Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 40 / 1 / i / Solution

Take $S_t$ to be the ex-dividend stock price: the date-$t$ output has already been paid when trading occurs. Receiving ${\theta }_t{d}_t$ units of fruit and selling ${\theta }_t-{\theta }_{t+1}$ shares gives the discrete dividend budget equationThe entering holding ${\theta }_t$ is chosen with date-$(t-1)$ information, while $C_t$ and the next holding ${\theta }_{t+1}$ may use date-$t$ information. The output is the consumption good and price numéraire; its stochastic law is exogenous to the individual's price-taking choice.

For the interior differentiable case, attach an adapted Lagrange multiplier ${\Lambda }_t$ to each budget constraint. On a finite horizon, with the terminal holding fixed or its continuation value included, the optimization Lagrangian isAn adapted variation in $C_t$ gives ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. An arbitrary date-$t$ measurable variation in ${\theta }_{t+1}$ affects two adjacent terms. Its coefficient must have zero conditional mean, soThese are the Euler equations of marginal utility pricing in a dividend economy. If the utility function is merely concave, a supporting slope replaces $U^{\prime }$; binding consumption or trading restrictions give the corresponding inequalities rather than unconstrained equalities.

With known initial information and ${\Lambda }_0>0$, normalize ${\zeta }_t={\Lambda }_t/{\Lambda }_0$. This is a state-price density for the traded asset. In particular,The ratio is the stochastic discount factor: payment in a state with greater marginal utility has a higher marginal value. For a priced contingent payoff $H$ at $t+1$, its price is ${\mathbb{E}}_t[({\zeta }_{t+1}/{\zeta }_t)H]$; pricing arbitrary claims uniquely also requires market completeness. The deflated cumulative gains ${\zeta }_t{S}_t+{\sum }_{s=1}^t{\zeta }_s{d}_s$ form a martingale whenever the displayed terms are integrable.

Iterating the Euler equation givesUnder the dividend-price transversality condition and the needed integrability, this becomes the fundamental priceThus an infinite-horizon price is not automatically just a dividend sum: its terminal term must vanish. Telescoping the individual budget constraints similarly givesA no-Ponzi lower restriction on this final term gives the consumption budget inequality; its vanishing gives equality. Together with the supporting-line inequality for a concave utility function, the Euler equations and these terminal conditions supply the usual sufficiency argument for an admissible candidate. In an incomplete market, each optimal agent's marginal utility can supply a different state-price density pricing the same traded tree.