Fundamental Theorem of Asset Pricing

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


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?


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.


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.


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.


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).


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.


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.


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


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.


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.


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)


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.


(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.


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.


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?


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?


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.


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.


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.


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.


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.


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?


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.


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.


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!

Fundamental Theorem of Asset Pricing

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