Artificial Financial Market

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".William F. Sharpe
"Financial Economics"
, in
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:Merton H. Miller, (1999). The History of Finance: An Eyewitness Account, ''Journal of Portfolio Management''. Summer 1999. 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".See Fama and Miller (1972), ''The Theory of Finance'', in Bibliography. It therefore centers on decision making under uncertainty in the context of the financial markets, and the resultant economic and financial models and principles, and is concerned with deriving testable or policy implications from acceptable assumptions.
It 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.

Underlying economics

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 arbitrage-free asset pricing.
The various "fundamental" valuation formulae result directly.

Present value, expectation and utility

Underlying all of financial economics are the concepts of present value and expectation.

Calculating their present value, $X_/r$ in the first formula, 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 then the starting point for financial decision making.
(Note that here, "$r$" represents a generic (or arbitrary) discount rate applied to the cash flows, whereas in the valuation formulae, the risk-free rate is applied once these have been "adjusted" for their riskiness; see below.)

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

This decision method, however, fails to consider risk aversion. In 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: $Y_$. 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 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 individual''s 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.

Arbitrage-free pricing and equilibrium

The concepts of arbitrage-free, "rational", pricing and equilibrium are then coupled

with the above to derive various of the "classical"See Rubinstein (2006), under "Bibliography". (or "neo-classical") financial economics models.

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, $s$, and that the complete set of possible bets on future states-of-the-world can therefore be constructed with existing assets (assuming no friction): essentially solving simultaneously for ''n'' (risk-neutral) probabilities, $q_$, given ''n'' prices. For a simplified example see , where the economy has only two possible states – up and down – and where $q_$ and $q_$ () are the two corresponding probabilities, and in turn, the derived distribution, or "measure".

The formal derivation will proceed by arbitrage arguments.Freddy Delbaen and Walter Schachermayer. (2004)
"What is... a Free Lunch?"
(pdf). Notices of the AMS 51 (5): 526–528 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 $X_$ and $X_$, are multiplied through by $q_$ and $q_$, 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 $Y$ and $r$ combined. This premium may be derived by the CAPM (or extensions) as will be seen under .

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.

(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 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'' $\pi_$ of this particular state of the world; the collection of these is also referred to as a "Risk Neutral Density".

In the above example, the state prices, $\pi_$, $\pi_$would equate to the present values of $$q_$ and $$q_$: i.e. what one would pay today, respectively, for the up- and down-state securities; the state price vector is the vector of state prices for all states. Applied to derivative valuation, the price today would simply be : 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".

State prices find immediate application as a conceptual tool ("contingent claim analysis"); but can also be applied to valuation problems.See de Matos, as well as Bossaerts and Ødegaard, under bibliography. Given the pricing mechanism described, one can decompose the derivative value – true in fact for "every security" – as a linear combination of its state-prices; i.e. back-solve for the state-prices corresponding to observed derivative prices.

These recovered state-prices can then be used for valuation of other instruments with exposure to the underlyer, or for other decision making relating to the underlyer itself.

Using the related stochastic discount factor - SDF; also called the pricing kernel - the asset price is computed by "discounting" the future cash flow by the stochastic factor $\tilde$, and then taking the expectation;See: David K. Backus] to represent the adaptive market hypothesis, adaptive behaviour of market participants".Katalin Boer, Arie De Bruin, Uzay Kaymak (2005)
"On the Design of Artificial Stock Markets"
''Research In Management'' ERIM Report Series

These 'bottom-up' models "start from first principals of agent behavior",LeBaron, B. (2002)
"Building the Santa Fe artificial stock market"
''Physica A'', 1, 20. with participants modifying their trading strategies having learned over time, and "are able to describe macro features .e. stylized facts">stylized_fact.html" ;"title=".e. stylized fact">.e. stylized factsEmergence#Economics">emerging from a soup of individual interacting strategies".
Agent-based models depart further from the classical approach — the representative agent, as outlined — in that they introduce Heterogeneity in economics">heterogeneity