Portfolio choice with permanent and temporary transaction ... · Portfolio choice with permanent and temporary transaction costs Ibrahim EKREN ETH Zurich Joint work with Johannes

IntroductionResultsProof

Portfolio choice with permanent and temporarytransaction costs

Ibrahim EKREN

ETH ZurichJoint work with Johannes Muhle-Karbe

November 2016, Financial/Actuarial Mathematics Seminar,University of Michigan

Ibrahim EKREN Portfolio choice with permanent and temporary transaction costs


Table of contents

1 IntroductionThe frictionless ProblemFrictionsAsymptotic

2 ResultsExpansion of the Value functionAsymptotically Optimal Policy

3 ProofSemilimitsObtaining upper and lower boundsExpansion of the Value function



The frictionless ProblemFrictionsAsymptotic

Optimal portfolio selection

Agent invests in a riskless asset and d risky assets withobjective

optimising the present value of the future stream of expectedexcess returns,penalising for risk, trading costs, and market impact.

The market is driven by Brownian motions.

Markovian Factors and affecting the prices process.

Temporary quadratic transaction costs and permanent meanreverting type frictions.

Asymptotics for small costs and fast mean reversion.




Frictionless problem

We assume the market is driven by a Markovian factor Y thatsatisfies

dYt = µY (Yt)dt+ σY (Yt)dWt

and the price process evolves according

dSt = µ(Yt, St)dt+ σ(Yt, St)dWt

For simplicity of notation we denote

µt := µ(Yt, St) and Σt := σ(Yt, St)2.

For notational simplicity we will consider S as additionaldimension in Y .




Frictionless problem

We assume that the agent can trade continuously without anycost.

For a given strategy xt that represents the position in eachrisky asset, the self-financing wealth process evolves accordingto

dVt = xt(µtdt+ σtdWt)

The agent optimises the present value of the future stream ofexpected excess returns penalised with risk of the portfoliowith risk aversion γ

max(xt)

E[∫ ∞

0e−ρt

(xtµt −

γ

2Σtx

2t

)dt

]The pointwise maximum is achieved at x∗t =Mt = µt

γΣt(Markowitz portfolio).




Literature

There are different types of frictions in the market.

Markowitz portfolio cannot be used in practice.

Proportional and fixed transaction costs literature for portfolioselection problem:

Magill and Constantinides 1976

Davis and Norman 1990

Shreve and Soner 1994 (asymptotics)

The objective is to find the non-trading region




Temporary price impact : Quadratic cost

We treat two different types of frictions.

For every trade of size ∆x the agent pays Λ(∆x)2 astransaction costs.

Equivalent to requiring that the limit order book is constantdepth 1

2Λ .

An order of size ∆x ”consumes” the order book from theunaffected prices St to St + 2Λ∆x.

The total cost of transaction is Λ(∆x)2 + St∆x.

Unless µtγΣt

is absolutely continuous the Markowitz protfoliohas infinite cost.

dxt = τtdt.

The natural question is for a given cost ε small what kind ofcontrol τ one should use.




Temporary price impact : Quadratic cost

We treat two different types of frictions.

For every trade of size ∆x the agent pays Λ(∆x)2 astransaction costs.

Equivalent to requiring that the limit order book is constantdepth 1

2Λ .

An order of size ∆x ”consumes” the order book from theunaffected prices St to St + 2Λ∆x.

The total cost of transaction is Λ(∆x)2 + St∆x.

Unless µtγΣt

is absolutely continuous the Markowitz protfoliohas infinite cost.

dxt = τtdt.

The natural question is for a given cost ε small what kind ofcontrol τ one should use.




Motivation for asymptotics

This gives rise to a nonlinear PDE in the Markovian case.

There are no explicit formulas.

For small ε > 0, the quantities are too large or too small. Anycomputation error might lead to instability of algorithms.

Find the asymptotics for small ε > 0 when the transactioncosts is ε2Λ.

The value of the problem with friction is an expansion aroundthe value of the Markowitz problem.

We want an asymptotically optimal strategy.




Review of the Literature for asymptotics

Soner and Touzi 2012 and Possamai, Soner, and Touzi 2013,proportional costs, viscosity approach.

