Financial economics 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): 526–528 For a simplified example see , where the economy has only two possible states – up and down – 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

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 – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM. In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are. 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. 2–20, 1984. and for other investors. The argument proceeds as follows: If one can construct an efficient frontier – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and the "

"Option Prices and Expected Returns"

Emanuel Derman

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

The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the fundamental result above – further explaining, and if required demonstrating, this unity.Mark Rubinstein, Rubinstein, Mark. (2005). "Great Moments in Financial Economics: IV. The Fundamental Theorem (Part I)", ''Journal of Investment Management'', Vol. 3, No. 4, Fourth Quarter 2005; ~ (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 Option style#Non-vanilla path-dependent "exotic" options, path dependent derivatives, Monte Carlo methods for option pricing are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various Option (finance)#Model implementation, other numeric techniques have also been developed. The theoretical framework too has been extended such that

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

, Bank of England – Prudential Regulation Authority (United Kingdom), Prudential Regulation Authority (Over-the-counter (finance), over the counter) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered when pricing, and a Credit Valuation Adjustment, or CVA – and potentially other ''valuation adjustments'', collectively xVA – 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, Overnight Index Swap (OIS) curve, as opposed to LIBOR as used previously. This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate". (Also, practically, the interest paid on cash collateral (finance), collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "Credit Support Annex, CSA discounting".) Swap (finance)#Valuation, Swap pricing – and, therefore, yield curve construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "multi-curve framework" where "forecast curves" are constructed for each floating-leg Libor#Maturities, LIBOR tenor, with discounting on the ''common'' OIS curve.

"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 many cases, following Williams #Certainty, 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#Accounting, Financial modeling § Accounting. In more modern treatments, then, it is the ''expected'' cashflows (in the Expected value, 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 – or extensions – the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows; (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. 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 . 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. 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, #State prices, as above, which is then applied to the analysis. For example, convertible bonds can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.See Kruschwitz and Löffler per Bibliography.

"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 funds) should, on average, serve better than any other active investing, active strategy. William F. Sharpe (2002)

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

. Presention: Monterey Institute of International Studies. Retrieved May 20, 2010. Burton Malkiel's ''A Random Walk Down Wall Street'' – first published in 1973, and in its 12th edition as of 2019 – is a widely read popularization of these arguments. (See also John C. Bogle's ''Common Sense on Mutual Funds''; but compare Warren Buffett's ''The Superinvestors of Graham-and-Doddsville''.) Relatedly, institutionally inherent ''limits to arbitrage'' – as opposed to factors directly contradictory to the theory – are sometimes proposed as an explanation for these departures from efficiency.

economics
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within those societies. The term was fo ...

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
In finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investm ...

s and exchange rates, as opposed to those concerning the real economyThe real economy concerns the production, purchase and flow of goods
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economi ...

.
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
:'' This article is theory focused: for the corporate finance
Corporate finance is the area of finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with ...

and corporate finance
Corporate finance is the area of finance that deals with sources of funding, the capital structure of corporations, the actions that managers take to increase the Value investing, value of the firm to the shareholders, and the tools and analysis ...

; the first being the perspective of providers of capital, i.e. investors, and the second of users of capital.
It thus provides the theoretical underpinning for much of finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available which could ...

.
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
Microeconomics is a branch of that studies the behavior of individuals and in making decisions regarding the allocation of and the interactions among these individuals and firms. Microeconomics focuses on the study of individual markets, se ...

and decision theory
Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent's choices. Decision theory can be broken into two branches: normative
Normative generally means relating to an evaluative standard. Normativi ...

