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".
Its concern is thus the interrelation of financial
variables, such as prices, interest rates
and shares, as opposed to those concerning
the real economy.
It has two main areas of focus: asset pricing
(or "investment theory") and corporate finance;
the first being the perspective of providers
of capital, i.e. investors, and the second
of users of capital.
The subject is concerned with "the allocation
and deployment of economic resources, both
spatially and across time, in an uncertain
environment".
It therefore centers on decision making under
uncertainty in the context of the financial
markets, and the resultant economic and financial
models and principles, and is concerned with
deriving testable or policy implications from
acceptable assumptions.
It is built on the foundations of microeconomics
and decision theory.
Financial econometrics is the branch of financial
economics that uses econometric techniques
to parameterise these relationships.
Mathematical finance is related in that it
will derive and extend the mathematical or
numerical models suggested by financial economics.
Note though that the emphasis there is mathematical
consistency, as opposed to compatibility with
economic theory.
Financial economics has a primarily microeconomic
focus, whereas monetary economics is primarily
macroeconomic in nature.
Financial economics is usually taught at the
postgraduate level; see Master of Financial
Economics.
Recently, specialist undergraduate degrees
are offered in the discipline.This article
provides an overview and survey of the field:
for derivations and more technical discussion,
see the specific articles linked.
== Underlying economics ==
As above, the discipline essentially explores
how rational investors would apply decision
theory to the problem of investment.
The subject is thus built on the foundations
of microeconomics and decision theory, and
derives several key results for the application
of decision making under uncertainty to the
financial markets.
=== Present value, expectation and utility
===
Underlying all of financial economics are
the concepts of present value and expectation.Calculating
their present value allows the decision maker
to aggregate the cashflows (or other returns)
to be produced by the asset in the future,
to a single value at the date in question,
and to thus more readily compare two opportunities;
this concept is therefore the starting point
for financial decision making.
(Its history is correspondingly early: Richard
Witt discusses compound interest in depth
already in 1613, in his book "Arithmeticall
Questions"; further developed by Johan de
Witt and Edmond Halley.)
An immediate extension is to combine probabilities
with present value, leading to the expected
value criterion which sets asset value as
a function of the sizes of the expected payouts
and the probabilities of their occurrence.
(These ideas originate with Blaise Pascal
and Pierre de Fermat.)
This decision method, however, fails to consider
risk aversion ("as any student of finance
knows").
In other words, since individuals receive
greater utility from an extra dollar when
they are poor and less utility when comparatively
rich, the approach is to therefore "adjust"
the weight assigned to the various outcomes
("states") correspondingly.
(Some investors may in fact be risk seeking
as opposed to risk averse, but the same logic
would apply).
Choice under uncertainty here may then be
characterized as the maximization of expected
utility.
More formally, the resulting expected utility
hypothesis states that, if certain axioms
are satisfied, the subjective value associated
with a gamble by an individual is that individual's
statistical expectation of the valuations
of the outcomes of that gamble.
The impetus for these ideas arise from various
inconsistencies observed under the expected
value framework, such as the St. Petersburg
paradox; see also Ellsberg paradox.
(The development here is originally due to
Daniel Bernoulli, and later formalized by
John von Neumann and Oskar Morgenstern.)
=== 
Arbitrage-free pricing and equilibrium ===
The concepts of arbitrage-free, "rational",
pricing and equilibrium are then coupled with
the above to derive "classical" (or "neo-classical")
financial economics.
Rational pricing is the assumption that asset
prices (and hence asset pricing models) will
reflect the arbitrage-free price of the asset,
as any deviation from this price will be "arbitraged
away".
This assumption is useful in pricing fixed
income securities, particularly bonds, and
is fundamental to the pricing of derivative
instruments.
Economic equilibrium is, in general, a state
in which economic forces such as supply and
demand are balanced, and, in the absence of
external influences these equilibrium values
of economic variables will not change.
General equilibrium deals with the behavior
of supply, demand, and prices in a whole economy
with several or many interacting markets,
by seeking to prove that a set of prices exists
that will result in an overall equilibrium.
(This is in contrast to partial equilibrium,
which only analyzes single markets.)
The two concepts are linked as follows: where
market prices do not allow for profitable
arbitrage, i.e. they comprise an arbitrage-free
market, then these prices are also said to
constitute an "arbitrage equilibrium".
