Financial economics is the branch of economics characterized by a "concentration on monetary activities", in which "money of one type or another is likely to appear on both sides of a trade".[1] Its concern is thus the interrelation of financial variables, such as share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus:[2] asset pricing and corporate finance; 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".[3] [4] 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 thus also includes a formal study of the financial markets themselves, especially market microstructure and market regulation.It is built on the foundations of microeconomics and decision theory.
Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise the relationships identified.Mathematical finance is related in that it will derive and extend the mathematical or numerical models suggested by financial economics.Whereas financial economics has a primarily microeconomic focus, monetary economics is primarily macroeconomic in nature.
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| Fundamental valuation equation | |
Underlying all of financial economics are the concepts of present value and expectation.
Calculating their present value,
Xsj/r
r
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,
Xs
ps
This decision method, however, fails to consider risk aversion. In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich, the approach is therefore to "adjust" the weight assigned to the various outcomes, i.e. "states", correspondingly:
Ys
Choice under uncertainty here may then be defined as the maximization of expected utility. More formally, the resulting expected utility hypothesis states that, if certain axioms are satisfied, the subjective value associated with a gamble by an individual is that individuals statistical expectation of the valuations of the outcomes of that gamble.
The impetus for these ideas arises from various inconsistencies observed under the expected value framework, such as the St. Petersburg paradox and the Ellsberg paradox.
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| JEL classification codes | |
| In the Journal of Economic Literature classification codes, Financial Economics is one of the 19 primary classifications, at JEL: G. It follows Monetary and International Economics and precedes Public Economics. The New Palgrave Dictionary of Economics also uses the JEL codes to classify its entries. The primary and secondary JEL categories are: JEL: G – Financial Economics (archived link) JEL: G0 – General JEL: G1 – General Financial Markets JEL: G2 – Financial institutions and Services JEL: G3 – Corporate finance and GovernanceEach is further divided into its tertiary categories. |
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: the "law of one price". This assumption is useful in pricing fixed income securities, particularly bonds, and is fundamental to the pricing of derivative instruments.
Economic equilibrium is 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 are complete and do not allow 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 they are therefore not in equilibrium. An arbitrage equilibrium is thus a precondition for a general economic equilibrium.
"Complete" here means that there is a price for every asset in every possible state of the world,
s
qs
qup
qdown
The fundamental theorem of asset pricing formalizes this. A market with "physical" probabilities, denoted by "P": is (i) considered arbitrage-free if, and only if, there exists at least one risk neutral probability measure "Q" that is equivalent to P; and (ii) is considered complete, if Q is unique and there is a risk-free bond which may act as the numeraire, or pricing reference.[7]
The formal derivation will proceed by arbitrage arguments.[8] [5] The analysis here is often undertaken to assume a representative agent, essentially treating all market participants, "agents", as identical (or, at least, assuming that they act in such a way that the sum of their choices is equivalent to the decision of one individual) with the effect that the problems are then mathematically tractable.
With this measure in place, the expected, i.e. required, return of any security (or portfolio) will then equal the risk-free 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.
Continuing the example, in pricing a derivative instrument, its forecasted cashflows in the abovementioned up- and down-states
Xup
Xdown
qup
qdown
Y
r
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 and a risk free "bond"; see (and 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.
Although mathematical finance separates into two analytic regimes, these - per the fundamental theorem - are at base consistent.Thus, risk and portfolio management generally use P, physical (or actual or actuarial) probabilities, while derivatives pricing uses Q, risk-neutral (or arbitrage-pricing) probabilities. (In specific applications the lower case symbols p and q are used, as in the above equations.) But here, "quants", in fact, construct the latter from the former (as outlined); and this as a modelling "convenience", allowing them to operate as if risk-neutral and to discount expected payoffs at the risk-free rate.
With the above relationship established, the further specialized Arrow–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 Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods.
A direct extension, then, is the concept of a state price security, also called an 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
\pis
In the above example, the state prices,
\piup
\pidown
$qup
$qdown
State prices find immediate application as a conceptual tool ("contingent claim analysis"); but can also be applied to valuation problems.[9] Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security"[2] – as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices.[9] [10] 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 - SDF; also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor
\tilde{m}
\tilde{m}s
Ys.
qs/ps.
