Financial economics, also known as finance, is the branch of

"Financial Economics"

, in Its concern is thus the interrelation of financial variables, such as share prices,

"What is... a Free Lunch?"

(pdf). Notices of the AMS 51 (5): 526â€“528 For a simplified example see , where the economy has only two possible states â€“ up and down â€“ and where $q\_$ and $q\_$ (=$1-q\_$) 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 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, $X\_$ and $X\_$, are multiplied through by $q\_$ and $q\_$, and are then

Fundamentals of Asset Pricing

Stern NYU the third equation above. Essentially, this factor divides expected utility at the relevant future period - a function of the possible asset values realized under each state - by the utility due to today's wealth, and is then also referred to as "the intertemporal

"Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management"

, in ''Modern Risk Management: A History''. Peter Field, ed. London: Risk Books, 2003. 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 â€“ in the sense of arbitrage-exploitation â€“ 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. This result will be independent of the investor's level of risk aversion and assumed utility function, thus providing a readily determined discount rate for corporate finance decision makers as above, 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. 2â€“20, 1984. and for other investors. The argument proceeds as follows: If one can construct an

"Option Prices and Expected Returns"

Emanuel Derman

''A Scientific Approach to CAPM and Options Valuation''

The Blackâ€“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 Arrowâ€“Debreu theory, and can be derived via state-pricing 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 . The Black-Scholes formula is found, in the limit, by attaching a binomial probability to each of numerous possible spot-prices (states) and then rearranging for the terms corresponding to $N(d\_1)$ and $N(d\_2)$, per the boxed description; see .

"Option Prices and State Prices"

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 on the same logic (using "

Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach

,

"Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations"

In ''Strategic Risk Taking: A Framework for Risk Management''. Prentice Hall. the correct discount rate here reflecting each point's "non-diversifiable risk looking forward." Related to this, is the treatment of forecasted cashflows in

. Ch 13 in Ivo Welch (2017). ''Corporate Finance'': 4th Edition And using the CAPM â€“ or extensions â€“ the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows (essentially, $Y$ and $r$ combined). Other developments here include 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; here emphasizing the issues interrelated with capital structure.

"The Arithmetic of Active Management"

. ''Financial Analysts Journal'' Vol. 47, No. 1, January/February The practical implication, therefore, is that passive investing (e.g. via low-cost

''Indexed Investing: A Prosaic Way to Beat the Average Investor''

. Presention:

economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...

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"

, in Its concern is thus the interrelation of financial variables, such as share prices,

interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...

s and exchange rates, as opposed to those concerning the real economy
The real economy concerns the production, purchase and flow of goods and services (like oil, bread and labour) within an economy. It is contrasted with the financial economy, which concerns the aspects of the economy that deal purely in transac ...

.
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
In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but correspon ...

, commonly known as "Investments", and corporate finance
Corporate finance is the area of finance that deals with the sources of funding, the capital structure of corporations, the actions that managers take to increase the value of the firm to the shareholders, and the tools and analysis used to ...

; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.
It thus provides the theoretical underpinning for much of finance.
The subject is concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment".See Fama and Miller (1972), ''The Theory of Finance'', in Bibliography. 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
Financial econometrics is the application of statistical methods to financial market data. Financial econometrics is a branch of financial economics, in the field of economics. Areas of study include capital markets, financial institutions, corp ...

is the branch of financial economics that uses econometric
Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8â ...

techniques to parameterise these relationships.
Underlying economics

Financial economics studies how rational investors would apply decision theory to investment management. The subject is thus built on the foundations of microeconomics and derives several key results for the application ofdecision making
In psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either ra ...

under uncertainty to the financial markets. The underlying economic logic yields the Fundamental theorem of asset pricing, which gives the conditions for arbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between ...

-free asset pricing.
Present value, expectation and utility

Underlying all of financial economics are the concepts ofpresent value
In economics and finance, present value (PV), also known as present discounted value, is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has in ...

and expectation.
Calculating their present value - $X\_/r$ - 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.
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, $X\_$ and $p\_$ respectively.
This decision method, however, fails to consider risk aversion ("as any student of finance knows"). In other words, since individuals receive greater utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosop ...

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, $Y\_$. See Indifference price In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The indifference price is also known as the reservation price or private valuation. In particular, the indifference price is the pr ...

. (Some investors may in fact be risk seeking
In accounting, finance, and economics, a risk-seeker or risk-lover is a person who has a preference ''for'' risk.
While most investors are considered risk ''averse'', one could view casino-goers as risk-seeking. A common example to explain risk ...

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 The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on th ...

. More formally, the resulting expected utility hypothesis The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on th ...

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 and the Ellsberg paradox.
Arbitrage-free pricing and equilibrium

The concepts ofarbitrage
In economics and finance, arbitrage (, ) is the practice of taking advantage of a difference in prices in two or more markets; striking a combination of matching deals to capitalise on the difference, the profit being the difference between ...

