Optimal Investments
Content
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Optimal portfolio under proportional transaction
costs
Petr Zahradn´ık
KPMS MFF UK; UTIA CAS
14th , 17th April 2014
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
1 Recapitulation why the size of investments matters
2 Optimal portfolio – dynamic programming
3 General market
4 Random coefficients – Results
5 Derivation
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Investments“alpha” considered crucial, the size of investments
much less worried about,
Since Markowitz portfolio selection evolved, variance
considered as a measure of risk,
Since Merton, there is a consistent theory of how “optimal”
investments in terms of investor’s utility of money or
consumption look like.
One should get investor’s utility function and then simply try
to maximize an expectation of utility at some point in future
(or over a time epoch). We are obviously not maximizing the
expectation of our payoff!
Utility functions are not arbitrary, it turns out that they are
increasing and concave. In many aspects a good class of utility
functions is the so–called CRRA (a subclass of HARA) class
p
of functions, especially power utility functions: U(x) = xp .
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Controlling a dynamic model
Suppose (Ω, Ft , F, P) and a random (in our case – “wealth”)
process Xtx,s,u which is a solution to
dXt = µ(t, Xt , ut )dt + σ(t, Xt , ut )dWt
Xs = x,
where Wt is an Ft –Brownian motion, µ and σ given functions and
ut is generally an Ft –progressively measurable real valued process
which “steers” the system. Typically, and from now on always, we
aim at Markovian control, which means ut (ω) = u(t, Xt (ω)) where
u(., .) is a real valued function of two variables.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
Our goal is to maximize
Z
J(s, x, u) = E
T
General market
Random coefficients – Results
f (t, Xtx,s,u , ut )dt
Derivation
+
s
g (XTs,x,u )
which means we search for an optimal value function
v (s, x) = suput ∈A J(s, x, u)
and the maximizing uˆt called optimal control. We assume f , g
reasonable enough and a set A of admissible controls such that
there exists a strong solution to the above mentioned equation
dXt = µ(t, Xt , ut )dt + σ(t, Xt , ut )dWt
Xs = x.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
We cope with such an optimization problem via classical Bellman
dynamic programming approach. We assume that there exists an
optimal control and that the optimal value function v (s, x) ∈ C 1,2 .
Fix (s, x) ∈ (0, T ) × R and h > 0 such that s + h < T . Fix an
arbitrary control u(t, y ) and define a new control u˜
u(t,
˜ y ) = u(t, y ) ∀t ∈ [s, s + h]
(1)
u(t,
˜ y ) = u(t,
ˆ y ) ∀t ∈ (s + h, T ].
(2)
The intuition behind this definition is clear: do whatever you come
up with for a small fraction of time, than switch to an optimal
control.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Now compare the two controls u˜t and uˆt :
the optimal control uˆt expected utility on [s, T ] is v (s, x)
u˜t : The expected utility is given by:
Z s+h
x,s,u
s,x,u
E
f (t, Xt
, ut )dt + v (s + h, Xs+h )
s
Clearly, because uˆt is optimal, it holds that
Z s+h
x,s,u
s,x,u
E
f (t, Xt
, ut )dt + v (s + h, Xs+h ) ≤ v (s, x)
s
s,x,u
v (s + h, Xs+h
) − v (s, x)
≤0
h→0
h
f (s, x, u) + E lim
where we subtracted v (s, x), divided by h and sent h → 0.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
By Ito formula:
1
dv (t, X ) =vt dt + vx dX + vxx d hX i
2
1
=vt dt + vx (µdt + σdW ) + vxx σ 2 dt,
2
Z s+h
s,x,u
v (s + h, Xs+h
) =v (s, x) +
vt (t, Xtx,s,u )dt
s
Z s+h
+
vx (t, Xtx,s,u )µ(t, Xtx,s,u , ut )dt
s
Z s+h
+
vx (t, Xtx,s,u )σ(t, Xtx,s,u , ut )dWt
s
Z
1 s+h
vxx (t, Xtx,s,u )σ 2 (t, Xtx,s,u , ut )dt.