.
Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise these relationships. Mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics
Physics is the natu ...

is related in that it will derive and extend the mathematical or numerical models suggested by financial economics. The emphasis there is mathematical consistency, as opposed to compatibility with economic theory. Whereas financial economics has a primarily microeconomic
Microeconomics is a branch of mainstream economics
Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that are generally accepted by economists as a basis for discussion. Also ...

focus, monetary economics
Monetary economics is the branch of economics that studies the different competing theories of money: it provides a framework for analyzing money and considers its functions (such as medium of exchange, store of value and unit of account), and it c ...

is primarily macroeconomic
Macroeconomics (from the Greek prefix ''makro-'' meaning "large" + ''economics'') is a branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), producti ...

in nature.
Underlying economics

As above, the discipline essentially explores how rational investors would applydecision theory
Decision theory (or the theory of choice not to be confused with choice theory) is the study of an agent's choices. Decision theory can be broken into two branches: normative
Normative generally means relating to an evaluative standard. Normativi ...

to the problem of investment
Investment is the dedication of an asset to attain an increase in value over a period of time. Investment requires a sacrifice of some present asset, such as time, money, or effort.
In , the purpose of investing is to generate a from the inve ...

. The subject is thus built on the foundations of microeconomics
Microeconomics is a branch of that studies the behavior of individuals and in making decisions regarding the allocation of and the interactions among these individuals and firms. Microeconomics focuses on the study of individual markets, se ...

and decision theory, and derives several key results for the application of decision making
In psychology
Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense s ...

under uncertainty to the financial market
A financial market is a market
Market may refer to:
*Market (economics)
*Market economy
*Marketplace, a physical marketplace or public market
Geography
*Märket, an island shared by Finland and Sweden
Art, entertainment, and media Films
*Ma ...

s. The underlying economic logic distills to a “fundamental valuation result”, as aside, which is developed in the following sections.
Present value, expectation and utility

Underlying all of financial economics are the concepts ofpresent value
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...

and expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* Expectation (alb ...

.
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.
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
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

("as any student of finance knows"). In other words, since individuals receive greater utility
As a topic of economics
Economics () is a social science
Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within thos ...

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 priceIn finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with the creation and management of money and investments. Savers and investors have money available w ...

. (Some investors may in fact be risk seeking as opposed to risk averse
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

, but the same logic would apply).
Choice under uncertainty here may then be characterized as the maximization of expected utilityThe expected utility hypothesis is a popular concept in economics, game theory and decision theory that serves as a reference guide for judging decisions involving uncertainty. The theory recommends which option a rational individual should choose in ...

. More formally, the resulting expected utility hypothesisThe expected utility hypothesis is a popular concept in economics, game theory
Game theory is the study of mathematical models of strategic interaction among Rational agent, rational decision-makers.Roger B. Myerson, Myerson, Roger B. (1991). '' ...

states that, if certain axioms are satisfied, the subjective
Subjective may refer to:
* Subjectivity, a subject's personal perspective, feelings, beliefs, desires or discovery, as opposed to those made from an independent, objective, point of view
** Subjective experience, the subjective quality of consciou ...

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
The Ellsberg paradox is a paradox
A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true prem ...

.
Arbitrage-free pricing and equilibrium

The concepts ofarbitrage
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

-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
Rational pricing is the assumption in financial economics
Financial economics is the branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), productio ...

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
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods ...

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
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...

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
The fundamental theorems of asset pricing (also: of arbitrage, of finance) provide necessary and sufficient conditions for a market to be arbitrage free and for a market to be complete. An arbitrage opportunity is a way of making money with no ...

, shows that where markets are as described – and are additionally (implicitly and correspondingly) complete – 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): 526–528 For a simplified example see , where the economy has only two possible states – up and down – 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
Overview
A risk premium is a measure of excess return that is required by an individual to compensate them for being subjected to an increased level of risk. It is used widely in finance and economics with the general definition being the expect ...

, 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
A creditor or lender is a party (e.g., person, organization, company, or government) that has a claim on the services of a second party. ...

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, the underlying of a derivative
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculu ...

and a risk free “bond”; see (and #Uncertainty below). Where the underlying is itself being priced, such "manufacturing" is of course not possible – the instrument being "fundamental", i.e. as opposed to "derivative" – and a premium is then required for risk.
State prices

With the above relationship established, the further specializedArrow–Debreu model
In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that Aggregate supply, aggregate supplies wi ...

