An Explicit Example of a Shadow Price Process
with Stochastic Investment Opportunity Set
Christoph Czichowsky
Faculty of Mathematics
University of Vienna
SIAM FM 12 — New Developments in Optimal Portfolio Choice
based on joint work with Philipp Deutsch and
Walter Schachermayer
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
1 / 18
Utility maximisation under transaction costs
Fix a strictly positive cádlág stock price process S = (St )0≤t≤T .
Buy at ask price S. Sell at lower bid price (1 − λ)S for fixed λ ∈ (0, 1).
Standard problem: Maximise
E U ϕ0T + (ϕ1T )+ (1 − λ)ST − (ϕ1 )−
T ST
over all self-financing and admissible strategies (ϕ0 , ϕ1 ) under transaction
costs starting from initial endowment (ϕ00 , ϕ10 ) = (x, 0).
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
2 / 18
Utility maximisation under transaction costs
Fix a strictly positive cádlág stock price process S = (St )0≤t≤T .
Buy at ask price S. Sell at lower bid price (1 − λ)S for fixed λ ∈ (0, 1).
Standard problem: Maximise
E U ϕ0T + (ϕ1T )+ (1 − λ)ST − (ϕ1 )−
T ST
over all self-financing and admissible strategies (ϕ0 , ϕ1 ) under transaction
costs starting from initial endowment (ϕ00 , ϕ10 ) = (x, 0).
How to obtain the solution to this problem?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
2 / 18
Utility maximisation under transaction costs
Fix a strictly positive cádlág stock price process S = (St )0≤t≤T .
Buy at ask price S. Sell at lower bid price (1 − λ)S for fixed λ ∈ (0, 1).
Standard problem: Maximise
E U ϕ0T + (ϕ1T )+ (1 − λ)ST − (ϕ1 )−
T ST
over all self-financing and admissible strategies (ϕ0 , ϕ1 ) under transaction
costs starting from initial endowment (ϕ00 , ϕ10 ) = (x, 0).
How to obtain the solution to this problem?
Classically: Try to find solution by solving HJB equation.
I
Davis and Norman (1992), Shreve and Soner (1994), . . .
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
2 / 18
Utility maximisation under transaction costs
Fix a strictly positive cádlág stock price process S = (St )0≤t≤T .
Buy at ask price S. Sell at lower bid price (1 − λ)S for fixed λ ∈ (0, 1).
Standard problem: Maximise
E U ϕ0T + (ϕ1T )+ (1 − λ)ST − (ϕ1 )−
T ST
over all self-financing and admissible strategies (ϕ0 , ϕ1 ) under transaction
costs starting from initial endowment (ϕ00 , ϕ10 ) = (x, 0).
How to obtain the solution to this problem?
Classically: Try to find solution by solving HJB equation.
I
Davis and Norman (1992), Shreve and Soner (1994), . . .
Alternatively: Try to find a shadow price.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
2 / 18
Shadow price
b = (S
bt )0≤t≤T is a price process valued in [(1 − λ)S, S]
A shadow price S
such that the frictionless utility maximisation problem for that price has the
same optimal strategy as the one under transaction costs.
e = (S
et )0≤t≤T valued in the bid-ask
Frictionless trading at any price process S
spread [(1 − λ)S, S] allows to generate higher terminal payoffs.
Hence, a shadow price corresponds to the least favourable frictionless market
evolving in the bid-ask spread.
The optimal strategy for the shadow price only buys, if the shadow price
equals the ask price, and sells, if the shadow price equals the bid price.
If such a shadow price exists,
I
I
I
obtain the optimal strategy by solving a frictionless problem.
apply all the techniques and knowledge from frictionless markets.
no qualitatively new effects arise due to transaction costs.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
3 / 18
Shadow price
b = (S
bt )0≤t≤T is a price process valued in [(1 − λ)S, S]
A shadow price S
such that the frictionless utility maximisation problem for that price has the
same optimal strategy as the one under transaction costs.
e = (S
et )0≤t≤T valued in the bid-ask
Frictionless trading at any price process S
spread [(1 − λ)S, S] allows to generate higher terminal payoffs.
