Regime Asset class | Equilibrium pricing | Risk neutral pricing |
|---|---|---|
Equities (and foreign exchange and commodities; interest rates for risk neutral pricing) | ||
Bonds, other interest rate instruments |
Infinancial economics,asset pricing refers to a formal treatment and development of two interrelatedpricing principles,[1][2] outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from eithergeneral equilibrium asset pricing orrational asset pricing,[3] the latter corresponding to risk neutral pricing.
Investment theory, which is near synonymous, encompasses the body of knowledge used to support thedecision-making process of choosinginvestments,[4][5] and the asset pricing models are then applied in determining theasset-specific required rate of return on the investment in question, and for hedging.
Undergeneral equilibrium theory prices are determined throughmarket pricing bysupply and demand.[6] Here asset prices jointly satisfy the requirement that the quantities of each asset supplied and the quantities demanded must be equal at that price - so calledmarket clearing. These models are born out ofmodern portfolio theory, with thecapital asset pricing model (CAPM) as the prototypical result. Prices here are determined with reference to macroeconomic variables–for the CAPM, the "overall market"; for theCCAPM, overall wealth– such that individual preferences are subsumed.
These models aim at modeling the statistically derivedprobability distribution of the market prices of "all" securities at a given future investment horizon; they are thus of "large dimension". See§ Risk and portfolio management: the P world underMathematical finance. General equilibrium pricing is then used when evaluating diverse portfolios, creating one asset price for many assets.[7]
Calculating an investment or share value here, entails:(i) afinancial forecast for the business or project in question; (ii) where theoutput cashflows are thendiscounted at the rate returned by the model selected; this rate in turn reflecting the "riskiness" - i.e. theidiosyncratic, orundiversifiable risk - of these cashflows;(iii) these present values are then aggregated, returning the value in question. See:Financial modeling § Accounting, andValuation using discounted cash flows. (Note that an alternate, although less common approach, is to apply a "fundamental valuation" method, such as theT-model, which instead relies on accounting information, attempting to model return based on the company's expected financial performance.)
UnderRational pricing,derivative prices are calculated such that they arearbitrage-free with respect tomore fundamental (equilibrium determined) securities prices;for an overview of the logic seeRational pricing § Pricing derivatives.
In general this approach does not group assets but rather creates a unique risk price for each asset; these models are then of "low dimension".For further discussion, see§ Derivatives pricing: the Q world under Mathematical finance.
Calculating option prices, and their"Greeks", i.e. sensitivities, combines:(i) a model of the underlying price behavior, or "process" - i.e. the asset pricing model selected, with its parameters having been calibrated to observed prices;and (ii) amathematical method which returns the premium (or sensitivity) as theexpected value of option payoffs over the range of prices of the underlying. SeeValuation of options § Pricing models.
The classical model here isBlack–Scholes which describes the dynamics of a market including derivatives (with itsoption pricing formula); leading more generally tomartingale pricing, as well as the above listed models. Black–Scholes assumes alog-normal process; the other models will, for example, incorporate features such asmean reversion, or will be "volatility surface aware", applyinglocal volatility orstochastic volatility.
Rational pricing is also applied to fixed income instruments such as bonds (that consist of just one asset), as well as to interest rate modeling in general, whereyield curves must be arbitrage freewith respect to the prices of individual instruments.SeeRational pricing § Fixed income securities,Bootstrapping (finance), andMulti-curve framework.For discussion as to how the models listed above are applied to options on these instruments, and otherinterest rate derivatives, seeshort-rate model andHeath–Jarrow–Morton framework.
These principles are interrelated[2]through thefundamental theorem of asset pricing.Here, "in the absence of arbitrage, the market imposes a probability distribution, called a risk-neutral or equilibrium measure, on the set of possible market scenarios, and... this probability measure determines market prices via discounted expectation".[8]Correspondingly, this essentially means that one may make financial decisions, using the risk neutral probability distribution consistent with (i.e. solved for) observed equilibrium prices. SeeFinancial economics § Arbitrage-free pricing and equilibrium.
Relatedly, both approaches are consistent[9][2] with what is called theArrow–Debreu theory.Here models can be derived as a function of "state prices" - contracts that pay one unit of anumeraire (a currency or a commodity) if a particular state occurs at a particular time, and zero otherwise. The approach taken is to recognize that since the price of a security can be returned as a linear combination of its state prices[2] (contingent claim analysis) so, conversely, pricing- or return-models can be backed-out, given state prices.[10][11]The CAPM, for example,can be derived by linkingrisk aversion to overall market return, and restating for price.[9] Black-Scholescan be derived by attaching abinomial probability to each of numerous possiblespot-prices (i.e. states) and then rearranging for the terms in its formula.SeeFinancial economics § Uncertainty.