}}
{{smallY_{s} risk aversion factors by state, normalized s.t. sum_{s}q_{s}=1}}
{{smallZequiv Y/r the stochastic discount factor}}
{{smallq_{s}equiv p_{s}Y_{s}, risk neutral probabilities}}
{{smallpi_{s}=q_{s}/r state prices; sum_{s}pi_{s} = 1/r}}

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 {{Webarchiveurl=https://web.archive.org/web/20161221163926weblink date=20161221 }}, 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.)">Arbitragefree pricing and equilibrium{ class"wikitable floatright"  width"250"
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 codesG. 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):
JEL: G â€“ Financial Economics
JEL: G0 â€“ General
JEL: G1 â€“ General Financial Markets
JEL: G2 â€“ Financial institutions and Services
JEL: G3 â€“ Corporate finance and Governance
Tertiary category entries can also be searched.For example,weblink {{Webarchiveurl=https://web.archive.org/web/20130529074942weblink date=20130529 }}. 
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 downstate 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 riskneutral 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 stateprices; i.e. backsolve for the stateprices corresponding to observed derivative prices. These recovered stateprices 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 StateContingent 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.
 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 riskfree asset is available. With a riskfree asset, the straight line is the efficient frontier. The graphic displays the CAL, Capital allocation line, formed when the risky asset is a singleasset rather than the market, in which case the line is the CML.
 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 riskfree asset is available. With a riskfree asset, the straight line is the efficient frontier. The graphic displays the CAL, Capital allocation line, formed when the risky asset is a singleasset rather than the market, in which case the line is the CML.
missing image!
 CMLplot.png 
The Capital market line is the tangent line drawn from the point of the riskfree 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 riskfree 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="textalign:center;" CMLplot.png 
The Capital market line is the tangent line drawn from the point of the riskfree 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 riskfree asset (the point L). Addition of leverage (the point R) creates levered portfolios that are also on the CML.
The capital asset pricing model (CAPM):
E(R_i) = R_f + beta_{i}(E(R_m)  R_f){edih}
{{smallThe expected return used when discounting cashflows on an asset i, is the riskfree 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!
 SMLchart.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, nondiversifiable risk.
 SMLchart.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, nondiversifiable risk.
missing image!
 Stockpricesimulation.jpg 
Simulated geometric Brownian motions with parameters from market data.
{ class="wikitable floatright"  width="250" style="textalign:center;" Stockpricesimulation.jpg 
Simulated geometric Brownian motions with parameters from market data.
(Blackâ€“Scholes equationThe 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_interpretationInterpretation:) 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.}} 
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)}
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.)}}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} 
 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).
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, longterm 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 {{sectionlinkOutline of financeDiscounted 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, Arbitragefree bond pricing, discounts each cashflow at the market derived rate â€” i.e. at each coupon's corresponding zerorate 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
For "choice under uncertainty" the twin assumptions of rationality and market efficiency, as more closely defined, lead to modern portfolio theory (MPT) with its capital asset pricing model (CAPM)â€”an equilibriumbased resultâ€”and to the Blackâ€“Scholesâ€“Merton theory (BSM; often, simply Blackâ€“Scholes) for option pricingâ€”an arbitragefree result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitragefree with respect to the more fundamental, equilibrium determined, securities prices; see asset pricing.Briefly, and intuitivelyâ€”and consistent with #Arbitragefree pricing and equilibrium aboveâ€”the linkage between rationality and efficiency is as follows.For a more formal treatment, see, for example: Eugene F. Fama. 1965. Random Walks in Stock Market Prices. Financial Analysts Journal, September/October 1965, Vol. 21, No. 5: 55â€“59. Given the ability to profit from private information, selfinterested 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 efficientmarket hypothesis, or EMH (Eugene Fama, 1965). 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 distributionâ€”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; and this, of course, could be "good" or "bad", minor or, less common, major : the random walk hypothesis. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See Earnings response coefficient.) (On Random walks in stock prices: Jules Regnault, 1863; Louis Bachelier, 1900; Maurice Kendall, 1953; Paul Cootner, 1964.)