Hence, a shadow price corresponds to the least favourable frictionless market
evolving in the bid-ask spread.
The optimal strategy for the shadow price only buys, if the shadow price
equals the ask price, and sells, if the shadow price equals the bid price.
If such a shadow price exists,
I
I
I
obtain the optimal strategy by solving a frictionless problem.
apply all the techniques and knowledge from frictionless markets.
no qualitatively new effects arise due to transaction costs.
Do these shadow prices exist in general?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
3 / 18
Previous literature: Shadow prices in general
Cvitanić and Karatzas (1996): Basic idea. In Brownian setting, if the
b 0, Z
b 1 ) to a suitable dual problem is a local martingale, then a
minimizer (Z
b1
shadow price exists and is given by Zb 0 .
Z
Cvitanić and Wang (2001): This dual minimizer is so far only guaranteed
to be a supermartingale.
Loewenstein (2000): Existence in Brownian setting, if no assets can be
sold short and a solution to the problem under transaction costs exists.
Kallsen and Muhle-Karbe (2011): Existence in finite probability spaces.
Benedetti, Campi, Kallsen and Muhle-Karbe (2011): Existence in a
general multi-currency model (jumps, random bid-ask spreads), if no assets
can be sold short and a solution exists. −→ Talk this afternoon.
Counter-example: unique candidate for shadow price admits arbitrage.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
4 / 18
Previous literature: Shadow prices in general
Cvitanić and Karatzas (1996): Basic idea. In Brownian setting, if the
b 0, Z
b 1 ) to a suitable dual problem is a local martingale, then a
minimizer (Z
b1
shadow price exists and is given by Zb 0 .
Z
Cvitanić and Wang (2001): This dual minimizer is so far only guaranteed
to be a supermartingale.
Loewenstein (2000): Existence in Brownian setting, if no assets can be
sold short and a solution to the problem under transaction costs exists.
Kallsen and Muhle-Karbe (2011): Existence in finite probability spaces.
Benedetti, Campi, Kallsen and Muhle-Karbe (2011): Existence in a
general multi-currency model (jumps, random bid-ask spreads), if no assets
can be sold short and a solution exists. −→ Talk this afternoon.
Counter-example: unique candidate for shadow price admits arbitrage.
Are there other conditions that ensure the existence of shadow prices?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
4 / 18
General result
Theorem (C./Schachermayer 2012)
Suppose that
i) S is continuous
ii) S satisfies (NFLVR)
(x)
and U : (0, ∞) → R satisfies lim sup xU
U(x) < 1 and u(x) := sup E [U(g )] < ∞.
0
x→∞
b 1 ) is a local martingale and S
b :=
b 0, Z
Then (Z
g ∈C(x)
b1
Z
b0
Z
a shadow price process.
Quite sharp: There exist counter-examples, if
i’) S is discontinuous and satisfies (NFLVR) and
I
b1
Z
b0
Z
satisfies (NFLVR).
C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices,
and Connections to Duality, 2012.
ii’) S is continuous and satisfies (CPS λ ) for all λ ∈ (0, 1) but not (NFLVR).
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
5 / 18
General result
Theorem (C./Schachermayer 2012)
Suppose that
i) S is continuous
ii) S satisfies (NFLVR)
(x)
and U : (0, ∞) → R satisfies lim sup xU
U(x) < 1 and u(x) := sup E [U(g )] < ∞.
0
x→∞
b 1 ) is a local martingale and S
b :=
b 0, Z
Then (Z
g ∈C(x)
b1
Z
b0
Z
a shadow price process.
Quite sharp: There exist counter-examples, if
i’) S is discontinuous and satisfies (NFLVR) and
I
b1
Z
b0
Z
satisfies (NFLVR).
C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices,
and Connections to Duality, 2012.
ii’) S is continuous and satisfies (CPS λ ) for all λ ∈ (0, 1) but not (NFLVR).
