Model
Uncertainty
The Geometry of Model Uncertainty
Mathias Beiglböck
M. Beiglböck
geometry of model uncertainty
1 / 10
Model
Uncertainty
The Geometry of Model Uncertainty
Mathias Beiglböck
1) background – model uncertainty and optimal transport
M. Beiglböck
geometry of model uncertainty
1 / 10
Model
Uncertainty
The Geometry of Model Uncertainty
Mathias Beiglböck
1) background – model uncertainty and optimal transport
2) particular aspect: Skorokhod embedding
M. Beiglböck
geometry of model uncertainty
1 / 10
Model
Uncertainty
Model Uncertainty – Overview
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
M. Beiglböck
. . . used since Black-Scholes ’73
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
. . . used since Black-Scholes ’73
model uncertainty:
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
M. Beiglböck
. . . used since Black-Scholes ’73
influence of P ?
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
M. Beiglböck
. . . used since Black-Scholes ’73
influence of P ?
geometry of model uncertainty
interval of possible prices?
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal:
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal: determine lower / upper extreme value for price
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal: determine lower / upper extreme value for price
p = inf EP [Φ(S)]
P
M. Beiglböck
p = sup EP [Φ(S)]
P
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal: determine lower / upper extreme value for price
p = inf EP [Φ(S)]
P
p = sup EP [Φ(S)]
P
market data: −→ restriction on admissible P
M. Beiglböck
geometry of model uncertainty
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal: determine lower / upper extreme value for price
p = inf EP [Φ(S)]
P
p = sup EP [Φ(S)]
P
market data: −→ restriction on admissible P e.g.
M. Beiglböck
geometry of model uncertainty
(Breeden-Litzenberger ’78)
2 / 10
Model
Uncertainty
Model Uncertainty – Overview
basic object: stock price (random process)
S = (St )t∈[0,T ]
model: probability P s.t. S is a P-martingale
derivative: payoff Φ = Φ((St )t∈[0,T ] )
price pΦ = EP [Φ(S)]
model uncertainty:
. . . used since Black-Scholes ’73
influence of P ?
interval of possible prices?
basic goal: determine lower / upper extreme value for price
p = inf EP [Φ(S)]
P
p = sup EP [Φ(S)]
P
market data: −→ restriction on admissible P e.g.
(Breeden-Litzenberger ’78)
all call prices known ⇐⇒ St ∼P µt
M. Beiglböck
geometry of model uncertainty
2 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
Φ = Realized Variance, S = Société Générale , T = 1.5 (years)
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
Φ = Realized Variance, S = Société Générale , T = 1.5 (years)
market data: calls (ST − K )+
lower price upper price
K = 0.2, 0.4, 0.6, 0.8, 0.9, 1, 1.1, 1.3, 1.55, 1.7, 3
54.7
58.5
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
Φ = Realized Variance, S = Société Générale , T = 1.5 (years)
market data: calls (ST − K )+
lower price upper price
K = 0.2, 0.4, 0.6, 0.8, 0.9, 1, 1.1, 1.3, 1.55, 1.7, 3
54.7
58.5
Example 2:
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
Φ = Realized Variance, S = Société Générale , T = 1.5 (years)
market data: calls (ST − K )+
lower price upper price
K = 0.2, 0.4, 0.6, 0.8, 0.9, 1, 1.1, 1.3, 1.55, 1.7, 3
54.7
58.5
Example 2:
Φ = Forward Start = (ST2 /ST1 − K )+ , S = DAX T1 = 1, T2 = 1.5
M. Beiglböck
geometry of model uncertainty
3 / 10
Good Cases
Bad Cases
Model Uncertainty – Examples
Example 1:
Φ = Realized Variance, S = Société Générale , T = 1.5 (years)
market data: calls (ST − K )+
lower price upper price
K = 0.2, 0.4, 0.6, 0.8, 0.9, 1, 1.1, 1.3, 1.55, 1.7, 3
54.7
58.5
Example 2:
Φ = Forward Start = (ST2 /ST1 − K )+ , S = DAX T1 = 1, T2 = 1.5
/9
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lower/upper prices versus (local) Bergomi and LV models
M. Beiglböck
geometry of model uncertainty
3 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
M. Beiglböck
geometry of model uncertainty
4 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
◦ Avellaneda-Levy-Paràs ’95
◦ Lyons ’95
◦ Hobson ’98
◦ Albrecher-Mayer-Schoutens ’98
◦ Brown-Hobson-Rogers ’01
◦ Madan-Yor ’02
◦ Dupire ’05
◦ Davis-Hobson ’07
◦ Hobson-Pederson ’08
M. Beiglböck
geometry of model uncertainty
4 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
◦ Avellaneda-Levy-Paràs ’95
◦ Cox-Obloj-Hobson ’09
◦ Lyons ’95
◦ Carr-Lee ’10
◦ Hobson ’98
◦ Cox-Obloj ’11,’12
◦ Albrecher-Mayer-Schoutens ’98
◦ Hobson-Neuberger ’12
◦ Brown-Hobson-Rogers ’01
◦ Cox-Wang ’13
◦ Madan-Yor ’02
◦ Hobson-Klimmek ’13
◦ Dupire ’05
◦ Davis-Obloj-Raval ’13
◦ Davis-Hobson ’07
◦ Hobson-Klimmek ’14
◦ Hobson-Pederson ’08
◦ ...
