An Explicit Example of a Shadow Price Process with ... · PDF fileAn Explicit Example of a Shadow Price Process with Stochastic Investment Opportunity Set Christoph Czichowsky Faculty

An Explicit Example of a Shadow Price Processwith Stochastic Investment Opportunity Set

Christoph Czichowsky

Faculty of MathematicsUniversity of Vienna

SIAM FM 12 — New Developments in Optimal Portfolio Choice

based on joint work with Philipp Deutsch andWalter 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 cadlag 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 + (ϕ1

T )+(1− λ)ST − (ϕ1)−T ST

)]over all self-financing and admissible strategies (ϕ0, ϕ1) under transactioncosts starting from initial endowment (ϕ0

0, 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.






E[U(ϕ0T + (ϕ1

T )+(1− λ)ST − (ϕ1)−T ST


0, ϕ10) = (x , 0).










E[U(ϕ0T + (ϕ1

T )+(1− λ)ST − (ϕ1)−T ST


0, ϕ10) = (x , 0).










E[U(ϕ0T + (ϕ1

T )+(1− λ)ST − (ϕ1)−T ST


0, ϕ10) = (x , 0).






Shadow price

A shadow price S = (St)0≤t≤T is a price process valued in [(1− λ)S ,S ]such that the frictionless utility maximisation problem for that price has thesame optimal strategy as the one under transaction costs.

Frictionless trading at any price process S = (St)0≤t≤T valued in the bid-askspread [(1− λ)S ,S ] allows to generate higher terminal payoffs.

Hence, a shadow price corresponds to the least favourable frictionless marketevolving in the bid-ask spread.

The optimal strategy for the shadow price only buys, if the shadow priceequals the ask price, and sells, if the shadow price equals the bid price.

If such a shadow price exists,

I obtain the optimal strategy by solving a frictionless problem.I apply all the techniques and knowledge from frictionless markets.I no qualitatively new effects arise due to transaction costs.

Do these shadow prices exist in general?


Shadow price

A shadow price S = (St)0≤t≤T is a price process valued in [(1− λ)S ,S ]such that the frictionless utility maximisation problem for that price has thesame optimal strategy as the one under transaction costs.

Frictionless trading at any price process S = (St)0≤t≤T valued in the bid-askspread [(1− λ)S ,S ] allows to generate higher terminal payoffs.

Hence, a shadow price corresponds to the least favourable frictionless marketevolving in the bid-ask spread.

The optimal strategy for the shadow price only buys, if the shadow priceequals the ask price, and sells, if the shadow price equals the bid price.

If such a shadow price exists,

I obtain the optimal strategy by solving a frictionless problem.I apply all the techniques and knowledge from frictionless markets.I no qualitatively new effects arise due to transaction costs.

Do these shadow prices exist in general?


Previous literature: Shadow prices in general

Cvitanic and Karatzas (1996): Basic idea. In Brownian setting, if the

minimizer (Z 0, Z 1) to a suitable dual problem is a local martingale, then a

shadow price exists and is given by Z 1

Z 0.

Cvitanic and Wang (2001): This dual minimizer is so far only guaranteedto be a supermartingale.

Loewenstein (2000): Existence in Brownian setting, if no assets can besold 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 ageneral multi-currency model (jumps, random bid-ask spreads), if no assetscan 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?


Previous literature: Shadow prices in general

Cvitanic and Karatzas (1996): Basic idea. In Brownian setting, if the

minimizer (Z 0, Z 1) to a suitable dual problem is a local martingale, then a

shadow price exists and is given by Z 1

Z 0.

Cvitanic and Wang (2001): This dual minimizer is so far only guaranteedto be a supermartingale.

Loewenstein (2000): Existence in Brownian setting, if no assets can besold 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 ageneral multi-currency model (jumps, random bid-ask spreads), if no assetscan 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?


General result

Theorem (C./Schachermayer 2012)

Suppose that

i) S is continuous

ii) S satisfies (NFLVR)

and U : (0,∞)→ R satisfies lim supx→∞

xU′(x)U(x) < 1 and u(x) := sup

g∈C(x)

E [U(g)] <∞.

