**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 share prices, interest rates and exchange rates, as opposed to those concerning the real economy. It has two main areas of focus:^{} 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”.^{}^{} 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.

## 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. The underlying economic logic distills to a “fundamental valuation result”,^{}^{} as aside, which is developed in the following sections.

Financial econometrics is the branch of financial economics that uses econometric techniques to parameterise these relationships. Whereas financial economics has a primarily microeconomic focus, monetary economics is primarily macroeconomic in nature.

### Present value, expectation and utility

Underlying all of financial economics are the concepts of present value and expectation. ^{}Calculating their present value – {\displaystyle 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, therefore, the starting point for financial decision making. ^{}

An immediate extension is to combine probabilities with present value, leading to the expected value criterion which sets asset value as a function of the sizes of the expected payouts and the probabilities of their occurrence, {\displaystyle X_{s}} and {\displaystyle p_{s}} respectively. ^{}

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, {\displaystyle 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.

### 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, {\displaystyle s}, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for *n* (risk-neutral) probabilities, {\displaystyle q_{s}}, given *n* prices. The formal derivation will proceed by arbitrage arguments.^{}^{} For a simplified example see Rational pricing § Risk neutral valuation, where the economy has only two possible states – up and down – and where {\displaystyle q_{up}} and {\displaystyle q_{down}} (={\displaystyle 1-q_{up}}) are the two corresponding (i.e. implied) probabilities, and in turn, the derived distribution, or “measure”.

With this measure in place, the expected, i.e. required, return of any security (or portfolio) will then equal the riskless return, plus an “adjustment for risk”,^{} i.e. a security-specific risk premium, compensating for the extent to which its cashflows are unpredictable. All pricing models are then essentially variants of this, given specific assumptions or conditions.^{} This approach is consistent with the above, but with the expectation based on “the market” (i.e. arbitrage-free, and, per the theorem, therefore in equilibrium) as opposed to individual preferences.

Thus, continuing the example, in pricing a derivative instrument its forecasted cashflows in the up- and down-states, {\displaystyle X_{up}} and {\displaystyle X_{down}}, are multiplied through by {\displaystyle q_{up}} and {\displaystyle q_{down}}, and are then discounted at the risk-free interest rate; per the second equation above. In pricing a “fundamental”, underlying, instrument (in equilibrium), on the other hand, a risk-appropriate premium over risk-free is required in the discounting, essentially employing the first equation with {\displaystyle Y} and {\displaystyle r} combined. In general, this may be derived by the CAPM (or extensions) as will be seen under #Uncertainty.

The difference is explained as follows: By construction, the value of the derivative will (must) grow at the risk free rate, and, by arbitrage arguments, its value must then be discounted correspondingly; in the case of an option, this is achieved by “manufacturing” the instrument as a combination of the underlying 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 risk-neutral probability (or arbitrage-pricing probability), denoted by “Q”. In specific applications the lower case is used, as in the above equations.)

### State prices

With the above relationship established, the further specialized 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 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.