-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
In economics, economic equilibrium is a situation in which economic forces such as supply and demand are balanced and in the absence of external influences the ( equilibrium) values of economic variables will not change. For example, in the ...

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) complete â€“ 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, $q\_$, given ''n'' prices. The formal derivation will proceed by arbitrage arguments.Freddy Delbaen and Walter Schachermayer. (2004)"What is... a Free Lunch?"

(pdf). Notices of the AMS 51 (5): 526â€“528 For a simplified example see , where the economy has only two possible states â€“ up and down â€“ and where $q\_$ and $q\_$ (=$1-q\_$) 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 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, $X\_$ and $X\_$, are multiplied through by $q\_$ and $q\_$, and are then

discounted
Discounting is a financial mechanism in which a debtor obtains the right to delay payments to a creditor, for a defined period of time, in exchange for a charge or fee.See "Time Value", "Discount", "Discount Yield", "Compound Interest", "Efficient ...

at the risk-free interest rate; per the second equation above. In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with $Y$ 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 construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by "manufacturing" the instrument as a combination of the underlying
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be u ...

and a risk free "bond"; see (and #Uncertainty below). Where the underlying is itself being priced, such "manufacturing" is of course not possible â€“ the instrument being "fundamental", i.e. as opposed to "derivative" â€“ and a premium is then required for risk.
(Correspondingly, mathematical finance separates into two analytic regimes:
risk and portfolio management (generally) use physical (or actual or actuarial) probability, denoted by "P";
while derivatives pricing uses risk-neutral probability (or arbitrage-pricing probability), denoted by "Q".
In specific applications the lower case is used, as in the above equations.)
State prices

With the above relationship established, the further specialized Arrowâ€“Debreu model may be derived. This result suggests that, under certain economic conditions, there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy. The analysis here is often undertaken assuming a ''representative agent
Economists use the term representative agent to refer to the typical decision-maker of a certain type (for example, the typical consumer, or the typical firm).
More technically, an economic model is said to have a representative agent if all age ...

''.
The Arrowâ€“Debreu model applies to economies with maximally complete market
In economics, a complete market (aka Arrow-Debreu market or complete system of markets) is a market with two conditions:
# Negligible transaction costs and therefore also perfect information,
# there is a price for every asset in every possible ...

s, in which there exists a market for every time period and forward prices for every commodity at all time periods.
A direct extension, then, is the concept of a state price
In financial economics, a state-price security, also called an Arrowâ€“Debreu security (from its origins in the Arrowâ€“Debreu model), a pure security, or a primitive security is a contract that agrees to pay one unit of a numeraire (a currency ...

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'' $\backslash pi\_$ of this particular state of the world; also referred to as a "Risk Neutral Density".
In the above example, the state prices, $\backslash pi\_$, $\backslash pi\_$would equate to the present values of $\$q\_$ and $\$q\_$: 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_Ã— + Ã—">math>\pi_Ã—$X\_$ + $\backslash pi\_$Ã—$X\_$ the fourth formula (see above regarding the absence of a risk premium here). For a continuous random variable
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

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
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or '' equivalent martingale measure'') is a probability measure such that each share price is exactly equal to the discounted expectation of the share price u ...

.
State prices find immediate application as a conceptual tool ("contingent claim analysis In finance, a contingent claim is a derivative whose future payoff depends on the value of another â€œunderlyingâ€ asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit ...

"); 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.
Using the related stochastic discount factor - also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor $\backslash tilde$, and then taking the expectation;See: David K. Backus (2015)Fundamentals of Asset Pricing

Stern NYU the third equation above. Essentially, this factor divides expected utility at the relevant future period - a function of the possible asset values realized under each state - by the utility due to today's wealth, and is then also referred to as "the intertemporal

marginal rate of substitution
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no exte ...

".
Resultant models

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
Option or Options may refer to:
Computing
*Option key, a key on Apple computer keyboards
*Option type, a polymorphic data type in programming languages
*Command-line option, an optional parameter to a command
*OPTIONS, an HTTP request method
...

: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money.
* Information
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely rando ...

: 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
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...

".)
Certainty

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 theModiglianiâ€“Miller theorem
The Modiglianiâ€“Miller theorem (of Franco Modigliani, Merton Miller) is an influential element of economic theory; it forms the basis for modern thinking on capital structure. The basic theorem states that in the absence of taxes, bankruptcy cos ...

, 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 (November 27, 1900 â€“ September 15, 1989) was an American economist, recognized as an important figure in the field of fundamental analysis, and for his analysis of stock prices as reflecting their " intrinsic value".
He is ...

'', 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
The net present value (NPV) or net present worth (NPW) applies to a series of cash flows occurring at different times. The present value of a cash flow depends on the interval of time between now and the cash flow. It also depends on the discount ...

(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 .
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. In many treatments bond valuation precedes equity valuation
In financial markets, stock valuation is the method of calculating theoretical values of companies and their stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit f ...

