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=ukck+ βkπ·ukck
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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