Intuitively, this may be seen by considering
that where an arbitrage opportunity does exist,
then prices can be expected to change, and
are therefore not in equilibrium.
An arbitrage equilibrium is thus a precondition
for a general economic equilibrium.
The immediate, and formal, extension of this
idea, the fundamental theorem of asset pricing,
shows that where markets are as described
—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, 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, given
n prices.
The formal derivation will proceed by arbitrage
arguments.
For a worked example see Rational pricing#Risk
neutral valuation, where, in a simplified
environment, the economy has only two possible
states—up and down—and where p and (1−p)
are the two corresponding (i.e. implied) probabilities,
and in turn, the derived distribution, or
"measure".
With this measure in place, the expected,
i.e. required, return of any security (or
portfolio) will then equal the riskless return,
plus an "adjustment for risk", i.e. a security-specific
risk premium, compensating for the extent
to which its cashflows are unpredictable.
All pricing models are then essentially variants
of this, given specific assumptions and/or
conditions.
This approach is consistent with the above,
but with the expectation based on "the market"
(i.e. arbitrage-free, and, per the theorem,
therefore in equilibrium) as opposed to individual
preferences.
Thus, continuing the example, to value a specific
security, its forecasted cashflows in the
up- and down-states are multiplied through
by p and (1-p) respectively, and are then
discounted at the risk-free interest rate
plus an appropriate premium.
In general, this premium may be derived by
the CAPM (or extensions) as will be seen under
#Uncertainty.
=== State prices ===
With the above relationship established, the
further specialized Arrow–Debreu model may
be derived.
This important result suggests that, under
certain economic conditions, there must be
a set of prices such that aggregate supplies
will equal aggregate demands for every commodity
in the economy.
The analysis here is often undertaken assuming
a representative agent.The 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
of this particular state of the world.
In the above example, the state prices would
equate to the present values of $p and $(1−p):
i.e. what one would pay today, respectively,
for the up- and down-state securities; the
state price vector is the vector of state
prices for all states.
Applied to valuation, the price of the derivative
today would simply be [up-state-price × up-state-payoff
+ down-state-price × down-state-payoff];
see below regarding the absence of any risk
premium here.
For a continuous random variable indicating
a continuum of possible states, the value
is found by integrating over the state price
density; see Stochastic discount factor.
These concepts are extended to martingale
pricing and the related risk-neutral measure.
State prices find immediate application as
a conceptual tool ("contingent claim analysis");
but can also be applied to valuation problems.
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.
(Breeden and Litzenberger's work in 1978 established
the use of state prices in financial economics.)
== Resultant models ==
Applying the preceding economic concepts,
we may then derive various economic- and financial
models and principles.
As above, the two usual areas of focus are
Asset Pricing and Corporate Finance, the first
being the perspective of providers of capital,
the second of users of capital.
Here, and for (almost) all other financial
economics models, the questions addressed
are typically framed in terms of "time, uncertainty,
options, and information", as will be seen
below.
Time: money now is traded for money in the
future.
Uncertainty (or risk): The amount of money
to be transferred in the future is uncertain.
Options: one party to the transaction can
make a decision at a later time that will
affect subsequent transfers of money.
Information: knowledge of the future can reduce,
or possibly eliminate, the uncertainty associated
with future monetary value (FMV).Applying
this framework, with the above concepts, leads
to the required models.
This derivation begins with the assumption
of "no uncertainty" and is then expanded to
incorporate the other considerations.
(This division sometimes denoted "deterministic"
and "random", or "stochastic".)
=== Certainty ===
The starting point here is “Investment under
certainty".
The Fisher separation theorem, asserts that
the objective of a corporation will be the
maximization of its present value, regardless
of the preferences of its shareholders.
Related is the Modigliani–Miller theorem,
which shows that, under certain conditions,
the value of a firm is unaffected by how that
firm is financed, and depends neither on its
dividend policy nor its decision to raise
capital by issuing stock or selling debt.
The proof here proceeds using arbitrage arguments,
and acts as a benchmark for evaluating the
effects of factors outside the model that
do affect value.
The mechanism for determining (corporate)
value is provided by The Theory of Investment
Value (John Burr Williams), which proposes
that the value of an asset should be calculated
using "evaluation by the rule of present worth".
Thus, for a common stock, the intrinsic, long-term
worth is the present value of its future net
cashflows, in the form of dividends.
What remains to be determined is the appropriate
discount rate.
