ECOLE POLYTECHNIQUE
CENTRE DE MATHÉMATIQUES APPLIQUÉES
UMR CNRS 7641
91128 PALAISEAU CEDEX (FRANCE). Tél: 01 69 33 41 50. Fax: 01 69 33 30 11
http://www.cmap.polytechnique.fr/
Model-free representation of
pricing rules as conditional
expectations
R.I. 600
July 2006
Model-free representation of pricing rules
as conditional expectations
∗
†
‡
Abstract
We introduce a distinction between model-based and model-free arbitrage
and formulate an operational denition for absence of model-free arbitrage in a
nancial market, in terms of a set of minimal requirements for the pricing rule
prevailing in the market. We show that any pricing rule verifying these properties can be represented as a conditional expectation operator with respect to
a probability measure under which prices of traded assets follow martingales.
Our result can be viewed as a model-free version of the fundamental theorem
of asset pricing, which does not require any notion of reference" probability
measure.
∗
†
Università degli Studi di Perugia (Italy). Email: [email protected]
Centre de Mathématiques Appliquées, Ecole Polytechnique (France).
[email protected]
Email:
We acknowledge nancial support from the European Network on Advanced Mathematical
Methods in Finance.
‡
Contents
1 Introduction
3
2 Denitions and notations
7
3 Pricing rules as conditional expectation operators
9
4 Discussion
12
1 Introduction
(Ω, (Ft )t≥0 , P)
P
(St )t≥0
R
(φt )t≥0
P
φdS
P admissible trading
P
strategies
Z t
t, P( φ dS ≥ −c) = 1
∃c ∈ R
φ
0
arbitrage opportunity
Z
P(
φ
T
φ dS ≥ 0) = 1
and
0
Z
P(
T
φ dS > 0) > 0,
0
P
Q
P
Vt (H)
H
Vt (H) = E Q [H|Ft ]
Q
fn =
fn → f ∗ P
RT
0
φn dS
fn−
P(f ∗ =0) = 1
0
Q
Vt (H)
P
H
Vt (H) = E Q [H|Ft ]
σ
P
model-based
P
P
P
P
(Qθ , θ ∈ E)
θ
(Qθ , θ ∈ E)
equivalent
•
•
absence of model-free arbitrage
P
Q
model-based
L∞ (Ω, P)
P
pricing rule
2 Denitions and notations
(Ω, (Ft )t∈[0,T ] )
F0
(Ft )t∈[0,T ]
L0
FT
L∞
R
Y
Y : Ω × [0, T ] → R ∪ {+∞, −∞}
t Yt
R ∪ {+∞, −∞}
Ft
Π : L0 → Y
H ∈ L0
H ∈ L0
Πt (H)
domain
Dom(Π)
Π
Dom(Π) , {G ∈ L0 | Π(G)
}
Denition 1. A pricing rule is a mapping
Π : L0 → Y
H 7→ (Πt (H))t∈[0,T ]
that satises the following properties:
A1 If G, H ∈ Dom(Π), then K = max(G, H) ∈ Dom(Π).
A2 Positivity. For any H ∈ L0 , if H ≥ 0, then Π(H) ≥ 0.
A3 Ft -linearity on Dom(Π): For any H1 , H2 ∈ Dom(Π) and any bounded Ft measurable variable λ, λH1 + H2 ∈ Dom(Π) and
Πt (λH1 + H2 ) = λΠt (H1 ) + Πt (H2 )
A4 Time consistency.
∀H ∈ L0 ,
Πs (Πt (H)) = Πs (H) 0 ≤ s ≤ t ≤ T
A5 Normalization. Π(1) = 1.
A6 Market consistency. If H is tradable at price (Vt )t∈[0,T ] in the market (whence
in particular H = VT ), then H ∈ Dom(Π) and
∀t ∈ [0, T ], ∀ω ∈ Ω,
Πt (H)(ω) = V (t, ω).
A7 Continuity. If (Hn )n≥1 is an increasing sequence in L0 , uniformly bounded
from below, with Hn ↑ H , then Π0 (Hn ) ↑ Π0 (H).
Π(H)
t
t
R∪{+∞, −∞}
Πt (H)
H
Π(H) = −∞
H
G
max(H, G)
S
S
Π
Ft
Dom(Π)
t
t
Dom(Π)
Ft
1
L∞
Π
⊂ Dom(Π)
(Hn )n≥1
(Π0 (H −Hn ))n≥0
H
Π0 (.)
Remark 1 (Vector lattice property).
Dom(Π)
∞
L
3 Pricing rules as conditional expectation operators
Q
Q
Proposition 1. Let Q be a probability measure dened on (Ω, (Ft )t∈[0,T ] ) such that
the prices Vt (H) of all traded assets are martingales with respect to Q. There exists
a pricing rule Λ such that
1. Dom(Λ) is the vector space L1 (Q) of Q-integrable payos ;
2. For any H ∈ Dom(Λ),
Λt (H) = E Q [H|Ft ]
1 One
Q − a.s.
could rewrite the whole formalism with the apparently (but not really) more general
condition 0 < Π(1) ≤ 1.
Proof.
