Adaptive mean-variance optimal order

execution and Dawson–Watanabe

superprocesses

Alexander SchiedUniversitat Mannheim

SIAM Conference on Financial Mathematics and Engineering 2012MinneapolisJuly 9, 2012

1

The Almgren–Chriss market impact model

Bertsimas & Lo (1998), Almgren & Chriss (1999, 2000), Almgren (2003)

x(t) = asset position of investor at time t

assumed to be an absolutely continuous function of time

realized price process:

Sxt = S0

t + �

Z t

0x(s) ds + ⌘x(t)

" " "una↵ected permanent temporary

price process impact impact

2

Optimization over adapted and absolutely continuous trajectories (x(t))0tT

with x0 given and liquidation time constraint x(T ) = 0

0.2 0.4 0.6 0.8 1.0

1

2

3

4

5

The costs of such a strategy are given by

costs(x) = x0S0 �Z T

0Sx

t (�x(t)) dt

=�

2x2

0 �Z T

0x(t) dS0

t + ⌘

Z T

0x(t)2 dt

3

Practitioners like to find adaptive strategies that minimize a mean-variancefunctional

E[ costs(x) ] + � var�costs(x)

�But this is not so easy due to the time inconsistency of the variance:

var(Z) = E[Z2 ]� E[Z ]2

(see e.g. Lorenz & Almgren (2011)).

Continuous re-optimization leads to the alternative time-consistent costfunction

Eh⌘

Z T

0|x(t)|2 dt + �

Z[0,T ]

|x(t)|2 d[S0]ti

This was observed, e.g., by Konishi & Makimoto (2001), Schoneborn (2008);see also Bjork et al. (2010), Ekeland & Lazrak (2008), Forsyth et al. (2011)

4

Almost all price processes S0 in financial mathematics are of the form

S0t = �(Zt)

For a (time-inhomogeneous) Markov process Z = (Zt, Pr,z).

Generalized finite-fuel control problem: minimize

E0,z

h Z T

0|x(s)|p⌘(Zs) ds +

Z[0,T ]

|x(s)|p A(ds)i

over progressively measurable and absolutely continuous strategies x withx(0) = x0 and x(T ) = 0

Here:• A is a nonnegative additive functional of Z. E.g., A(dt) = d[�(Z)]t = d[S0]t• ⌘ is a strictly positive function

W.l.o.g.: x is monotone

5

Heuristic solution

Assume A(du) = ↵(Zu) du and let

V (t, z, x0) := infx(·)

Et,z

h Z T

t|x(u)|p⌘(Zu) du +

Z T

t|x(u)|pa(Zu) du

i.

Optimal control suggest that V should satisfy the following HJB equation

@V

@t(t, z, x0) + inf

⇠

n⌘(z)|⇠|p +

@V

@x0(t, z, x0)⇠

o+ ↵(z)|x0|p + LtV (t, z, x0) = 0

with singular terminal condition

V (T, z, x0) =

8<:

0 if x0 = 0,

+1 otherwise.

6

We make the ansatzV (t, z, x0) = |x0|pv(t, z)

for some function v. Plugging this ansatz into the HJB eqn, minimizing over⇠, and dividing by |x0|p, yields the PDE

vt �1�⌘�

v1+� + ↵+ Ltv = 0,

v(T, z) = +1,

where� =

1p� 1

.

Moreover, the minimizing ⇠ in the HJB eqn is given by ⇠ = �xv�/⌘� , whichsuggestes that the optimal strategy is a solution of the o.d.e.

x(u) = �x(u)v(u,Zu)�

⌘(Zu)�,

i.e.,

x(u) = x0 exp⇣Z u

t�v(s, Zs)�

⌘(Zs)�ds

⌘.

7

Dynkin (1991, 1992): for uniformly elliptic di↵usions on Rd, constant �, and� 2 (0, 1] (() p � 2), singular PDEs of the form

vt � �v1+� + ↵+ Ltv = 0,

v(T, z) = +1,

can be solved by means of Dawson–Watanabe superprocesses (and hence viaMonte Carlo techniques . . . )

Dawson–Watanabe superprocesses are measure-valued Markov processesmotivated by describing the spatial evolution of colonies of bacteria . . .