, 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. Fisher's formulation of the theory here - developing an intertemporal equilibrium model - underpins also the below applications to uncertainty.
See for the development.
Uncertainty

For "choice under uncertainty" the twin assumptions of rationality and market efficiency, as more closely defined, lead tomodern portfolio theory
Modern portfolio theory (MPT), or mean-variance analysis, is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversificat ...

(MPT) with its capital asset pricing model
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio.
The model takes into ac ...

(CAPM) â€“ an ''equilibrium-based'' result â€“ and to the Blackâ€“Scholesâ€“Merton theory (BSM; often, simply Blackâ€“Scholes) for option pricing â€“ 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 .
Briefly, and intuitively â€“ and consistent with #Arbitrage-free pricing and equilibrium above â€“ the relationship between rationality and efficiency is as follows.
Given the ability to profit from 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
The efficient-market hypothesis (EMH) is a hypothesis in financial economics that states that asset prices reflect all available information. A direct implication is that it is impossible to "beat the market" consistently on a risk-adjusted bas ...

, 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
In financial economics, finance, and accounting, the earnings response coefficient, or ERC, is the estimated relationship between equity returns and the unexpected portion of (i.e., new information in) companies' earnings announcements.
Developme ...

.)
The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the ''best guess of the future'': the assumption of rational expectations.
The EMH does allow that when faced with new information, some investors may overreact and some may underreact, but what is required, however, is that investors' reactions follow a normal distribution â€“ 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: the random walk hypothesis The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.
History
The concept can be traced to French broker Jules Regnault who p ...

.
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.
Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium.
Here, as under the certainty-case above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends,
Christopher L. Culp and John H. Cochrane. (2003)."Equilibrium Asset Pricing and Discount Factors: Overview and Implications for Derivatives Valuation and Risk Management"

, in ''Modern Risk Management: A History''. Peter Field, ed. London: Risk Books, 2003. 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 â€“ in the sense of arbitrage-exploitation â€“ 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. This result will be independent of the investor's level of risk aversion and assumed utility function, thus providing a readily determined discount rate for corporate finance decision makers as above, 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. 2â€“20, 1984. and for other investors. The argument proceeds as follows: If one can construct an

efficient frontier
In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the "efficient" parts of the riskâ€“return spectrum.
Formally, it is the set of portfolios which satisfy the condition that no o ...

â€“ i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram â€“ then mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and the "market portfolio
Market portfolio is a portfolio consisting of a weighted sum of every asset in the market, with weights in the proportions that they exist in the market, with the necessary assumption that these assets are infinitely divisible.
Richard Roll's cr ...

" (the Mutual fund separation theorem), with the combinations here plotting as the capital market line
Capital market line (CML) 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, so named since all rational investors (minimum variance criter ...

, or CML.
Then, given this CML, the required return on a risky security will be independent of the investor's utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosop ...

, and solely determined by its covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...

("beta") with aggregate, i.e. market, risk.
This is because investors here can then maximize utility through leverage as opposed to pricing; see Separation property (finance), 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 can be generalized to derive other pricing models.
Blackâ€“Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of European-styled options.
The model is expressed as the Blackâ€“Scholes equation, a partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...

describing the changing price of the option over time; it is derived assuming log-normal, geometric Brownian motion
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...

(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 risk-free rate).
This hedge, in turn, implies that there is only one right price â€“ in an arbitrage-free sense â€“ 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 #Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation.
ItÃ´'s lemma
In mathematics, ItÃ´'s lemma or ItÃ´'s formula (also called the ItÃ´-Doeblin formula, especially in French literature) is an identity used in ItÃ´ calculus to find the differential of a time-dependent function of a stochastic process. It serves ...

provides the underlying mathematics, and, with ItÃ´ calculus
ItÃ´ calculus, named after Kiyosi ItÃ´, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.
The centra ...

more generally, remains fundamental in quantitative finance.
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"

Emanuel Derman

''A Scientific Approach to CAPM and Options Valuation''

The Blackâ€“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 Arrowâ€“Debreu theory, and can be derived via state-pricing 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 . The Black-Scholes formula is found, in the limit, by attaching a binomial probability to each of numerous possible spot-prices (states) and then rearranging for the terms corresponding to $N(d\_1)$ and $N(d\_2)$, per the boxed description; see .

Extensions

More recent work further generalizes and extends these models. As regardsasset pricing
In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below, together with the resultant models.
There have been many models developed for different situations, but correspon ...

, 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

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 theCarhart four-factor model
In portfolio management, the Carhart four-factor model is an extra factor addition in the Famaâ€“French three-factor model, proposed by Mark Carhart. The Fama-French model, developed in the 1990, argued most stock market returns are explained by ...