Applying the above economic concepts, we may then derive various economic- and financial models and principles. As outlined, the two 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",[1] as will be seen below.
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.[4] (This division sometimes denoted "deterministic" and "random", or "stochastic".)
The mechanism for determining (corporate) value is provided by[13] [14] John Burr Williams' The Theory of Investment Value, which proposes that the value of an asset should be calculated using "evaluation by the rule of present worth". Thus, for a common stock, the "intrinsic", long-term worth is the present value of its future net cashflows, in the form of dividends; in the corporate context, "free cash flow" as aside. What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. Net present value (NPV), the result of this discounted cashflow valuation, 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, or "Face value") are deterministic, may proceed in the same fashion.[15] An immediate extension, Arbitrage-free bond pricing, discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding zero rate, and of equivalent credit worthiness – as opposed to an overall rate.In many treatments bond valuation precedes equity valuation, under which cashflows (dividends) are not "known" per se. Williams and onward allow for forecasting as to these – based on historic ratios or published dividend policy – and cashflows are then treated as essentially deterministic; see below under .
For both stocks and bonds, "under certainty, with the focus on cash flows from securities over time," valuation based on a term structure of interest rates is in fact consistent with arbitrage-free pricing.[16] Indeed, a corollary of the above is that "the law of one price implies the existence of a discount factor".[17] Corresponding to these: ; and equivalently, for the above "numeraire bond", . In practice, the latter is interchangeable with the risk-free yield curve: this ensures consistency between the theoretical no-arbitrage framework and applied valuation, providing a common benchmark for discounting future cash flows.
Whereas 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[13] the below applications to uncertainty; see[18] for the development.
Briefly, and intuitively – and consistent with above – the relationship between rationality and efficiency is as follows.[19] 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, or EMH. Thus, if prices of financial assets are (broadly) efficient, then deviations from these (equilibrium) values could not last for long. (See earnings response coefficient.)The EMH (implicitly) assumes that average expectations constitute an "optimal forecast", i.e. prices using all available information are identical to the best guess of the future: the assumption of rational expectations. The EMH does allow that when faced with new information, some investors may overreact and some may underreact,[20] but what is required, however, is that investors' reactions follow a normal distribution – so that the net effect on market prices cannot be reliably exploited[20] 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. 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;[20] 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,[21] [22] 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,[23] and for other investors.The argument proceeds as follows:[24] 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" (the Mutual fund separation theorem), with the combinations here plotting as the capital market line, or CML. Then, given this CML, the required return on a risky security will be independent of the investor's utility function, and solely determined by its covariance ("beta") with aggregate, i.e. market, risk. This is because investors here can then maximize utility through leverage as opposed to stock selection; see Separation property (finance), 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.[21] A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other equilibrium-pricing models.
Black–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 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
r
As implied by the Fundamental Theorem, the two major results are consistent. Here, the Black-Scholes equation can alternatively be derived from the CAPM, and the price obtained from the Black–Scholes model is thus consistent with the assumptions of the CAPM.[25] [26] The Black–Scholes theory, although built on Arbitrage-free pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via state-pricing – essentially, by expanding the above fundamental equations – further explaining, and if required demonstrating, this consistency.[27] Here, the CAPM is derived[11] by linking
Y
j
Xj/Pricej
N(d1)
N(d2)
More recent work further generalizes and extends these models. As regards asset pricing, developments in equilibrium-based pricing are discussed under "Portfolio theory" below, while "Derivative pricing" relates to risk-neutral, i.e. arbitrage-free, pricing. As regards the use of capital, "Corporate finance theory" relates, mainly, to the application of these models.
Much of the development here relates 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 CAPM similarly extend the model. With intertemporal portfolio choice, the investor now repeatedly optimizes her portfolio; while the inclusion of consumption (in the economic sense) then incorporates all sources of wealth, and not just market-based investments, into the investor's calculation of required return.