Then (Z 0, Z 1) is a local martingale and S := Z 1

Z 0a shadow price process.

Quite sharp: There exist counter-examples, if

i’) S is discontinuous and satisfies (NFLVR) and Z 1

Z 0satisfies (NFLVR).

I C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices,and Connections to Duality, 2012.

ii’) S is continuous and satisfies (CPSλ) for all λ ∈ (0, 1) but not (NFLVR).

Do shadow prices allow to actually compute the solution in particular models?


General result

Theorem (C./Schachermayer 2012)

Suppose that

i) S is continuous

ii) S satisfies (NFLVR)

and U : (0,∞)→ R satisfies lim supx→∞

xU′(x)U(x) < 1 and u(x) := sup

g∈C(x)

E [U(g)] <∞.

Then (Z 0, Z 1) is a local martingale and S := Z 1

Z 0a shadow price process.

Quite sharp: There exist counter-examples, if

i’) S is discontinuous and satisfies (NFLVR) and Z 1

Z 0satisfies (NFLVR).

I C./Muhle-Karbe/Schachermayer: Transaction Costs, Shadow Prices,and Connections to Duality, 2012.

ii’) S is continuous and satisfies (CPSλ) for all λ ∈ (0, 1) but not (NFLVR).

Do shadow prices allow to actually compute the solution in particular models?


Previous literature: Shadow prices in particular

Shadow prices in Black-Scholes model: Various optimisation problems

I Kallsen and Muhle-Karbe (2009)I Gerhold, Muhle-Karbe and Schachermayer (2011)I Guasoni, Gerhold, Muhle-Karbe and Schachermayer (2011)I Herczegh and Prokaj (2011)I Choi, Sirbu and Zitkovic (2012) . . .

Shadow prices for Ito processes:

I Kallsen and Muhle-Karbe (2012): asymptotics for exponential utility

Results for general diffusion models (without shadow prices):

I Martin and Schoneborn (2011): local utilityI Martin (2012): multi-dimensional diffusions and local utilityI 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?






















Example: Shadow price for geometric OU process

Ornstein-Uhlenbeck process: dXt = κ(x − Xt)dt + σdWt , X0 = x0.

Stock price: St = exp (Xt), i.e.,

dSt

St=

(κ(x − log(St)

)+σ2

2

)dt + σdWt =: µ(St)dt + σdWt .

Stochastic investment opportunity set, i.e., random coefficients.

Basic problem: Maximise the asymptotic logarithmic growth-rate

lim supT→∞

1

TE[

log(ϕ0T + (ϕ1

T )+(1− λ)ST − (ϕ1)−T ST

)]over all self-financing, admissible strategies (ϕ0, ϕ1) under transaction costs.

Black-Scholes model:

I Taksar, Klass and Assaff (1988): Solving HJB equation.I Gerhold, Muhle-Karbe and Schachermayer (2011): Shadow price.


Qualitative behaviour of the optimal strategy

Without transaction costs:

Invest fraction θ(St) := µ(St)σ2 =

(κ(x−log(St))+ σ2

2 )

σ2 of wealth in stock.

Trading in number of shares:

s

0 a0 b0

dϕ1

ds > 0 dϕ1

ds < 0 dϕ1

ds > 0

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?






2 )



s

0 a0 b0

dϕ1

ds > 0 dϕ1

ds < 0 dϕ1

ds > 0










2 )



s

0 a0 b0

dϕ1

ds > 0 dϕ1

ds < 0 dϕ1

ds > 0






Ansatz for the shadow price


Ansatz for the shadow price

Ansatz St = 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 ′(a) = 1 at buying boundary

g(b) = (1− λ)b and g ′(b) = (1− λ) at selling boundary

Ito’s formula: dg(St)/g(St) = µtdt + σtdWt

Frictionless log-optimizer for S given by

ϕ1t0

St

ϕ0t0

+ ϕ1t0

St

=πg(St)

(a− π) + πg(St)=µt

σ2t

Yields ODE for g :

g ′′(s) =2πg ′(s)2

(a− π) + πg(s)− 2θ(s)g ′(s)

s


Computing the candidate

General solution to ODE with g(a) = a and g ′(a) = 1:

g(s; a, π) = aah(a) + (1− π)H(a, s)

ah(a)− πH(a, s),

where h(s) := exp(κσ2

(x − log(s) + σ2

2κ

)2)

and H(a, s) :=∫ s

ah(u)du.

