[PPT] - Portfolio optimisation with small transaction costs an engineers PowerPoint Presentation

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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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SLIDE 2

Outline

1

Introduction

2

Asymptotically optimal portfolios for small costs

3

Multivariate extension

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SLIDE 3

Outline

1

Introduction

2

Asymptotically optimal portfolios for small costs

3

Multivariate extension

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SLIDE 4

Portfolio choice under transaction costs

using duality

Cvitani´ c & Karatzas (1996): Loewenstein (2000):

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SLIDE 5

Shadow price processes in optimisation

The basic principle

Goal: max

ϕ

E(u(VT(ϕ))) V(ϕ) wealth under transaction costs Ex. S in [S, S] such that

◮ optimiser ϕ⋆ maximises ϕ → E(u(v0 + ϕ •

ST)),

◮ VT(ϕ⋆) = v0 + ϕ⋆ •

ST.

0.0 0.2 0.4 0.6 0.8 1.0 0.80 0.85 0.90 0.95 1.00 1.05 1.10 1.15 t bid

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SLIDE 6

What about explicit results —

using shadow prices?

Cvitani´ c & 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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SLIDE 7

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, Janeˇ cek & Shreve 2004, Rogers 2004, . . . )

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SLIDE 8

Outline

1

Introduction

2

Asymptotically optimal portfolios for small costs

3

Multivariate extension

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SLIDE 9

Our setup

The portfolio optimisation problem

Want to maximise ϕ → 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)?

◮ 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. 9 / 26

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Our setup

The traded asset

Traded asset dSt = btdt + σtdWt

◮ univariate ◮ continuous ◮ otherwise rather arbitrary

Frictionless optimiser ϕ is assumed to be known.

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SLIDE 11

Shadow price approach

Look for

ptimal strategy ϕε,

shadow price process Sε, dual martingale Z ε, which satisfy ZT = u′(v0 + ϕε • Sε

T),

Z ε martingale, Z ε Sε martingale, ϕε changes only when Sε

t ∈ {St, St}.

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SLIDE 12

Optimality of ϕε

from an engineer’s perspective

For any competitor ψ we have E(u(VT(ψ))) ≤ E(u(v0 + ψ • Sε

T))

≤ E(u(v0 + ϕε • Sε

T)) + E

u′(v0 + ϕε •

Sε

T)((ψ − ϕε) •

Sε

T)

=

E(u(VT(ϕε))) + E

ZT((ψ − ϕε) •

Sε

T)

=

E(u(VT(ϕε))). (Second inequality follows from u(y) ≤ u(x) + u′(x)(y − x).)

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SLIDE 13

Approximate solution

Look for approximately optimal strategy ϕε, approximate shadow price process Sε, approximate dual martingale Z ε, which satisfy ZT = u′(v0 + ϕε • Sε

T) + O(ε),

Z ε has drift o(ε2/3), Z ε Sε has drift O(ε2/3), ϕε changes only when Sε

t ∈ {St, St}.

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SLIDE 14

Approximate optimality of ϕε

from an engineer’s perspective

For any competitor ψε with ψε − ϕ = o(1) we have E(u(VT(ψε))) ≤ E(u(v0 + ψε • Sε

T))

≤ E(u(v0 + ϕε • Sε

T)) + E

u′(v0 + ϕε •

Sε

T)((ψε − ϕε) •

Sε

T)

=

E(u(VT(ϕε))) + E

ZT((ψε − ϕε) •

Sε

T)

+ o(ε2/3)

= E(u(VT(ϕε))) + o(ε2/3). Compare with E(u(VT(ϕε))) − E(u(v0 + ϕ • ST)) = O(ε2/3).

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SLIDE 15

How to find an approximate solution

The ansatz

How to find ϕε, Sε, Z ε, no-trade bounds ∆ϕ±? Write ϕε = ϕ + ∆ϕ,

Sε

= S + ∆S, Z ε = Z(1 + K), where ϕ, Z are optimal for the frictionless problem. Ansatz: ∆S = f(∆ϕ, . . . ), f(x) = αx3 − γx

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SLIDE 16

How to find an approximate solution

The solution

This works for ∆St =

(γt∆ϕ)3 − 3γtx)

εSt 2 with γt =

3

2p

3εSt cS

t

cϕ

t

, where dS, St = cS

t dt and dϕ, ϕt = cϕ t dt.

no-trade bounds ∆ϕ±

t = ± 1

γt = ± 3

3εSt

2p cϕ

t

cS

t

certainty equivalent of utility loss −E

ZT

T p 2(∆ϕ±

t )2dS, ST

(2 3 due to transaction costs, 1 3 due to displacement )