+
2 s
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
These results yield
R s+h
vt (t, Xtx,s,u )dt
f (s, x, u) + E lim ( s
h→0
h
R s+h
x,s,u
vx (t, Xt
)µ(t, Xtx,s,u , ut )dt
+ s
h
R s+h
x,s,u
vx (t, Xt
)σ(t, Xtx,s,u , ut )dWt
+ s
h
R
x,s,u 2
1 s+h
)σ (t, Xtx,s,u , ut )dt
vxx (t, Xt
2 s
)
+
h
≤0
if v (., .) is nicely behaved, we have a true martingale above and
after switching limit and expectation one gets:
1
sup(f (s, x, u) + vt + vx µ + vxx σ 2 ) = 0.
2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
The nonlinear PDE
1
2
supu f (s, x, u) + vt + vx (s, x)µ(s, x, u) + vxx (s, x)σ (s, x, u) = 0
2
is called the Hamilton–Jacobi–Bellman equation. In its most
notorious form in the classical frictionless BS market, such an
equation becomes: (zero interest rates: dXt = γdSt )
1
2 2 2
supθ f (s, x, u) + vt + vx αxθ + vxx σ x θ = 0.
2
With one boundary condition v (x, T ) = U(x). In the classical
Merton problem maximizing utility at T , f = 0 and it is easy to
x (x,t)
compute the maximizing θˆ = σ−αv
2 xv (x,t) . The PDE can be solved
xx
analytically by guess and verify method to get:
2
x 1−p
α 1−p
α
v (t, x) =
exp
(T
−
t)
→θ=
1−p
2σ 2 p
pσ 2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
When considering maximization of utility at time T from the
probabilistic side of view, we arrive at a need to satisfy a
verification lemma:
Let v (y , t) be such that
v (y , T ) = U(y )
v (Xt , t) is a super–martingale for any controlled self–financing
portfolio process, and there exists a process Xˆt such that
v (Xˆt , t) is a martingale.
Then
v (x, s) = sup EU(XTx,s,u ).
u
Remark: The above developed optimal control posseses unrealistic
behaviour – rebalancing infinitely often.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
I concentrate on random or “stochastic” coefficients of a linear
Brownian motion, in a one–dimensional case only. Standard
Brownian motion rather than Geometric is not an original approach
and appears in Janecek, Shreve (2011) for example, or in yet
unpublished works by Petr Dostal, who reason that arithmetic
Brownian motion is even a more realistic model for futures for
example. I stick to “futures” onwards.
I postulate the following easiest possible setup. Let Xt be the
investors capital, Lt , St the cumulative number of bought, sold
futures respectively:
dFt = αt dt + σt dWt ;
dαt = at0 dt + at dWt1
dσt = bt0 dt + bt dWt2
Yt = Lt − St ; dYt = dLt − dSt = (lt − st )dt;
dXt = γt dFt − δlt dt − δst dt.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
It should be noted that Lt , St have finite variation (they are
positive nondecreasing) and hence must have zero quadratic
variation. We have a sufficiently rich filtration so that three
mutually independent Brownian motions exist, above–mentioned
W , W 1 and W 2 are however generally correlated.
Inspired by the verification lemma in the frictionless case, it is
straightforward to come up with an analogous verification lemma
for transaction costs. To know the optimal control and optimal
value and utility, it suffices to find a (value) function v (x, y , a, s, t)
such that
v (x, y , a, s, T ) = U(x)
v (Xt , Yt , αt , σt , t) is a super–martingale for any
(Xt , Yt , αt , σt , t) and there exists (an optimal control) Yˆt
such that it is a martingale.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
This lemma results in a theorem which to be proved needs only
some more Ito lemma application. Here I stick to random volatility
only to save notation, i.e. only two Brownian motions W and W 2
with instantaneous correlation ρ. The function v should then solve:
maxθ (Lv , vy − vx δ, vy + vx δ) = 0, with boundary condition
v (x, y , a, s, T ) = U(x).
Lv = vt + vxy α + 12 vxx s 2 y 2 + 21 vss a2 s 2 + vxs ys 2 aρs 2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Shreve, Soner (1994) have a similar HJB and thoroughly argue
why a solution in a general viscosity sense exists. I conjecture that
such a solution exists here too. Being interested in the optimal
policy, hence in the optimal wedge, Janecek, Shreve (2011) have in
a similar setup proved that once coefficients are constant, via
decomposing the loss in utility caused by transaction costs and
t
displacement, there is an optimal wedge too. Denote πt = Y
Xt ,
α
θˆ = pσ
2 , the wedge is of the following form:
π ∈ (θˆ − xˆt , θˆt + xˆt )
where
xˆt =
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
3δ ˆ4
θ
2p
1/3
.
