Equilibrium Pricing in Incomplete Markets
under Translation Invariant Preferences
Patrick Cheridito
Princeton University
joint work with
Ulrich Horst, Michael Kupper, Traian Pirvu
September 2012
Motivation
Price processes are martingales under Q
Motivation
Price processes are martingales under Q
Harrison–Kreps (1979). Martingales and arbitrage in
multiperiod security markets.
Harrison–Pliska. (1981). Martingales and stochastic
integrals in the theory of continuous trading.
Dalang–Morton–Willinger (1989). Equivalent martingale
measures and no-arbitrage in stochastic securities market
models.
Delbaen (1992). Representing martingale measures when
asset prices are continuous and bounded.
Schachermayer (1993). Martingale measures for discrete
time processes with infinite horizon.
Delbaen–Schachermayer (1994). A general version of the
fundamental theorem of asset pricing.
Delbaen–Schachermayer (1998). The fundamental theorem
of asset pricing for unbounded stochastic processes.
Motivation
Usual approach to derivatives pricing:
1
Model the underlying securities as a J-dimensional
stochastic process (Rt ) on a probability space (Ω, F, P)
2
Price derivatives by EQ [ . ] for some equivalent martingale
measure Q ∼ P
Motivation
Usual approach to derivatives pricing:
1
Model the underlying securities as a J-dimensional
stochastic process (Rt ) on a probability space (Ω, F, P)
2
Price derivatives by EQ [ . ] for some equivalent martingale
measure Q ∼ P
In complete markets: Q is unique
binomial tree models, Black–Scholes model ...
In incomplete markets: Q is not unique
trinomial tree models, GARCH-type models,
stochastic volatility models, jump-diffusion models,
Levy-process models, more general semimartingale models
...
Motivation
Problem:
Choose a pricing measure Q̂ among all equivalent martingale
measures
Some commonly used methods:
1
Parameterize Q̂θ , θ ∈ Θ and calibrate to market data of
Q̂θ
traded derivatives ... via ddP
or without P.
E.g. build a stoch vol model or local vol model directly
under Q̂
2
Choose Q̂ so that it minimizes some distance to P,
e.g. Lp -distance, relative entropy, f -divergence ...
3
Indifference pricing
4
...
Motivation
Our goal:
derive Q̂ from equilibrium considerations
Some motivating Examples
Horst and Müller (2007).
On the spanning property of risk bonds priced by equilibrium
Bakshi, Kapadia and Madan (2003).
Stock return characteristics, skew laws, and the differential
pricing of individual equity options
Garleanu, Pedersen and Poteshman (2009).
Demand-based option pricing
Carmona, Fehr, Hinz and Porchet (2010).
Market design for emission trading schemes.
Outline
1
Model
2
Existence of equilibrium
3
Uniqueness of equilibrium
4
Random walks and BS∆Es
5
Brownian motion and BSDEs
6
Option pricing under demand pressure
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
exogenous asset (Rt )Tt=0 satisfying (NA)
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
exogenous asset (Rt )Tt=0 satisfying (NA)
structured product in external supply n with final payoff
S ∈ L∞ (FT )
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
exogenous asset (Rt )Tt=0 satisfying (NA)
structured product in external supply n with final payoff
S ∈ L∞ (FT )
a group of finitely many agents A
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
exogenous asset (Rt )Tt=0 satisfying (NA)
structured product in external supply n with final payoff
S ∈ L∞ (FT )
a group of finitely many agents A
agent a ∈ A is endowed with an uncertain payoff
H a = g a,R RT + g a,S ST + Ga
Model
Ingredients
filtered probability space (Ω, F, (Ft )Tt=0 , P)
money market account with r ≡ 0
exogenous asset (Rt )Tt=0 satisfying (NA)
structured product in external supply n with final payoff
S ∈ L∞ (FT )
a group of finitely many agents A