, propose factors other than market return as relevant in pricing. The intertemporal CAPM
Within mathematical finance, the Intertemporal Capital Asset Pricing Model, or ICAPM, is an alternative to the CAPM provided by Robert Merton. It is a linear factor model with wealth as state variable that forecast changes in the distribution of ...

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
In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely beli ...

(APT) similarly differs as regards its assumptions. APT "gives up the notion that there is one right portfolio for everyone in the world, and ...replaces it with an explanatory model of what drives asset returns." 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 Portfolio optimization is the process of selecting the best portfolio (asset distribution), out of the set of all portfolios being considered, according to some objective. The objective typically maximizes factors such as expected return, and minim ...

, the Blackâ€“Litterman model departs from the original Markowitz model â€“ i.e. of constructing portfolios via an efficient frontier
In modern portfolio theory, the efficient frontier (or portfolio frontier) is an investment portfolio which occupies the "efficient" parts of the riskâ€“return spectrum.
Formally, it is the set of portfolios which satisfy the condition that no o ...

. 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
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision making (both in daily life and in settings ...

can be applied; here deriving a Pareto efficient
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848â€“1923), Italian civil engine ...

portfolio. The universal portfolio algorithm The universal portfolio algorithm is a portfolio selection algorithm from the field of machine learning and information theory. The algorithm learns adaptively from historical data and maximizes the log-optimal growth rate in the long run. It was in ...

applies machine learning to asset selection, learning adaptively from historical data. Behavioral portfolio theory
Behavioral portfolio theory (BPT), put forth in 2000 by Shefrin and Statman,SHEFRIN, H., AND M. STATMAN (2000): "Behavioral Portfolio Theory," ''Journal of Financial and Quantitative Analysis'', 35(2), 127â€“151. provides an alternative to the assu ...

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 and Machine learning, more generally.
(Tail) risk parity
Risk parity (or risk premia parity) is an approach to investment management which focuses on allocation of risk, usually defined as volatility, rather than allocation of capital. The risk parity approach asserts that when asset allocations are ad ...

focuses on allocation of risk, rather than allocation of capital.
See for other techniques and objectives, and for discussion.
Derivative pricing

In pricing derivatives, thebinomial options pricing model
In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation of options. Essentially, the model uses a "discrete-time" ( lattice based) model of the varying price over time of the underlying fin ...

provides a discretized version of Blackâ€“Scholes, useful for the valuation of American styled options. Discretized models of this type are built â€“ at least implicitly â€“ 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"

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 on the same logic (using "

contingent claim analysis In finance, a contingent claim is a derivative whose future payoff depends on the value of another â€œunderlyingâ€ asset,Dale F. Gray, Robert C. Merton and Zvi Bodie. (2007). Contingent Claims Approach to Measuring and Managing Sovereign Credit ...

"). Real options valuation
Real options valuation, also often termed real options analysis,Adam Borison ( Stanford University)''Real Options Analysis: Where are the Emperor's Clothes?''
(ROV or ROA) applies option valuation techniques to capital budgeting decisions.Campb ...

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 derivative
In finance, a credit derivative refers to any one of "various instruments and techniques designed to separate and then transfer the ''credit risk''" The Economist ''Passing on the risks'' 2 November 1996 or the risk of an event of default of a co ...

s allow that payment obligations or delivery requirements might not be honored. Exotic derivative
An exotic derivative, in finance, is a derivative which is more complex than commonly traded "vanilla" products. This complexity usually relates to determination of payoff; see option style.
The category may also include derivatives with a non ...

s are now routinely valued. Multi-asset underlyers are handled via simulation or copula based analysis.
Similarly, the various short-rate model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a s ...

s 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.) The more general HJM Framework describes the dynamics of the full forward-rate curve â€“ as opposed to working with short rates â€“ and is then more widely applied. The valuation of the underlying instrument â€“ additional to its derivatives â€“ is relatedly extended, particularly for hybrid securities, where credit risk is combined with uncertainty re future rates; see and .
Following the Crash of 1987, equity options traded in American markets began to exhibit what is known as a "volatility smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Blackâ€“Scholes formula to fit market prices. In particular for a given exp ...

"; 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.
After the financial crisis of 2007â€“2008
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline o ...

, a further development:Didier Kouokap Youmbi (2017).Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach

,

Bank of England
The Bank of England is the central bank of the United Kingdom and the model on which most modern central banks have been based. Established in 1694 to act as the English Government's banker, and still one of the bankers for the Government of ...

â€“ Prudential Regulation Authority (over the counter
Over-the-counter (OTC) drugs are medicines sold directly to a consumer without a requirement for a prescription from a healthcare professional, as opposed to prescription drugs, which may be supplied only to consumers possessing a valid presc ...

) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment â€“ called into question during the crisis.
Addressing this, therefore, issues such as counterparty credit risk
A credit risk is risk of default on a debt that may arise from a borrower failing to make required payments. In the first resort, the risk is that of the lender and includes lost principal and interest, disruption to cash flows, and increased ...