Guasoni and Weber 2015 nonlinear impact.

Moreau, Muhle-Karbe and Soner 2015, quadratic costs,viscosity approach. They also provide with an asymptoticallyoptimal policy.

Altarovici, Muhle-Karbe, and Soner 2013, fixed costs, viscosityapproach.

Kallsen and Muhle-Karbe 2014, high resilience limit.

Cai, Rosenbaum and Tankov, Bichuch and Shreve, Janecekand Shreve, Korn, Feodoria,. Survey of Muhle-Karbe, Reppenand Soner.

There is only temporary costs and there is no permanent.




Microstructure foundation for permanent impact

Permanent price impact is commonly used in optimalexecution literature.

This is in addition to some temporary costs.

Jaimungal et al, Obizawa and Wang, Almgren and Chris,Alfonsi et al.

In Garleanu and Pedersen 2016 the authors gives amicrostructural foundation to these frictions.

Three different kind of market participants.Optimising traders, market makers and end users.Frictions as a consequence of inventory risk of market makers.The resilience is related to the inventory depletion rate of themarket makers.

The continuous time model with friction can be obtained as alimit of discrete time model.




Permanent impact

The trades of the agent has a lasting impact on the price processthat he faces.Besides the unaffected price St the agent trades with a priceSt +Dt where Dt is the price distortion.Starting from the state (D0, S0, Y0), a trade τt∆t during ∆t

increases D from D0 to D0 +Cτt∆t where C is the distortionimpact parameter

D reverts back to 0 at a rate R (mean reversion parameter).

The equation for D is

dDt

dt= −RDt + Cτt.




Admissible strategies

The agent should not have the possibility to delay the risk tofuture times.

We need to have an admissibility condition on the controls.Aρ(D,x, Y ) is the set of controls τ such that

limt→∞

e−ρt(|xt|2 + |Dt|2) = 0.

This condition allows integration by parts formulas.

Recall our convention that Y also contains the elements of S.




Value function

In the market described above, an agent optimises the presentvalue of the future stream of expected excess returns.

The optimisation criterion penalises the risk of the portfolioand transaction costs.

The value J (D,x, Y, τ) associated to a strategy τ

E[∫ ∞

0e−ρt

(xt(µt −RDt + Cτt)−

γ

2Σx2

t −1

2Λτ2

t

)dt

]where (Dt, xt, Yt) starts at (D,x, Y ) and evolves as above.

Our value function of interest is

V (D,x, Y ) := supτ∈Aρ(D,x,y)

J (D,x, Y, τ)




Viscosity property of the Value function

Theorem

V is a viscosity solution of the PDE

ρV = −γ2

Σx2 + µx−RD(∂DV + x) + LY (V )

+ supτ

{−1

2Λτ2 + τ(∂xV + C(x+ ∂DV ))

}

Elliptic degenerate PDE.

The statement even holds for path-dependent problems wherethe value functional is a viscosity solution to thecorrespondant path-dependent PDE.




FBSDE for the control

The control can be obtained by the solution of the followingFBSDE

dXt = b(Xt, Yt)dt ∈ R2

dYt = −F (t,Xt, Yt)dt+ ZdWt ∈ R2

YT = G(XT )

where X = (D,x) and Y allows us to obtain τ∗.

This is a linear coupled multidimensional FBSDE.

There is no wellposedness result in the general case.




Reasons for Asymptotics

There is no closed form formula of the value function forgeneral dynamics.The quadratic transaction cost and the permanent priceimpact should be small.Obtain a simple tractable formulas that allows us tounderstand the structure of the solution.V should be an expansion around the frictionless problem.

V ε(D,x, Y ) = V 0(Y ) + εαV 1(D,x, y) + o(εα)

V 0 is the frictionless problem.V 1 simple and allows us to understand the effect of differentparameters.

Numerical methods are problematic if some parameters aresmall.Additionally we want an asymptotically optimal control.




Correct scaling of the parameters

We need to find the correct critical regime where we obtain aformula that exposes the effect of different parameters on thevalue and the dynamics.

We assume that the quadratic transaction costs scale in ε asfollows

Λε = ε2Λ.

We can write formal asymptotics for a simple linear quadraticmodel in Garleanu and Pedersen (2016).