M. Beiglböck
geometry of model uncertainty
4 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
◦ Avellaneda-Levy-Paràs ’95
◦ Cox-Obloj-Hobson ’09
◦ Lyons ’95
◦ Carr-Lee ’10
◦ Hobson ’98
◦ Cox-Obloj ’11,’12
◦ Albrecher-Mayer-Schoutens ’98
◦ Hobson-Neuberger ’12
◦ Brown-Hobson-Rogers ’01
◦ Cox-Wang ’13
◦ Madan-Yor ’02
◦ Hobson-Klimmek ’13
◦ Dupire ’05
◦ Davis-Obloj-Raval ’13
◦ Davis-Hobson ’07
◦ Hobson-Klimmek ’14
◦ Hobson-Pederson ’08
◦ ...
important intuition:
M. Beiglböck
geometry of model uncertainty
4 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
◦ Avellaneda-Levy-Paràs ’95
◦ Cox-Obloj-Hobson ’09
◦ Lyons ’95
◦ Carr-Lee ’10
◦ Hobson ’98
◦ Cox-Obloj ’11,’12
◦ Albrecher-Mayer-Schoutens ’98
◦ Hobson-Neuberger ’12
◦ Brown-Hobson-Rogers ’01
◦ Cox-Wang ’13
◦ Madan-Yor ’02
◦ Hobson-Klimmek ’13
◦ Dupire ’05
◦ Davis-Obloj-Raval ’13
◦ Davis-Hobson ’07
◦ Hobson-Klimmek ’14
◦ Hobson-Pederson ’08
◦ ...
important intuition:
supP EP [Φ(S)] = inf{robust bounds}
M. Beiglböck
geometry of model uncertainty
4 / 10
Skorokhod
Optimal Transport
Model Uncertainty – Literature
◦ Avellaneda-Levy-Paràs ’95
◦ Cox-Obloj-Hobson ’09
◦ Lyons ’95
◦ Carr-Lee ’10
◦ Hobson ’98
◦ Cox-Obloj ’11,’12
◦ Albrecher-Mayer-Schoutens ’98
◦ Hobson-Neuberger ’12
◦ Brown-Hobson-Rogers ’01
◦ Cox-Wang ’13
◦ Madan-Yor ’02
◦ Hobson-Klimmek ’13
◦ Dupire ’05
◦ Davis-Obloj-Raval ’13
◦ Davis-Hobson ’07
◦ Hobson-Klimmek ’14
◦ Hobson-Pederson ’08
◦ ...
important intuition:
supP EP [Φ(S)] = inf{robust bounds}
B., Henry-Labordere, Penkner / Galichon, Touzi (’13)
M. Beiglböck
geometry of model uncertainty
−→ transport approach
4 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
k!
.
t.tt
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
Hill
Hi
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.
it's
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
Hill
Hi
"
.
it's
principal idea: model P transports distribution through time
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.fi
Hi
.
it's
principal idea: model P transports distribution through time
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
t.ee#f*
Hi
.
it's
principal idea: model P transports distribution through time
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
M. Beiglböck
B., Henry-Labordere, Penkner ’13
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
M. Beiglböck
B., Julliet ’14
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
M. Beiglböck
B., Julliet ’14, B., Cox, Huesmann ’15
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
B., Julliet ’14, B., Cox, Huesmann ’15
geometric description of extreme models
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
B., Julliet ’14, B., Cox, Huesmann ’15
geometric description of extreme models
cont. time
M. Beiglböck
geometry of model uncertainty
5 / 10
transport
uncertainty
Transport Approach to Model Uncertainty
call prices known
m
St ∼P µt
(Breeden-Litzenberger ’78)
ftp.t.tt#i*Hi
it
principal idea: model P transports distribution through time
(I) DUALITY
B., Henry-Labordere, Penkner ’13
Thm: Let St ∼ µt , t = 1, . . . , T , Φ u.s.c.