8

Xt =X

�Zit

9

Xt =X

�Zit

The superprocess with � = 1 is obtained by starting with individual N

particles, speeding up branching by a factor N , assigning mass 1N to each

particle, and then sending N to infinity

9

The superprocess X = (Xt, Pr,µ) can be characterized by its Laplacefunctionals:

Er,µ[ e�hf,XT i ] = e�hv(r,·),µi, f � 0 and bounded,

where hf, µi is shorthand forR

f dµ and v is the unique positive mild solutionof the quasilinear PDE

vt + Ltv � �v1+� = 0

v(T, z) = f(z)

That is

v(r, z) = Er,z[ f(ZT ) ]�Er,z

h�

Z T

rv(t, Zt)1+� dt

i

Remark: Note that the superprocess is an a�ne process

10

Example: Taking f equal to a constant k > 0, we see that

v(r, z) = k �Er,z

h�

Z T

rv(t, Zt)1+� dt

i

is solved by

vk(r, z) =1

(k�� + ��(T � t))1/�

Sending k to infinity yields

� log Pr,�z [XT = 0 ] = � limk"1

log Er,�z [ e�kh1,XT i ]

= limk"1

vk(r, z)

=1

(��(T � r))1/�.

11

The characterization of Laplace functionals extends to nonnegative additivefunctionals A of Z of the form

A[r, t] =X

rtit

fi(Zti)

by means of the J-functional

IA =X

i

hfi,Xtii

and to suitable limits thereof

� log Er,µ[ e�IA ] = hv(r, ·), µi,

where v(r, z) solves

v(r, z) = Er,z[A[r, T ] ]� �Er,z

h Z T

rv(s, Zs)1+� ds

i, 0 r T.

12

Problem: For St = �(Zt), the additive functional A(dt) = d[S]t may not beof this form unless Z is a path process, e.g.,

Zt = (Ss^t)s�0

In this case, A(dt) = d[S]t can be approximated by additive functionals

An[r, t] =X

rti�1, tit

�Zti(ti)� Zti(ti�1)

�2

| {z }=fi(Zti )

The corresponding superprocess is the historical superprocess (Dawson &Perkins (1991), Dynkin (1991))

But in this situation there is no hope to get smoothness of the solutions of thecorresponding log–Laplace equations ; we will work only with mild solutionshere

13

Theorem 1 (Case ⌘ = 1). For p � 2 let q be such that 1p + 1

q = 1 and take A

such that Er,z[A[r, T ]q ] <1 for all r, z. Let IA be the correspondingJ-functional of the superprocess with parameters Z, � := p� 1, and � := 1

� ,and define the function

v1(r, z) = � log Er,�z [ e�IA · 1{XT =0} ].

For�(⇠, ⌘) := ⇠p � p⌘p�1⇠ + (p� 1)⌘p, ⇠, ⌘ � 0

the costs of any admissible strategy x are

E0,z

h Z T

0|x(s)|p ds +

Z[0,T ]

|x(s)|p A(ds)i

= |x0|pv1(0, z) + E0,z

h Z T

0��|x(t)|, |x(t)|v1(t, Zt)1/(p�1)

�dt

i

The unique strategy minimizing these costs is

x⇤(t) := x0 exp⇣�

Z t

0v1(s, Zs)� ds

⌘

and the minimal costs are |x0|pv1(0, z)14

Idea of proof:

First step: probabilistic verification argument without smoothnessFor k 2 N,

(1) vk(r, z) = � log Er,�z [ e�IA�k<1,XT i ].

Then vk % v1 and vk uniquely solves

vk(r, z) = k + Er,z

⇥A[r, T ]

⇤� �Er,z

h Z T

rvk(s, Zs)1+� ds

i.

For an admissible strategy x � 0, let

(2) Ckt :=

Z t

0|x(s)|p ds +

Z[0,t]

x(s)p A(ds) + x(t)pvk(t, Zt).

Show then

(3) Ct �! CT :=Z T

0|x(s)|p ds +

Z[0,T ]

x(s)p A(ds) P0,z-a.s. as t " T .

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We next let

(4) Mkt := k + E0,z

hA[0, T ]� �

Z T

0vk(s, Zs)1+� ds

���Ft

i

so that

(5) Vt := vk(t, Zt) = Mkt �A[0, t) + �

Z t

0vk(s, Zs)1+� ds

Ito’s formula yields

dCkt = |x(t)|p dt + x(t)p A(dt) + px(t)p�1x(t)Vt dt + x(t)p dVt

=⇣|x(t)|p + px(t)p�1x(t)Vt + �x(t)pV 1+�

t

⌘dt + x(t)p dMk

t

= ��|x(t)|, x(t)vk(t, Zt)1/(p�1)

�dt + x(t)p dMk

t ,

where, in the last step, we have used the relations 1 + � = p/(p� 1) and� = ��1 = p� 1

16

Using (3), we obtain in the limit t " T thatZ T

0|x(s)|p ds +

Z[0,T ]

x(s)p A(ds)� xp0vk(0, z)

= CkT � Ck

0

=Z T

0��|x(t)|, x(t)vk(t, Zt)1/(p�1)

�dt +

Z T

0x(t)p dMk

t .