The only scaling where both quadratic costs and thepermanent price impact are both non negligible.

Rε = Rε−1.

Cε = Cε.




Limit order book meaning

There are two limit order books that are being used in parallel.

One book of constant height 12Λε2

It mean reverts instantaneously to the unaffected price St.

Another book of constant depth 1Cε .

That mean reverts with a mean-reversion speed Rε−1.

We want the high mean-reversion and small impact asymptotics.




Scaling of D

According to the formal asymptotics of Garleanu and Pedersen2016 to have non-trivial limits we need to plug Dε−1.

If Rε = Rε−1 then for a given τ and D0 = 0 we have

Dtε−1 = C

∫ t

0e−Rε(t−s)τsds ∼

Cε

Rτt.

Optimal portfolio is of order ε−1(xt −Mt).

→ Dtε−1 is of the order of (xt −Mt)




Asymptotics without permanent impact

In Moreau, Muhle-Karbe and Moreau 2016, the problem withoutpermanent price impact is studied.The expansion

V ε(x, Y ) = V 0(Y ) + ε (u(y) + w(x−M(y))) + o(ε)

is proven.M(y) is the Markowitz portfolio corresponding to the state y.An asymptotically optimal portfolio is

τ ε = −αε

(x−M(y))

This is a portfolio tracking the frictionless position at an optimalrate α.




The function V ε

We need to study the function

V ε(D,x, Y ) = V ε(Dε, x, Y )

which solves the PDE

ρV ε = −γ2

Σx2 + µx−RD(ε−1∂DVε + x) + LY (V ε)

+ supτ

{−1

2Λετ

2 + τ(∂xV + Cε(x+ ε−1∂DVε))

}




A condition for wellposedness

In Garleanu and Pedersen 2016, the authors require aninequality between γ,Σ and C to write the problem in alinear-quadratic form.

In the asymptotic regime, the condition of Garleanu andPedersen 2016 is satisfied.

We don’t need this assumption. The value function is finite.

One can find a finite supersolution to the PDE.

By comparison principle, we obtain that value function isfinite.



Expansion of the Value functionAsymptotically Optimal Policy

Expansion of the Value function 1

Theorem (Main Result)

We have the following expansion for the value function

V ε(D,x, Y ) = V 0(Y ) + εu(Y )

+ εw(D −M1(Y ), x−M2(Y )) + o(ε)

where w is a quadratic form solving the first order correctorequation and u solves the second corrector equations.

• This is an expansion around V0.




The function w

The function w(ξ1, ξ2, y) is a quadratic form in (ξ1, ξ2) thatsolves the following PDE

0 = −γξT2 Σξ2

2−RξT1 (ξ2 + ∂1w)

+1

2(C(ξ2 + ∂1w) + ∂2w)TΛ−1(C(ξ2 + ∂1w) + ∂2w)

It can be written under the form

w(ξ, y) = ξTA(y)ξ

where A(y) solves an algebraic Riccati equation that can becompletely solved in 1d.




The function w cont’d

Let Qx = −√

ΛγΣ and QD be the unique negative solution of

0 = −1−QD(

1− QxΛR

)+CQ2

D

2ΛR

Then A can be written as

A =

Q2D

4ΛR12

(1 + QDQx

R

)12

(1 + QDQx

R

)(Λ− CQD

R )Qx4

Positive in ξ1

Negative in ξ2




Expansion around which controls?

The expansion is around some Mi,

ξ1 =D −M1(y)

ε1/2and ξ2 =

x−M2(y)

ε1/2

where Mi solves the linear system

Q2D

2ΛM1 + (R− CQD

Λ)M2 = 0

QDM1 + (Qx − C)M2 = ΛµQx

The determinant is always negative. The solution exists.

The definition of Mi is need to obtain that the value functionconverges V 0.




The function u

Theorem

The function u solves the linear PDE

ρu(y) = LY u(Y ) + a(Y )

where a depends on Mi.

Linear PDE.

Feynmann Kac representation.

a can be written as a covariation using Mi.

After simplification it can be shown only depend on d〈M〉.The gamma of the Markowitz portfolio is the main statisticsfor losses due to frictions.




