Portfolio optimisation with small transaction costs
— an engineer’s approach
Jan Kallsen
based on joint work with
Johannes Muhle-Karbe and Paul Krühner
New York, June 5, 2012
dedicated to Ioannis Karatzas
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Outline
1
Introduction
2
Asymptotically optimal portfolios for small costs
3
Multivariate extension
2 / 26
Outline
1
Introduction
2
Asymptotically optimal portfolios for small costs
3
Multivariate extension
3 / 26
Portfolio choice under transaction costs
using duality
Cvitanić & Karatzas (1996):
Loewenstein (2000):
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Shadow price processes in optimisation
The basic principle
Goal:
max E(u(VT (ϕ)))
ϕ
V (ϕ) wealth under transaction costs
e in [S, S] such that
Ex. S
1.15
1.10
1.05
1.00
bid
0.95
0.90
0.85
I
e T )),
optimiser ϕ? maximises ϕ 7→ E(u(v0 + ϕ • S
?
? • e
VT (ϕ ) = v0 + ϕ ST .
0.80
I
0.0
0.2
0.4
0.6
t
0.8
1.0
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What about explicit results —
using shadow prices?
Cvitanić & Karatzas (1996):
but cf. K. & Muhle-Karbe (2009), Kühn & Stroh (2010), Gerhold et
al. (2011), Herczegh & Prokaj (2011), Choi et al. (2012)
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What about explicit results —
using shadow prices?
Problem: explicit results are very rare.
(e.g. Davis & Norman 1990, . . . )
Way out: look at asymptotics for small transaction costs.
(Constantinides 1986, Whalley & Wilmott 1997, Atkinson & Al-Ali
1997, Janeček & Shreve 2004, Rogers 2004, . . . )
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Outline
1
Introduction
2
Asymptotically optimal portfolios for small costs
3
Multivariate extension
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Our setup
The portfolio optimisation problem
Want to maximise
ϕ 7→ E(− exp(−pVT (ϕ)))
V (ϕ) wealth process under transaction costs
bid price S = (1 − ε)S, ask price S = (1 + ε)S
Why exponential utility u(x) = − exp(−px)?
I
I
I
I
It allows for random endowment (= hedging problem)
by a measure change.
Solution does not depend on initial wealth.
Utility on R better suited for hedging than utilities on R+ .
Exponential utility often leads to simple structure.
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Our setup
The traded asset
Traded asset
dSt = bt dt + σt dWt
I
I
I
univariate
continuous
otherwise rather arbitrary
Frictionless optimiser ϕ is assumed to be known.
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Shadow price approach
Look for
optimal strategy ϕε ,
e ε,
shadow price process S
dual martingale Z ε ,
which satisfy
e ε ),
ZT = u 0 (v0 + ϕε • S
T
Z ε martingale,
e ε martingale,
Z εS
e ε ∈ {S , S t }.
ϕε changes only when S
t
t
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Optimality of ϕε
from an engineer’s perspective
For any competitor ψ we have
e ε ))
E(u(VT (ψ))) ≤ E(u(v0 + ψ • S
T
e ε )) + E u 0 (v0 + ϕε • S
e ε )((ψ − ϕε ) • S
eε )
≤ E(u(v0 + ϕε • S
T
T
T
eε )
= E(u(VT (ϕε ))) + E ZT ((ψ − ϕε ) • S
T
= E(u(VT (ϕε ))).
(Second inequality follows from u(y ) ≤ u(x) + u 0 (x)(y − x).)
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Approximate solution
Look for
approximately optimal strategy ϕε ,
e ε,
approximate shadow price process S
approximate dual martingale Z ε ,
which satisfy
e ε ) + O(ε),
ZT = u 0 (v0 + ϕε • S
T
Z ε has drift o(ε2/3 ),
e ε has drift O(ε2/3 ),
Z εS
e ε ∈ {S , S t }.
ϕε changes only when S
t
t
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Approximate optimality of ϕε
from an engineer’s perspective
For any competitor ψ ε with ψ ε − ϕ = o(1) we have
e ε ))
E(u(VT (ψ ε ))) ≤ E(u(v0 + ψ ε • S
T
e ε )) + E u 0 (v0 + ϕε • S
e ε )((ψ ε − ϕε ) • S
eε )
≤ E(u(v0 + ϕε • S
T
T
T
e ε ) + o(ε2/3 )
= E(u(VT (ϕε ))) + E ZT ((ψ ε − ϕε ) • S
T
= E(u(VT (ϕε ))) + o(ε2/3 ).
Compare with
e T )) = O(ε2/3 ).
E(u(VT (ϕε ))) − E(u(v0 + ϕ • S
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How to find an approximate solution
The ansatz
e ε , Z ε , no-trade bounds ∆ϕ± ?
How to find ϕε , S
Write
ϕε = ϕ + ∆ϕ,
e ε = S + ∆S,
S
Z ε = Z (1 + K ),
where ϕ, Z are optimal for the frictionless problem.
Ansatz:
∆S = f (∆ϕ, . . . ),
f (x) = αx 3 − γx
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How to find an approximate solution
The solution
This works for
with
εS
t
∆St = (γt ∆ϕ)3 − 3γt x)
2
s
2p ctS
γt = 3
,
3εSt ctϕ
where dhS, Sit = ctS dt and dhϕ, ϕit = ctϕ dt.
no-trade bounds
s
ϕ
3εSt ct
1
3
∆ϕ±
=
±
=
±
t
γt
2p ctS
certainty equivalent of utility loss
!