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 Model (economics), economic model is said to have a representative ...

''.
The Arrow–Debreu model applies to economies with maximally complete market
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods an ...

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 priceIn financial economics
Financial economics is the branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribut ...

security (also called an Arrow–Debreu security), a contract that agrees to pay one unit of a numeraire (a currency or a commodity) if a particular state occurs ("up" and "down" in the simplified example above) at a particular time in the future and pays zero numeraire in all the other states. The price of this security is the state price $\backslash pi\_$ of this particular state of the world.
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 is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be \pi_× + ×">math>\pi_×$X\_$ + $\backslash pi\_$×$X\_$ the fourth formula (see above regarding the absence of a risk premium here). For a continuous random variable
In probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

indicating a continuum of possible states, the value is found by integrating over the state price "density". These concepts are extended to martingale pricingMartingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. option ...

and the related risk-neutral measure
In mathematical financeMathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. Generally, mathematical finance will derive an ...

.
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 Ødegaard, under bibliography. Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security" – 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 factorThe concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow \tilde_i by the stochastic factor \tilde, ...

- 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
As a topic of economics, utility is used to model w ...

".
Resultant models

Applying the above economic concepts, we may then derive various and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information", as will be seen below. * Time: money now is traded for money in the future. * Uncertainty (or risk): The amount of money to be transferred in the future is uncertain. * Options: one party to the transaction can make a decision at a later time that will affect subsequent transfers of money. *Information
Information is processed, organised and structured data
Data (; ) are individual facts
A fact is something that is truth, true. The usual test for a statement of fact is verifiability—that is whether it can be demonstrated to c ...

: 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
Determinism is the philosophical
Philosophy (from , ) is the study of general and fundamental questions, such as those about existence
Existence is the ability of an entity to interact with physical or mental reality
Reality is the ...

" and "random", or "stochastic
Stochastic () refers to the property of being well described by a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...

".)
Certainty

The starting point here is “Investment under certainty", and usually framed in the context of a corporation. TheFisher separation theorem
In economics
Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...

, asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders.
Related is the Modigliani–Miller theoremThe Modigliani–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 costs, ...

, 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 – September 15, 1989) was an American economist
An economist is a professional and practitioner in the social science
Social science is the Branches of science, branch of science devoted to the s ...

'', 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 dividend
A dividend is a distribution of profit
Profit may refer to:
Business and law
* Profit (accounting), the difference between the purchase price and the costs of bringing to market
* Profit (economics), normal profit and economic profit
* Profit ...

s. What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. 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 ra ...

(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
Bond valuation is the determination of the fair price of a bond. As with any security or capital investment, the theoretical fair value of a bond is the present value
In economics
Economics () is the social science that studies how pe ...

, 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 – i.e. at each coupon's corresponding zero-rate – as opposed to an overall rate. In many treatments bond valuation precedes equity valuation, under which cashflows (dividends) are not "known" ''per se''. Williams and onward allow for forecasting as to these – based on historic ratios or published policy – and cashflows are then treated as essentially deterministic; see below under #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 andmarket efficiency
The efficient-market hypothesis (EMH) is a hypothesis in financial economics
Financial economics is the branch of economics
Economics () is the social science that studies how people interact with value; in particular, the Production (ec ...

, as more closely defined, lead to modern 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 Diversification ...

(MPT) with its capital asset pricing model
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, i ...

(CAPM) – an ''equilibrium-based'' result – and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing
In finance
Finance is a term for the management, creation, and study of money
In a 1786 James Gillray caricature, the plentiful money bags handed to King George III are contrasted with the beggar whose legs and arms were amputated, in t ...

– 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 – and consistent with #Arbitrage-free pricing and equilibrium above – 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
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 o ...