Do shadow prices allow to actually compute the solution in particular models?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
5 / 18
Previous literature: Shadow prices in particular
Shadow prices in Black-Scholes model: Various optimisation problems
I
I
I
I
I
Kallsen and Muhle-Karbe (2009)
Gerhold, Muhle-Karbe and Schachermayer (2011)
Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011)
Herczegh and Prokaj (2011)
Choi, Sirbu and Zitkovic (2012) . . .
Shadow prices for Itô processes:
I
Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility
Results for general diffusion models (without shadow prices):
I
I
I
Martin and Schöneborn (2011): local utility
Martin (2012): multi-dimensional diffusions and local utility
Soner and Touzi (2012): asymptotics for general utilities
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
6 / 18
Previous literature: Shadow prices in particular
Shadow prices in Black-Scholes model: Various optimisation problems
I
I
I
I
I
Kallsen and Muhle-Karbe (2009)
Gerhold, Muhle-Karbe and Schachermayer (2011)
Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011)
Herczegh and Prokaj (2011)
Choi, Sirbu and Zitkovic (2012) . . .
Shadow prices for Itô processes:
I
Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility
Results for general diffusion models (without shadow prices):
I
I
I
Martin and Schöneborn (2011): local utility
Martin (2012): multi-dimensional diffusions and local utility
Soner and Touzi (2012): asymptotics for general utilities
How do these shadow prices look like in a particular diffusion model?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
6 / 18
Previous literature: Shadow prices in particular
Shadow prices in Black-Scholes model: Various optimisation problems
I
I
I
I
I
Kallsen and Muhle-Karbe (2009)
Gerhold, Muhle-Karbe and Schachermayer (2011)
Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011)
Herczegh and Prokaj (2011)
Choi, Sirbu and Zitkovic (2012) . . .
Shadow prices for Itô processes:
I
Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility
Results for general diffusion models (without shadow prices):
I
I
I
Martin and Schöneborn (2011): local utility
Martin (2012): multi-dimensional diffusions and local utility
Soner and Touzi (2012): asymptotics for general utilities
How do these shadow prices look like in a particular diffusion model?
Do they allow us to actually compute the solution there?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
6 / 18
Example: Shadow price for geometric OU process
Ornstein-Uhlenbeck process: dXt = κ(x̄ − Xt )dt + σdWt ,
X0 = x0 .
Stock price: St = exp (Xt ), i.e.,
σ2
dSt
= κ x̄ − log(St ) +
dt + σdWt =: µ(St )dt + σdWt .
St
2
Stochastic investment opportunity set, i.e., random coefficients.
Basic problem: Maximise the asymptotic logarithmic growth-rate
lim sup
T →∞
1 E log ϕ0T + (ϕ1T )+ (1 − λ)ST − (ϕ1 )−
T ST
T
over all self-financing, admissible strategies (ϕ0 , ϕ1 ) under transaction costs.
Black-Scholes model:
I
I
Taksar, Klass and Assaff (1988): Solving HJB equation.
Gerhold, Muhle-Karbe and Schachermayer (2011): Shadow price.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
7 / 18
Qualitative behaviour of the optimal strategy
Without transaction costs:
Invest fraction θ(St ) :=
µ(St )
σ2
2
=
(κ(x̄−log(St ))+ σ2 )
σ2
of wealth in stock.
Trading in number of shares:
dϕ1
ds
dϕ1
ds
>0
dϕ1
ds
<0
>0
s
0
Christoph Czichowsky (Uni Wien)
a0
b0
Explicit example of a shadow price
Minneapolis, July 11, 2012
8 / 18
Qualitative behaviour of the optimal strategy
Without transaction costs:
Invest fraction θ(St ) :=
µ(St )
σ2
2
=
(κ(x̄−log(St ))+ σ2 )
σ2
of wealth in stock.
Trading in number of shares:
dϕ1
ds
dϕ1
ds
>0
dϕ1
ds
<0
>0
s
0
a0
b0
With transaction costs it is ‘folklore’:
Do nothing in the interior of some no-trade region.