H
Q
Λ(H)
Q
H
L0
Λ
G EQ [ G | Ft ]
H
Ft
{+∞}
t ≤ T, k ∈ N
R∪
(EQ [ |H| | Ft ])t
(αt )t
Ak,t = {k ≤ αt < k + 1}
t
Ak,t
fk,t
Λt (H) = fk,t
EQ [HIAk,t | Ft ]
Ak,t
Ω − ∪k Ak,t ,
Λt (H) = +∞
Dom(Λ) = L1 (Q)
Y
Λ(H)
Λt (H)
Q
Λ
H
E Q [H|Ft ]
Λt (H)
Vt (H)
Q
Theorem 1. Given a pricing rule Π, there exists a probability measure Q dened
on (Ω, FT ) such that Π coincides with the conditional expectation with respect to Q.
More precisely:
1. Dom(Π) is the vector space L1 (Q) of Q-integrable payos ;
2. For any H ∈ Dom(Π),
Πt (H) = E Q [H|Ft ]
Q − a.s.
3. Prices of traded assets are Q-martingales.
Proof.
FT
Q
∀A ∈ FT ,
Q(A) = Π0 (1A )
Q
Π
Q
Π
Π0
Q
∞
H ∈L
H=
n
X
ci 1Ai ,
Ai ∈ FT ,
ci ∈ R.
H
Π0 (H) = E Q [H]
i=1
Π
H ∈ L0 , H ≥ 0
(Hn )n≥1
Hn ↑ H
Q
E Q [Hn ] = Π0 (Hn )
Π
Π0 (H) = E Q [H]
H
L1 (Q)
Q
Π0 (H ) = E [H + ]
+
Q
H ∈ L1 (Q)
A ∈ Ft
A ∈ Ft
H +, H −
Π
Π0 (H) = E Q [H]
H
−
Q
−
Π0 (H ) = E [H ]
Dom(Π) ⊆ L1 (Q)
Πt
t>0
t ∈ [0, T ]
Ft
Π0 (1A H)
Q
Π0 (H)
Π0 (1A Πt (H))
E Q [1A H] = E Q [1A Πt (H)]
Πt (H)
Ft
Π
Q
H
Q
Dom(Π)
V
1
L (Q)
H
V
H
Vt = Πt (H) = E Q [H | Ft ]
Remark 2 (Continuity of Π).
Π
Q
ψ : L∞ → R
ψ(H) = Π0 (H)
L∞ ⊆ Dom(Π)
L∞
Q
ψ
(Ω, FT )
Q
ψ
4 Discussion
L0
Q
Q
P
equivalent
Q
P
Q
P
(Qθ , θ ∈ E)
θ
(Qθ , θ ∈ E)
Qσ
σ
σ1 6= σ2
Qσ1
Qσ2
P
L∞ (Ω, P
P
all
L0
L∞ (P)
Dom(Π)
1
L (Q)
d
σ−
Π
S1 , · · · , Sd
underlyings
S = (S 1 , · · · , S d )
Rd
S
P
S
Q
S
Denition 2. Given a pricing rule Π on the market, represented by a martingale
measure Q and an Rd -valued Q-semimartingale S , a payo H ∈ L0 is said to be
S -replicable if there exist a x ∈ R and a predictable process (
) ϕ such that:
1. ϕ is S -integrable under Q.
2. Q( Πt (H) = x +
Rt
0
ϕdS ) = 1.
Remark 3. In the above denition and in what follows probabilistic notions are
induced by the pricing rule through its representing Q.
P
S
P
σ
σ
σ
φ ∈
L(S)(Q)
Q
φ
S
Proposition 2. [11, Proposition 2] Let S be a d-dimensional semimartingale on
(Ω, F, (Ft )t∈[0,T ] , Q). The following assertions are equivalent:
1. there exist a d-dimensional Q- martingale N and a positive (scalar) process
R
ψ ∈ ∩1≤i≤d L(N i )(Q) such that S i = ψ dN i ;
2. there exists a countable predictable partition (Bn )n of Ω×R+ such that IBn dS i
is a Q-martingale for every i, n;
R
3. there exist (scalar) processes ηi with paths that Q − a.s. never touch zero, such
R
that η i ∈ L(S i )(Q) and η i dS i is a Q-local martingale.
Denition 3. We say that S is a σ-martingale under Q if it satises any of the
equivalent conditions of the above Proposition.
Remark 4.
Tn
(Bn )n
]Tn , Tn+1 ]
+∞
i
Si
not necessarily a martingale
Si
σ
S
the market spanned by S .
H
S
Q
S
Proposition 3. Suppose that for all i there exists an S i -replicable derivative H i
traded in the market with a strategy (ϕit )t∈[0,T ] that Q-a.s. never touches zero. Then
S is a σ -martingale under Q.
Proof.
Hi
R i i
ϕ dS
σ
V i = Π(H i )
ϕi
Q
S
Q
Remark 5 (The 'No Free Lunch with Vanishing Risk' property).
σ−
S
Q
P∼Q
Q
Q
ϕ
Q
S
Z
∃c > 0, Q( ϕdS ≥ −c) = 1
S
Q
σ
Q
RT
EQ [ 0 ϕdS] ≤ 0
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