Asset Pricing and No-Arbitrage

Asset pricing theory unites the microeconomics of optimal intertemporal choice with the mathematics of martingales. The fundamental theorem connects the absence of arbitrage to the existence of a measure under which discounted prices are martingales. This is the foundation of all modern financial engineering.

The Markowitz Portfolio Selection Problem

Formulation: n risky assets, expected returns μ ∈ ℝⁿ, covariance matrix Σ. The problem is to minimize risk for a given expected return:

min_w wᵀΣw subject to wᵀμ ≥ μ₀, wᵀ1 = 1

Here, w is the portfolio weights vector. The KKT conditions yield: w* = λΣ⁻¹μ + γΣ⁻¹1, where λ, γ are Lagrange multipliers. The efficient frontier is a parabola in the (σₚ, μₚ) space, σₚ = √(wᵀΣw).

Two Fund Theorem: All efficient portfolios are linear combinations of any two efficient portfolios. In practice: construct the frontier knowing only two points.

Tangency Portfolio with a risk-free asset r_f: Maximizes the Sharpe ratio SR = (μₚ − r_f)/σₚ. Weight: T = Σ⁻¹(μ − r_f·1)/cᵀΣ⁻¹(μ − r_f·1). The Capital Market Line (CML): μₚ = r_f + SR·σₚ (linear, above the efficient frontier).

Diversification: For the equilibrium portfolio (n assets, σᵢ² = σ², correlation ρ): σₑw² = σ²/n + ρσ²(1−1/n). When n→∞: σₑw² → ρσ². Unsystematic risk disappears, systematic risk (the ρ component) does not.

CAPM

Equilibrium under homogeneous expectations: All investors hold T = the market portfolio M. CAPM:

E[Rᵢ] = r_f + βᵢ(E[Rₘ] − r_f), βᵢ = Cov(Rᵢ, Rₘ)/Var(Rₘ)

Interpretation: βᵢ is the “systematic risk” of asset i—how much the asset “moves with the market.” E[Rₘ] − r_f is the “market risk premium” (on average ≈ 6% per annum for the US). Only systematic risk (β) is compensated by the premium; idiosyncratic risk is diversified away.

SDF derivation of CAPM: The optimal consumer plan yields: M = βu'(c₁)/u'(c₀) — the stochastic discount factor. E[M·Rᵢ] = 1 for all i. For quadratic utility: M is linear in Rₘ → CAPM.

No-Arbitrage and Martingale Measures

Arbitrage: a strategy with zero cost today and non-negative payoffs in the future (with positive probability — strictly positive). If arbitrage exists, the market is “broken.”

Fundamental Theorem of Asset Pricing: No arbitrage ⟺ there exists a measure Q, equivalent to P, such that discounted prices are Q-martingales: E^Q[e^{−rT} Sₜ] = S₀. In a complete market Q is unique.

Derivative pricing under measure Q:

π₀(F) = E^Q[e^{−rT} F]

F is the derivative's payoff at T. The expectation is taken under the “risk-neutral” measure Q, discounted at the risk-free rate r. Q exists (no arbitrage), is unique (complete market).

Binomial model: S₀ = 100, S_u = 120 (up), S_d = 80 (down), r = 5%. The risk-neutral probability: q = (S₀(1+r) − S_d)/(S_u − S_d) = (105 − 80)/40 = 0.625. Call with K = 110: C₀ = e^{−r}[q·max(120−110,0) + (1−q)·max(80−110,0)] = (1/1.05)[0.625·10 + 0.375·0] = 6.25/1.05 ≈ 5.95.

Numerical Example

Three assets: μ = (10%, 15%, 12%), σ = (20%, 30%, 25%), ρ₁₂ = 0.4, ρ₁₃ = 0.2, ρ₂₃ = 0.3, r_f = 5%.

Covariance matrix Σ: σ₁₂ = 0.4·0.20·0.30 = 0.024, σ₁₃ = 0.01, σ₂₃ = 0.0225. Tangency portfolio (approximately): w ≈ (0.3, 0.5, 0.2). μ_T ≈ 13%, σ_T ≈ 22%. SR = (13−5)/22 ≈ 0.36.

Beta of each asset: β₁ = Cov(R₁,Rₘ)/Var(Rₘ) ≈ 0.75, β₂ ≈ 1.4, β₃ ≈ 0.9. CAPM: E[R₁] = 5 + 0.75·8 = 11% (actual 10% → slightly below SML). E[R₂] = 5 + 1.4·8 = 16.2% (actual 15% → below SML). Minor anomalies are normal for CAPM.

Exercise: Two-period binomial tree: S₀ = 100, each period u = 1.2, d = 0.9, r = 5%. Price a European put K = 105. (1) Compute q at each node. (2) Find the price at each node by backward induction. (3) Construct the replicating strategy (Δ, B) at each node. (4) Check: does the initial portfolio (Δ₀ shares + B₀ rubles) replicate the put's payoff?