Minimal trading on the boundary to stay within this region.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
8 / 18
Qualitative behaviour of the optimal strategy
Without transaction costs:
Invest fraction θ(St ) :=
µ(St )
σ2
2
=
(κ(x̄−log(St ))+ σ2 )
σ2
of wealth in stock.
Trading in number of shares:
dϕ1
ds
dϕ1
ds
>0
dϕ1
ds
<0
>0
s
0
a0
b0
With transaction costs it is ‘folklore’:
Do nothing in the interior of some no-trade region.
Minimal trading on the boundary to stay within this region.
But how does this look like?
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
8 / 18
Ansatz for the shadow price
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
9 / 18
Ansatz for the shadow price
Ansatz Ŝt = g (St ) during this excursion from St0 = a to St1 = b
(1 − λ)s ≤ g (s) ≤ s for all s between a and b
g (a) = a and g 0 (a) = 1 at buying boundary
g (b) = (1 − λ)b and g 0 (b) = (1 − λ) at selling boundary
Itô’s formula: dg (St )/g (St ) = µ̂t dt + σ̂t dWt
Frictionless log-optimizer for Ŝ given by
ϕ1t0 Ŝt
ϕ0t0 + ϕ1t0 Ŝt
=
πg (St )
µ̂t
= 2
(a − π) + πg (St )
σ̂t
Yields ODE for g :
g 00 (s) =
Christoph Czichowsky (Uni Wien)
2θ(s)g 0 (s)
2πg 0 (s)2
−
(a − π) + πg (s)
s
Explicit example of a shadow price
Minneapolis, July 11, 2012
10 / 18
Computing the candidate
General solution to ODE with g (a) = a and g 0 (a) = 1:
ah(a) + (1 − π)H(a, s)
,
ah(a) − πH(a, s)
Rs
σ2 2
x̄ − log(s) + 2κ
and H(a, s) := a h(u)du.
g (s; a, π) = a
where h(s) := exp
κ
σ2
Plugging this into g (b) = (1 − λ)b, g 0 (b) = 1 − λ we obtain
π(a, b, λ) := a
H(a, b) + λbh(a) − bh(a) + ah(a)
(a + λb − b)H(a, b)
and
F (a, b, λ) := H(a, b)2 (λ − 1) + (a + b(λ − 1))2 h(a)h(b) = 0,
which gives two equations λ1,2 (a, b) = λ.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
11 / 18
Computing the candidate
General solution to ODE with g (a) = a and g 0 (a) = 1:
ah(a) + (1 − π)H(a, s)
,
ah(a) − πH(a, s)
Rs
σ2 2
x̄ − log(s) + 2κ
and H(a, s) := a h(u)du.
g (s; a, π) = a
where h(s) := exp
κ
σ2
Plugging this into g (b) = (1 − λ)b, g 0 (b) = 1 − λ we obtain
π(a, b, λ) := a
H(a, b) + λbh(a) − bh(a) + ah(a)
(a + λb − b)H(a, b)
and
F (a, b, λ) := H(a, b)2 (λ − 1) + (a + b(λ − 1))2 h(a)h(b) = 0,
which gives two equations λ1,2 (a, b) = λ.
Only need λ1 (a, b) = λ that can, however, not be solved explicitly.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
11 / 18
Computing the candidate
General solution to ODE with g (a) = a and g 0 (a) = 1:
ah(a) + (1 − π)H(a, s)
,
ah(a) − πH(a, s)
Rs
σ2 2
x̄ − log(s) + 2κ
and H(a, s) := a h(u)du.
g (s; a, π) = a
where h(s) := exp
κ
σ2
Plugging this into g (b) = (1 − λ)b, g 0 (b) = 1 − λ we obtain
π(a, b, λ) := a
H(a, b) + λbh(a) − bh(a) + ah(a)
(a + λb − b)H(a, b)
and
F (a, b, λ) := H(a, b)2 (λ − 1) + (a + b(λ − 1))2 h(a)h(b) = 0,
which gives two equations λ1,2 (a, b) = λ.
Only need λ1 (a, b) = λ that can, however, not be solved explicitly.