Whereas the above extend the CAPM, the single-index model is a more simple model. It assumes, only, a correlation between security and market returns, without (numerous) other economic assumptions. It is useful in that it simplifies the estimation of correlation between securities, significantly reducing the inputs for building the correlation matrix required for portfolio optimization.
The arbitrage pricing theory (APT) 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."[29] 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.
The linear factor model structure of the APT is used as the basis for many of the commercial risk and fund management systems employed by asset managers. Here,[30] [31] managers apply various of the abovementioned multi-factor models - often bespoke extensions - such that their portfolio has the desired exposure ("tilt") to macroeconomic, market and / or fundamental risk factors;[32] respectively: Macro-, Factor-, and Style Funds.Research here, and application (e.g. "Smart Beta"), is ongoing, and over the years "hundreds of factors attempt to explain the cross-section of expected returns"; [33] with this "factor zoo", [34] there is a risk of data mining, and researchers have proposed various criteria for establishing significance.
At the same time, "classic" mean-variance optimization — i.e. building an efficient portfolio as described above — is still widely used by Asset Allocation Funds.[35] Here, given issues noted with this approach, the application is typically in combination with other techniques.The Black–Litterman model[36] is often employed.Black–Litterman departs from the original Markowitz model approach: it starts with an equilibrium assumption, as for the latter, but this 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 [37] asset allocation. A further modification often used [38] relates to the subsequent optimization: under (Tail) risk parity, the focus is on the allocation of risk, rather than the allocation of capital.
Other developments re portfolio optimization include the following. 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 identifies the "growth optimal portfolio" per the Kelly criterion, applying information theory to asset selection. Behavioral portfolio theory recognizes that investors have varied aims and create an investment portfolio that meets a broad range of goals. Copulas have lately been applied here; recently this is the case also for genetic algorithms and machine learning, more generally; see below.[39]
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 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 based analysis.
Similarly, the various short-rate models allow for an extension of these techniques to fixed income- and interest rate derivatives. (The Vasicek and CIR models are equilibrium-based, while Ho–Lee and subsequent models are based on arbitrage-free pricing.) The more general HJM Framework describes the dynamics of the full 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 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"; 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 – as well as implications re the standard theory – are discussed in the next section.
After the 2008 financial crisis, a further development:[41] as outlined, (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,[42] and a credit valuation adjustment, or CVA – and potentially other valuation adjustments, collectively xVA – is generally added to the risk-neutral derivative value.The standard economic arguments can be extended to incorporate these various adjustments.[43]
A related, and perhaps more fundamental change, is that discounting is now on the Overnight Index Swap (OIS) curve, as opposed to LIBOR as used previously.[41] This is because post-crisis, the overnight rate is considered a better proxy for the "risk-free rate".[44] (Also, practically, the interest paid on cash collateral is usually the overnight rate; OIS discounting is then, sometimes, referred to as "CSA discounting".) Swap pricing – and, therefore, 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 tenor, with discounting on the common OIS curve.
Mirroring the above developments, corporate finance valuations and decisioning no longer need assume "certainty".Monte Carlo methods in finance allow financial analysts to construct "stochastic" or probabilistic corporate finance models, as opposed to the traditional static and deterministic models; see . 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) possible events (or states) and consequent management decisions;[45] [46] the correct discount rate here reflecting each decision-point's "non-diversifiable risk looking forward."[46]
Related to this, is the treatment of forecasted cashflows in equity valuation. In many cases, following Williams above, the average (or most likely) cash-flows were discounted,[47] as opposed to a theoretically correct state-by-state treatment under uncertainty; see comments under Financial modeling § 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.[48] [49] [46] 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[46] (essentially,
Y
r
Further as regards valuation, various approaches focus on economic profit as opposed to dividends; the main of these are EVA, MVA and RIV. Price here is returned as a function of earnings, expected returns, and / or changes in book value; see Clean surplus accounting. These approaches address - to some extent - the contradiction of seeing value as a function of dividends while [50] recognizing Modigliani and Miller's Irrelevance principle.