Later developments show that, "rationally",
i.e. in the formal sense, the appropriate
discount rate here will (should) depend on
the asset's riskiness relative to the overall
market, as opposed to its owners' preferences;
see below.
Net present value (NPV) is the direct extension
of these ideas typically applied to Corporate
Finance decisioning (introduced by Joel Dean
in 1951).
For other results, as well as specific models
developed here, see the list of "Equity valuation"
topics under Outline of finance#Discounted
cash flow valuation.
Bond valuation, in that cashflows (coupons
and return of principal) are deterministic,
may proceed in the same fashion.
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.
Note that in many treatments bond valuation
precedes equity valuation, under which cashflows
(dividends) are not "known" per se. Williams
and onward allow for forecasting as to these
— 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.
=== Uncertainty ===
For "choice under uncertainty" the twin assumptions
of rationality and market efficiency, as more
closely defined, lead to modern portfolio
theory (MPT) with its capital asset pricing
model (CAPM)—an equilibrium-based result—and
to the Black–Scholes–Merton theory (BSM;
often, simply Black–Scholes) for option
pricing—an arbitrage-free result.
Note that the latter derivative prices are
calculated such that they are arbitrage-free
with respect to the more fundamental, equilibrium
determined, securities prices; see asset pricing.
Briefly, and intuitively—and consistent
with #Arbitrage-free pricing and equilibrium
above—the linkage 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, or EMH (Eugene
Fama, 1965).
The EMH (implicitly) assumes that average
expectations constitute an "optimal forecast",
i.e. prices using all available information,
are identical to the best guess of the future:
the assumption of rational expectations.
The EMH does allow that when faced with new
information, some investors may overreact
and some may underreact, but what is required,
however, is that investors' reactions follow
a normal distribution—so that the net effect
on market prices cannot be reliably exploited
to make an abnormal profit.
In the competitive limit, then, market prices
will reflect all available information and
prices can only move in response to news;
and this, of course, could be "good" or "bad",
major or minor: the random walk hypothesis.
Thus, if prices of financial assets are (broadly)
efficient, then deviations from these (equilibrium)
values could not last for long.
(On Random walks in stock prices: Jules Regnault,
1863; Louis Bachelier, 1900; Maurice Kendall,
1953; Paul Cootner, 1964.)
Under these conditions investors can then
be assumed to act rationally: their investment
decision must be calculated or a loss is sure
to follow; correspondingly, where an arbitrage
opportunity presents itself, then arbitrageurs
will exploit it, reinforcing this equilibrium.
Here, as under the certainty-case above, the
specific assumption as to pricing is that
prices are calculated as the present value
of expected future dividends, 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.
Importantly, this result will be independent
of the investor's level of risk aversion,
and / or assumed utility function, thus providing
a readily determined discount rate for corporate
finance decision makers as above, 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" (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 risky securities will be independent of
the investor's utility function, and solely
determined by their covariance ("beta") with
aggregate, i.e. market, risk.
This is because investors here can then maximize
utility through leverage as opposed to pricing;
see CML diagram.
As can be seen in the formula aside, this
result is consistent with the preceding, equaling
the riskless return plus an adjustment for
risk.
(The efficient frontier was introduced by
Harry Markowitz in 1952.
The CAPM was derived by Jack Treynor (1961,
1962), William F. Sharpe (1964), John Lintner
(1965) and Jan Mossin (1966) independently.)
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
{\displaystyle V}
, the value, or price, of the option, grows
at
r
{\displaystyle r}
, the risk-free rate; see Black–Scholes
equation § Financial interpretation).
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 the solution
to the equation.)
Since the formula is without reference to
the share's expected return, Black–Scholes
entails (assumes) risk neutrality, consistent
with the "elimination of risk" here.
Relatedly, therefore, the pricing formula
may also be derived directly via risk neutral
expectation.
(BSM - two seminal 1973 papers
- is consistent with "previous versions of
the formula" of Louis Bachelier and Edward
O. Thorp; although these were more "actuarial"
in flavor, and had not established risk-neutral
discounting.
See also Paul Samuelson (1965).
Vinzenz Bronzin (1908) produced very early
results.)
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 may 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.
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 may be
derived via state-pricing, further explaining,
and if required demonstrating, this unity.
== Extensions ==
More recent work further generalizes and / or
extends these models.
