Financial economics
Academic discipline concerned with the exchange of money / From Wikipedia, the free encyclopedia
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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.
Fundamental valuation equation ^{[5]} 
$Price_{j}=\sum _{s}(p_{s}Y_{s}X_{sj})/r$
Four equivalent formulations,^{[6]} where:

Financial economics studies how rational investors would apply decision theory to investment management. The subject is thus built on the foundations of microeconomics and derives several key results for the application of decision making under uncertainty to the financial markets. The underlying economic logic yields the fundamental theorem of asset pricing, which gives the conditions for arbitragefree asset pricing.^{[6]}^{[5]} The aside formulae result directly.
Present value, expectation and utility
Underlying all of financial economics are the concepts of present value and expectation.^{[6]}
Calculating their present value $X_{sj}/r$ allows the decision maker to aggregate the cashflows (or other returns) to be produced by the asset in the future to a single value at the date in question, and to thus more readily compare two opportunities; this concept is the starting point for financial decision making. ^{[note 1]}
An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, $X_{s}$ and $p_{s}$ respectively. ^{[note 2]}
This decision method, however, fails to consider risk aversion ("as any student of finance knows"^{[6]}). In other words, since individuals receive greater utility from an extra dollar when they are poor and less utility when comparatively rich, the approach is to therefore "adjust" the weight assigned to the various outcomes ("states") correspondingly, $Y_{s}$. See indifference price. (Some investors may in fact be risk seeking as opposed to risk averse, but the same logic would apply).
Choice under uncertainty here may then be characterized as the maximization of expected utility. More formally, the resulting expected utility hypothesis states that, if certain axioms are satisfied, the subjective value associated with a gamble by an individual is that individual's statistical expectation of the valuations of the outcomes of that gamble.
The impetus for these ideas arise from various inconsistencies observed under the expected value framework, such as the St. Petersburg paradox and the Ellsberg paradox. ^{[note 3]}
Arbitragefree pricing and equilibrium
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. For detailed subclassifications see JEL classification codes § G. Financial Economics.
The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries in v. 8, Subject Index, including Financial Economics at pp. 863–64. The below have links to entry abstracts of The New Palgrave Online for each primary or secondary JEL category (10 or fewer per page, similar to Google searches):
Tertiary category entries can also be searched.^{[10]} 
The concepts of arbitragefree, "rational", pricing and equilibrium are then coupled with the above to derive "classical"^{[11]} (or "neoclassical"^{[12]}) financial economics.
Rational pricing is the assumption that asset prices (and hence asset pricing models) will reflect the arbitragefree 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 arbitragefree 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.^{[13]} 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, $s$, and that the complete set of possible bets on future statesoftheworld can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for n (riskneutral) probabilities, $q_{s}$, given n prices. For a simplified example see Rational pricing § Risk neutral valuation, where the economy has only two possible states – up and down – and where $q_{up}$ and $q_{down}$ (=$1q_{up}$) are the two corresponding probabilities, and in turn, the derived distribution, or "measure".
The formal derivation will proceed by arbitrage arguments.^{[6]}^{[13]} The analysis here is often undertaken assuming a representative agent, ^{[14]} essentially treating all marketparticipants, "agents", as identical (or, at least, 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 riskless return, plus an "adjustment for risk",^{[6]} i.e. a securityspecific 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.^{[6]}^{[5]}^{[15]} This approach is consistent with the above, but with the expectation based on "the market" (i.e. arbitragefree, 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 downstates, $X_{up}$ and $X_{down}$, are multiplied through by $q_{up}$ and $q_{down}$, and are then discounted at the riskfree interest rate; per the second equation above. In pricing a "fundamental", underlying, instrument (in equilibrium), on the other hand, a riskappropriate premium over riskfree is required in the discounting, essentially employing the first equation with $Y$ and $r$ combined. In general, this premium 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 and a risk free "bond"; see Rational pricing § Delta hedging (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.
(Correspondingly, mathematical finance separates into two analytic regimes: risk and portfolio management (generally) use physical (or actual or actuarial) probability, denoted by "P"; while derivatives pricing uses riskneutral probability (or arbitragepricing probability), denoted by "Q". In specific applications the lower case is used, as in the above equations.)
State prices
With the above relationship established, the further specialized Arrow–Debreu model may be derived. ^{[note 4]} 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 $\pi _{s}$ of this particular state of the world; also referred to as a "Risk Neutral Density".^{[19]}
In the above example, the state prices, $\pi _{up}$, $\pi _{down}$would equate to the present values of $\$q_{up}$ and $\$q_{down}$: i.e. what one would pay today, respectively, for the up and downstate securities; the state price vector is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be [$\pi _{up}$×$X_{up}$ + $\pi _{down}$×$X_{down}$]: the fourth formula (see above regarding the absence of a risk premium here). For a continuous random variable indicating a continuum of possible states, the value is found by integrating over the state price "density". These concepts are extended to martingale pricing and the related riskneutral measure.
State prices find immediate application as a conceptual tool ("contingent claim analysis");^{[6]} but can also be applied to valuation problems.^{[20]} Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security"^{[2]} – as a linear combination of its stateprices; i.e. backsolve for the stateprices corresponding to observed derivative prices.^{[21]}^{[20]} ^{[19]} These recovered stateprices can then be used for valuation of other instruments with exposure to the underlyer, or for other decision making relating to the underlyer itself.
Using the related stochastic discount factor  also called the pricing kernel  the asset price is computed by "discounting" the future cash flow by the stochastic factor ${\tilde {m}}$, and then taking the expectation;^{[15]} 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".
DCF valuation formula, where the value of the firm, is its forecasted free cash flows discounted to the present using the weighted average cost of capital. For share valuation investors use the related dividend discount model. 
The capital asset pricing model (CAPM):
The expected return used when discounting cashflows on an asset $i$, is the riskfree rate plus the market premium multiplied by beta ($\rho _{i,m}{\frac {\sigma _{i}}{\sigma _{m}}}$), the asset's correlated volatility relative to the overall market $m$. 
The Black–Scholes equation:

