Utility maximization problem with transaction costs- Shadow price approach

Jin Hyuk Choijoint work with Mihai Sîrbu and Gordan !itkovic

Department of MathematicsUniversity of Texas at Austin

Carnegie Mellon UniversityPittsburgh, Sep 12th, 2011

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Outline

Problem settings and related work! Merton problem! Shadow price approach

Heuristic derivation of the free boundary ODE

Our result (for power utility)

Value function < !"# $ Shadow price process"# Explicit condition for market parameters"# Existence of C2 solution to the free boundary ODE

Sketch of the proof! Existence of C2 solution! Construction and Verification of the Shadow process.

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Market Model : Davis and Norman (’90), Shreve and Soner (’94)

Stock : dSt = St(µdt + !dWt), µ > 0, ! > 0.

Bond : B % 1.

Transaction costs : " > 0, " & (0, 1),

Bid St " (1 + ")St, Ask St " (1 ' ")St

Portfolio :

!"""#"""$

#0t : # of shares of B#t : # of shares of Sct : consumption rate

.

Initial position : (#00,#0)=($B, $S)

Admissibility : (#0,#, c) & A (S,",") i!

Self-financing d#0t = Std#

(t ' Std#)t ' ctdt

No-bankrupcy #0t + St#

+t ' St#'t * 0

where # = #)t ' #(t is the pathwide minimal decomposition of # into adi!erence of two non-decreasing processes. (#)t = Lt ,#

(t =Mt in Shreve and Soner (’94))

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Investor : Davis and Norman (’90), Shreve and Soner (’94)

Utility functions (CRRA) : Up(x) =%

ln x, p = 0xp

p , p & ('!, 1) \ {0} , for x > 0.

Investor’s goal : to maximize expected utility by consumption

u(S,",") " sup(#0,#,c)&A (S,",")

E& ' !

0e'%tUp(ct)dt

(,

where % > 0 is a discount factor.

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Related Work - no transaction cost, Melton(’71)

Proof : There exists an explicit solution to the HJB equation, we can writedown optimal strategy and do the verification.

Remark : For 0 < p < 1, u(S, 0, 0) < ! "# µ <)

2(1'p)!2%p .

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Related Work - transaction cost

Magill and Constantinides (’76) : Optimal behavior (heuristic)Davis and Norman (’90) : Analytic proof, several technical conditions.Shreve and Soner (’94) : Rigorous proof, only assumed u(S,",") < !.

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Shadow price approach

Motivation : In the frictionless market, we can use duality. Can weconstruct a stock price process which somehow absorbs transactioncosts? (S,",")! (S+, 0, 0).

Consistent price processes : S " {S : St , St , St , t * 0}Shadow price process : We call S+ & S a shadow price if

u(S,",") = u(S+, 0, 0) < !

Observations :

For S & S , u(S,",") , u(S, 0, 0)

u(S,",") = u(S+, 0, 0) = infS&S

u(S, 0, 0)

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Related work - Construction of Shadow price

Kallsen and Muhle-Karbe (’10) showed that if p = 0 (log utility) andµ < !2, the shadow price process can be constructed.

They derived a free boundary ODE based on the following observation -the optimal strategy does not trade when St < S+t < St.

In log utility case (p=0), their methods work, becasue the explicitexpression for the optimal strategy exists.

But their methods only work for p = 0 (log utility).

How can we do for p " 0 ?

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Our approach - derivation of free boundary problem

If the Shadow price process S+ exists,

u(S+, 0, 0) = infS&S

u(S, 0, 0)

Let’s parametrize S & S with an Ito-process Y :

St = eYt St,

dYt = *µtdt +*!tdWt,

with y , Yt , y, y " ln (1 ' "), y " ln (1 + ")

In the complete market, by convex duality,

u(S, 0, 0) = ($B+$SS0)p

p

+E& ' !

0E ('& ·W)

pp'1

t e%

p'1 tdt(,1'p,

where &t "µ+*µs+!*!s+

12*!

2s

*!s+!.