As regards asset pricing, developments in
equilibrium-based pricing are discussed under
"Portfolio theory" below, while "Derivative
pricing" relates to risk-neutral, i.e. arbitrage-free,
pricing.
As regards the use of capital, "Corporate
finance theory" relates, mainly, to the application
of these models.
=== Portfolio theory ===
See also: Post-modern portfolio theory and
Mathematical finance § Risk and portfolio
management: the P world.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
CAPM similarly extend the model.
With intertemporal portfolio choice, the investor
now repeatedly optimizes her portfolio; while
the inclusion of consumption (in the economic
sense) then incorporates all sources of wealth,
and not just market-based investments, into
the investor's calculation of required return.
Whereas the above extend the CAPM, the single-index
model is a more simple model.
It assumes, only, a correlation between security
and market returns, without (numerous) other
economic assumptions.
It is useful in that it simplifies the estimation
of correlation between securities, significantly
reducing the inputs for building the correlation
matrix required for portfolio optimization.
The arbitrage pricing theory (APT; Stephen
Ross, 1976) similarly differs as regards its
assumptions.
APT "gives up the notion that there is one
right portfolio for everyone in the world,
and ...replaces it with an explanatory model
of what drives asset returns."
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
approach 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 (Thomas
M. Cover) applies machine learning to asset
selection, learning adaptively from historical
data.
Behavioral portfolio theory recognizes that
investors have varied aims and create an investment
portfolio that meets a broad range of goals.
Copulas have lately been applied here.
See Portfolio optimization § Improving portfolio
optimization for other techniques and / or
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 (as
above); relatedly, a large number of researchers
have used options to extract state-prices
for a variety of other applications in financial
economics.
For path dependent derivatives, Monte Carlo
methods for option pricing are employed; here
the modelling is in continuous time, but similarly
uses risk neutral expected value.
Various other numeric techniques have also
been developed.
The theoretical framework too has been extended
such that martingale pricing is now the standard
approach.
Developments relating to complexities in return
and / or volatility are discussed below.
Drawing on these techniques, derivative models
for various other underlyings and applications
have also been developed, all based off the
same logic (using "contingent claim analysis").
Real options valuation allows that option
holders can influence the option's underlying;
models for employee stock option valuation
explicitly assume non-rationality on the part
of option holders; Credit derivatives allow
that payment obligations and / or delivery
requirements might not be honored.
Exotic derivatives are now routinely valued.
Multi-asset underlyers are handled via simulation
or copula based analysis.
Similarly, beginning with Oldrich Vasicek
(1977), various short rate models, as well
as the HJM and BGM forward rate-based techniques,
allow for an extension of these techniques
to fixed income- and interest rate derivatives.
(The Vasicek and CIR models are equilibrium-based,
while Ho–Lee and subsequent models are based
on arbitrage-free pricing.)
Bond valuation is relatedly extended: the
Stochastic calculus approach, employing these
methods, allows for rates that are "random"
(while returning a price that is arbitrage
free, as above); lattice models for hybrid
securities allow for non-deterministic cashflows
(and stochastic rates).
As above, (OTC) derivative pricing has relied
on the BSM risk neutral pricing framework,
under the assumptions of funding at the risk
free rate and the ability to perfectly replicate
cashflows so as to fully hedge.
This, in turn, is built on the assumption
of a credit-risk-free environment.
Post the financial crisis of 2008, therefore,
issues such as counterparty credit risk, funding
costs and costs of capital are additionally
considered, 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 (OIS) curve, as opposed to LIBOR
as used previously.
This is because post-crisis, OIS is considered
a better proxy for the "risk-free rate".
(Also, practically, the interest paid on cash
collateral is usually the overnight rate;
OIS discounting is then, sometimes, referred
to as "CSA discounting".)
Swap pricing 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 a common OIS curve; see Interest rate swap
§ Valuation and pricing.
=== Corporate finance theory ===
Corporate finance theory has also been extended:
mirroring the above developments, asset-valuation
and decisioning no longer need assume "certainty".
As discussed, Monte Carlo methods in finance,
introduced by David B. Hertz in 1964, allow
financial analysts to construct "stochastic"
or probabilistic corporate finance models,
as opposed to the traditional static and deterministic
models; see Corporate finance § Quantifying
uncertainty.
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;
the correct discount rate here reflecting
each point's "non-diversifiable risk looking
forward."
(This technique predates the use of real options
in corporate finance; it is borrowed from
operations research, and is not a "financial
economics development" per se.)
