CS代写 Lesson 6: Arrow-Debreu Pricing: Multiple States, Stochastic Discount Factor

Lesson 6: Arrow-Debreu Pricing: Multiple States, Stochastic Discount Factor, Heterogeneity, Pareto Optimality Economics of Finance
School of Economics, UNSW

Pricing state-contingent claims

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• Using the atomic state prices, often called, pricing kernel: p = q · c,
q – row vector of atomic state prices or pricing kernel,
c – column vector of state-contingent payments • Using risk-neutral measure:
p = d f · E􏰅 ( c ) ,
c – random variable, realised value depends on a state, E􏰅(·) – expectation taken with respect to risk-neutral measure using risk-neutral probabilities π􏰅
• Using stochastic discount factor:
p = E(m1c),
c – random variable, realised value depends on a state, m1 – stochastic discount factor,
E(·) – expectation taken with respect to physical probability measure using actual probabilities π

Heterogeneous agents: consumers’ problem
• Each agent k maximises expected utility, Uk, given by Uk=uk􏰂ck􏰃+ βk􏰄π·uk􏰂ck􏰃
0 s1 s1 s1 ∈S1
expected discounted future utility
• subject to period-0 constraint
c k0 + 􏰄 q s 1 · a ks 1 = e k0 ,
• and a series of period-1 constraints for every possible state:
cks1 = aks1 + eks1 , for all s1 ∈ S1 • Market clearing (now makes more sense)
􏰄ck0 =􏰄ek0; 􏰄cks1 =􏰄eks1,∀s1 k=1 k=1 k=1 k=1

Characterisation of the Equilibrium
• From the first order conditions the prices of the atomic (Arrow-Debreu) securities
uk′􏰀ck 􏰁 k s1
q s 1 = β π s 1 u k ′ 􏰀 c k0 􏰁 =
=βπs1 uk′􏰀ek0+ak0􏰁 for all k and s1 ∈ S.
• also impose market clearing which implies that
􏰄 a ks 1 = 0 , ∀ s 1 ∈ S . k=1
uk′􏰀ek +ak􏰁 k s1 s1

Pareto optimality
• Allocation of atomic (Arrow-Debreu) securities in which it is impossible to make any one consumer better off without making at least one consumer worse off.
• Under the first welfare theorem (we do not prove it here) competitive equilibrium (prices are taken as given) is equivalent to Pareto optimality.
• Some conditions: completeness – existence of atomic (Arrow-Debreu) securities for all states, no transaction costs, no externalities (utilities are independent).

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