Final form

The final expansion is

V ε(D,x, Y ) = V 0(Y ) + εu(Y )

+ εw(D −M1(Y ), x−M2(Y )) + o(ε)

Mi solves a linear system,

w is quadratic in ξ,

w solves a Ricatti equation,

u solves linear PDE.




Qualitative properties

In Moreau, Muhle-Karbe and Soner 2015, the term D doesnot appear and w is negative.

In our case the quadratic w is negative in ξ2 and positive in ξ1.

The sign in ξ2 express the fact that this deviation is a frictionand the agent needs to pay extra to achieve the targetposition.

The sign in ξ1 expresses the fact ξ1 gives an anticipation onthe future evolution of the prices.

The initial conjecture would be an expansion of the valuefunction around (M, 0) for the value function where M is theMarkowitz portfolio.

The expansion for the value is around (M1,M2) where thecouple satisfies a linear equation given by the data of theproblem and w.




Qualitative properties

In Moreau, Muhle-Karbe and Soner 2015, the term D doesnot appear and w is negative.

In our case the quadratic w is negative in ξ2 and positive in ξ1.

The sign in ξ2 express the fact that this deviation is a frictionand the agent needs to pay extra to achieve the targetposition.

The sign in ξ1 expresses the fact ξ1 gives an anticipation onthe future evolution of the prices.

The initial conjecture would be an expansion of the valuefunction around (M, 0) for the value function where M is theMarkowitz portfolio.

The expansion for the value is around (M1,M2) where thecouple satisfies a linear equation given by the data of theproblem and w.




Asymptotically Optimal Policy

Theorem

An asymptotically optimal policy for the investment problem isgiven by

τ ε :=Λ−1

ε(QD(D −M1) +Qx(x−M2) + CΛM2) .

It is a scaled version of the optimal control for the linearquadratic model with impatience 0.Since D can be inferred from τ one could have guessed that

τ = −αε(x−M).

By definition of Mi, we can write

τ ε = Λ−1

(Qxε

(x−M) +QDεD

).




Asymptotically Optimal Policy cont’d

D was scaled. For the initial problem an asymptoticallyoptimal control is

τ ε =Λ−1Qxε

(x−M0) +Λ−1QDε2

D.

whereQx = −

√γΣΛ

and QD < 0 solves a second order polynomial equation.

In a multidimensional problem, Qx, QD comes from thesolution of a Riccati equation.

It is also a scaled linear-quadratic problem in themultidimensional case.



SemilimitsObtaining upper and lower boundsExpansion of the Value function

Semi-limits

We define the ratio

uε(D,x, y) :=V ε(D,x, y)− V 0(y)

ε

and the adjusted semi limits

uε(D,x, y) := uε(D,x, y)− εw(D −M1(y), x−M2(y), y)

The difficult part of the proof is to show that this function definedafter some algebraic computations converges to u.

Theorem

” lim sup ”uε(D,x, y) and ” lim inf ”uε(D,x, y)

does not depend on D,x.

We use a first order PDE.Ibrahim EKREN Portfolio choice with permanent and temporary transaction costs



Semi-limits, cont’d

Theorem

” lim sup ”uε(M1(y),M2(y), y) and ” lim inf ”uε(M1(y),M2(y), y)

are respectively viscosity subsolution and supersolution of thesecond corrector equation.

By definition

” lim sup ”uε(M1(y),M2(y), y) ≥ ” lim inf ”uε(M1(y),M2(y), y).

By comparison of viscosity solution of the linear equation

” lim sup ”uε(M1(y),M2(y), y) ≤ ” lim inf ”uε(M1(y),M2(y), y).

Independence in D,x implies the result.




Upper bound for uε

We need uε to be boundeduniformly in εlocally uniformly in (D,x)

In Moreau, Muhle-Karbe and Soner (2015), the followinginequality holds

uε(D,x, y) :=V ε(D,x, y)− V 0(y)

ε≤ 0

In our case this inequality does not hold.

Integration by parts and admissibility condition implies

uε(D,x, y) ≤ −xD +CD2

2.

” lim sup ” well-defined.




Obtaining lower bound

V is defined with a supremum.