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Now we get back to our verification lemma and proceed to solving:
maxy (Lv , vy − vx δ, vy + vx δ) = 0, with boundary condition
v (x, y , s, T ) = U(x).
Lv = vt + vxy α + 12 vxx s 2 y 2 + 21 vss a2 σ 2 + vxs ys 2 aρσ 2
This can be tried via asymptotic expansion - a method known in
fluid mechanics etc. where it is used extensively to conjecture the
qualities of generally unknown solutions. Other approaches include
various probabilistic approaches in which a shape of the value
function is “guessed” and one tries to make the “martingale
property” hold almost – Landau symbols are used here too.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
An interesting possibility is proposed by Petr Dostal, who opts for
maximizing certainty equivalent. The goal of such an approach is
ˆ
to find a process νt and an optimal strategy such that e γ Ut is a
martingale and for any other strategy e γUt is a super–martingale,
where:
Z t
Ut = log Xt −
νs ds.
0
Intuition:
assume log Xn −
assume
P
e γ(log Xn −
νk
Pn
ν
n k)
a martingale
a martingale
Interestingly enough, several formulations, as much as they differ
in the value of the task, lead to the same optimal behaviour:
WHICH IS WHAT WE WOULD HOPE FOR IN REAL WORLD
PROBLEMS!
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
General Result:
π ∈ (θˆ − xˆt , θˆt + xˆt )
1
3δ ˆ4
d[π]t 3
2 d[π, F ]t
ˆ
xˆt =
(θ + 2 θ
+
)
2p
d[F ]t
d[F ]t
In other words, if we make a (reality driven) assumption
dθt = φdt + ΦdWt + Ψ1t dWt1 + Ψ2t dWt2 :
1
3
3δ
K (θ) , where:
xˆt =
2p
2
2
2
σt θ2 + Φt + Ψ1 + Ψ2
K (θ) =
.
σt2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Some sanity checks:
2
for nonrandom coefficients: K (θ) =
(σt θ2 )
σ2
.
Φ < 0 leads to smaller wedges.
Ψ1 6= 0,Ψ2 6= 0 lead to greater wedges.
Ψ1 → ∞,Ψ2 → ∞ lead to no trading.
...
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Remarks on the structure:
transaction costs lead to reduced wealth, so the wedge is not
centered around the no transaction-costs position ceteris
paribus. However, it is centered around the same relative
position.
transaction costs increase the width of the interval, higher
costs lead to less activity.
risk tolerance plays a role in the width of the wedge as
expected - more risk aversion leads to closer tracking.
risk tolerance plays role in the optimal relative position which decreases with increasing aversion. This is as expected
- certainty equivalent scales in the same way.
quadratic variation of the risky asset is in the denominator –
more volatility leads to closer tracking.
...
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
First step - homotheticity of the value function
Proposition (Homotheticity of type I)
v (t, kx, ky , s) = k 1−p v (t, x, y , s)
Proposition (Homotheticity of type II)
v (t, x, y , s; α, δ) = v (t, x, ky , s/k; α/k, δ/k)
Proofs: The first is deduced from the properties of a power utility
function. The second is independent of the choice, comes from the
arithmetic Brownian motion setup.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Dimensions reduction
From the type I homotheticity:
v (t, x, y , s) = x 1−p v (t, 1, y /x, s) = x 1−p u(t, z, s)
Hence it might be useful to drop one dimension and consider the
variable y /x instead. This is motivated not only by technical
advantages but by resemblance to the no transaction case too. We
are onwards using the function u, often dropping variables which
remain fixed in the computation. Subscripts onwards denote
derivatives with respect to the respective variable.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Solutions in buy and sell regions
v (x, y , s, t) = x 1−p u(z, s, t)
vx (x, y ) = x −p ((1 − p)u(z) − uz (z)z)
vy (x, y ) = x −p uz (z)
In the buy resp. sell regions, with such a dimension reduction the
PDEs actually reduce to ODEs, let us solve it for the buy region,
for z < zl :
−δ(1 − p)u(z) = uz (z)(1 − δz)
u(z) = u(zl )(1 + δz)1−p
Similarly for the sell region u(z) = u(zs )(1 − δz)1−p (For risk
averse investors (p > 1) it is worse to oversize.)