For sufficiently small λ, there exists b(a, λ) with λ1 (a, b(a, λ)) = λ for ‘all’ a.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
11 / 18
Computing the candidate
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
12 / 18
Theorem (Fractional Taylor expansions in terms of λ1/3 )
For a, b ∈ (0, ∞) \ {a0 , b0 }, we have expansions (of arbitrary order)
2
1/3
3Γ(a) + 2κ
6λ
σ 4 x̄ − log(a)
b(a, λ) = a + a
+a
λ2/3 + O(λ),
Γ(a)
61/3 Γ(a)5/3
1/3
2κ2
x̄ − log(a) 2/3
3
4
π(a, λ) = θ(a) −
Γ(a)2 λ
− σ 1/3
λ + O(λ),
4
6 Γ(a)2/3
1/3
2κ2
3
2
σ 4 x̄ − log(b)
Γ(b) λ
−
λ2/3 + O(λ),
π(b, λ) = θ(b) +
4
61/3 Γ(b)2/3
where Γ(s) denotes the sensitivity of the displacement from the optimal fraction
Γ(s) = θ(s)(1 − θ(s)) − θ0 (s)s
=
4κσ 2 + σ 4 − 4κ2 log(s)2 + 8κ2 x̄ log(s) − 4κ2 x̄ 2
.
4σ 4
Compare Gerhold, Muhle-Karbe and Schachermayer (2011) for the
Black-Scholes model and Soner and Touzi (2012) for first terms.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
13 / 18
Theorem (Fractional Taylor expansions in terms of λ1/4 )
For a = a0 and b = b0 , we have expansions (of arbitrary order)
√
b1,2 (a, λ) = a ± a 2
√
a1,2 (b, λ) = b ± b 2
π(a, λ) = θ(a) −
π(b, λ) = θ(b) +
Christoph Czichowsky (Uni Wien)
3σ 4
p
!1/4
λ1/4 + O(λ1/2 ),
κ2 σ 2 (4κ + σ 2 )
3σ 4
!1/4
λ1/4 + O(λ1/2 ),
κ2 σ 2 (4κ + σ 2 )
!1/2
p
κ2 σ 2 (4κ + σ 2 )
λ1/2 + O(λ3/4 ),
3σ 4
!1/2
p
κ2 σ 2 (4κ + σ 2 )
λ1/2 + O(λ3/4 ).
3σ 4
p
Explicit example of a shadow price
Minneapolis, July 11, 2012
14 / 18
Verification
Up to now: Only one excursion starting from St0 = a.
Need to define a process (At )0≤t<∞ such that everything fits together.
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
15 / 18
Verification
Defined continuous process Ŝ = g S; A, π(A, λ)
Moves between [(1 − λ)S, S]
This is even a nice process.
Proposition
Ŝ = g S; A, π(A, λ) is an Itô process, which satisfies the SDE
d Ŝt = g 0 St ; At , π(At , λ) dSt + 12 g 00 St ; At , π(At , λ) dhS, Sit
Similar arguments as in Gerhold, Muhle-Karbe and Schachermayer (2011).
Frictionless log-optimal portfolio is well-known
Number of stocks only increases resp. decreases when Ŝ = S resp.
Ŝ = (1 − λ)S by construction
Hence, Ŝ is a shadow price!
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
16 / 18
Summary
Existence of shadow prices
Sufficient condition: S is continuous and satisfies (NFLVR).
Quite sharp: Counter-examples.
Explicit construction of shadow price:
Growth-optimal portfolio for geometric Ornstein-Uhlenbeck process.
Sufficiently small but fixed transaction costs λ.
Shadow price is an Itô process.
Function of ask price S and a truncation of its running minima resp. maxima
during excursions of an OU process.
Explicitly determined up to one implicitly defined function b(a, λ).
Asymptotic expansions of arbitrary order in terms of λ1/3 and λ1/4 .
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
17 / 18
Thank you for your attention!
http://www.mat.univie.ac.at/∼czichoc2
Christoph Czichowsky (Uni Wien)
Explicit example of a shadow price
Minneapolis, July 11, 2012
18 / 18