, or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See Earnings response coefficient.)
The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information, are identical to the ''best guess of the future'': the assumption of rational expectations
In economics, "rational expectations" are model-consistent expectations, in that agent (economics), agents inside the model (economics), model are assumed to "know the model" and on average take the model's predictions as valid. Rational expectat ...

.
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
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

– 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 hypothesisThe random walk hypothesis is a financial theory stating that stock market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on busine ...

.
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 – in the sense of arbitrage-exploitation – gives rise to Black–Scholes; option values here ultimately consistent with the CAPM. In general, then, while portfolio theory studies how investors should balance risk and return when investing in many assets or securities, the CAPM is more focused, describing how, in equilibrium, markets set the prices of assets in relation to how risky they are. 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. 2–20, 1984. and for other investors. The argument proceeds as follows: If one can construct an efficient frontier – i.e. each combination of assets offering the best possible expected level of return for its level of risk, see diagram – then mean-variance efficient portfolios can be formed simply as a combination of holdings of the risk-free asset and the "

market portfolio
Market portfolio is a portfolio consisting of a weighted sum of every asset
In financial accountancy, financial accounting, an asset is any resource owned or controlled by a business or an economic entity. It is anything (tangible or intangible) ...

" (the Mutual fund separation theoremIn 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 extensio ...

), 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 Separation property (finance), and CML diagram aside.
As can be seen in the formula aside, this result is consistent with #Arbitrage-free pricing and equilibrium, the preceding, equaling the riskless return plus an adjustment for risk.
A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other pricing models.
Black–Scholes provides a mathematical model of a financial market containing Derivative (finance), derivative instruments, and the resultant formula for the price of option style, European-styled options.
The model is expressed as the Black–Scholes equation, a partial differential equation describing the changing price of the option over time; it is derived assuming log-normal, geometric Brownian motion (see Brownian model of financial markets).
The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing ($V$, 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 – in an arbitrage-free sense – for the option. And this price is returned by the Black–Scholes option pricing formula. (The formula, and hence the price, is consistent with the equation, as the formula is Partial differential equation#Analytical solutions, the solution to the equation.)
Since the formula is without reference to the share's expected return, Black–Scholes inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with #Arbitrage-free pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation.
Itô's lemma provides Itô's lemma#Black–Scholes formula, the underlying mathematics, and, with Itô calculus more generally, remains fundamental in quantitative finance.
As mentioned, it can be shown that the two models are consistent; then, as is to be expected, "classical" financial economics is thus unified.
Here, the Black Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the 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 Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the fundamental result above – further explaining, and if required demonstrating, this unity.Mark Rubinstein, Rubinstein, Mark. (2005). "Great Moments in Financial Economics: IV. The Fundamental Theorem (Part I)", ''Journal of Investment Management'', Vol. 3, No. 4, Fourth Quarter 2005; ~ (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
:'' This article is theory focused: for the corporate finance
Corporate finance is the area of finance
Finance is the study of financial institutions, financial markets and how they operate within the financial system. It is concerned with ...

, 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 Fama–French three-factor model and the Carhart four-factor model, propose factors other than market return as relevant in pricing. The intertemporal CAPM and Consumption-based capital asset pricing model, consumption-based CAPM similarly extend the model. With intertemporal portfolio choice, the investor now repeatedly optimizes her portfolio; while the inclusion of Consumption (economics), consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return. Whereas the above extend the CAPM, the single-index model is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization. The arbitrage pricing theory (APT) 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, the Black–Litterman model departs from the original Markowitz model – i.e. of constructing portfolios via an efficient frontier. Black–Litterman instead starts with an equilibrium assumption, and is then modified to take into account the 'views' (i.e., the specific opinions about asset returns) of the investor in question to arrive at a bespoke asset allocation. Where factors additional to volatility are considered (kurtosis, skew...) then multiple-criteria decision analysis can be applied; here deriving a Pareto efficient portfolio. The universal portfolio algorithm 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 Copula (probability theory)#Quantitative finance, lately been applied here; recently this is the case also List of genetic algorithm applications#Finance and Economics, for genetic algorithms and Machine learning#Applications, Machine learning, more generally. See for other techniques and objectives.Derivative pricing

