DO ARBITRAGE FREE PRICES COME FROM UTILITY MAXIMIZATION? Pietro Siorpaes

Ann Arbor, May 2011

SHOULD I BUY OR SELL?

ARBITRAGE FREE PRICES ALWAYS BUY

IT DEPENDS

ALWAYS SELL

SHOULD I BUY OR SELL?

MARKET ARBITRAGE FREE PRICES ALWAYS BUY

IT DEPENDS

ALWAYS SELL

MARKET + AGENT MARGINAL PRICES BUY

DO NOTHING

SELL

MARGINAL PRICES

Agent u(x, q)

maximal expected utility achievable

x

initial cash wealth

q

initial number of cont. claims

Marginal Prices Intuitive definition p is a marginal price for the agent with utility u and initial endowment (x, q) if his optimal demand of cont. claims at price p is zero.

MARGINAL PRICES Agent u(x, q)

maximal expected utility

x

cash wealth

q

number of cont. claims

Marginal Prices Definition of MP(x, q; u) p is a marginal price at (x, q) relative to u if u(x − pq 0 , q + q 0 ) ≤ u(x, q)

for all q 0 ∈ Rn ,

i.e. if (x, q) maximizes u over {(x − pq 0 , q + q 0 ) : q 0 ∈ Rn } =: A

QUESTIONS

1

Are marginal prices always arbitrage free ? MP(x, q; u) ⊆ AFP

?

QUESTIONS

1

Are marginal prices always arbitrage free ? MP(x, q; u) ⊆ AFP K ARATZAS

AND

KOU (1996)

?

QUESTIONS

1

Are marginal prices always arbitrage free ? MP(x, q; u) ⊆ AFP K ARATZAS

2

AND

?

KOU (1996)

Do all arbitrage free prices come from utility maximization? [

Union over what ?

MP(x, q; u) ⊇ AFP

?

THE MODEL Market S Stocks Bank account with no interest Exist martingale measures No constraints on strategy H

THE MODEL Market S Stocks Bank account with no interest Exist martingale measures No constraints on strategy H Agent Maximal expected utility u(x, q) := sup E[U(x + qf + H

U : (0, ∞) → R

Z

T 0

HdS)]

Utility: strictly concave, increasing, differentiable, Inada conditions

ARBITRAGE FREE PRICES Contingent claims f (ω) ∈ Rn |f | ≤ c +

RT 0

random payoff HdS

qf is not replicable

for some c, H for any q 6= 0

Arbitrage free prices Definition of AFP p is an arbitrage free price if q 0 (f − p) +

RT 0

HdS ≥ 0

implies

q 0 (f − p) +

RT 0

HdS = 0

MAIN THEOREM Theorem If supx (u(x, 0) − xy ) < ∞ [

for all

y >0

then

MP(x, q; u) = AFP

(x,q)∈{u>−∞}

u(x, 0) = u(x) as in Kramkov and Schachermayer (1999) Marginal prices are always arbitrage free, and vice versa Any U is enough to reconstruct AFP

TECHNICAL POINTS

Not enough to consider (x, q) in the interior of {u > −∞} Study u on the boundary of {u > −∞}, and prove its upper semi-continuity New geometrical characterization of AFP:

Lemma If u(x, q) > −∞, then p ∈ AFP iff A ∩ {u > −∞} is bounded, where A = {(x − pq 0 , q + q 0 ) : q 0 ∈ Rn }.

DOMAIN OF UTILITY u

q u∈R

u = −∞

x

P ARBITRAGE FREE PRICE

B

u = −∞

u∈R (x,q)

A := {(x − pq 0 , q + q 0 ) : q 0 ∈ Rn } p ∈ MP(x, q; u) if (x, q) is maximizer of u on B

P ARBITRAGE PRICE

B u∈R u = −∞ 

(x,q)

A := {(x − pq 0 , q + q 0 ) : q 0 ∈ Rn } p ∈ MP(x, q; u) if (x, q) is maximizer of u on B

do arbitrage free prices come from utility maximization?

Bank account with no interest. Exist martingale measures. No constraints on strategy H. Agent. Maximal expected utility u(x,q) := sup. H. E[U(x + qf + ∫ T. 0.

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