State-price density

A state-price density is a positive random variable or process that converts future payoffs into present values by expectation. In a one-period zero-cost return model $X$, its normalization is commonly $\mathbb{E}\rho =1$ and $\mathbb{E}[\rho X]=0$.

At an interior consumption and share-holding optimum, the Lagrange multiplier on the discrete dividend budget equation is ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. Varying the next holding by a variable measurable at date $t$ gives the displayed Euler equation. Thus the positive normalized process ${\zeta }_t={\Lambda }_t/{\Lambda }_0$ is a state-price density for traded dividend-inclusive gains. In an incomplete market, different investors can have different such densities; equality of their marginal-utility densities is an additional risk-sharing condition, not a consequence of trading one asset alone.

A positive state-price density prices a productive stock through a conditional dividend sum plus a future deflated-price term. The displayed condition removes that terminal term, yielding transversality and fundamental dividend prices. An individual consumption budget instead has terminal term $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$, so its vanishing must be imposed or proved for the particular holdings as well.

Iteration of the pricing Euler equation givesThe displayed fundamental price follows when the future dividend series is integrable and ${\mathbb{E}}_t[{\zeta }_N{S}_N]\to 0$. Without this dividend-price transversality condition, a deflated price martingale component can survive. For individual holdings, the analogous terminal term is $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$. Telescoping the discrete dividend budget equation gives the initial wealth cost of consumption minus this terminal term; its vanishing is needed for budget equality.

For an Ornstein-Uhlenbeck process $dX=\sigma dB-\lambda Xdt$ under a reference measure, this is a positive supermartingale when $\alpha a\ge {\sigma }^2/2$. Its short rate is $[\alpha a-{\sigma }^2/2+(\lambda +\alpha /2)X^2]/(a+X^2/2)$. Conditional zero-coupon bond prices are ratios of the OU second moments of the quadratic factor. The money-market density has diffusion coefficient $\sigma X/(a+X^2/2)$, bounded by $\sigma /\sqrt{2a}$, so the Novikov condition validates the pricing measure. The reference OU dynamics must not be silently treated as the risk-neutral dynamics.

On a finite state space, the Arrow–Debreu state price is the current price of a payoff equal to one in state $s$ and zero elsewhere. It equals the physical state probability times the state-price density in that state. With a riskless asset of deterministic prices $B_0,B_1$, the Arrow state prices sum to $B_0/B_1$. Normalizing them gives the risk-neutral probabilities, while division by the physical probabilities instead recovers the pricing density.

If a traded payoff matrix $A$ has initial price vector $q=A^{T}p$ with $p\ge 0$, there cannot be a negative-cost portfolio with nonnegative terminal payoff. Nevertheless a nonzero nonnegative attainable payoff supported only in zero-price states can cost zero and give an arbitrage. When every state has positive physical probability, excluding this possibility requires strictly positive Arrow state prices, not merely a nonnegative supporting vector.

A local martingale deflator only forces deflated asset prices to be local martingales. True expectation pricing requires additional martingale and integrability properties. In a Brownian one-factor market, continuity of the coefficients ensures a pathwise finite market-price-of-risk square integral, but not expectation-one of its stochastic exponential. The reciprocal three-dimensional Bessel strict local martingale gives a counterexample to promoting this local conclusion automatically.

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

If wealth and consumption are nonnegative, this local martingale is bounded below by minus the finite initial deflated capital. Adding that capital gives a nonnegative local martingale, hence a supermartingale by the Conditional Fatou lemma. Consequently expected discounted consumption cannot exceed initial deflated wealth. The consumption sign and predictable stochastic integrability are part of the hypotheses.

A state-price density converts terminal wealth and consumption into initial cost. For a nonnegative admissible self-financing portfolio with consumption, the deflated wealth plus cumulative deflated consumption is a supermartingale, giving $\mathbb{E}[{\zeta }_{T}X]+\mathbb{E}{\int }_0^T{\zeta }_t{c}_{t}dt\le w$. In a complete market, an integrable nonnegative terminal claim with full budget equality is replicable.

For $0<R<1$, nonnegative consumption with state-price budget $\mathbb{E}\int Z_t{c}_{t}dt\le x$ satisfies this bound under CRRA utility. Apply Holder inequality on probability-times-time measure to $(Zc{)}^{1-R}$ and $e^{-bt}{Z}^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}{Z}_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.

For a bounded payoff vector $P$, the condition that $h\cdot p\le 0\le h\cdot P$ almost surely forces $h=0$ makes $F(h)=\mathbb{E}{e}^{-h\cdot P}+h\cdot p$ coercive. At a minimizer $h_{\ast }$, differentiation gives $\mathbb{E}[P{e}^{-h_{\ast }\cdot P}]=p$. The resulting positive pricing kernel is bounded above and away from zero. A direct coercivity proof uses compactness of the unit sphere and separates directions with negative-payoff probability from directions with nonnegative payoffs.