As regards derivative pricing, the binomial options pricing model provides a discretized version of Black–Scholes, useful for the valuation of American styled options. Discretized models of this type are built – at least implicitly – using state-prices (#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 Option style#Non-vanilla path-dependent "exotic" options, path dependent derivatives, Monte Carlo methods for option pricing are employed; here the modelling is in continuous time, but similarly uses risk neutral expected value. Various Option (finance)#Model implementation, other numeric techniques have also been developed. The theoretical framework too has been extended such that

martingale pricingMartingale pricing is a pricing approach based on the notions of martingale and risk neutrality. The martingale pricing approach is a cornerstone of modern quantitative finance and can be applied to a variety of derivatives contracts, e.g. option ...

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"). Real options valuation allows that option holders can influence the option's underlying; models for Employee stock option#Valuation, employee stock option valuation explicitly assume non-rationality on the part of option holders; Credit derivatives allow that payment obligations or delivery requirements might not be honored. Exotic derivatives are now routinely valued. Multi-asset underlyers are handled via simulation or Copula (probability theory)#Quantitative finance, copula based analysis.
Similarly, the various short-rate models allow for an extension of these techniques to Fixed income#Derivatives, fixed income- and interest rate derivatives. (The Vasicek model, Vasicek and Cox–Ingersoll–Ross model, CIR models are equilibrium-based, while Ho–Lee model, Ho–Lee and subsequent models are based on arbitrage-free pricing.) The more general Heath–Jarrow–Morton framework, HJM Framework describes the dynamics of the full forward rate, forward-rate curve – as opposed to working with short rates – and is then more widely applied. The valuation of the underlying instrument – additional to its derivatives – is relatedly extended, particularly for Hybrid security, hybrid securities, where credit risk is combined with uncertainty re future rates; see and .
Following the Black Monday (1987), 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 volatility, implied volatilities, than what is suggested by BSM. (The pattern differs across various markets.) Modelling the volatility smile is an active area of research, and developments here – as well as implications re the standard theory – are discussed #Departures from normality, in the next section.
After the financial crisis of 2007–2008, a further development:Didier Kouokap Youmbi (2017).Derivatives Pricing after the 2007-2008 Crisis: How the Crisis Changed the Pricing Approach

, Bank of England – Prudential Regulation Authority (United Kingdom), Prudential Regulation Authority (Over-the-counter (finance), over the counter) derivative pricing had relied on the BSM risk neutral pricing framework, under the assumptions of funding at the risk free rate and the ability to perfectly replicate cashflows so as to fully hedge. This, in turn, is built on the assumption of a credit-risk-free environment – called into question during the crisis. Addressing this, therefore, issues such as counterparty credit risk, funding costs and costs of capital are now additionally considered when pricing, and a Credit Valuation Adjustment, or CVA – and potentially other ''valuation adjustments'', collectively xVA – 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, Overnight Index Swap (OIS) curve, as opposed to LIBOR as used previously. This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate". (Also, practically, the interest paid on cash collateral (finance), collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "Credit Support Annex, CSA discounting".) Swap (finance)#Valuation, Swap pricing – and, therefore, yield curve construction – is further modified: previously, swaps were valued off a single "self discounting" interest rate curve; whereas post crisis, to accommodate OIS discounting, valuation is now under a "multi-curve framework" where "forecast curves" are constructed for each floating-leg Libor#Maturities, LIBOR tenor, with discounting on the ''common'' OIS curve.

Corporate finance theory

Corporate finance theory has also been extended: mirroring the #Certainty, 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
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wi ...