agent a ∈ A is endowed with an uncertain payoff
H a = g a,R RT + g a,S ST + Ga
at time t agent a invests to optimize a preference functional
Uta : L∞ (FT ) → L∞ (Ft )
Model
We assume Uta has the following properties:
(N) Normalization
Uta (0) = 0
Model
We assume Uta has the following properties:
(N) Normalization
Uta (0) = 0
(M) Monotonicity
Uta (X) ≥ Uta (Y ) for all X, Y ∈ L∞ (FT ) such that X ≥ Y
Model
We assume Uta has the following properties:
(N) Normalization
Uta (0) = 0
(M) Monotonicity
Uta (X) ≥ Uta (Y ) for all X, Y ∈ L∞ (FT ) such that X ≥ Y
(C) Ft -Concavity
Uta (λX + (1 − λ)Y ) ≥ λUta (X) + (1 − λ)Uta (Y ) for all
X, Y ∈ L∞ (FT ) and λ ∈ L∞ (Ft ) such that 0 ≤ λ ≤ 1
Model
We assume Uta has the following properties:
(N) Normalization
Uta (0) = 0
(M) Monotonicity
Uta (X) ≥ Uta (Y ) for all X, Y ∈ L∞ (FT ) such that X ≥ Y
(C) Ft -Concavity
Uta (λX + (1 − λ)Y ) ≥ λUta (X) + (1 − λ)Uta (Y ) for all
X, Y ∈ L∞ (FT ) and λ ∈ L∞ (Ft ) such that 0 ≤ λ ≤ 1
(T) Translation property
Uta (X + Y ) = Uta (X) + Y for all X ∈ L∞ (FT ) and
Y ∈ L∞ (Ft )
Model
We assume Uta has the following properties:
(N) Normalization
Uta (0) = 0
(M) Monotonicity
Uta (X) ≥ Uta (Y ) for all X, Y ∈ L∞ (FT ) such that X ≥ Y
(C) Ft -Concavity
Uta (λX + (1 − λ)Y ) ≥ λUta (X) + (1 − λ)Uta (Y ) for all
X, Y ∈ L∞ (FT ) and λ ∈ L∞ (Ft ) such that 0 ≤ λ ≤ 1
(T) Translation property
Uta (X + Y ) = Uta (X) + Y for all X ∈ L∞ (FT ) and
Y ∈ L∞ (Ft )
(TC) Time-consistency
a (X) ≥ U a (Y )
Ut+1
t+1
⇔
implies
Uta (X) ≥ Uta (Y )
a (X))
Uta (X) = Uta (Ut+1
Model
Related to coherent and convex risk measures
Artzner–Delbaen–Eber–Heath (1999).
Coherent measures of risk.
Föllmer–Schied (2002).
Convex measures of risk and trading constraints
Frittelli–Rosazza Gianin (2002).
Putting order in risk measures.
Model
Examples
1) Uta (X) = − γ1 log E e−γX | Ft
2) Uta (X) = E [X | Ft ] − λE (X − E [X | Ft ])2 | Ft
3) Uta (X) = (1 − λ)E [X | Ft ] − λρt (X)
where ρt is a conditional convex risk measure
Model
An equilibrium of plans, prices and price expectations à la
Radner (1972) consists of
an adapted process (St )Tt=0 with ST = S
trading strategies (ϑ̂at )Tt=1
such that the following hold:
(i) individual optimality
Uta
Ha +
T
X
!
a,2
ϑ̂a,1
s ∆Rs + ϑ̂s ∆Ss
s=t+1
≥
Uta
a
H +
T
X
!
ϑa,1
s ∆Rs
+
ϑa,2
s ∆Ss
s=t+1
for every t and all possible strategies (ϑas )
P
a,2
(ii) market clearing
a∈A ϑ̂t = n
Model
Hart (1975) On the optimality of equilibrium when the market
structure is incomplete:
In general, a Radner equilibrium does not exist, and if there is
one, it is not unique.
Existence
One-step representative agents
Set HTa = H aand
a
a
Ht+1
= Ut+1
Ha +
a,1
a,2
s=t+2 ϑ̂s ∆Rs + ϑ̂s ∆Ss
PT
the true representative agent would be
X
a
ût (x) =
ess sup
Uta Ht+1
+ ϑa,1 ∆Rt+1 + ϑa,2 ∆St+1
ϑa ∈ L∞ (Ft )2 a∈A
P
a,2 = x
a∈A ϑ
Existence
One-step representative agents
Set HTa = H aand
a
a
Ht+1
= Ut+1
Ha +
a,1
a,2
s=t+2 ϑ̂s ∆Rs + ϑ̂s ∆Ss
PT
the true representative agent would be
X
a
ût (x) =
ess sup
Uta Ht+1
+ ϑa,1 ∆Rt+1 + ϑa,2 ∆St+1
ϑa ∈ L∞ (Ft )2 a∈A
P
a,2 = x
a∈A ϑ
But St is not known. So define
X
a
ût (x) =
ess sup
Uta Ht+1
+ ϑa,1 ∆Rt+1 + ϑa,2 St+1
a ∈ L∞ (F )2
a
ϑP
t
a,2
=x
a∈ ϑ
ût is Ft -concave
Existence
Convex dual characterization of equilibrium
Theorem A bounded, adapted process (St )Tt=0 satisfying
ST = S together with trading strategies (ϑ̂at )Tt=1 , a ∈ A, form an
equilibrium ⇐⇒ for all t:
(i) St ∈ ∂ ût (n)
a
a
a∈A Ut (Ht+1
(ii)
P
(iii)
P
a,2
a∈A ϑ̂t+1
a,2
+ ϑ̂a,1
t+1 ∆Rt+1 + ϑ̂t+1 St+1 ) = ût (n)
=n
Existence
Assumption (A)
For all t = 0, . . . , T − 1, V a ∈ L∞ (Ft+1 ), W ∈ L∞ (Ft+1 ),
there exist ϑ̂at+1 ∈ L∞ (Ft )2 , a ∈ A, such that
X
ϑ̂a,2
t+1 = 0
a∈A
and
X
a,2
Uta V a + ϑ̂a,1
∆R
+
ϑ̂
W
t+1
t+1
t+1
a∈A
=
ess sup
ϑat+1 ∈ L∞ (Ft )2
P
a,2
a∈A ϑt+1 = 0
X
a,2
Uta V a + ϑa,1
t+1 ∆Rt+1 + ϑt+1 W .