, funding costs and costs of capital are now additionally considered when pricing, and a credit valuation adjustment, or CVA â€“ and potentially other ''valuation adjustments'', collectively xVA
An X-Value Adjustment (XVA, xVA) is an umbrella term referring to a number of different â€œvaluation adjustmentsâ€ that banks must make when assessing the value of derivative contracts that they have entered into. The purpose of these is twofold: ...

â€“ 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
The London Inter-Bank Offered Rate is an interest-rate average calculated from estimates submitted by the leading banks in London. Each bank estimates what it would be charged were it to borrow from other banks. The resulting average rate is ...

as used previously. This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate". (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 â€“ and, therefore, yield curve construction â€“ 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
In finance, an interest rate swap (IRS) is an interest rate derivative (IRD). It involves exchange of interest rates between two parties. In particular it is a "linear" IRD and one of the most liquid, benchmark products. It has associations wit ...

" where "forecast curves" are constructed for each floating-leg LIBOR tenor, with discounting on the ''common'' OIS curve.
Corporate finance theory

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
Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...

" or probabilistic corporate finance models, as opposed to the traditional static and deterministic models; see .
Relatedly, Real Options theory allows for owner â€“ i.e. managerial â€“ 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.
More traditionally, decision tree
A decision tree is a decision support tool that uses a tree-like model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm that only contains condi ...

s â€“ which are complementary â€“ have been used to evaluate projects, by incorporating in the valuation (all) possible events (or states) and consequent management decisions; Aswath Damodaran (2007)"Probabilistic Approaches: Scenario Analysis, Decision Trees and Simulations"

In ''Strategic Risk Taking: A Framework for Risk Management''. Prentice Hall. the correct discount rate here reflecting each point's "non-diversifiable risk looking forward." Related to this, is the treatment of forecasted cashflows in

equity valuation
In financial markets, stock valuation is the method of calculating theoretical values of companies and their stocks. The main use of these methods is to predict future market prices, or more generally, potential market prices, and thus to profit f ...

. In many cases, following Williams above, the average (or most likely) cash-flows were discounted, as opposed to a more correct state-by-state treatment under uncertainty; see comments under Financial modeling Â§ Accounting.
In more modern treatments, then, it is the ''expected'' cashflows (in the mathematical sense: ) combined into an overall value per forecast period which are discounted.
"Capital Budgeting Applications and Pitfalls". Ch 13 in Ivo Welch (2017). ''Corporate Finance'': 4th Edition And using the CAPM â€“ or extensions â€“ the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows (essentially, $Y$ and $r$ combined). Other developments here include 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; here emphasizing the issues interrelated with capital structure.

Clean surplus accounting
The clean surplus accounting method provides elements of a forecasting model that yields price as a function of earnings, expected returns, and change in book value.
Ohlson, J. A. (1995)"Earnings, Book Values and Dividends in Equity Valuation" Con ...

and the related residual income valuation provide a model that returns price as a function of earnings, expected returns, and change in book value
In accounting, book value is the value of an asset according to its balance sheet account balance. For assets, the value is based on the original cost of the asset less any depreciation, amortization or impairment costs made against the asset. ...

, 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 .
"Corporate finance" as a discipline more generally, per Fisher above, relates to the long term objective of maximizing the value of the firm - and its return to shareholders - and thus also incorporates the areas of capital structure and dividend policy.
Extensions of the theory here then also consider these latter, as follows:
(i) optimization re capitalization structure, and theories here as to corporate choices and behavior: Capital structure substitution theory In finance, the capital structure substitution theory (CSS) describes the relationship between earnings, stock price and capital structure of public companies. The CSS theory hypothesizes that managements of public companies manipulate capital stru ...

, Pecking order theory In corporate finance, the pecking order theory (or pecking order model) postulates that the cost of financing increases with asymmetric information.
Financing comes from three sources, internal funds, debt and new equity. Companies prioritize their ...

, Market timing hypothesis, Trade-off theory
The trade-off theory of capital structure is the idea that a company chooses how much debt finance and how much equity finance to use by balancing the costs and benefits. The classical version of the hypothesis goes back to Kraus and Litzenberger ...

;
(ii) considerations and analysis re dividend policy, additional to - and sometimes contrasting with - Modigliani-Miller, include:
the Walter model
Otto Moritz Walter Model (; 24 January 1891 â€“ 21 April 1945) was a German field marshal during World War II. Although he was a hard-driving, aggressive panzer commander early in the war, Model became best known as a practitioner of defe ...

, Lintner model, and Residuals theory, as well as discussion re the observed clientele effect and dividend puzzle {{More footnotes, date=May 2021
The dividend puzzle is a concept in finance in which companies that pay dividends are rewarded by investors with higher valuations, even though, according to many economists, it should not matter to investors whether ...