Related to this, is the treatment of forecasted
cashflows in equity valuation.
In many cases, following Williams above, the
average (or most likely) cash-flows were discounted,
as opposed to a more 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.
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.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 Dividend policy § Irrelevance of dividend
policy.
The typical application of real options is
to capital budgeting type problems as described.
However, they are also applied to questions
of capital structure and dividend policy,
and to the related design of corporate securities;
and since stockholder and bondholders have
different objective functions, in the analysis
of the related agency problems.
In all of these cases, state-prices can provide
the market-implied information relating to
the corporate, as above, which is then applied
to the analysis.
For example, convertible bonds can (must)
be priced consistent with the state-prices
of the corporate's equity.
== Challenges and criticism ==
As above, there is a very close link between
(i) the random walk hypothesis, with the associated
expectation that price changes should follow
a normal distribution, on the one hand, and
(ii) market efficiency and rational expectations,
on the other.
Note, however, that (wide) departures from
these are commonly observed, and there are
thus, respectively, two main sets of challenges.
=== Departures from normality ===
As discussed, the assumptions that market
prices follow a random walk and / or that
asset returns are normally distributed are
fundamental.
Empirical evidence, however, suggests that
these assumptions may not hold (see Kurtosis
risk, Skewness risk, Long tail) and that in
practice, traders, analysts and risk managers
frequently modify the "standard models" (see
Model risk).
In fact, Benoît Mandelbrot had discovered
already in the 1960s that changes in financial
prices do not follow a Gaussian distribution,
the basis for much option pricing theory,
although this observation was slow to find
its way into mainstream financial economics.
Financial models with 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.
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.
For further discussion see Fat-tailed distribution
§ Applications in economics, and Value at
risk § Criticism.
Portfolio managers, likewise, have modified
their optimization criteria and algorithms;
see #Portfolio theory above.
Closely related is the volatility smile, where
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; see Black–Scholes
model § The volatility smile.
In consequence traders (and risk managers)
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
in question; the most common are Heston, SABR
and CEV.
This approach addresses certain problems identified
with hedging under local volatility.Related
to local volatility are the lattice-based
implied-binomial and -trinomial trees — essentially
a discretization of the approach — which
are similarly used for pricing; these are
built on state-prices recovered from the surface.
Edgeworth binomial trees allow for a specified
(i.e. non-Gaussian) skew and kurtosis in the
spot price; priced here, options with differing
strikes will return differing implied volatilities,
and the tree can be calibrated to the smile
as required.
Similarly purposed closed-form models have
also been developed.
As above, additional to log-normality in returns,
BSM—and, typically, other derivative models—assume(d)
the ability to perfectly replicate cashflows
so as to fully hedge, and hence to discount
at the risk-free rate.
This, in turn, is built on the assumption
of a credit-risk-free environment.
Post crisis, then, various x-value adjustments
are made to the risk-neutral derivative value.
Note that these are additional to any smile
or surface effect: this is valid as the surface
is built on price data relating to fully collateralized
positions, and there is therefore no "double
counting" of credit risk (etc.) when including
xVA.
(Also, were this not the case, then each counterparty
would have its own surface...)
=== 
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 empirically.
These assumptions are also challenged theoretically,
by behavioral finance, a discipline primarily
concerned with the limits to rationality of
economic agents.
Consistent with, and complementary to these
findings, various persistent market anomalies
have been documented, these being price and/or
return distortions—e.g. size premiums—which
appear to contradict the efficient-market
hypothesis; 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 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 particularly following
the financial crisis of 2007–2010, 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 studied in recent
years has thus been financial crises, and
the failure of financial economics to model
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; Cascades in financial networks;
Flight-to-quality.)
Areas of research attempting to explain (or
at least model) these phenomena, and crises,
include noise trading, market microstructure,
and Heterogeneous agent models.
The latter is extended to agent-based computational
economics, 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 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-based modelling has similarly been
applied.
See also Hyman Minsky's "financial instability
hypothesis", as well as George Soros' approach,
§ 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 ("alpha").
The practical implication, therefore, is that
passive investing (e.g. via low-cost index
funds) should, on average, serve better than
any other active strategy.
Burton Malkiel's A Random Walk Down Wall Street—first
published in 1973, and in its 11th edition
as of 2015—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.)
Note also that 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 ==
== References ==
== Bibliography ==
== 
External links ==
