DO ARBITRAGE FREE PRICES COME FROM UTILITY MAXIMIZATION?

Transcription

1 DO ARBITRAGE FREE PRICES COME FROM UTILITY MAXIMIZATION? Pietro Siorpaes University of Vienna, Austria Warsaw, June 2013

2 SHOULD I BUY OR SELL? ARBITRAGE FREE PRICES ALWAYS BUY IT DEPENDS ALWAYS SELL

3 SHOULD I BUY OR SELL? MARKET ARBITRAGE FREE PRICES ALWAYS BUY IT DEPENDS ALWAYS SELL MARKET + AGENT BUY MARGINAL PRICES DO NOTHING SELL

4 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.

5 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 Õ, q + q Õ ) Æ u(x, q) for all q Õ œ R n,

6 QUESTIONS 1 Are marginal prices always arbitrage free? MP(x, q; u) AFP?

7 QUESTIONS 1 Are marginal prices always arbitrage free? KARATZAS AND KOU (1996) MP(x, q; u) AFP?

8 QUESTIONS 1 Are marginal prices always arbitrage free? KARATZAS AND KOU (1996) MP(x, q; u) AFP? 2 Do all arbitrage free prices come from utility maximization? MP(x, q; u) AFP? Union over what?

9 THE MARKET Liquid frictionless market Bank account, with no interest Stocks: semimartingale S, admitting ELMMs (Almost) no constraints on strategy H

10 THE MARKET Liquid frictionless market Bank account, with no interest Stocks: semimartingale S, admitting ELMMs (Almost) no constraints on strategy H Illiquid contingent claims f (Ê) œ R n random payoff f Æ c + s T 0 HdS for some c, H qf is not replicable for any q = 0

11 THE MARKET Liquid frictionless market Bank account, with no interest Stocks: semimartingale S, admitting ELMMs (Almost) no constraints on strategy H Illiquid contingent claims f (Ê) œ R n random payoff f Æ c + s T 0 HdS for some c, H qf is not replicable for any q = 0 Definition of AFP p is an arbitrage free price if q Õ (f p)+ s T 0 HdS Ø 0 implies qõ (f p)+ s T 0 HdS = 0

12 UTILITY, MARGINAL PRICES Agent Maximal expected utility u(x, q) :=sup H E[U(x + qf + T 0 HdS)] U :(0, Œ) æ R Utility: strictly concave, increasing, differentiable, Inada conditions

13 UTILITY, MARGINAL PRICES Agent Maximal expected utility u(x, q) :=sup H E[U(x + qf + T 0 HdS)] U :(0, Œ) æ R Utility: strictly concave, increasing, differentiable, Inada conditions Definition of MP(x, q; u) p is a marginal price at (x, q) relative to u if u(x pq Õ, q + q Õ ) Æ u(x, q) for all q Õ œ R n,

14 UTILITY, MARGINAL PRICES Agent Maximal expected utility u(x, q) :=sup H E[U(x + qf + T 0 HdS)] U :(0, Œ) æ R Utility: strictly concave, increasing, differentiable, Inada conditions Definition of MP(x, q; u) p is a marginal price at (x, q) relative to u if u(x pq Õ, q + q Õ ) Æ u(x, q) for all q Õ œ R n, i.e. if (x, q) maximizes u over {(x pq Õ, q + q Õ ):q Õ œ R n } =: A Setting as in HUGONNIER AND KRAMKOV (2004)

15 MAIN THEOREM Theorem If sup x (u(x, 0) xy) < Œ for all y > 0 then (x,q)œ{u> Œ} MP(x, q; u) =AFP

16 MAIN THEOREM Theorem If sup x (u(x, 0) xy) < Œ for all y > 0 then (x,q)œ{u> Œ} MP(x, q; u) =AFP u(x, 0) =u(x) as in Kramkov and Schachermayer (1999) Any U is enough to reconstruct AFP Enough to consider small (x, q) Always we need (x, q) close to ˆ{u > Œ} In general we need (x, q) œ ˆ{u > Œ}

17 BOUNDARY POINTS ARE ILL-BEHAVED Technical reasons The multi-function MP : int{u > Œ} æ R n (x, q) æ MP(x, q; u) has compact, non-empty values and is upper-hemicontinuous...none of this is true on the boundary!

