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# Fundamental Theorem of Asset Pricing

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### Fundamental Theorem of Asset Pricing

1. 1. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems The Fundamental Theorem of Asset Pricing under Transaction Costs Paolo Guasoni (joint work with Miklós Rásonyi) Boston University Department of Mathematics and Statistics
2. 2. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Overview Model Bid and Ask Prices in continuous time. Jumps allowed. Theorem (Robust No Free Lunch with Vanishing Risk) (Exists Strictly Consistent Price System) Getting there: what is an admissible strategy? Consequences (RNFLVR) ⇒ Finite variation strategies. No stochastic integrals. Do we need a probability?
3. 3. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Model One risky and one risk-free asset. Risk-free asset as numeraire. Risky asset: Bid price St − κt , Ask price St + κt . Prices may become negative. Numeraire does matter. Assumption (Ω, F, (F)0≤t≤T , P) ﬁltered probability space. Usual conditions. (S, κ) càdlàg adapted locally bounded. κ ≥ 0.
4. 4. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Simple Strategies Deﬁnition Simple strategy: θ predictable, θ0 = θT = 0, and: ∞ θσ n 1 + θσ n 1 θ= + σn σn ,σn+1 n=1 (σn )n≥1 strictly increasing stopping times. supn≥1 σn > T a.s., that is P(∪n≥1 {σn > T }) = 1. Finite number of transactions. May depend on ω. Doubling Strategies? Left and Right Transactions.
5. 5. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Left and Right Transactions Right transaction at a stopping time σ and price (S ± κ)σ . Trade “when market opens”. a q Left transaction at a predictable time σ and price (S ± κ)σ− . q Trade “before market closes”. a In general two transactions: a q a Both right and left transactions considered simple.
6. 6. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Cost Deﬁnition Cost of a simple strategy θ: ∞ (S + κ)σ− (θσn − θσ+ )+ + (S + κ)σn (θσn − θσn )+ C(θ) = + n n−1 n=1 ∞ (S − κ)σ− (θσn − θσ+ )− + (S − κ)σn (θσn − θσn )− − + n n−1 n=1 Purchases minus sales, for left and right transactions. Terminal value V (θ) = −C(θ).
7. 7. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems What is an Admissible Strategy? Numeraire-free version. For some c > 0: V (θ) ≥ −c(1 + ST ) Too loose: Not the usual deﬁnition. Martingales vs. Local martingales. Leverage without collateral. c(ST − S0 ) admissible. Many banks still alive... Naïve deﬁnition. For some c > 0: V (θ1[0,t] ) ≥ −c for all t ∈ [0, T ] Too strict: Payoff space not closed. Forget separation arguments. No leverage with markets closed. All banks dead.
8. 8. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Freeze, Wait, Close You cannot trade your way out of losses. Anytime, the broker can freeze the account, and wait for a good time to close risky positions, for a bounded loss. A simple strategy θ is admissible if and only if, after every transaction, there exists a liquidation time. Continuous prices (or totally inaccessible jumps): for all t, there exists a stopping time t ≤ τ ≤ T such that V (θ1[0,t] + θt 1 t,τ ) + x ≥ 0 for some x > 0. Accessible jumps allowed: Both freeze and liquidation either left or right. Four cases.
9. 9. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Four Cases Right Freeze and Right Close. a q q a a Right Freeze and Left Close. a a q a q Left Freeze and Right Close. q q a a Left Freeze and Left Close. q a a q
10. 10. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Freeze and Close, Left or Right ˆ Discrete ﬁltration F = (F0 , Fσ− , Fσ1 , Fσ− , Fσ2 , . . . ) 1 2 ˆˆ (S, κ)n≥0 deﬁned analogously. ˆ (θt )0≤t≤T induces (θn )n≥0 deﬁned as ˆ ˆˆ θ = (0, θσ , θ + , θσ , θ + , . . . ). θ is F-adapted. σ1 σ2 1 2 Deﬁnition θ simple x-admissible if, for all k ≥ 0, there exists a liquidation strategy k θ, such that: ˆ ˆ i) k θ = θ·∧k 1{·<λk } for some F-stopping time λk > k a.s. (liquidation time). ii) x + V (k θ) ≥ 0. Reduces to frictionless deﬁnition for κ = 0.
11. 11. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems No Simple Arbitrage Deﬁnition Simple arbitrage: θ ∈ As such that P(V (θ) ≥ 0) = 1 and P(V (θ) > 0) > 0. (NA-S): θ ∈ As and P(V (θ) ≥ 0) = 1 implies that V (θ) = 0. Proposition If (NA-S) holds, then As = {θ ∈ As : x + V (θ) ≥ 0 a.s.}. x Admissibility of θ depends on ﬁnal payoff only. Key property to obtain closedness of admissible payoffs. ⊂ easy. ⊃ far less so.
12. 12. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems The Frictionless Story Frictionless markets: κ = 0. (1) (NFLVR) for Simple Strategies ⇓ S is a semimartingale ⇓ Payoffs of general strategies as stochastic integrals θdS (2) (NFLVR) for General Strategies ⇓ Equivalent Local Martingale Measure “The use of general integrands however seems more difﬁcult to interpret and their use can be questioned in economic models” (Delbaen and Schachemayer, 1994)
13. 13. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Payoffs as Integrals Frictionless payoffs: θdS stochastic integrals. Approximations. θ is x-admissible. (x + ε)-admissible θn with |θ − θn | < ε? No, in general. Model misspeciﬁcations. If S and S are close, are dS and dS close? θ θ No, again. Needs underlying probability. Why? Troubling properties. Only simple strategies concrete. No probability in accounting.