a∈A
Lemma Under assumption (A) an equilibrium exists
Existence
Definition
U0a is sensitive to large losses if
lim U0a (λX) = −∞
λ→∞
for all X ∈ L∞ (FT ) such that P[X < 0] > 0.
Theorem
If all U0a are sensitive to large losses,
then condition (A) is satisfied and an equilibrium exists.
Remark
The theorem also works with convex trading constraints.
Existence
Proposition
If the market is in equilibrium and at least one agent has
strictly monotone preferences and open trading constraints,
then there exists a probability measure Q ∼ P such that
Rt = EQ [RT | Ft ] and St = EQ [RT | Ft ].
Uniqueness
Differentiable preferences
We say Uta satisfies the differentiability condition (D) if for all
X, Y ∈ L∞ (Ft+1 ), there exists Z ∈ L1 (Ft+1 ) such that
Y
a
a
− Ut (X) = E [Y Z | Ft ].
lim k Ut X +
k→∞
k
If such a Z exists, it has to be unique, and we denote it by
∇Uta (X).
Uniqueness
Differentiable preferences
We say Uta satisfies the differentiability condition (D) if for all
X, Y ∈ L∞ (Ft+1 ), there exists Z ∈ L1 (Ft+1 ) such that
Y
a
a
− Ut (X) = E [Y Z | Ft ].
lim k Ut X +
k→∞
k
If such a Z exists, it has to be unique, and we denote it by
∇Uta (X).
Theorem If at least one Uta satisfies (D), then there can exist
at most one equilibrium price process (St )Tt=0 , and if the market
is in equilibrium, then
!
T
X
dQat
a
a
a,1
a,2
:= ∇Ut H +
ϑ̂s ∆Rs + ϑ̂s ∆Ss
dP
s=1
defines a pricing measure.
Random walks and BS∆Es
Fix h > 0 and N ∈ N
Denote T = {0, h, . . . , T = N h}
b1t , . . . , bdt d independent
random walks with
√
P [∆bit+h = ± h] = 1/2
bd+1
, . . . , bD
2d − (d + 1) random walks orthogonal to b1t , . . . , bdt
t
t
Every X ∈ L∞ (Ft+h ) can be represented as
X = E [X | Ft ] + πt (X) · ∆bt+h
for
πt (X) · ∆bt+h =
D
X
i=1
πti (X)∆bit+h
and πti (X) =
1 E X∆bit+h | Ft .
h
Random walks and BS∆Es
Uta (X) = Uta (E [X|Ft ] + πt (X) · ∆bt+h ) = E [X | Ft ]−fta (πt (X))h
for the Ft -convex function fta : L∞ (Ft )D → L∞ (Ft ) given by
1
fta (z) := − Uta (z · ∆bt+h ) .
h
Assume condition (A) is satisfied and all Uta satisfy the
differentiability condition (D).
Then there exists ∇fta (z) ∈ L∞ (Ft )D such that
lim k fta (z + z 0 /k) − fta (z) = z 0 · ∇fta (z)
k→∞
Random walks and BS∆Es
a
For given Rt+h , St+h , Ht+h
denote
R
Zt+h
:= πt (Rt+h )
S
Zt+h
:= πt (St+h )
a
a
Zt+h
:= πt (Ht+h
)
Zt+h
=
R
S
a
(Zt+h
, Zt+h
, Zt+h
, a ∈ A) .
and define the function ft : L∞ (Ft )(3+|A|)D → L∞ (Ft ) by
ft (v, Zt+h )
=
ess inf
a ∈ L(F )2
ϑ
t
P
a,2 = 0
ϑ
a∈A
X
a∈A
fta
v
R
S
a
Zt+h
+ ϑa,2
+ Zt+h
+ ϑa,1
t+h
t+h Zt+h
|A|
−ϑa,1
t+h
E [∆Rt+h | Ft ]
.
h
Random walks and BS∆Es
Set
S
S
gtS (Zt+h ) := Zt+h
· ∇v ft (nZt+h
, Zt+h )
a,1
a,2
a
a
a
R
S
gt (Zt+h ) := ft Zt+h + ϑ̂t+h Zt+h + ϑ̂t+h Zt+h
1
a,2 S
−ϑ̂a,1
t+h h E [∆Rt+h | Ft ] − ϑ̂t+h gt (Zt+h ).