18 BOUNDARY POINTS ARE ILL-BEHAVED Technical reasons The multi-function MP : int{u > Œ} æ R n (x, q) æ MP(x, q; u) has compact, non-empty values and is upper-hemicontinuous...none of this is true on the boundary! Need to extend HUGONNIER AND KRAMKOV (2004)

19 BOUNDARY POINTS ARE ILL-BEHAVED Technical reasons The multi-function MP : int{u > Œ} æ R n (x, q) æ MP(x, q; u) has compact, non-empty values and is upper-hemicontinuous...none of this is true on the boundary! Need to extend HUGONNIER AND KRAMKOV (2004) Economic reasons Theorem If p 0 œ P(x, q) for some non-zero (x, q) œ ˆ{u > Œ}, then p œ R n \ AFP such that [p 0, p) MP(x, q; u)

20 DOMAIN OF UTILITY u q u R u = x

21 P ARBITRAGE PRICE B u R u = (x,q) A := {(x pq, q + q ) : q R n } p MP(x, q; u) if (x, q) is maximizer of u on B

22 SKETCH OF PROOF New geometric characterization of AFP The following are equivalent: 1 p œ AFP 2 B is bounded 3 If (x Õ, q Õ ) œ cl{u > Œ} satisfies x Õ + q Õ p = 0 then (x Õ, q Õ )=(0, 0) 4 There exists an ELMM Q such that p = E Q [f ] etc.

23 SKETCH OF PROOF New geometric characterization of AFP The following are equivalent: 1 p œ AFP 2 B is bounded 3 If (x Õ, q Õ ) œ cl{u > Œ} satisfies x Õ + q Õ p = 0 then (x Õ, q Õ )=(0, 0) 4 There exists an ELMM Q such that p = E Q [f ] etc. PROOF OF MP(u) AFP : Fix p /œ AFP, (x, q) œ {u > Œ}, let s show p /œ MP(x, q; u).

24 SKETCH OF PROOF New geometric characterization of AFP The following are equivalent: 1 p œ AFP 2 B is bounded 3 If (x Õ, q Õ ) œ cl{u > Œ} satisfies x Õ + q Õ p = 0 then (x Õ, q Õ )=(0, 0) 4 There exists an ELMM Q such that p = E Q [f ] etc. PROOF OF MP(u) AFP : Fix p /œ AFP, (x, q) œ {u > Œ}, let s show p /œ MP(x, q; u). Since u(x, q) < u(x + x Õ, q + q Õ ) holds for any non-zero (x Õ, q Õ ) œ cl{u > Œ},

25 SKETCH OF PROOF New geometric characterization of AFP The following are equivalent: 1 p œ AFP 2 B is bounded 3 If (x Õ, q Õ ) œ cl{u > Œ} satisfies x Õ + q Õ p = 0 then (x Õ, q Õ )=(0, 0) 4 There exists an ELMM Q such that p = E Q [f ] etc. PROOF OF MP(u) AFP : Fix p /œ AFP, (x, q) œ {u > Œ}, let s show p /œ MP(x, q; u). Since u(x, q) < u(x + x Õ, q + q Õ ) holds for any non-zero (x Õ, q Õ ) œ cl{u > Œ}, taking (x Õ, q Õ )=( q Õ p, q Õ ) as in item (4) gives u(x, q) < u(x q Õ p, q + q Õ )

26 P ARBITRAGE FREE PRICE B u R u = (x,q) A := {(x pq, q + q ) : q R n } p MP(x, q; u) if (x, q) is maximizer of u on B

27 PROOF OF AFP MP(u) We need that maximizer of u of B. Since B is compact, it s enough to show that u is upper semi-continuous

28 PROOF OF AFP MP(u) We need that maximizer of u of B. Since B is compact, it s enough to show that u is upper semi-continuous SKETCH OF PROOF Take (x k, q k ) æ (x, q), H k s.t. W k := x k + q k f +(H k S) T satisfies E[U(W k )] = u(x k, q k ) æ s œ R

29 PROOF OF AFP MP(u) We need that maximizer of u of B. Since B is compact, it s enough to show that u is upper semi-continuous SKETCH OF PROOF Take (x k, q k ) æ (x, q), H k s.t. W k := x k + q k f +(H k S) T satisfies E[U(W k )] = u(x k, q k ) æ s œ R By Kolmos lemma V k œ conv{(w n ) nøk } which converges a.s. to some r.v. V Use duality theory to show that H s.t. V Æ W := x + qf +(H S) T, so E[U(V )] Æ u(x, q)

30 PROOF OF AFP MP(u) We need that maximizer of u of B. Since B is compact, it s enough to show that u is upper semi-continuous SKETCH OF PROOF Take (x k, q k ) æ (x, q), H k s.t. W k := x k + q k f +(H k S) T satisfies E[U(W k )] = u(x k, q k ) æ s œ R By Kolmos lemma V k œ conv{(w n ) nøk } which converges a.s. to some r.v. V Use duality theory to show that H s.t. V Æ W := x + qf +(H S) T, so E[U(V )] Æ u(x, q) By Jensen inequality E[U(V k )] Ø inf nøk E[U(W n )] Show that U(V k ) + is uniformly integrable, so by Fatou lim k E[U(V k )] Æ E[U(V )], so lim k u(x k, q k ) Æ u(x, q)

31 SUMMARY Arbitrage free prices come from utility maximization (x,q)œ{u> Œ} MP(x, q; u) =AFP In general we need also (x, q) œ ˆ{u > Œ} The corresponding p 0 œ MP(x, q) are quirky p œ R n \ AFP such that [p 0, p) MP(x, q)