The Black–Scholes formula for the value of a call option:

Applying the above economic concepts, we may then derive various economic and financial models and principles. As above, the two usual areas of focus are Asset Pricing and Corporate Finance, the first being the perspective of providers of capital, the second of users of capital. Here, and for (almost) all other financial economics models, the questions addressed are typically framed in terms of "time, uncertainty, options, and information",^{[1]}^{[14]} 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.^{[4]} (This division sometimes denoted "deterministic" and "random",^{[22]} or "stochastic".)
Certainty
The starting point here is "Investment under certainty", and usually framed in the context of a corporation. The Fisher separation theorem, asserts that the objective of the corporation will be the maximization of its present value, regardless of the preferences of its shareholders. Related is 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. ^{[note 5]}
The mechanism for determining (corporate) value is provided by ^{[25]} ^{[26]} 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", longterm worth is the present value of its future net cashflows, in the form of dividends. What remains to be determined is the appropriate discount rate. Later developments show that, "rationally", i.e. in the formal sense, the appropriate discount rate here will (should) depend on the asset's riskiness relative to the overall market, as opposed to its owners' preferences; see below. Net present value (NPV) is the direct extension of these ideas typically applied to Corporate Finance decisioning. 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. ^{[note 6]}
Bond valuation, in that cashflows (coupons and return of principal) are deterministic, may proceed in the same fashion.^{[22]} An immediate extension, Arbitragefree bond pricing, discounts each cashflow at the market derived rate – i.e. at each coupon's corresponding zerorate – 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 ^{[25]} the below applications to uncertainty. ^{[note 7]} See ^{[27]} for the development.
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 equilibriumbased result – and to the Black–Scholes–Merton theory (BSM; often, simply Black–Scholes) for option pricing – an arbitragefree result. As above, the (intuitive) link between these, is that the latter derivative prices are calculated such that they are arbitragefree with respect to the more fundamental, equilibrium determined, securities prices; see Asset pricing § Interrelationship.
Briefly, and intuitively – and consistent with § Arbitragefree pricing and equilibrium above – the relationship between rationality and efficiency is as follows.^{[28]} Given the ability to profit from private information, selfinterested traders are motivated to acquire and act on their private information. In doing so, traders contribute to more and more "correct", i.e. efficient, prices: the efficientmarket hypothesis, or EMH. 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, 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:^{[29]} 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 lognormal distribution. ^{[note 8]}
Under these conditions, investors can then be assumed to act rationally: their investment decision must be calculated or a loss is sure to follow; correspondingly, where an arbitrage opportunity presents itself, then arbitrageurs will exploit it, reinforcing this equilibrium. Here, as under the certaintycase above, the specific assumption as to pricing is that prices are calculated as the present value of expected future dividends, ^{[5]} ^{[29]} ^{[14]} 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 arbitrageexploitation – 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. ^{[note 9]} 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,^{[31]} 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 meanvariance efficient portfolios can be formed simply as a combination of holdings of the riskfree asset and the "market portfolio" (the Mutual fund separation theorem), with the combinations here plotting as the capital market line, or CML. Then, given this CML, the required return on a risky security will be independent of the investor's utility function, and solely determined by its covariance ("beta") with aggregate, i.e. market, risk. This is because investors here can then maximize utility through leverage as opposed to pricing; see Separation property (finance), Markowitz model § Choosing the best portfolio and CML diagram aside. As can be seen in the formula aside, this result is consistent with the preceding, equaling the riskless return plus an adjustment for risk.^{[5]} A more modern, direct, derivation is as described at the bottom of this section; which can be generalized to derive other equilibriumpricing models.
Black–Scholes provides a mathematical model of a financial market containing derivative instruments, and the resultant formula for the price of Europeanstyled options. ^{[note 10]} 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 lognormal, geometric Brownian motion (see Brownian model of financial markets). The key financial insight behind the model is that one can perfectly hedge the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk", absenting the risk adjustment from the pricing ($V$, the value, or price, of the option, grows at $r$, the riskfree rate).^{[6]}^{[5]} This hedge, in turn, implies that there is only one right price – in an arbitragefree 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 inheres risk neutrality; intuitively consistent with the "elimination of risk" here, and mathematically consistent with § Arbitragefree pricing and equilibrium above. Relatedly, therefore, the pricing formula may also be derived directly via risk neutral expectation. Itô's lemma provides the underlying mathematics, and, with Itô calculus more generally, remains fundamental in quantitative finance. ^{[note 11]}
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 assumptions of the CAPM.^{[37]}^{[12]} The Black–Scholes theory, although built on Arbitragefree pricing, is therefore consistent with the equilibrium based capital asset pricing. Both models, in turn, are ultimately consistent with the Arrow–Debreu theory, and can be derived via statepricing – essentially, by expanding the fundamental result above – further explaining, and if required demonstrating, this unity.^{[6]} 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 Stochastic discount factor § Properties. The BlackScholes formula is found, in the limit, by attaching a binomial probability to each of numerous possible spotprices (states) and then rearranging for the terms corresponding to $N(d_{1})$ and $N(d_{2})$, per the boxed description; see Binomial options pricing model § Relationship with Black–Scholes.