14. 14. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems (Robust) No Free Lunch with Vanishing Risk Deﬁnition (S, κ) satisﬁes i) (NFLVR) if, for any sequence (θn )n≥1 such that θn ∈ As 1/n and V (θn ) converges a.s. to some limit V , then V = 0 a.s. ii) (RNFLVR) if, there exists (S , κ ) satisfying (NFLVR), and the bid-ask spread of (S , κ ) is within that of (S, κ): inf (κt − κt − |St − St |) > 0 a.s. t∈[0,T ] (RNFLVR) ⇒ efﬁcient friction: inft∈[0,T ] κt > 0 a.s. Only simple strategies.
15. 15. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems General Admissible Strategies Deﬁnition (θn )n≥1 ⊂ As converges admissibly to (θt )t∈[0,T ] : θn ∈ As n x+1/n for some x > 0, and θ converge to θ a.s. Any such limit is an x-admissible strategy. Ax : x-admissible strategies. A := ∪x>0 Ax admissible strategies. Cost C(θ) of θ ∈ A (limits in a.s. sense): adm C(θ) = ess inf lim inf C(θn ) : θn −→ θ n→∞ x-admissible as limit of simple, almost x-admissible. Cost of θ as the lowest cost of its simple approximations.
16. 16. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Admissible implies Finite Variation Proposition If (RNFLVR) holds, any admissible strategy has ﬁnite variation. Finite variation derived, not assumed. Explicit expression for C(θ)? Interpretation? Properties?
17. 17. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Predictable Stieltjes Integrals Deﬁnition S càdlàg. θ predictable ﬁnite variation. Integral: Sdθ− − (θs − θs− )∆Ss IT (S, θ) = [0,T ] s≤T Stieltjes integral plus correction term. No probability. Look at Sdθ, not θdS! Why this deﬁnition?
18. 18. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Simple Strategies Proposition ∞ i) If θ = θ τn 1 + θ τn 1 predictable, then + τn τn ,τn+1 n=1 Sτ − (θτi − θτ − ) + Sτi (θτ + − θτi ) IT (S, θ) = i i i τi ≤T τi <T ∗ ii) IT is linear both in S and in θ, and |IT (S, θ)| ≤ θ T ST Consistent with simple strategies. Robust for misspeciﬁcations.
19. 19. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Convergence Theorem i) supn≥1 θn T < ∞. θn → θ pointwise ⇒ I(S, θn ) → I(S, θ) pointwise. ii) supn≥1 θn T < ∞ and S ≥ 0. θn → θ pointwise ⇒ lim infn I(S, θn ) ≥ I(S, θ ) pointwise. Lebesgue and Fatou properties... ...but for the integrator. Still no probability.
20. 20. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Approximations Theorem S càdlàg adapted locally bounded. θ predictable ﬁnite variation. For all ε > 0 there exists a simple strategy: ∞ θσ n 1 + θσ n 1 θ= + σn σn ,σn+1 n=0 satisfying θ ∈ PV , |θ − θ| ≤ ε, | Sdθ − Sdθ| ≤ ε and θ ≤ θ pointwise on [0, T ] (outside a P-zero set). If θ x-admissible, there exists (x + ε)-admissible θε . Simple approximations for any ﬁnite variation strategy. Approximation depends on probability.
21. 21. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Compatible with Stochastic Integral Proposition θ predictable ﬁnite variation. S càdlàg semimartingale. T T Sdθ = θT ST − θ0 S0 − θdS, 0 0 Left: predictable Stieltjes integral. Right: usual stochastic integral. Linked by integration by parts.
22. 22. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Representation for Cost adm Cost: C(θ) = ess inf lim infn→∞ C(θn ) : θn −→ θ Explicit formula with predictable Stieltjes integrals: C(θ) = Sdθ + κd θ [0,T ] [0,T ] Simple approximations with simple strategies. For all ε > 0 there exists θε simple such that: |θ − θε |, |C(θ) − C(θε )| < ε a.s. Crucial consequence: payoff space C = {V (θ) : θ ∈ A} − L0 Fatou closed. + Separation works. Kreps-Yan Theorem.
23. 23. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems A Path Downhill Understanding admissibility and value as main problems. Kreps-Yan theorem: separating measure. Sandwich martingale within bid and ask. Well-known path (Jouini-Kallal, Cherny, Choulli-Stricker) New admissibility: supermartingale property?
24. 24. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Consistent Price Systems Deﬁnition Strictly Consistent Price System (SCPS): pair (M, Q) of probability Q equivalent to P and Q-local martingale M within bid-ask spread: inf (κt − |St − Mt |) > 0 a.s. t∈[0,T ] Consistent Price System (CPS) if inequality not strict. Proposition EQ [V (M,0) (θ)] ≤ 0 for any CPS (M, Q) and θ ∈ A. Analogue of supermartingale property. (SCPS) ⇒ (RNFLVR) clear.
25. 25. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems From Separating Measure to CPS Lemma (Xt )t∈[0,T ] and (Yt )t∈[0,T ] be two càdlàg processes. The following conditions are equivalent: i) There exists a càdlàg martingale (Mt )t∈[0,T ] such that: X ≤M≤Y a.s. ii) For all stopping times σ, τ such that 0 ≤ σ ≤ τ ≤ T a.s.: E [ Xτ | Fσ ] ≤ Yσ E [ Yτ | Fσ ] ≥ Xσ and a.s. ii) ⇒ i) delivers CPS from separating measure.
26. 26. Outline Simple Strategies (RNFLVR) Predictable Stieltjes Integrals Consistent Price Systems Conclusion Bid and ask prices moving freely. Value? Admissibility? Arbitrage? Finite Variation? The Fundamental Theorem as a tool to understand. Left and Right Transactions. Admissibility: freeze, wait and close. Anytime. Robust no free lunches and ﬁnite variation. Thank You!