=⇒
supP EP [Φ(S)] = inf{robust bounds}
(II) MONOTONICITY
B., Julliet ’14, B., Cox, Huesmann ’15
geometric description of extreme models
cont. time – geometry of Skorokhod embedding
M. Beiglböck
geometry of model uncertainty
5 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
given:
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
given: µ,
R
x 2 dµ < ∞,
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task:
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
◦ Skorokhod ’61
◦ Root ’69
◦ Rost ’71
◦ Azema-Yor ’79
◦ Bass ’83
◦ Vallois I ’83
◦ Perkins ’85
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
◦ Skorokhod ’61
◦ Jacka ’88
◦ Root ’69
◦ Valois II ’93
◦ Rost ’71
◦ Hobson ’98
◦ Azema-Yor ’79
◦ Madan-Yor ’02
◦ Bass ’83
◦ Cox-Hobson ’07
◦ Vallois I ’83
◦ Eldan ’15
◦ Perkins ’85
◦ ...
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
◦ Skorokhod ’61
◦ Jacka ’88
◦ Root ’69
◦ Valois II ’93
◦ Rost ’71
◦ Hobson ’98
◦ Azema-Yor ’79
◦ Madan-Yor ’02
◦ Obloj ’04
◦ Bass ’83
◦ Cox-Hobson ’07
◦ Hobson ’11
◦ Vallois I ’83
◦ Eldan ’15
◦ Perkins ’85
◦ ...
M. Beiglböck
geometry of model uncertainty
recent
surveys:
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root:
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R
'
HE
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R
"
#¥
M. Beiglböck
.
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R
"
¥¥E
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
¥¥E
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
¥¥E
Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
Rost
¥¥E
Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
Rost
KEI
f÷
"
.
Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
...
¥a
¥¥
"
.
-
Eτ 2
Rost
x
→ min
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
Rost
...
i¥€
¥#
"
.
.
Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
Rost
...
i¥€
¥#
"
.
.
Eτ 2 → min
M. Beiglböck
Eτ 2 → max
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
"
Rost
Azema-Yor
...
i¥€
¥#
"
.
.
Eτ 2 → min
M. Beiglböck
Eτ 2 → max
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
,
.
Eτ 2 → min
M. Beiglböck
Rost
Azema-Yor
... f€±
.tk#**a
¥#
if", " "
Eτ 2 → max
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
,
.
Eτ 2 → min
M. Beiglböck
Rost
Azema-Yor
... f€±
.tk#**a
¥#
if", " "
Eτ 2 → max
geometry of model uncertainty
EB̄τ → max
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
Rost
Azema-Yor
...
'ak¥*÷⇒
f€±e
¥#s
if" "
"
.
Eτ 2 → min
M. Beiglböck
Eτ 2 → max
geometry of model uncertainty
.
.
.
.
EB̄τ → max
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
Rost
Azema-Yor
...
'ak¥*÷⇒
f€±e
¥#s
if" "
"
.
Eτ 2 → min
Eτ 2 → max
.
.
.
.
EB̄τ → max
Hobson (’98):
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
Rost
Azema-Yor
...
'ak¥*÷⇒
f€±e
¥#s
if" "
"
.
Eτ 2 → min
Hobson (’98):
M. Beiglböck
Eτ 2 → max
.
.
.
.
EB̄τ → max
p̄Φ = sup{EP [Φ(S)] : P mart. measure, ST ∼ µT }
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
Rost
Azema-Yor
...
'ak¥*÷⇒
f€±e
¥#s
if" "
"
.
Eτ 2 → min
Hobson (’98):
Eτ 2 → max
.
.
.
.
EB̄τ → max
p̄Φ = sup{EP [Φ(S)] : P mart. measure, ST ∼ µT }
= sup{E[Φ(Bs )s≤τ ] : Bτ ∼ µT }
M. Beiglböck
geometry of model uncertainty
6 / 10
Skorokhod
embedding
Skorokhod embedding problem (SEP)
R
R
given: µ, x 2 dµ < ∞, B...BM with B0 = x dµ
task: determine τ s.t. Bτ ∼ µ, Eτ < ∞
Root: ∃ barrier R, BτR ∼ µ
Rost
Azema-Yor
...
'ak¥*÷⇒
f€±e
¥#s
if" "
"
.
Eτ 2 → min
Hobson (’98):
Eτ 2 → max
.
.
.
.
