Asset Pricing Models

In financial economics, asset pricing refers to a formal treatment and development of two interrelated pricing principles, outlined below, together with the resultant models. There have been many models developed for different situations, but correspondingly, these stem from either general equilibrium asset pricing or rational asset pricing, the latter corresponding to risk neutral pricing.

Investment theory, which is near synonymous, encompasses the body of knowledge used to support the decision-making process of choosing investments, and the asset pricing models are then applied in determining the asset-specific required rate of return on the investment in question, and for hedging.

General equilibrium asset pricing

Under general equilibrium theory prices are determined through market pricing by supply and demand.
See, e.g., Tim Bollerslev (2019)
"Risk and Return in Equilibrium: The Capital Asset Pricing Model (CAPM)"
/ref>
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 called market clearing. These models are born out of modern portfolio theory, with the capital asset pricing model (CAPM) as the prototypical result. Prices here are determined with reference to macroeconomic variables–for the CAPM, the "overall market"; for the Consumption-based capital asset pricing model, CCAPM, overall wealth– such that individual preferences are subsumed.

These models aim at modeling the statistically derived probability distribution of the market prices of "all" securities at a given future investment horizon; they are thus of "large dimension". See Mathematical finance#Risk and portfolio management: the P world, § Risk and portfolio management: the P world under Mathematical finance. General equilibrium pricing is then used when evaluating diverse portfolios, creating one asset price for many assets.

Calculating an investment or share value here, entails:
(i) a financial forecast for the business or project in question;
(ii) where the cash flow forecast, output cashflows are then present value, discounted at the rate returned by the model selected; this rate in turn reflecting the "riskiness" - i.e. the Idiosyncratic risk, idiosyncratic, or Systematic risk, undiversifiable risk - of these cashflows;
(iii) these present values are then aggregated, returning the value in question.
See: , and Valuation using discounted cash flows.
(Note that an alternate, although less common approach, is to apply a "fundamental valuation" method, such as the T-model, which instead relies on accounting information, attempting to model return based on the company's expected financial performance.)

Rational pricing

Under Rational pricing, derivative (finance), derivative prices are calculated such that they are arbitrage-free with respect to Underlying, more fundamental (equilibrium determined) securities prices;
for an overview of the logic see .

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 Mathematical finance#Derivatives pricing: the Q world, § Derivatives pricing: the Q world under Mathematical finance.

Calculating option prices, and their Greeks (finance), "Greeks", i.e. sensitivities, combines:
(i) a model of the underlying price behavior, or "stochastic process, process" - i.e. the asset pricing model selected, with its parameters having been calibrated to observed prices;
and
(ii) a numerical method, mathematical method which returns the premium (or sensitivity) as the expected value of option payoffs over the range of prices of the underlying.
See .

The classical model here is Black–Scholes model, Black–Scholes which describes the dynamics of a market including derivatives (with its Black–Scholes model#Black–Scholes formula, option pricing formula); leading more generally to martingale pricing, as well as the above listed models. Black–Scholes assumes a log-normal process; the other models will, for example, incorporate features such as Mean reversion (finance), mean reversion, or will be "volatility surface aware", applying local volatility or stochastic 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, where yield curves must be arbitrage free Yield curve#Construction of the full yield curve from market data, with respect to the prices of individual instruments.
See , Bootstrapping (finance), and Multi-curve framework.
For discussion as to how the models listed above are applied to options on these instruments, and other interest rate derivatives, see short-rate model and Heath–Jarrow–Morton framework.

Interrelationship

These principles are interrelated

through the fundamental 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".
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. See .

Relatedly, both approaches are consistent Mark Rubinstein (2005). "Great Moments in Financial Economics: IV. The Fundamental Theore
Part I
, ''Journal of Investment Management'', Vol. 3, No. 4, Fourth Quarter 2005;

~ (2006)
Part II
Vol. 4, No. 1, First Quarter 2006. with what is called the Arrow–Debreu model, Arrow–Debreu theory.
Here models can be derived as a function of "state prices" - contracts that pay one unit of a numeraire (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 (contingent claim analysis) so, conversely, pricing- or return-models can be backed-out, given state prices.
Edwin H. Neave and Frank Fabozzi, Frank J. Fabozzi (2012). Introduction to Contingent Claims Analysis, in Encyclopedia of Financial Models, Frank Fabozzi ed. Wiley (2012)
Bhupinder Bahra (1997)
Risk-neutral probability density functions from option prices: theory and application
Bank of England
The CAPM, for example, Stochastic discount factor #Properties, can be derived by linking risk aversion to overall market return, and restating for price. Black-Scholes Binomial_options_pricing_model#Relationship_with_Black–Scholes, can be derived by attaching a binomial probability to each of numerous possible spot price, spot-prices (i.e. states) and then rearranging for the terms in its formula.
See .

See also

*:Financial economics, List of financial economics articles
*
*

References

{{Reflist

Financial economics
Asset
Pricing
Financial models
Finance theories