Z T
p
2
−E ZT
(∆ϕ±
t ) dhS, SiT
0 2
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How to find an approximate solution
The solution
This works for
with
εS
t
∆St = (γt ∆ϕ)3 − 3γt x)
2
s
2p ctS
γt = 3
,
3εSt ctϕ
where dhS, Sit = ctS dt and dhϕ, ϕit = ctϕ dt.
no-trade bounds
s
ϕ
3εSt ct
1
3
∆ϕ±
=
±
=
±
t
γt
2p ctS
certainty equivalent of utility loss
!
Z T
p
2
−E ZT
(∆ϕ±
t ) dhS, SiT
0 2
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How to find an approximate solution
The solution
This works for
with
εS
t
∆St = (γt ∆ϕ)3 − 3γt x)
2
s
2p ctS
γt = 3
,
3εSt ctϕ
where dhS, Sit = ctS dt and dhϕ, ϕit = ctϕ dt.
no-trade bounds
s
ϕ
3εSt ct
1
3
∆ϕ±
=
±
=
±
t
γt
2p ctS
certainty equivalent of utility loss
!
Z T
p
2
−E ZT
(∆ϕ±
t ) dhS, SiT
0 2
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Outline
1
Introduction
2
Asymptotically optimal portfolios for small costs
3
Multivariate extension
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The multivariate case
The setup
Traded asset
dSt = bt dt + σt dWt
with Rd -valued S and multivariate Wiener process W
(e.g. Akian et al. 1996, Liu 2004, Lynch & Tan 2006, Muthuraman
& Kumar 2006, Atkinson & Ingpochai 2007, Goodman & Ostrov
2007, Law et al. 2009, Bichuch & Shreve 2012)
Frictionless optimiser ϕ is assumed to be known.
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The multivariate case
The ansatz
e ε , Z ε , no-trade region?
How to find ϕε , S
Write
ϕε = ϕ + ∆ϕ,
e ε = S + ∆S,
S
Z ε = Z (1 + K ),
where ϕ, Z are optimal for frictionless problem.
Ansatz:
∆S i
= fi (∆ϕ, . . . ), i = 1, . . . , d
εS i
fi (x) =
(γi> x)3 − 3γi> x
2
No-trade region is parallelotop spanned by d vectors
±a1 . . . , ±ad ∈ Rd .
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The multivariate case
The solution
εS i
∆S i = (γi> x)3 − 3γi> x
2
d×d
works for γ = (γ1 , . . . , γd ) ∈ R
given by
s
2p
1
cS ,
γij = 3
i
S
3εS (c c ϕ c S )ii ij
where dhS i , S j it = (cijS )t dt and dhϕi , ϕj it = (cijϕ )t dt.
no-trade parallelotop is spanned by a1 , . . . , ad ∈ Rd , where
a = (a1 , . . . , ad ) ∈ Rd×d is given by
s
3εS i S ϕ S
aij = (γ −1 )ij = 3
(c c c )jj (c S )−1
ij
2p
certainty equivalent of utility loss more complicated
than in univariate case
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The multivariate case
The solution
εS i
∆S i = (γi> x)3 − 3γi> x
2
d×d
works for γ = (γ1 , . . . , γd ) ∈ R
given by
s
2p
1
cS ,
γij = 3
i
S
3εS (c c ϕ c S )ii ij
where dhS i , S j it = (cijS )t dt and dhϕi , ϕj it = (cijϕ )t dt.
no-trade parallelotop is spanned by a1 , . . . , ad ∈ Rd , where
a = (a1 , . . . , ad ) ∈ Rd×d is given by
s
3εS i S ϕ S
aij = (γ −1 )ij = 3
(c c c )jj (c S )−1
ij
2p
certainty equivalent of utility loss more complicated
than in univariate case
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The multivariate case
The solution
εS i
∆S i = (γi> x)3 − 3γi> x
2
d×d
works for γ = (γ1 , . . . , γd ) ∈ R
given by
s
2p
1
cS ,
γij = 3
i
S
3εS (c c ϕ c S )ii ij
where dhS i , S j it = (cijS )t dt and dhϕi , ϕj it = (cijϕ )t dt.
no-trade parallelotop is spanned by a1 , . . . , ad ∈ Rd , where
a = (a1 , . . . , ad ) ∈ Rd×d is given by
s
3εS i S ϕ S
aij = (γ −1 )ij = 3
(c c c )jj (c S )−1
ij
2p
certainty equivalent of utility loss more complicated
than in univariate case
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Ingredient of the certainty equivalent of utility loss
0.0
−1.0
−0.5
x[, 2]
0.5
1.0
Covariance matrix of correlated Brownian motion
with oblique reflection at the boundary
−1.0
−0.5
0.0
0.5
1.0
x[, 1]
What is the covariance matrix of the stationary law?
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Conclusion
Shadow price approach is useful to the engineer as well.
One can get quite far with regards to explicit formulas for
asyptotically optimal portfolios.
Rigorous theorems still left to future research.
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