EB̄τ → max
p̄Φ = sup{EP [Φ(S)] : P mart. measure, ST ∼ µT }
= sup{E[Φ(Bs )s≤τ ] : Bτ ∼ µT }
optimal embeddings: basis for all known extremal models
M. Beiglböck
geometry of model uncertainty
6 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
Φ upper semi-continuous =⇒ dual theory of Skorokhod embededding
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
Φ upper semi-continuous =⇒ dual theory of Skorokhod embededding
Thm – Monotonicity Principle
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
Φ upper semi-continuous =⇒ dual theory of Skorokhod embededding
Thm – Monotonicity Principle
Φ Borel, τ optimal
M. Beiglböck
=⇒ support of τ is pathwise optimal
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
Φ upper semi-continuous =⇒ dual theory of Skorokhod embededding
Thm – Monotonicity Principle
Φ Borel, τ optimal
=⇒ support of τ is pathwise optimal
|
{z
}
∃ monotone Γ s.t. P((Bs )s≤τ ) ∈ Γ) = 1
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Transport Approach to Skorokhod Embedding [BCH15]
optimal Skorokhod embedding problem:
sup {E [Φ((Bs )s≤τ )] : Bτ ∼ µ}
Thm – Existence
Φ upper semi-continuous =⇒ ∃ optimizer τ
Thm – Duality
Φ upper semi-continuous =⇒ dual theory of Skorokhod embededding
Thm – Monotonicity Principle
Φ Borel, τ optimal
=⇒ support of τ is pathwise optimal
|
{z
}
∃ monotone Γ s.t. P((Bs )s≤τ ) ∈ Γ) = 1
Γ monotone :⇐⇒ Γ cannot be improved by pathwise modifications
M. Beiglböck
geometry of model uncertainty
7 / 10
Transport
Skorokhod
Thm [Root69]:
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
M. Beiglböck
Skorokhod
#. - type, BτR ∼ µ
.
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
M. Beiglböck
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]:
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
mmrhn
afferent
←
aft
1i.
←
=.
.
hhhwu
A
A
'
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
f÷E¥E#
:
a
r
(
M. Beiglböck
mmrhn
aft
1i.
←
=.
r
"
A
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
r
f÷E¥E#
:
a
r
(
M. Beiglböck
"
yteefo
whnnnf
=÷
xfjt
a
a
'
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
r
HE€It¥#÷
(
M. Beiglböck
.
yteefo
whnnnf
=÷
xfjt
a
a
'
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
r
HE€It¥#÷
(
M. Beiglböck
.
fa÷#tt
=
÷
mnennnf
•
a
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
#
Et
M. Beiglböck
0
0
small
,
"
fa÷#tt
=
÷
mnennnf
•
a
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
#
Et
M. Beiglböck
0
0
small
ifieeetti
an
←:
.
,
"
-
E
geometry of model uncertainty
Ti
o
large
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
#
Et
M. Beiglböck
0
0
,
"
←
*-0
.
afeetettnti
.
small
E
geometry of model uncertainty
t
•
large
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
0
0
#
Et
,
"
←
*-0
.
afeetettnti
.
small
E
t
•
large
hence:
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
xHtEEt#ef
0
.
,
"
hence:
.
.
afeetettnti
#
Et
M. Beiglböck
0
0
←
*-0
o
.
small
E
÷¥i
geometry of model uncertainty
t
•
large
.
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
xHtEEt#ef
0
.
,
#
Et
hence:
.
M. Beiglböck
0
0
←
*-0
o
.
"
.
afeetettnti
.
small
E
÷¥÷i
geometry of model uncertainty
t
•
large
.
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
o
.
xHtEEt#ef
0
.
0
0
,
#
Et
hence:
.
"
←
*-0
.
afeetettnti
.
small
E
t
•
large
#¥e
.
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
o
.
xHtEEt#ef
0
.
,
#
Et
hence:
.
M. Beiglböck
0
0
"
←
*-0
.
afeetettnti
.
small
E
t
•
large
:#¥E
hwm#
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
xHtEEt#ef
0
.
0
0
,
#
Et
hence:
.
←
*-0
o
"
.
afeetettnti
.
small
E
eke
÷~
t
•
large
.
hmmmw
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
Thm [Root69]: given µ ⇒ ∃τR of
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
,
←
*-0
afeetettnti
#
Et
hence:
0
0
"
.
.
small
E
t
•
large
#
⇒tE
hmmm#
Mhwmm
.
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
#
Et
hence:
M. Beiglböck
0
0
,
"
←
*-0
.
afeetettnti
.
small
E
t
•
large
ihunt
⇒
¥E
mhryL
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
.
o
xHtEEt#ef
0
.
,
←
*-0
afeetettnti
#
Et
hence:
0
0
"
.