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Our approach - derivation of free boundary problem

u(S+, 0, 0) = infS&S

u(S, 0, 0)

= infY

u(S, 0, 0)

= infy&[y,y]

inf*µ,*!

u(S, 0, 0)

= infy&[y,y]

($B+$SeyS0)p

p w(y)1'p

where w(y) " inf*µ,*!E& ' !

0E ('& ·W)

pp'1

t e%

p'1 tdt(.

We set dP = E ( p1'p& ·W)dP, then,

w(y) = inf*µ,*!EP& ' !

0e%

p'1 tep

2(1'p)2

- t0 &

2udu

dt(.

We can write down the HJB-equation for this optimal stochastic control.

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Our approach - derivation of free boundary problem

HJB equation : For y & (y, y),

inf*µ,*!

.( p

2(1'p)2&2 ' %

1'p )w(y) + (*µ + p1'p*!&)w-(y) + 1

2*!w--(y) + 1

/= 0

Boundary condition : At y and y, turn o! the di!usion, which correspond to

w--(y) = w--(y) = !

Order reduction : Set w(y) = g(w-(y)) for g : [x, x] ./ R, (x = w-(y), x = w-(y)),

inf*µ,*!

.( p

2(1'p)2&2 ' %

1'p )g(x) + (*µ + p1'p*!&)x + 1

2*!x

g-(x) + 1/= 0,

with g-(x) = g-(x) = 0,' x

x

g-(x)x dx = y ' y.

(boundary conditions from g-(x) = xI-(x) = xw--(I(x)) , where I(x) = (w-)'1(x).)

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Free boundary ODE

Finally, after simple change of variable, we obtain a free boundary problem :!""#""$

g-(x) = L(x, g(x)) on [x, x]g-(x) = g-(x) = 0,

- xx

g-(x)x dx = ±(y ' y)

where

L(x, y) =

!""""#""""$

'2µx+%!2x2+2% y(%x'1)(2% y'(2µ'!2)x) , p = 0

'(1'p)3!2x2+2q(1+µx)y'2p%y2

(1'p)x(2+2µx+(p2'1)!2x)+.

2q+(q(2µ'!2)'2%)x/

y+2p%y2, p " 0

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Main result

Theorem. The followings are equivalent :

1 . u(S,",") < !.

2 . The Shadow price process exists.

3 . $ x, x, g & C2([x, x]) with

!""#""$g-(x) = L(x, g(x)) on [x, x]g-(x) = g-(x) = 0,

- xx

g-(x)x dx = ± ln

.1+"1'"/ ,

where L(x, y) =

!"""#"""$

'2µx+%!2x2+2% y(%x'1)(2% y'(2µ'!2)x) , p = 0

'(1'p)3!2x2+2q(1+µx)y'2p%y2

(1'p)x(2+2µx+(p2'1)!2x)+(2q+(q(2µ'!2)'2%)x)y+2p%y2 , p " 0.

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Main result

4 . If 0 < p < 1 and)

2(1'p)!2%p , µ < %p +

(1'p)!2

2 , ln.

1+"1'"/> C(µ, !, %, p).

where C(µ, !, %, p) =' k1

0

. kh-1(k)k'h1(k) '

kh-2(k)k'h2(k)

/dk =# Explicit!

k1 "2q(pµ'

02q%!2)

p(2%'2qµ+q!2'2p0

2q%!2), q " p(1 ' p)

h1(k) "2qµ+(2%+q(2µ'!2))k+

04%2k2+q2(!2k'2µ(1+k))2'4q%(2µk(1+k)+!2(k2(2p2'1)+2(1'p)2'4qk))

4p%(1+k)

h2(k) "2qµ+(2%+q(2µ'!2))k'

04%2k2+q2(!2k'2µ(1+k))2'4q%(2µk(1+k)+!2(k2(2p2'1)+2(1'p)2'4qk))

4p%(1+k)14 / 25

Main result

Remarks :

We get a direct proof to the original control problem, by constructing theshadow price process.

We get an explicit necessary and su"cient condition for finiteness of thevalue function, this was only known for two bonds market.