.
As described, the typical application of real options is to capital budgeting
Capital budgeting in corporate finance is the planning process used to determine whether an organization's long term capital investments such as new machinery, replacement of machinery, new plants, new products, and research development projects ...

type problems.
However, here, they are also applied to problems 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.
and since stockholder and bondholders have different objective functions, in the analysis of the related agency problems.
In all of these cases, state-prices can provide the market-implied information relating to the corporate, as above, which is then applied to the analysis. For example, convertible bond
In finance, a convertible bond or convertible note or convertible debt (or a convertible debenture if it has a maturity of greater than 10 years) is a type of bond that the holder can convert into a specified number of shares of common stock i ...

s can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.See Kruschwitz and LÃ¶ffler under Bibliography.
Challenges and criticism

As above, there is a very close link between (i) therandom walk hypothesis The random walk hypothesis is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.
History
The concept can be traced to French broker Jules Regnault who p ...

, with the associated belief that price changes should follow a normal distribution, on the one hand, and (ii) market efficiency and rational expectations, on the other. Wide departures from these are commonly observed, and there are thus, respectively, two main sets of challenges.
Departures from normality

As discussed, the assumptions that market prices follow a random walk and that asset returns are normally distributed are fundamental. Empirical evidence, however, suggests that these assumptions may not hold, and that in practice, traders, analysts and risk managers frequently modify the "standard models" (seeKurtosis risk
In statistics and decision theory, kurtosis risk is the risk that results when a statistical model assumes the normal distribution, but is applied to observations that have a tendency to occasionally be much farther (in terms of number of stan ...

, Skewness risk
Skewness risk in financial modeling is the risk that results when observations are not spread symmetrically around an average value, but instead have a skewed distribution. As a result, the mean and the median can be different. Skewness risk ...

, Long tail
In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random ...

, Model risk
In finance, model risk is the risk of loss resulting from using insufficiently accurate models to make decisions, originally and frequently in the context of valuing financial securities. However, model risk is more and more prevalent in activitie ...

).
In fact, Benoit Mandelbrot had discovered already in the 1960s
that changes in financial prices do not follow a normal distribution, the basis for much option pricing theory, although this observation was slow to find its way into mainstream financial economics.
Financial models with long-tailed distributions and volatility clustering have been introduced to overcome problems with the realism of the above "classical" financial models; while jump diffusion models allow for (option) pricing incorporating "jumps" in the spot price
In finance, a spot contract, spot transaction, or simply spot, is a contract of buying or selling a commodity, security or currency for immediate settlement (payment and delivery) on the spot date, which is normally two business days after th ...

.
Risk managers, similarly, complement (or substitute) the standard value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...

models with historical simulations, mixture models, principal component analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...

, extreme value theory
Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the p ...

, as well as models for volatility clustering.
For further discussion see , and .
Portfolio managers, likewise, have modified their optimization criteria and algorithms; see #Portfolio theory above.
Closely related is the volatility smile
Volatility smiles are implied volatility patterns that arise in pricing financial options. It is a parameter (implied volatility) that is needed to be modified for the Blackâ€“Scholes formula to fit market prices. In particular for a given exp ...

, 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
In finance, moneyness is the relative position of the current price (or future price) of an underlying asset (e.g., a stock) with respect to the strike price of a derivative, most commonly a call option or a put option. Moneyness is firstly a ...

), true only if the price-change distribution is non-normal, unlike that assumed by BSM.
The term structure of volatility describes how (implied) volatility differs for related options with different maturities.
An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure.
These empirical phenomena negate the assumption of constant volatility â€“ and log-normality â€“ upon which Blackâ€“Scholes is built.
Within institutions, the function of Black-Scholes is now, largely, to ''communicate'' prices via implied volatilities, much like bond prices are communicated via YTM; see .
In consequence traders ( and risk managers) now, instead, use "smile-consistent" models, firstly, when valuing derivatives not directly mapped to the surface, facilitating the pricing of other, i.e. non-quoted, strike/maturity combinations, or of non-European derivatives, and generally for hedging purposes.
The two main approaches are local volatility A local volatility model, in mathematical finance and financial engineering, is an option pricing model that treats volatility as a function of both the current asset level S_t and of time t . As such, it is a generalisation of the Blackâ€“Scho ...

and stochastic volatility
In statistics, stochastic volatility models are those in which the variance of a stochastic process is itself randomly distributed. They are used in the field of mathematical finance to evaluate derivative securities, such as options. The name ...