Random walks and BS∆Es
The processes (St ) and (Hta ) satisfy the following
coupled system of BS∆Es
S
∆St+h = gtS (Zt+h )h + Zt+h
· ∆bt+h ,
a
∆Ht+h
=
gta (Zt+h ) h
+
a
Zt+h
· ∆bt+h ,
ST = S
HTa = H.
Random walks and BS∆Es
Example
Assume that the price of the exogenous asset is given by
∆Rt+h = Rt (µh + σ∆b1t+h ) ,
R0 > 0
and agent a’s preference functional is
Uta (X) = −
1
log E [exp(−γ a X) | Ft ]
γa
for some
Then
Uta (X) = E [X | Ft ] − fta (πt (X))h
for
fta (z) =
1
log E [exp(−γ a z · ∆bt+h )] .
hγ a
γa > 0 .
Random walks and BS∆Es
Neglect the random walks bd+1 , . . . , bD
and use the approximation
d
d
√
γa X
1 X
a i
log
cosh
hγ
z
≈
(z i )2
hγ a
2
i=1
i=1
Then the BS∆E of the last theorem yields ...
Random walks and BS∆Es
... the recursive algorithm
St
Hta
= E [St+1 | Ft ] − gtS h ,
a
= E Ht+1
| Ft − gta h ,
ST = S
HTa = H a ,
where
gtS
=
gta
=
ϑ̂a,1
t+h
=
1 RS
c µSt + γ n cRR cSS − cRS cRS + cRA cRR − cSR cRA
2
γ a
a,1
a,1
a,2 S
R
S + ϑ̂a,2
Z
Zt+h + ϑ̂t+h Zt+h
t+h t+h − ϑ̂t+h µRt − ϑ̂t+h gt
2
2
cSR cSa − cRa cSS
cRS γ
cRR cSA − cRS cAR
µSt
+ RR SS
− RR a n + RR SS
γ a cRR
c c − cRS cRS
c
γ
c c − cRS cRS
cRR
a cRS cRa − cSa cRR − γγa cRS cRA − cRR cSA
γ
= n a+
γ
cRR cSS − cRS cRS
P
for γ := ( a (γ a )−1 )−1 ,
P a
S
S
S
R
S
cRR := Zt+h
·Zt+h
, cSR := Zt+h
·Zt+h
, cSA := Zt+h
· a Zt+h
,
ϑ̂a,2
t+h
...
Brownian motion and BSDEs
Example
Let BtR , BtS , Bta , a ∈ A, be independent Brownian motions
dRt = µRt dt + σRt dBtR ,
R0 > 0
and suppose agent a’s preference functional is
Uta (X) = −
1
log E [exp(−γ a X) | Ft ]
γa
for some
γa > 0 .
Brownian motion and BSDEs
The BSDE corresponding to the above BSDE is
dSt = gtS dt + ZtS · dBt ,
dHta
=
gta dt
+
Zta
· dBt ,
ST = S
HTa = H a ,
where
gtS
=
gta
=
ϑ̂a,1
t
=
ϑ̂a,2
t
=
1 RS
c µSt + γ n cRR cSS − cRS cRS + cRA cRR − cSR cRA
2
γ a
a,1
a,2
a,1
a,2
Zt + ϑ̂t ZtR + ϑ̂t ZtS − ϑ̂t µRt − ϑ̂t gtS
2
2
cSR cSa − cRa cSS
cRS γ
cRR cSA − cRS cAR
µSt
+
−
n
+
γ a cRR
cRR cSS − cRS cRS
cRR γ a
cRR cSS − cRS cRS
cRR
a cRS cRa − cSa cRR − γγa cRS cRA − cRR cSA
γ
n a+
γ
cRR cSS − cRS cRS
for
cRR := ZtR · ZtR ,
cRS := ZtR · ZtS ,
cRA := ZtR ·
X
a
Zta ,
...
Equilibrium prices of out-of-the-money put options
Zero endowments and stochastic volatility
(I) No demand pressure
frequent hedging
infrequent hedging
Equilibrium prices of out-of-the-money put options
Zero endowments and stochastic volatility
(II) Positive demand pressure
frequent hedging
infrequent hedging
Equilibrium prices of out-of-the-money put options
Zero endowments and stochastic volatility
(III) Positive demand pressure and short selling
constraints