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 certaintycase above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends,Christopher L. Culp and John H. Cochrane. (2003). ""Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management" {{Webarchiveurl=https://web.archive.org/web/20160304190225weblink date=20160304 }}, in Modern Risk Management: A History. Peter Field, ed. London: Risk Books, 2003. {{ISBN1904339050}}JOURNAL, Shiller, Robert J., Robert J. Shiller, 2003, From Efficient Markets Theory to Behavioral Finance, Journal of Economic Perspectives, 17, 1 (Winter 2003), 83â€“104,weblink 10.1257/089533003321164967,weblink" title="web.archive.org/web/20150412081613weblink">weblink 20150412, no, 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, rationalityâ€”in the sense of arbitrageexploitationâ€”gives rise to Blackâ€“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.Importantly, this result will be independent of the investor's level of risk aversion, and / or assumed utility function, thus providing a readily determined discount rate for corporate finance decision makers as above,Jensen, Michael C. and Smith, Clifford W., "The Theory of Corporate Finance: A Historical Overview". In: The Modern Theory of Corporate Finance, New York: McGrawHill Inc., pp. 2â€“20, 1984. and for other investors.The argument proceeds as follows: If one can construct an efficient frontierâ€”i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagramâ€”then meanvariance efficient portfolios can be formed simply as a combination of holdings of the riskfree 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 pricing; see {{sectionlinkMarkowitz modelChoosing 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. A more modern, direct, derivation is as described at the bottom of this section; which may be generalized to derive other pricing models.(The efficient frontier was introduced by Harry Markowitz in 1952. The CAPM was derived by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently. )Blackâ€“Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of Europeanstyled options.The model is expressed as the Blackâ€“Scholes equation, a partial differential equation describing the changing price of the option over time; it is derived assuming lognormal, 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 (V, the value, or price, of the option, grows at r, the riskfree rate).This hedge, in turn, implies that there is only one right priceâ€”in an arbitragefree senseâ€”for the option. And this price is returned by the Blackâ€“Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is the solution to the equation.)Since the formula is without reference to the share's expected return, Blackâ€“Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with #Arbitragefree pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation.(BSM  two seminal 1973 papersJOURNAL, The Pricing of Options and Corporate Liabilities, Black, Fischer, Myron Scholes, Journal of Political Economy, 1973, 81, 3, 637â€“654, 10.1086/260062, weblink
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
 is consistent with "previous versions of the formula" of Louis Bachelier (1900) and Edward O. Thorp (1967);Haug, E. G. and Taleb, N. N. (2008). Why We Have Never Used the BlackScholesMerton Option Pricing Formula, Wilmott Magazine January 2008 although these were more "actuarial" in flavor, and had not established riskneutral discounting. See also Paul Samuelson (1965).JOURNAL, Samuelson Paul, Paul Samuelson, 1965, A Rational Theory of Warrant Pricing,weblink Industrial Management Review, 6, 2, 20170228,weblink" title="web.archive.org/web/20170301092720weblink">weblink 20170301, no, Vinzenz Bronzin (1908) produced very early results, also.)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 Blackâ€“Scholes model is thus consistent with the expected return from the CAPM.Don M. Chance (2008). "Option Prices and Expected Returns" {{Webarchiveurl=https://web.archive.org/web/20150923195335weblink date=20150923 }}Emanuel Derman, A Scientific Approach to CAPM and Options Valuation {{Webarchiveurl=https://web.archive.org/web/20160330002200weblink date=20160330 }} The Blackâ€“Scholes theory, although built on Arbitragefree pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrowâ€“Debreu theory, and can be derived via statepricing â€” essentially, by expanding the fundamental result above â€” further explaining, and if required demonstrating, this unity.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". Here, the CAPM is derived by linking Y, risk aversion, to overall market return, and setting the return on security j as X_j/Price_j; see {{sectionlinkStochastic discount factorProperties}}.The BlackScholes formula is found, in the limit, by attaching a binomial probability to each of numerous possible spotprices (states) and then rearranging for the terms corresponding to N(d_1) and N(d_2), per the boxed description; see {{sectionlinkBinomial options pricing modelRelationship with Blackâ€“Scholes}}.Extensions
More recent work further generalizes and / or extends these models. As regards asset pricing, developments in equilibriumbased pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to riskneutral, i.e. arbitragefree, 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 (Paretooptimal points in red)
(File:Four Correlations.pngthumbrightalt=Examples of bivariate copulÃ¦ used in finance.Examples of bivariate copulÃ¦ used in finance.)