"Corporate finance" as a discipline more broadly,[51] building on 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.[52] 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, Pecking order theory, Market timing hypothesis, Trade-off theory.(ii) Considerations and analysis re dividend policy, additional to - and sometimes contrasting with - Modigliani-Miller, include: the Walter model, Lintner model, Residuals theory and signaling hypothesis, as well as discussion re the observed clientele effect and dividend puzzle.
A significant development is the application of Agency theory to the corporate finance setting. Here, the analysis focuses on the difficulties in motivating corporate management (the "agent"; in a different sense to the above) to act in the best interests of shareholders (the "principal"), rather than in their own interests; [53] see contract theory.The resultant (unseen) "agency costs" may impact firm value - as well as overall market efficiency.[54] A related emphasis is placed on the agency issues interrelated with capital structure and with dividends.Given this scope, all corporate finance decisions may (must), in fact, be viewed through an agency lens. [55]
Although, as described, the typical "use case" for real options is capital budgeting, they may also be applied [56] to problems of capital structure and dividend policy (here it is stockholders and bondholders who have different objective functions) and to the related design of corporate securities.[57] Further, they are often used in the analysis of the related agency problems. The basis for these analyses is that equity may be viewed as a call option on the firm, with bonds, i.e. debt, as a put option (and put-call parity true overall[9]); see Merton model.In all of these settings, state-prices can provide the market-implied information relating to the corporate, as above, which is then applied to the analysis. For example, convertible bonds can (must) be priced consistent with the (recovered) state-prices of the corporate's equity.[9] [58]
The discipline, as outlined, also includes a formal study of financial markets. Of interest especially are market regulation and market microstructure, and their relationship to price efficiency.
Regulatory economics studies, in general, the economics of regulation. In the context of finance, it will address the impact of financial regulation on the functioning of markets and the efficiency of prices, while also weighing the corresponding increases in market confidence and financial stability.Research here considers how, and to what extent, regulations relating to disclosure (earnings guidance, annual reports), insider trading, and short-selling will impact price efficiency, the cost of equity, and market liquidity.[59]
Market microstructure is concerned with the details of how exchange occurs in markets (with Walrasian-, matching-, Fisher-, and Arrow-Debreu markets as prototypes), and "analyzes how specific trading mechanisms affect the price formation process",[60] examining the ways in which the processes of a market affect determinants of transaction costs, prices, quotes, volume, and trading behavior.It has been used, for example, in providing explanations for long-standing exchange rate puzzles,[61] and for the equity premium puzzle.[62] In contrast to the above classical approach, models here explicitly allow for (testing the impact of) market frictions and other imperfections;see also market design.
For both regulation[63] and microstructure,[64] and generally, agent-based models can be developed to examine any impact due to a change in structure or policy - or to make inferences re market dynamics - by testing these in an artificial financial market, or AFM.This approach, essentially simulated trade between numerous agents, "typically uses artificial intelligence technologies [often [[genetic algorithms]] and neural nets] to represent the adaptive behaviour of market participants".[65]
These 'bottom-up' models "start from first principals of agent behavior",[66] with participants modifying their trading strategies having learned over time, and "are able to describe macro features [i.e. [[stylized fact]]s] emerging from a soup of individual interacting strategies".[66] Agent-based models depart further from the classical approach — the representative agent, as outlined — in that they introduce heterogeneity into the environment (thereby addressing, also, the aggregation problem).
More recent research [39] focuses on the potential impact of machine learning on market functioning and efficiency.As these methods become more prevalent in financial markets, economists would expect greater information acquisition and improved price efficiency.[67] In fact, an apparent rejection of market efficiency (see below) might simply represent "the unsurprising consequence of investors not having precise knowledge of the parameters of a data-generating process that involves thousands of predictor variables".[68] At the same time, it is acknowledged that a potential downside of these methods, in this context, is their lack of interpretability "which translates into difficulties in attaching economic meaning to the results found."[39]
As above, there is a very close link between:the random walk hypothesis, with the associated belief that price changes should follow a normal distribution, on the one hand; and 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.