.
small
E
t
•
large
⇒
€#€i
hummed
M. Beiglböck
geometry of model uncertainty
8 / 10
Transport
Thm [Root69]: given µ ⇒ ∃τR of
Skorokhod
#. - type, BτR ∼ µ, EτR2 → min
.
Proof [BCH15]: 1) fix τ s.t. Bτ ∼ µ, Eτ 2 → min 2) find geometry of τ
←
±
o
.
xHtEEt#ef
0
.
M. Beiglböck
,
#
Et
hence:
0
0
"
←
*-0
.
afeetettnti
.
small
E
⇒
hunt
.¥#E
geometry of model uncertainty
t
•
large
.
8 / 10
Extensions
Applications
Consequences of [BCH15]
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps
p̄Φ = sup{EP [Φ(S)] : P mart.,
M. Beiglböck
geometry of model uncertainty
ST ∼ µT }
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps
p̄Φ = sup{EP [Φ(S)] : P mart., St1 ∼ µ1 , . . . , Stn ∼ µn , ST ∼ µT }
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps – multi-marginal SEP
p̄Φ = sup{EP [Φ(S)] : P mart., St1 ∼ µ1 , . . . , Stn ∼ µn , ST ∼ µT }
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps – multi-marginal SEP
p̄Φ = sup{EP [Φ(S)] : P mart., St1 ∼ µ1 , . . . , Stn ∼ µn , ST ∼ µT }
◦ Hobson ’98
◦ Obloj, Spoida ’13
◦ Brown, Hobson, Rogers ’01
◦ Henry-Labordere, Obloj, Touzi ’14
◦ Madan, Yor ’02
◦ Cox, Obloj, Touzi ’16
◦ Hobson, Pederson ’02
◦ Claisse, Guo, Henry-Labordere, ’16
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps – multi-marginal SEP
p̄Φ = sup{EP [Φ(S)] : P mart., St1 ∼ µ1 , . . . , Stn ∼ µn , ST ∼ µT }
◦ Hobson ’98
◦ Obloj, Spoida ’13
◦ Brown, Hobson, Rogers ’01
◦ Henry-Labordere, Obloj, Touzi ’14
◦ Madan, Yor ’02
◦ Cox, Obloj, Touzi ’16
◦ Hobson, Pederson ’02
◦ Claisse, Guo, Henry-Labordere, ’16
transport approach [BCH16]:
M. Beiglböck
geometry of model uncertainty
9 / 10
Extensions
Applications
Consequences of [BCH15]
unified approach to all optimal embeddings
results extend to higher dimensions / continuous Markov processes
systematic approach to construct optimal embeddings
−→ many new embeddings
systematic approach to construct extreme models
systematic way to calculate upper/lower prices
market data at multiple time steps – multi-marginal SEP
p̄Φ = sup{EP [Φ(S)] : P mart., St1 ∼ µ1 , . . . , Stn ∼ µn , ST ∼ µT }
◦ Hobson ’98
◦ Obloj, Spoida ’13
◦ Brown, Hobson, Rogers ’01
◦ Henry-Labordere, Obloj, Touzi ’14
◦ Madan, Yor ’02
◦ Cox, Obloj, Touzi ’16
◦ Hobson, Pederson ’02
◦ Claisse, Guo, Henry-Labordere, ’16
transport approach [BCH16]: all optimal emb. extend in full generality
M. Beiglböck
geometry of model uncertainty
9 / 10
Summary
References
Model Uncertainty and Skorokhod Embedding:
Hobson, Robust hedging of the lookback option, Finance and Stochastics ’98
Obloj, The Skorokhod embedding problem and its offspring, Probab. Surv., ’04
Hobson, The Skorokhod embedding problem and model-independent bounds for option
prices, Springer, Berlin, ’11
Connection to (Martingale) Optimal Transport:
B/Henry-Labordere/Penkner, Model-independent Bounds for Option Prices: A Mass
Transport Approach, Finance and Stochastics ’13
Galichon/Henry-Laborder/Touzi, A stochastic control approach to no-arbitrage bounds
given marginals ’14, AAP
Dolinsky/Soner, Martingale optimal transport in continuous time, PTRF ’14
Acciaio/B/Penkner/Schachermayer, A model-free version of the FTAP, Math. Fin. ’16
Monotonicity Principle:
B/Julliet, Optimal transport under marginal martingale constraints, Ann. Prob. ’16
B/Nutz/Touzi, Complete Duality for MOT on the Line, Ann. Prob., to appear
B/Cox/Huesmann, Optimal Transport and Skorokhod embedding, Inventiones Math.,
to appear
M. Beiglböck
geometry of model uncertainty
10 / 10