AXIOMS in asset pricing

1. Mean-Variance Optimization (Markowitz, 1952)

Axioms:

Derived From:
Utility theory and statistical analysis of diversification.

Impact:
Introduced the concept of efficient frontiers, forming the basis for all subsequent equilibrium models.

2. Capital Asset Pricing Model (CAPM) (Sharpe, Lintner, Mossin, 1964)

Axioms:

  1. Homogeneous expectations: All investors share identical return and risk estimates.
  2. Existence of a risk-free asset: Unlimited borrowing/lending at a risk-free rate.
  3. Single-period investment horizon.
  4. Perfectly divisible and liquid markets.

Derived From:
Markowitz's mean-variance framework, with added equilibrium assumptions.

Key Contribution:
Expressed expected returns as $$ \mathbb{E}[R_i] = R_f + \beta_i (\mathbb{E}[R_m] - R_f) $$, linking returns to systematic risk (beta).

3. Arbitrage Pricing Theory (APT) (Ross, 1976)

Axioms:

  1. No-arbitrage principle: Markets eliminate risk-free profit opportunities.
  2. Linear factor structure: Returns driven by exposure to macroeconomic factors.

Derived From:
Critique of CAPM’s single-factor limitation. APT relaxed CAPM’s assumptions (e.g., no need for a market portfolio).

Impact:
Pioneered multi-factor models (e.g., Fama-French 3/5/6 factors) to explain cross-sectional returns.

4. Fundamental Theorem of Asset Pricing (FTAP) (Harrison-Kreps, 1979)

Axioms:

  1. No free lunch with vanishing risk (NFLVR): Stricter no-arbitrage condition for infinite markets.
  2. Existence of an equivalent martingale measure: Prices are expectations under a risk-neutral measure $$ \mathbb{Q} $$.

Derived From:
Mathematical formalization of Black-Scholes hedging arguments.

Key Contribution:
Established $$ \text{No arbitrage} \iff \exists \mathbb{Q} \text{ s.t. } S_t = \mathbb{E}^\mathbb{Q}[e^{-rT}S_T] $$, foundational for derivatives pricing.

5. Consumption-Based Models (Breeden, 1979; Lucas, 1978)

Axioms:

Derived From:
General equilibrium theory and critique of CAPM’s static framework.

Formula:

\text{Price} = \mathbb{E}[M_{t+1} \cdot \text{Payoff}_{t+1}] $$, where $$ M_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_t)} $$. --- ### 6. **Behavioral Asset Pricing (Post-1980s)** **Axioms**: - **Investor irrationality**: Psychological biases (overconfidence, loss aversion) affect prices. - **Market anomalies**: Limits to arbitrage allow mispricing persistence. **Derived From**: Empirical failures of CAPM/APT (e.g., momentum, value effects). **Impact**: Integrated psychology into pricing models (e.g., prospect theory). --- ### Evolutionary Logic: 1. **CAPM vs. APT**: CAPM’s reliance on unobservable market portfolios led to APT’s factor-based flexibility. 2. **FTAP’s Universality**: Unified discrete and continuous-time models under a no-arbitrage umbrella. 3. **Consumption Models**: Addressed CAPM’s static limitations by linking pricing to macroeconomic fundamentals. 4. **Behavioral Models**: Reacted to neoclassical assumptions by incorporating cognitive biases. --- ### Current Axiomatic Hierarchy: ```mermaid graph TD A[Mean-Variance Optimization] --> B[CAPM] A --> C[APT] B --> D[FTAP] C --> D D --> E[Consumption-Based Models] E --> F[Behavioral Models] ``` **Why This Order Matters**: Each layer addressed prior limitations: CAPM introduced systematic risk, APT added multi-factor flexibility, FTAP mathematically formalized arbitrage, and behavioral models humanized decision-making. The progression mirrors finance’s shift from idealized rationality to empirical realism and mathematical rigor.