Plugging this into g(b) = (1− λ)b, g ′(b) = 1− λ we obtain

π(a, b, λ) := aH(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))2h(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.




g(s; a, π) = aah(a) + (1− π)H(a, s)

ah(a)− πH(a, s),


(x − log(s) + σ2

2κ

)2)

and H(a, s) :=∫ s

ah(u)du.



(a + λb − b)H(a, b)

and








g(s; a, π) = aah(a) + (1− π)H(a, s)

ah(a)− πH(a, s),


(x − log(s) + σ2

2κ

)2)

and H(a, s) :=∫ s

ah(u)du.



(a + λb − b)H(a, b)

and








Theorem (Fractional Taylor expansions in terms of λ1/3)

For a, b ∈ (0,∞) \ {a0, b0}, we have expansions (of arbitrary order)

b(a, λ) = a + a

(6λ

Γ(a)

)1/3

+ a3Γ(a) + 2κ2

σ4

(x − log(a)

)61/3Γ(a)5/3

λ2/3 + O(λ),

π(a, λ) = θ(a)−(

3

4Γ(a)2λ

)1/3

−2κ2

σ4

(x − log(a)

)61/3Γ(a)2/3

λ2/3 + O(λ),

π(b, λ) = θ(b) +

(3

4Γ(b)2λ

)1/3

−2κ2

σ4

(x − log(b)

)61/3Γ(b)2/3

λ2/3 + O(λ),

where Γ(s) denotes the sensitivity of the displacement from the optimal fraction

Γ(s) = θ(s)(1− θ(s))− θ′(s)s

=4κσ2 + σ4 − 4κ2 log(s)2 + 8κ2x log(s)− 4κ2x2

4σ4.

Compare Gerhold, Muhle-Karbe and Schachermayer (2011) for theBlack-Scholes model and Soner and Touzi (2012) for first terms.


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

(3σ4√

κ2σ2 (4κ+ σ2)

)1/4

λ1/4 + O(λ1/2),

a1,2(b, λ) = b ± b√

2

(3σ4√

κ2σ2 (4κ+ σ2)

)1/4

λ1/4 + O(λ1/2),

π(a, λ) = θ(a)−

(√κ2σ2 (4κ+ σ2)

3σ4

)1/2

λ1/2 + O(λ3/4),

π(b, λ) = θ(b) +

(√κ2σ2 (4κ+ σ2)

3σ4

)1/2

λ1/2 + O(λ3/4).


Verification

Up to now: Only one excursion starting from St0 = a.

Need to define a process (At)0≤t<∞ such that everything fits together.


Verification

Defined continuous process S = g(S ; A, π(A, λ)

)Moves between [(1− λ)S ,S ]

This is even a nice process.

Proposition

S = g(S ; A, π(A, λ)

)is an Ito process, which satisfies the SDE

dSt = g ′(St ; At , π(At , λ)

)dSt + 1

2 g ′′(St ; At , π(At , λ)

)d〈S ,S〉t

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 S = S resp.S = (1− λ)S by construction

Hence, S is a shadow price!


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 Ito process.

Function of ask price S and a truncation of its running minima resp. maximaduring 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.


Thank you for your attention!

http://www.mat.univie.ac.at/∼czichoc2


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An Explicit Example of a Shadow Price Process with ... · PDF fileAn Explicit Example of a Shadow Price Process with Stochastic Investment Opportunity Set Christoph Czichowsky Faculty