It is sufficient to find τ ε such that

J (Dε−1, x, Y, τ ε)− V 0(Y )

ε

is uniformly bounded from below.

We use a general method to obtain this lower.

It also allow us to obtain an optimal policy in a restrictedclass policy.

It is based on the so called Laplace method.




Laplace method

For f, g smooth function find an expansion for∫ t

0eλf(s)g(s)ds when λ→∞.

Only the values of f and g around the maximum of f matters.

The distortion solves a linear, mean reverting ODE

Dt = D0e−Rε−1t +

∫ t

0e−Rε

−1(t−s)τsds

The difference with the classical Laplace method is the factthat τ is not smooth but a semi-martingale.




Laplace method

For f, g smooth function find an expansion for∫ t

0eλf(s)g(s)ds when λ→∞.

Only the values of f and g around the maximum of f matters.

The distortion solves a linear, mean reverting ODE

Dt = D0e−Rε−1t +

∫ t

0e−Rε

−1(t−s)τsds

The difference with the classical Laplace method is the factthat τ is not smooth but a semi-martingale.




An Ansatz

We only need one policy to have the lower bound. We startwith the following family

τ ε = −αε(x−M0(y))− βεD.

The couple (Dt, xt) solves a linear mean-reverting type ODE.

We need approximation for∫ t

0εα(t−s)Msds

where M is a semi martingale satisfying

dMs = asds+ rsdWs.




Laplace method

Theorem (Laplace Method for Semi-martingales)

Under assumptions we have the following expansion∫ t

0e−α(t−s)Msds ∼α→∞

Mt

α− αtα2− rtGα(t)

α√

2α+ negligible

whereGα(t) =

√2α∫ t

0 e−α(t−s)dWs ∼ N(0, 1− e−2αt)→ N(0, 1).

We also need asymptotics for

E[Gα(t)Mt] ∼ E

[d〈M〉dt

]√

2α−1

for M semimartingale with absolutely continuous quadraticvariation.




Expansion for the Ansatz

Under regularity assumption on M we can compute thefollowing expansion for the ansatz

J (Dε−1, x, Y, τ ε) = V 0(Y )

−εα2R(α+2C+R)+(αR+(βC+R)2)

4αR(α+βC+R) E[∫∞

0 e−ρtγΣtd〈M〉tdt dt

]+ o(ε)

One can choose any α and β to obtain the boundedness frombelow of uε.

”Liminf” well-defined. All assumption for the viscosityapproach works.

Thus, there is an asymptotically optimal policy of the form ofthe ansatz.




Final Remarks

A posteriori

α2R(α+ 2C +R) + (αR+ (βC +R)2)

2αR(α+ βC +R)

is indeed the best rate.

β = 0 does not give the maximum.

The function has a maximum at α = −Qx and β = −QD.

Without viscosity approach this is only an optimiser within theprescribed class.

With viscosity approach, we obtained an asymptoticallyoptimal portfolio.

The viscosity approach is necessary to get the structuralresults on the shape of the optimal policy.




THANK YOU!




Why does it work?

Let Gε be the generator of Eε, and

ψε(D,x, y) = v(y) + εuφ(D,x, y) + εω(X −M(y))

with v, u, ω smooth. Then the following expansion holds,

Gε(D, x, y, ∂Dψε, ∂xψ

ε, ∂yψε, ∂yyψ

ε) = LY (v) + µTΣ−1µ2γ

+ E2(∂Dφ(D, x, y), ∂xφ(D, x, y), y)

+E1(y,D −M1, x−M2, ∂1ω(X −M), ∂2ω(X −M)) + B(X −M,∂1ω(X −M), ∂2ω(X −M))

−R(D −M1)T ∂Dφ + (∂xφ + C∂Dφ)Λ−1(∂2ω(X −M(y)) + C(x−M2 + ∂1ω(X −M)))

+εΓ(D, x, y).

where

E1 is the generator of the equation solved by w,

E2 is the PDE giving us the fact that u does not depend onD,x

B is 0 if (D,x) = (M1,M2).

Γ contains the second corrector equation at the highest order.


Documents

Portfolio choice with permanent and temporary transaction ... · Portfolio choice with permanent and temporary transaction costs Ibrahim EKREN ETH Zurich Joint work with Johannes