Taking expectations yields

E0,z

h Z T

0|x(s)|p ds +

Z[0,T ]

x(s)p A(ds)i

= xp0vk(0, z) + E0,z

h Z T

0��|x(t)|, x(t)vk(t, Zt)1/(p�1)

�dt

i.

Then argue that we may pass to the limit k " 1 on the right-hand side.

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Second step: The more di�cult part of the proof is to show that

x⇤(t) := x0 exp⇣�

Z t

0v1(s, Zs)� ds

⌘

is an admissible strategy, i.e., x⇤(t)! 0 as t " T and that x⇤ has a finite cost.

Based on auxiliary results that are of independent interest. E.g.,

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Theorem 2 (Estimates for Laplace functionals of J-functionals).For the superprocess with homogeneous branching rate �, let IA be aJ-functional. Define moreover the Borel measure ↵r,z(ds) on [r, T ] byZ

f(s)↵r,z(ds) = Er,z

h Z[r,T ]

f(s)A(ds)i

for bounded measurable f : [r, T ]! R. Then

(6) Er,�z

⇥IA

�� h1,Xti, r t T⇤

=Z

[r,T ]h1,Xti↵r,z(dt)

and

(7) � log Er,�z [ e�IA ] � log Er,�z

⇥e�

R[r,T ]h1,Xti↵r,z(dt) ⇤

=: u(r)

and u solves the ordinary integral equation

u(t) = ↵r,z([t, T ])� �Z T

tu(s)1+� ds, r t T.

19

The case of nonconstant ⌘(·)

Need a superprocess with Laplace functionals

� log Er,�z [ e�IA ] = v(r, ·)

where v solves

v(r, z) = Er,z

hA[r, T ]�

Z t

rv(s, Zs)1+� 1

�⌘(Zs)�ds

i

Such superprocesses were constructed in A.S. (1999) by means of the followingh-transform technique that was also introduced independently by Englanderand Pinsky (1999)

20

Assume that

(8)1cT⌘(z) Er,z[ ⌘(Zt) ] cT ⌘(z), 0 r t T

and define the space-time harmonic function

(9) h(r, z) := Er,z[ ⌘(ZT ) ]

Let Zh = (Zt, Phr,z) be the h-transform of Z, i.e.,

Phr,z[A ] =

1h(r, z)

Er,z[ ⌘(ZT )1A

], A 2 FT ,

For an additive functional B of Z define an additive functional Bh of Zh by

Bh(ds) =1

h(s, Zs)B(ds).

With this notation,

Kh(ds) =1

h(s, Zs)K(ds) =

⌘(Zs)�h(s, Zs)

ds

is a bounded and continuous additive functional of Zh.21

For

(z, ⇠) =⇣ ⇠

⌘(z)

⌘1+�

the function h(t, z, ⇠) = (z, h(t, z)⇠)

is bounded in t and z for each ⇠. Therefore we can define the(Zh, h,Kh)-superprocess Xh = (Xh

t , Phr,µ). The superprocess X under Pr,µ is

then defined as the law of

Xt(dz) :=1

h(t, z)Xh

t (dz)

Since for any additive functional B of Z

Er,z

⇥B[r, t]

⇤= Er,z

h Z[r,t]

h(s, Zs)Bh(ds)i

= Er,z

⇥Bh[r, t] · ⌘(ZT )

⇤

= h(r, z) · Ehr,z[Bh[r, t] ]

one checks that X has the desired Laplace functionals.

22

Then get estimates for the Laplace functionals for Xh by comparison to thecase of constant branching by using:

Proposition 1 (Parabolic maximum principle for Laplace eqn).Suppose that A, eA and K, eK are nonnegative additive functionals of Z withK, eK continuous. Suppose furthermore that A[r, t] eA[r, t] andK[r, t] � eK[r, t] for r t T . Let v and ev be finite and nonnegative solutionsof the integral equations

v(r, z) = Er,z

hA[r, T ]�

Z T

rv(t, Zt)1+� K(dt)

i

ev(r, z) = Er,z

h eA[r, T ]�Z T

rev(t, Zt)1+� eK(dt)

i

Then v ev.

This then gives estimates for the Laplace functionals of X via the“h-transform” for superprocesses

23

Thank you

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