.
The first returns the volatility which is "local" to each spot-time point of the finite difference- or simulation-based valuation; i.e. as opposed to implied volatility, which holds overall. In this way calculated prices â€“ and numeric structures â€“ are market-consistent in an arbitrage-free sense. The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first 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.
Related to local volatility are the lattice-based implied-binomial and -trinomial trees â€“ essentially a discretization of the approach â€“ which are similarly, but less commonly, used for pricing; these are built on state-prices recovered from the surface. Edgeworth binomial trees allow for a specified (i.e. non-Gaussian) 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.
Similarly purposed (and derived) closed-form models were also developed.
As discussed, additional to assuming log-normality in returns, "classical" BSM-type models also (implicitly) assume the existence of a credit-risk-free environment, where one can perfectly replicate cashflows so as to fully hedge, and then discount at "the" risk-free-rate.
And therefore, post crisis, the various x-value adjustments must be employed, effectively correcting the risk-neutral value for counterparty- and funding-related risk.
These xVA are ''additional'' to any smile or surface effect. This is valid as the surface is built on price data relating to fully collateralized positions, and there is therefore no " 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
Financial engineering is a multidisciplinary field involving financial theory, methods of engineering, tools of mathematics and the practice of programming. It has also been defined as the application of technical methods, especially from mathem ...

) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, James Rickards, amongst other critics of financial economics, suggests that, instead, the theory needs revisiting almost entirely:
:"The current system, based on the idea that risk is distributed in the shape of a bell curve, is flawed... The problem is hat 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."
Departures from rationality

As seen, a common assumption is that financial decision makers act rationally; seeHomo economicus
The term ''Homo economicus'', or economic man, is the portrayal of humans as agents who are consistently rational and narrowly self-interested, and who pursue their subjectively defined ends optimally. It is a word play on ''Homo sapiens'', u ...

. Recently, however, researchers in experimental economics and experimental finance The goals of experimental finance are to understand human and market behavior in settings relevant to finance. Experiments are synthetic economic environments created by researchers specifically to answer research questions. This might involve, for ...

have challenged this assumption empirically
In philosophy, empiricism is an epistemological theory that holds that knowledge or justification comes only or primarily from sensory experience. It is one of several views within epistemology, along with rationalism and skepticism. Empir ...

. These assumptions are also challenged theoretically, by behavioral finance, a discipline primarily concerned with the limits to rationality of economic agents.
For related criticisms re corporate finance theory vs its practice see: .
Consistent with, and complementary to these findings, various persistent market anomalies have been documented, these being price or return distortions â€“ e.g. size premium The size premium is the historical tendency for the stocks of firms with smaller market capitalizations to outperform the stocks of firms with larger market capitalizations. It is one of the factors in the Famaâ€“French three-factor model
In asse ...

s â€“ which appear to contradict the efficient-market hypothesis
The efficient-market hypothesis (EMH) is a hypothesis in financial economics that states that asset prices reflect all available information. A direct implication is that it is impossible to "beat the market" consistently on a risk-adjusted bas ...

; calendar effect
A calendar effect (or calendar anomaly) is any market anomaly, different behaviour of stock markets, or economic effect which appears to be related to the calendar, such as the day of the week, time of the month, time of the year, time within the ...

s are the best known group here. Related to these are various of the economic puzzles, concerning phenomena similarly contradicting the theory. The ''equity premium puzzle
The equity premium puzzle refers to the inability of an important class of economic models to explain the average equity risk premium (ERP) provided by a diversified portfolio of U.S. equities over that of U.S. Treasury Bills, which has been obser ...

'', as one example, arises in that the difference between the observed returns on stocks as compared to government bonds is consistently higher than the risk premium rational equity investors should demand, an " abnormal return". For further context see Random walk hypothesis Â§ A non-random walk hypothesis, and sidebar for specific instances.
More generally, and particularly following the financial crisis of 2007â€“2008
Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline o ...

, financial economics and mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that requi ...

have been subjected to deeper criticism; notable here is Nassim Nicholas Taleb
Nassim Nicholas Taleb (; alternatively ''Nessim ''or'' Nissim''; born 12 September 1960) is a Lebanese-American essayist, mathematical statistician, former option trader, risk analyst, and aphorist whose work concerns problems of randomness, ...

, who claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading; see Black swan theory
The black swan theory or theory of black swan events is a metaphor that describes an event that comes as a surprise, has a major effect, and is often inappropriately rationalized after the fact with the benefit of hindsight. The term is based o ...

, Taleb distribution.
A topic of general interest has thus been financial crises,
and the failure of (financial) economics to model (and predict) these.
A related problem is systemic risk: where companies hold securities in each other then this interconnectedness may entail a "valuation chain" â€“ and the performance of one company, or security, here will impact all, a phenomenon not easily modeled, regardless of whether the individual models are correct. See: Systemic risk Â§ Inadequacy of classic valuation models; Cascades in financial networks; Flight-to-quality
A flight-to-quality, or flight-to-safety, is a financial market phenomenon occurring when investors sell what they perceive to be higher-risk investments and purchase safer investments, such as gold and other precious metals. This is considered a ...

.
Areas of research attempting to explain (or at least model) these phenomena, and crises, include noise trading, market microstructure, and Heterogeneous agent models. The latter is extended to agent-based computational economics
Agent-based computational economics (ACE) is the area of computational economics that studies economic processes, including whole economies, as dynamic systems of interacting agents. As such, it falls in the paradigm of complex adaptive systems. ...