" or probabilistic corporate finance models, as opposed to the traditional static and deterministic
Determinism is the philosophical
Philosophy (from , ) is the study of general and fundamental questions, such as those about existence
Existence is the ability of an entity to interact with physical or mental reality
Reality is the ...

models; see .
Relatedly, Real Options theory allows for owner – i.e. managerial – actions that impact underlying value: by incorporating option pricing logic, these actions are then applied to a distribution of future outcomes, changing with time, which then determine the "project's" valuation today.
More traditionally, decision trees – which are complementary – have been used to evaluate projects, by incorporating in the valuation (all) Event (probability theory), possible events (or states) and consequent Decision making#Decision making in business and management, 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 many cases, following Williams #Certainty, 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#Accounting, Financial modeling § Accounting. In more modern treatments, then, it is the ''expected'' cashflows (in the Expected value, 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 – or extensions – the discounting here is at the risk-free rate plus a premium linked to the uncertainty of the entity or project cash flows; (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. 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 . 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. 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, #State prices, as above, which is then applied to the analysis. For example, convertible bonds can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.See Kruschwitz and Löffler per Bibliography.

Challenges and criticism

As above, there is a very close link between (i) therandom walk hypothesisThe random walk hypothesis is a financial theory stating that stock market
A stock market, equity market, or share market is the aggregation of buyers and sellers of stocks (also called shares), which represent ownership claims on busine ...

, with the associated belief that price changes should follow a normal distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

, on the one hand, and (ii) market efficiency and rational expectations
In economics, "rational expectations" are model-consistent expectations, in that agent (economics), agents inside the model (economics), model are assumed to "know the model" and on average take the model's predictions as valid. Rational expectat ...

, 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 financial risk management, risk managers frequently modify the "standard models" (see Kurtosis risk, Skewness risk, Long tail, Model risk). In fact, Benoit Mandelbrot had discovered already in the 1960s that changes in financial prices do not follow anormal distribution
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ...

, 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#In economics and finance, jump diffusion models allow for (option) pricing incorporating jump process, "jumps" in the spot price.
Risk managers, similarly, complement (or substitute) the standard value at risk models with Historical simulation (finance), historical simulations, Mixture model#A financial model, mixture models, principal component analysis, extreme value theory, 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, where, as above, implied volatility – the volatility corresponding to the BSM price – is observed to ''differ'' as a function of strike price (i.e. moneyness), true only if the price-change distribution is non-normal, unlike that assumed by BSM.
The term structure of volatility describes how (implied) volatility differs for related options with different maturities.
An implied volatility surface is then a three-dimensional surface plot of volatility smile and term structure.
These empirical phenomena negate the assumption of constant volatility – and log-normality – upon which Black–Scholes is built.
Within institutions, the function of Black-Scholes is now, largely, to ''communicate'' prices via implied volatilities, much like bond prices are communicated via yield to maturity, 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 and stochastic volatility.
The first returns the volatility which is “local” to each spot-time point of the Finite difference methods for option pricing, finite difference- or Monte Carlo methods for option pricing, simulation-based valuation; i.e. as opposed to implied volatility, which holds overall. In this way calculated prices – and numeric structures – are market-consistent in an arbitrage-free sense. The second approach assumes that the volatility of the underlying price is a stochastic process rather than a constant. Models here are first Stochastic volatility#Calibration and estimation, calibrated to observed prices, and are then applied to the valuation or hedging in question; the most common are Heston model, Heston, SABR volatility model, SABR and Constant elasticity of variance model, CEV. This approach addresses certain problems identified with hedging under local volatility.
Related to local volatility are the Lattice model (finance), lattice-based Implied binomial tree, implied-binomial and Implied trinomial tree, -trinomial trees – essentially a discretization of the approach – 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) Skewness, 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 expression, closed-form models have also been 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 credit risk, counterparty- and XVA#Valuation adjustments, funding-related risk.
These xVA are ''additional'' to any smile or surface effect. This is valid as the surface is built on price data relating to fully collateralized positions, and there is therefore no "double counting (accounting), double counting" of credit risk (etc.) when appending xVA. (Were this not the case, then each counterparty would have its own surface...)
As mentioned at top, mathematical finance (and particularly financial engineering) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, James Rickards, amongst other critics of financial economics, suggests that, instead, the theory needs revisiting almost entirely:
:"The current system, based on the idea that risk is distributed in the shape of a bell curve, is flawed... The problem is [that economists and practitioners] never abandon the bell curve. They are like medieval astronomers who believe the sun revolves around the earth and are Geocentric model#Ptolemaic system, furiously tweaking their geo-centric math in the face of contrary evidence. They will never get this right; Copernican Revolution, they need their Copernicus."
Departures from rationality