 Pareto Efficient Frontier for the Markowitz Portfolio selection problem..png 
Plot of two criteria when maximizing return and minimizing risk in financial portfolios (Paretooptimal points in red)
See also: Postmodern portfolio theory and {{sectionlinkMathematical financeRisk and portfolio management: the P world}}.
The majority of developments here relate to required return, i.e. pricing, extending the basic CAPM. Multifactor models such as the Famaâ€“French threefactor model and the Carhart fourfactor model, propose factors other than market return as relevant in pricing. The intertemporal CAPM and consumptionbased 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 marketbased investments, into the investor's calculation of required return.Whereas the above extend the CAPM, the singleindex 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 macroeconomic 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 multiplecriteria 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 {{sectionlinkPortfolio optimizationImproving portfolio optimization}} for other techniques and / or objectives.Derivative pricing
File:Arbre Binomial Options Reelles.pngthumbright Binomial Lattice with (Binomial options pricing model#STEP 1: Create the binomial price treeCRR formulae) ]]{{See alsoMathematical finance#Derivatives pricing: the Q world}}{ class="wikitable floatright"  width="250" style="textalign:center;"PDE for a zerocoupon 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 BlackScholes, arbitrage arguments describe the instantaneous change in the bond price P for changes in the (riskfree) short rate r; the analyst selects the specific shortrate model to be employed.}} 
missing image!
 volatility smile.svgthumbrightStylized volatility smile: showing the (implied) volatility by strikeprice, where the BlackScholes formulaBlackScholes 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 stateprices (as above); relatedly, a large number of researchers have used options to extract stateprices for a variety of other applications in financial economics.Don M. Chance (2008). "Option Prices and State Prices" {{Webarchiveurl=https://web.archive.org/web/20120209215717weblink date=20120209 }} 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 nonrationality 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. Multiasset 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 ratebased techniques, allow for an extension of these techniques to fixed income and interest rate derivatives. (The Vasicek and CIR models are equilibriumbased, while Hoâ€“Lee and subsequent models are based on arbitragefree 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 nondeterministic 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 creditriskfree 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,"PostCrisis Pricing of Swaps using xVAs" {{Webarchiveurl=https://web.archive.org/web/20160917015231weblink date=20160917 }}, 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 riskneutral 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 postcrisis, the overnight rate is considered a better proxy for the "riskfree 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 "multicurve" framework where "forecast curves" are constructed for each floatingleg LIBOR tenor, with discounting on the common OIS curve; see {{sectionlinkInterest rate swapValuation and pricing}}.
Corporate finance theory has also been extended: mirroring the above developments, assetvaluation 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 {{sectionlinkCorporate financeQuantifying 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, 20170817,weblink" title="web.archive.org/web/20100612170613weblink">weblink 20100612, no, Aswath Damodaran (2007). "Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations". In Strategic Risk Taking: A Framework for Risk Management. Prentice Hall. {{ISBN0137043775}} the correct discount rate here reflecting each point's "nondiversifiable 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, 20170816,weblink 20170816, 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) cashflows were discounted,JOURNAL, Kritzman, Mark, An Interview with Nobel Laureate Harry M. Markowitz
 volatility smile.svgthumbrightStylized volatility smile: showing the (implied) volatility by strikeprice, where the BlackScholes formulaBlackScholes 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 stateprices (as above); relatedly, a large number of researchers have used options to extract stateprices for a variety of other applications in financial economics.Don M. Chance (2008). "Option Prices and State Prices" {{Webarchiveurl=https://web.archive.org/web/20120209215717weblink date=20120209 }} 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 nonrationality 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. Multiasset 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 ratebased techniques, allow for an extension of these techniques to fixed income and interest rate derivatives. (The Vasicek and CIR models are equilibriumbased, while Hoâ€“Lee and subsequent models are based on arbitragefree 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 nondeterministic 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 creditriskfree 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,"PostCrisis Pricing of Swaps using xVAs" {{Webarchiveurl=https://web.archive.org/web/20160917015231weblink date=20160917 }}, 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 riskneutral 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 postcrisis, the overnight rate is considered a better proxy for the "riskfree 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 "multicurve" framework where "forecast curves" are constructed for each floatingleg LIBOR tenor, with discounting on the common OIS curve; see {{sectionlinkInterest rate swapValuation and pricing}}.