, where price is treated as an emergent phenomenon, resulting from the interaction of the various market participants (agents). The noisy market hypothesis
In finance, the noisy market hypothesis contrasts the efficient-market hypothesis in that it claims that the prices of securities are not always the best estimate of the true underlying value of the firm. It argues that prices can be influenced by ...

argues that prices can be influenced by speculators and momentum traders, as well as by insiders and institutions that often buy and sell stocks for reasons unrelated to fundamental value
In finance, the intrinsic value of an asset usually refers to a value calculated on simplified assumptions. For example, the intrinsic value of an option is based on the current market value of the underlying instrument, but ignores the possib ...

; see Noise (economic)
Economic noise, or simply noise, describes a theory of pricing developed by Fischer Black. Black describes noise as the opposite of information: hype, inaccurate ideas, and inaccurate data. His theory states that noise is everywhere in the economy ...

. The adaptive market hypothesis is an attempt to reconcile the efficient market hypothesis with behavioral economics, by applying the principles of evolution to financial interactions. An information cascade
An Information cascade or informational cascade is a phenomenon described in behavioral economics and network theory in which a number of people make the same decision in a sequential fashion. It is similar to, but distinct from herd behavior.
An ...

, alternatively, shows market participants engaging in the same acts as others ("herd behavior
Herd behavior is the behavior of individuals in a group acting collectively without centralized direction. Herd behavior occurs in animals in herds, packs, bird flocks, fish schools and so on, as well as in humans. Voting, demonstrations, ri ...

"), despite contradictions with their private information. Copula-based modelling has similarly been applied. See also Hyman Minsky
Hyman Philip Minsky (September 23, 1919 â€“ October 24, 1996) was an American economist, a professor of economics at Washington University in St. Louis, and a distinguished scholar at the Levy Economics Institute of Bard College. His research a ...

's "financial instability hypothesis", as well as George Soros' approach under Â§ Reflexivity, financial markets, and economic theory.
On the obverse, however, various studies have shown that despite these departures from efficiency, asset prices do typically exhibit a random walk and that one cannot therefore consistently outperform market averages, i.e. attain "alpha". William F. Sharpe (1991)"The Arithmetic of Active Management"

. ''Financial Analysts Journal'' Vol. 47, No. 1, January/February The practical implication, therefore, is that passive investing (e.g. via low-cost

index fund
An index fund (also index tracker) is a mutual fund or exchange-traded fund (ETF) designed to follow certain preset rules so that the fund can a specified basket of underlying investments.Reasonable Investor(s), Boston University Law Review, avail ...

s) should, on average, serve better than any other active strategy. William F. Sharpe (2002)''Indexed Investing: A Prosaic Way to Beat the Average Investor''

. Presention:

Monterey Institute of International Studies
The Middlebury Institute of International Studies at Monterey (MIIS), formerly known as the Monterey Institute of International Studies, is an American graduate school of Middlebury College, a private college in Middlebury, Vermont.
Established ...

. Retrieved May 20, 2010. Burton Malkiel's '' A Random Walk Down Wall Street'' â€“ first published in 1973, and in its 12th edition as of 2019 â€“ is a widely read popularization of these arguments.
(See also John C. Bogle's '' Common Sense on Mutual Funds''; but compare Warren Buffett
Warren Edward Buffett ( ; born August 30, 1930) is an American business magnate, investor, and philanthropist. He is currently the chairman and CEO of Berkshire Hathaway. He is one of the most successful investors in the world and has a net ...

's ''The Superinvestors of Graham-and-Doddsville
"The Superinvestors of Graham-and-Doddsville" is an article by Warren Buffett promoting value investing, published in the Fall, 1984 issue of ''Hermes'', Columbia Business School magazine. It was based on a speech given on May 17, 1984, at the Col ...

''.)
Relatedly, institutionally inherent '' limits to arbitrage'' â€“ as opposed to factors directly contradictory to the theory â€“ are sometimes proposed as an explanation for these departures from efficiency.
See also

* :Finance theories * :Financial models * Deutsche Bank Prize in Financial Economics *Economic model
In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework de ...

*
* Financial modeling
Financial modeling is the task of building an abstract representation (a model) of a real world financial situation. This is a mathematical model designed to represent (a simplified version of) the performance of a financial asset or portfolio ...

* Fischer Black Prize
* List of financial economics articles
* List of financial economists
*
* Master of Financial Economics
* Monetary economics
* Outline of economics
* Outline of finance
The following outline is provided as an overview of and topical guide to finance:
Finance – addresses the ways in which individuals and organizations raise and allocate monetary resources over time, taking into account the risks entailed ...

Historical notes

References

Bibliography

Financial economics * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Volume I ; Volume II . * Asset pricing * * * * * * * * * * * * * * Corporate finance * * * * * * * * * * * * *External links

{{Financial risk Actuarial science