As seen, a common assumption is that financial decision makers act rationally; see Homo economicus. Recently, however, researchers in experimental economics and experimental finance have challenged this assumption Empirical evidence, empirically. These assumptions are also challenged Theory, theoretically, by behavioral finance, a discipline primarily concerned with the limits to rationality of economic agents. For related criticisms re corporate finance theory vs its practice see: . Consistent with, and complementary to these findings, various persistent Market anomaly, market anomalies have been documented, these being price or return distortions – e.g. size premiums – which appear to contradict theefficient-market hypothesis
The efficient-market hypothesis (EMH) is a hypothesis in financial economics
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 o ...

; calendar effects 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'', 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
Overview
A risk premium is a measure of excess return that is required by an individual to compensate them for being subjected to an increased level of risk. It is used widely in finance and economics with the general definition being the expect ...

rational equity investors should demand, an "abnormal return". For further context see Random walk hypothesis#A non-random walk hypothesis, Random walk hypothesis § A non-random walk hypothesis, and sidebar for specific instances.
More generally, and particularly following the financial crisis of 2007–2008, financial economics and mathematical finance have been subjected to deeper criticism; notable here is Nassim Nicholas Taleb, who claims that the prices of financial assets cannot be characterized by the simple models currently in use, rendering much of current practice at best irrelevant, and, at worst, dangerously misleading; see Black swan theory, 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" – 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#Inadequacy of classic valuation models, Systemic risk § Inadequacy of classic valuation models; Cascades in financial networks; Flight-to-quality.
Areas of research attempting to explain (or at least model) these phenomena, and crises, include Noise trader, noise trading, market microstructure, and Heterogeneous agent models. The latter is extended to agent-based computational economics, where price is treated as an emergent phenomenon, resulting from the interaction of the various market participants (agents). The noisy market hypothesis argues that prices can be influenced by speculators and momentum traders, as well as by insider trading, insiders and institutions that often buy and sell stocks for reasons unrelated to fundamental value; see Noise (economic). 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, alternatively, shows market participants engaging in the same acts as others ("herd behavior"), despite contradictions with their private information. Copula (probability theory)#Quantitative finance, Copula-based modelling has similarly been applied. See also Hyman Minsky's Hyman Minsky#Minsky's financial instability-hypothesis, "financial instability hypothesis", as well as George Soros' approach under George Soros#Reflexivity, financial markets, and economic theory, § 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 (attain Alpha (investment), "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 funds) should, on average, serve better than any other active investing, active strategy. William F. Sharpe (2002)

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

. Presention: Monterey Institute of International Studies. Retrieved May 20, 2010. Burton Malkiel's ''A Random Walk Down Wall Street'' – first published in 1973, and in its 12th edition as of 2019 – is a widely read popularization of these arguments. (See also John C. Bogle's ''Common Sense on Mutual Funds''; but compare Warren Buffett's ''The Superinvestors of Graham-and-Doddsville''.) Relatedly, institutionally inherent ''limits to arbitrage'' – as opposed to factors directly contradictory to the theory – are sometimes proposed as an explanation for these departures from efficiency.

See also

* :Finance theories * :Financial models * Deutsche Bank Prize in Financial Economics * Economic model * * Financial modeling * Fischer Black Prize * Outline_of_finance#Economics_and_finance, List of financial economics articles * List of financial economists * * Master of Financial Economics * Monetary economics * Outline of economics * Outline of financeHistorical notes

References

Bibliography

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

{{Financial risk Financial economics, Actuarial science