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" (see kurtosis risk, skewness risk, long tail, model risk). In fact, Benoit Mandelbrot had discovered already in the 1960s[69] 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.[70]
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.[71] Risk managers, similarly, complement (or substitute) the standard value at risk models with historical simulations, mixture models, principal component analysis, extreme value theory, as well as models for volatility clustering.[72] For further discussion see, and . Portfolio managers, likewise, have modified their optimization criteria and algorithms; see 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 (i.e.
N(d1)
N(d2)
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- or 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 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.[73]
Related to local volatility are the lattice-based implied-binomial and -trinomial trees – essentially a discretization of the approach – which are similarly, but less commonly,[10] 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.[74] Similarly purposed (and derived) closed-form models were also developed.[75]
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: with the surface built on price data for fully-collateralized positions, there is therefore no "double counting" of credit risk (etc.) when appending xVA. (Were this not the case, then each counterparty would have its own surface...)
As mentioned at top, mathematical finance (and particularly financial engineering) is more concerned with mathematical consistency (and market realities) than compatibility with economic theory, and the above "extreme event" approaches, smile-consistent modeling, and valuation adjustments should then be seen in this light. Recognizing this, critics of financial economics - especially vocal since the 2008 financial crisis - suggest that instead, the theory needs revisiting almost entirely:
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| Market anomalies and economic puzzles |
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Various market anomalies have been documented in parallel. These comprise price "distortions", e.g. size premiums, or return "predictability" [78] as exemplified by the various calendar effects, and are anomalous in the sense that [78] they cannot be explained by traditional economic theories, (apparently) contradicting the efficient-market hypothesis. Related to these are various of the economic puzzles, 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 rational equity investors should demand, an "abnormal return". For further context see Random walk hypothesis § A non-random walk hypothesis, and sidebar for specific instances.
More generally, and, again, particularly following the 2008 financial crisis, financial economics (and mathematical finance) has been subjected to deeper criticism.Notable here is Nassim Taleb, whose critique overlaps the above, but extends[79] also to the institutional[80] aspects of finance - including academic.[81] His Black swan theory posits that although events of large magnitude and consequence play a major role in finance, since these are (statistically) unexpected, they are "ignored" by economists and traders - who often have no skin in the game. Thus, although a "Taleb distribution" - which normally provides a payoff of small positive returns, while carrying a small but significant risk of catastrophic losses - more realistically describes markets than current models, the latter continue to be preferred; and even with professionals here acknowledging that it only "generally works" or only "works on average".[82]
Here,[83] financial crises have been a topic of interest[84] and, in particular, the failure[80] of (financial) economists - as well as[83] bankers and regulators - to model and predict these.See .The related problem of systemic risk, has also received attention. 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; Cascades in financial networks; Flight-to-quality.
Areas of research attempting to explain (or at least model) these phenomena, and crises, include[85] market microstructure and Heterogeneous agent models, as above. The latter is extended to agent-based computational models; here,[86] as mentioned, 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 insiders and institutions that often buy and sell stocks for reasons unrelated to fundamental value; see Noise (economic) and Noise trader. 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-based modelling has similarly been applied. See also Hyman Minsky's "financial instability hypothesis", as well as George Soros' application of "reflexivity".In the alternative, institutionally inherent limits to arbitrage - i.e. as opposed to factors directly contradictory to the theory - are sometimes referenced.
Note however, that despite the above inefficiencies, asset prices do effectively[20] follow a random walk - i.e. (at least) in the sense that "changes in the stock market are unpredictable, lacking any pattern that can be used by an investor to beat the overall market".[87] Thus after fund costs - and given other considerations - it is difficult to consistently outperform market averages[88] and achieve "alpha".The practical implication[89] is that passive investing, i.e. via low-cost index funds, should, on average, serve better than any other active strategy - and, in fact, this practice is now widely adopted.Here, however, the following concern is posited:although in concept, it is "the research undertaken by active managers [that] keeps prices closer to value... [and] thus there is a fragile equilibrium in which some investors choose to index while the rest continue to search for mispriced securities";[89] in practice, as more investors "pour money into index funds tracking the same stocks, valuations for those companies become inflated",[90] potentially leading to asset bubbles.See Grossman-Stiglitz paradox.
See also: List of quantitative analysts.
Financial economics
Asset pricing
Corporate finance