C(µ, 0, %, p) = ln( %%'pµ ) < ln

.1+"1'"/=# % > pµ

1' 1'"1+"

,

coinside with the result in Shreve & Soner (’94) for the two bonds market.

Sensitivity analysis for small transaction cost possible.

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Solution of the free boundary ODE (µ <)

2(1'p)!2%p )

Pick ', evolve g' until it meets blue line (at ('). Then, g-'(') = g-'((') = 0.

' ./' ('

'

g-'(x)x dx is continous

lim')x1

' ('

'

g-'(x)x dx = 0, lim

'(0

' ('

'

g-'(x)x dx = !

=# $x, x' x

x

g-(x)x dx = y ' y

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Solution of the free boundary ODE (µ <)

2(1'p)!2%p )

$x, x such that- x

xg-(x)

x dx = y ' y (by patching curves at x0).

Need to show g & C2.

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Solution of the free boundary ODE ()

2(1'p)!2%p , µ < %p +

(1'p)!2

2 )

' ./' ('

'

g-'(x)x dx is continous

lim'(0

' ('

'

g-'(x)x dx = !, lim

')!

' ('

'

g-'(x)x dx = C(µ, !, %, p)

# If y ' y > C(µ, !, %, p), then $x, x' x

x

g-(x)x dx = y ' y

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Solution of the free boundary ODE (µ > %p +(1'p)!2

2 )

g' does NOT hit the blue line, there is no solution.

Remark : In this case, we can explicity write down the trading/consumptionstrategy which leads to the infinite value function.

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Backward Construction of Shadow price (p = 0)

Let’s define f & C2([x, x]) and x0 as

f (x) " y +' x

x

g-(t)t

dt

minx&[x,x]

.g(x) + ln ($B+$Sef (x)S0)+ln %'1

%

/at x0

Let {Xt}t, {#t}t be the solution of the following Skorokhod SDE

Xt = x0 +

' t

0%Xsds +

' t

0

2%g(Xs)+(!2'2µ)Xs! dWs +#t,

where #t is the "instantaneous inward reflection" at the boundary x, x.

Shadow price : ef (Xt)St will be a Shadow price process.

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Verification (p = 0)

Let’s define

Shadow price : St " ef (Xt)St

Optimal wealth : Vt " ($B + $SS0)E. - ·

0 %XtdSs

Ss'- ·

0 %ds/

tOptimal strategy : (#0

t , #t, ct) ".(1 ' %Xt)Vt,

%XtVt

St, %Vt/

with (#00, #0) = ($B, $S)

Check (#0, #, c) is optimal for (S, frictionless) market.

d(ln #t) = 1Xt

d#t =# d#t = 1{St=St}d#)t ' 1{St=St}d#

(t .

Thus (#0, #, c) is also optimal for (S, transaction costs) market.

u(S,",") = u(S, 0, 0), conclude that S is the Shadow price process!

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Sensitivity analysis (currently working on this)

" ! ln (1 + ") =' x

x

g-(x)x dx ! C(g(x) ' g(x)) = Ch3 + o(h3)

=# u(S, 0, 0) ' u(S, 0,") = h2 +O(h3) = C"23 +O("),

Size of the no-trade region : C"13 +O("

23 )

By using the series expension of g, we may find a fractional tailor expansionfor any order.

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Summary

We get a direct proof to the original control problem, by constructing theshadow price process.

We get an explicit necessary and su"cient condition for finiteness of thevalue function.

We may do sensitivity analysis for small transaction cost, any order.

We may extract more financial interpretations from the structure of theODE.

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References

A. Skorokhod. Stochastic equations for di!usion processes in a boundedregion. Theory of Probability and its Applications, 6:264-274, 1961.

J. Kallsen and J. Muhle-Karbe. On using shadow prices in portfoliooptimization with transaction costs. Annals of Applied Probability,20:1341âAS1358, 2010.

M. Davis and A. Norman. Portfolio selection with transaction costs.Mathematics of Operations Research, 15:676-713, 1990.

S. Shreve and M. Soner. Optimal investment and consumption withtransaction costs. The Annals of Applied Probability, 4:609-692, 1994.

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Thank you!

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