Corporate finance theory
Manual decision tree.jpg , Financial Analysts Journal, 73, 4, 2017, 16â€“21, 10.2469/faj.v73.n4.3, as opposed to a more correct statebystate treatment under uncertainty; see comments under Financial modeling Â§ Accounting.
In more modern treatments, then, it is the expected cashflows (in the mathematical sense: {{smallsum_{s}p_{s}X_{sj}}}) combined into an overall value per forecast period which are discounted."Capital Budgeting Applications and Pitfalls" {{Webarchiveurl=https://web.archive.org/web/20170815234404weblink date=20170815 }}. Ch 13 in Ivo Welch (2017). Corporate Finance: 4th EditionGeorge Chacko and Carolyn Evans (2014). Valuation: Methods and Models in Applied Corporate Finance. FT Press. {{ISBN0132905221}}And using the CAPMâ€”or extensionsâ€”the discounting here is at the riskfree rate plus a premium linked to the uncertainty of the entity or project cash flows;(essentially, Y and r combined).Other developments here includeSee Jensen and Smith under "External links", as well as Rubinstein under "Bibliography". agency theory, which analyses the difficulties in motivating corporate management (the "agent") to act in the best interests of shareholders (the "principal"), rather than in their own interests. Clean surplus accounting and the related residual income valuation provide a model that returns price as a function of earnings, expected returns, and change in book value, as opposed to dividends. This approach, to some extent, arises due to the implicit contradiction of seeing value as a function of dividends, while also holding that dividend policy cannot influence value per Modigliani and Miller's "Irrelevance principle"; see {{sectionlinkDividend policyIrrelevance of dividend policy}}.The typical application of real options is to capital budgeting type problems as described. However, they are also applied to questions of capital structure and dividend policy, and to the related design of corporate securities;Kenneth D. Garbade (2001). Pricing Corporate Securities as Contingent Claims. MIT Press. {{ISBN9780262072236}} and since stockholder and bondholders have different objective functions, in the analysis of the related agency problems.JOURNAL, Damodaran, Aswath, Aswath Damodaran, The Promise and Peril of Real Options, NYU Working Paper, SDRP0502, 2005,weblink 20161214,weblink" title="web.archive.org/web/20010613082802weblink">weblink 20010613, no, In all of these cases, stateprices can provide the marketimplied information relating to the corporate, as above, which is then applied to the analysis. For example, convertible bonds can (must) be priced consistent with the stateprices of the corporate's equity.See Kruschwitz and LÃ¶ffler per Bibliography.Challenges and criticism
{{see alsoFinancial mathematics#CriticismFinancial engineering#CriticismsFinancial Modelers' ManifestoUnreasonable ineffectiveness of mathematics#Economics and financePhysics 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 Zaxis represents implied volatility in percent, and X and Y axes represent the option delta, and the days to maturity.