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SLIDE 17

How to find an approximate solution

The solution

This works for ∆St =

(γt∆ϕ)3 − 3γtx)

εSt 2 with γt =

3

2p

3εSt cS

t

cϕ

t

, where dS, St = cS

t dt and dϕ, ϕt = cϕ t dt.

no-trade bounds ∆ϕ±

t = ± 1

γt = ± 3

3εSt

2p cϕ

t

cS

t

certainty equivalent of utility loss −E

ZT

T p 2(∆ϕ±

t )2dS, ST

(2 3 due to transaction costs, 1 3 due to displacement )

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SLIDE 18

How to find an approximate solution

The solution

This works for ∆St =

(γt∆ϕ)3 − 3γtx)

εSt 2 with γt =

3

2p

3εSt cS

t

cϕ

t

, where dS, St = cS

t dt and dϕ, ϕt = cϕ t dt.

no-trade bounds ∆ϕ±

t = ± 1

γt = ± 3

3εSt

2p cϕ

t

cS

t

certainty equivalent of utility loss −E

ZT

T p 2(∆ϕ±

t )2dS, ST

(2 3 due to transaction costs, 1 3 due to displacement )

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SLIDE 19

Outline

1

Introduction

2

Asymptotically optimal portfolios for small costs

3

Multivariate extension

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SLIDE 20

The multivariate case

The setup

Traded asset dSt = btdt + σtdWt 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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SLIDE 21

The multivariate case

The ansatz

How to find ϕε, Sε, Z ε, no-trade region? Write ϕε = ϕ + ∆ϕ,

Sε

= S + ∆S, Z ε = Z(1 + K), where ϕ, Z are optimal for frictionless problem. Ansatz: ∆Si = fi(∆ϕ, . . . ), i = 1, . . . , d fi(x) =

(γ⊤

i x)3 − 3γ⊤ i x

εSi 2 No-trade region is parallelotop spanned by d vectors ±a1 . . . , ±ad ∈ Rd.

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SLIDE 22

The multivariate case

The solution

∆Si =

(γ⊤

i x)3 − 3γ⊤ i x

εSi 2 works for γ = (γ1, . . . , γd) ∈ Rd×d given by γij =

3

2p

3εSi 1 (cScϕcS)ii cS

ij ,

where dSi, Sjt = (cS

ij )tdt and dϕi, ϕjt = (cϕ ij )tdt.

no-trade parallelotop is spanned by a1, . . . , ad ∈ Rd, where a = (a1, . . . , ad) ∈ Rd×d is given by aij = (γ−1)ij =

3

3εSi

2p (cScϕcS)jj(cS)−1

ij

certainty equivalent of utility loss more complicated than in univariate case

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SLIDE 23

The multivariate case

The solution

∆Si =

(γ⊤

i x)3 − 3γ⊤ i x

εSi 2 works for γ = (γ1, . . . , γd) ∈ Rd×d given by γij =

3

2p

3εSi 1 (cScϕcS)ii cS

ij ,

where dSi, Sjt = (cS

ij )tdt and dϕi, ϕjt = (cϕ ij )tdt.

no-trade parallelotop is spanned by a1, . . . , ad ∈ Rd, where a = (a1, . . . , ad) ∈ Rd×d is given by aij = (γ−1)ij =

3

3εSi

2p (cScϕcS)jj(cS)−1

ij

certainty equivalent of utility loss more complicated than in univariate case

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SLIDE 24

The multivariate case

The solution

∆Si =

(γ⊤

i x)3 − 3γ⊤ i x

εSi 2 works for γ = (γ1, . . . , γd) ∈ Rd×d given by γij =

3

2p

3εSi 1 (cScϕcS)ii cS

ij ,

where dSi, Sjt = (cS

ij )tdt and dϕi, ϕjt = (cϕ ij )tdt.

no-trade parallelotop is spanned by a1, . . . , ad ∈ Rd, where a = (a1, . . . , ad) ∈ Rd×d is given by aij = (γ−1)ij =

3

3εSi

2p (cScϕcS)jj(cS)−1

ij

certainty equivalent of utility loss more complicated than in univariate case

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SLIDE 25

Ingredient of the certainty equivalent of utility loss

Covariance matrix of correlated Brownian motion with oblique reflection at the boundary

−1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 x[, 1] x[, 2] −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

What is the covariance matrix of the stationary law?

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SLIDE 26

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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