{{See alsoCapital asset pricing model#ProblemsModern portfolio theory#CriticismsBlackâ€“Scholes model#Criticism and comments}}As discussed, the assumptions that market prices follow a random walk and / or that asset returns are normally distributed are fundamental. Empirical evidence, however, suggests that these assumptions may not hold (see Kurtosis risk, Skewness risk, Long tail) and that in practice, traders, analysts and risk managers frequently modify the "standard models" (see Model risk). In fact, BenoÃ®t Mandelbrot had discovered already in the 1960s that changes in financial prices do not follow a Gaussian distribution, the basis for much option pricing theory, although this observation was slow to find its way into mainstream financial economics.Financial models with longtailed distributions and volatility clustering have been introduced to overcome problems with the realism of the above "classical" financial models; while jump diffusion models allow for (option) pricing incorporating "jumps" in the spot price.JOURNAL, How to use the holes in BlackScholes, Fischer, Black, Fischer Black, Journal of Applied Corporate Finance, 1, Jan, 1989, 67â€“73, 10.1111/j.17456622.1989.tb00175.x, Risk managers, similarly, complement (or substitute) the standard value at risk models with historical simulations, mixture models, principal component analysis, extreme value theory, as well as models for volatility clustering.See for example III.A.3, in Carol Alexander, ed. (January 2005). The Professional Risk Managers' Handbook. PRMIA Publications. {{ISBN9780976609704}} For further discussion see {{sectionlinkFattailed distributionApplications in economics}}, and {{sectionlinkValue at riskCriticism}}.Portfolio managers, likewise, have modified their optimization criteria and algorithms; see #Portfolio theory above. Closely related is the volatility smile, where, as above, implied volatility â€” the volatility corresponding to the BSM price â€” is observed to differ as a function of strike price (i.e. moneyness), true only if the pricechange distribution is nonnormal, 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 threedimensional surface plot of volatility smile and term structure. These empirical phenomena negate the assumption of constant volatilityâ€”and lognormalityâ€”upon which Blackâ€“Scholes is built. Within institutions, the function of BlackScholes is now, largely, to communicate prices via implied volatilities, much like bond prices are communicated via YTM; see {{sectionlinkBlackâ€“Scholes modelThe volatility smile}}.In consequence traders (and risk managers) now, instead, use "smileconsistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. nonquoted, strike/maturity combinations, or of nonEuropean derivatives, and generally for hedging purposes. The two main approaches are local volatility and stochastic volatility. The first returns the volatility which is â€œlocalâ€ to each spottime point of the finite difference or simulationbased valuation â€” i.e. as opposed to implied volatility, which holds overall. In this way calculated prices â€” and numeric structures â€” are marketconsistent in an arbitragefree sense. The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first calibrated to observed prices, and are then applied to the valuation or hedging in question; the most common are Heston, SABR and CEV. This approach addresses certain problems identified with hedging under local volatility.JOURNAL, Managing smile risk, Patrick, Hagan, etal, Wilmott Magazine, Sep, 2002, 84â€“108, Related to local volatility are the latticebased impliedbinomial and trinomial trees â€” essentially a discretization of the approach â€” which are similarly (but less commonly) used for pricing; these are built on stateprices recovered from the surface. Edgeworth binomial trees allow for a specified (i.e. nonGaussian) skew and kurtosis in the spot price; priced here, options with differing strikes will return differing implied volatilities, and the tree can be calibrated to the smile as required.See for example Pg 217 of: Jackson, Mary; Mike Staunton (2001). Advanced modelling in finance using Excel and VBA. New Jersey: Wiley. {{ISBN0471499226}}. Similarly purposed (and derived) closedform models have also been developed. These include: Jarrow and Rudd (1982); Corrado and Su (1996); Brown and Robinson (2002); Backus, Foresi, and Wu (2004). See: Emmanuel Jurczenko, Bertrand Maillet & Bogdan Negrea, 2002. "Revisited multimoment approximate option pricing models: a general comparison (Part 1)". Working paper, London School of Economics and Political Science.As discussed, additional to assuming lognormality in returns, "classical" BSMtype models also (implicitly) assume the existence of a creditriskfree environment, where one can perfectly replicate cashflows so as to fully hedge, and then discount at "the" riskfreerate. And therefore, post crisis, the various xvalue adjustments must be employed, effectively correcting the riskneutral value for counterparty and fundingrelated risk.Note that these xVA are, of course, 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 "double counting" of credit risk (etc.) when appending xVA. (Were this not the case, then each counterparty would have its own surface...) As mentioned at top, mathematical finance (and particularly financial engineering) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smileconsistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, Nassim Taleb (further below), amongst other critics of financial economics, suggests that, instead, the theory needs revisiting almost entirely:
 Ivsrf.gif 
Implied volatility surface. The Zaxis 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 geocentric 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"
Market anomalies and Economic puzzles 
See also
{{Wikipedia booksFinance}}{{div col}} (:Category:Finance theories)
 Deutsche Bank Prize in Financial Economics
 Financial modeling
 Fischer Black Prize
 List of financial economists
 {{sectionlinkList of unsolved problems in economicsFinancial economics}}
 Monetary economics
 Outline of economics
 Outline of finance
References
{{Reflist20em}}Bibliography
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