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
NT region: asymptotic expansion
These were the exact solutions in the NT region.
In the NT region, the solution can’t be found in a closed form,
one of the possibilities to tackle this is the asymptotic
expansion. We can employ continuity arguments of the value
function and of its derivatives (we used Itos formula!) and the
knowledge of the solutions in the buy and sell regions.
An interesting question arises, in what powers is it reasonable
to perform the expansion? The powers of δ 1/3 are to be found
Janecek, Shreve(2004) and Rogers(2004) argues, why it must
be that way. His intuition is actually included in Janecek,
Shreve(2004) and Janecek, Shreve(2011) partially too.
Instead of z we use a more natural scaled Z from
z = z ∗ (t, s) + δ 1/3 Z .
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
The expansion may be tried as follows. Notice how Z appears. This
is because of the continutity arguments from the buy sell regions.
u(t, Z , s; δ) = [H0 (t, s) + δ 1/3 ZH1 (t, s) + δ 2/3 H2 (t, s) + δH3 (t, s) . . .
+ δ 4/3 H4 (t, Z , s) + δ 5/3 H5 (t, Z , s) . . .]
Now we must compute derivatives and do some miracles with the
PDEs connected with respective powers. I haven’t succeeded in
deriving the optimal wedge bounds for random volatility yet. I have
however succeeded in deliberately deriving the optimal wedge
bounds for constant volatility via such an expansion and they
coincide with the result of Janecek, Shreve (2011)and others:
3δ ∗ 4
∗
(z ) + O(δ 2/3 ).
zb , zs = z ±
2p
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
I am however quite confident that this approach should lead to the
desired outcome. The reformulation of Petr Dostal also works well
– but the proof techniques are not ideal in my opinion.
The message is: we have a result which seems ok by several means
of derivation, so we probably have a rule for investors. Now we
need to find a good enough formal language to make it publishable
in a scientific paper.
Thank you for attention!
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Optimal portfolio under proportional transaction
costs
Petr Zahradn´ık
KPMS MFF UK; UTIA CAS
14th , 17th April 2014
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
1 Recapitulation why the size of investments matters
2 Optimal portfolio – dynamic programming
3 General market
4 Random coefficients – Results
5 Derivation
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Investments“alpha” considered crucial, the size of investments
much less worried about,
Since Markowitz portfolio selection evolved, variance
considered as a measure of risk,
Since Merton, there is a consistent theory of how “optimal”
investments in terms of investor’s utility of money or
consumption look like.
One should get investor’s utility function and then simply try
to maximize an expectation of utility at some point in future
(or over a time epoch). We are obviously not maximizing the
expectation of our payoff!
Utility functions are not arbitrary, it turns out that they are
increasing and concave. In many aspects a good class of utility
functions is the so–called CRRA (a subclass of HARA) class
p
of functions, especially power utility functions: U(x) = xp .
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Controlling a dynamic model
Suppose (Ω, Ft , F, P) and a random (in our case – “wealth”)
process Xtx,s,u which is a solution to
dXt = µ(t, Xt , ut )dt + σ(t, Xt , ut )dWt
Xs = x,
where Wt is an Ft –Brownian motion, µ and σ given functions and
ut is generally an Ft –progressively measurable real valued process
which “steers” the system. Typically, and from now on always, we
aim at Markovian control, which means ut (ω) = u(t, Xt (ω)) where
u(., .) is a real valued function of two variables.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
Our goal is to maximize
Z
J(s, x, u) = E
T
General market
Random coefficients – Results
f (t, Xtx,s,u , ut )dt
Derivation
+
s
g (XTs,x,u )
which means we search for an optimal value function
v (s, x) = suput ∈A J(s, x, u)
and the maximizing uˆt called optimal control. We assume f , g
reasonable enough and a set A of admissible controls such that
there exists a strong solution to the above mentioned equation
dXt = µ(t, Xt , ut )dt + σ(t, Xt , ut )dWt
Xs = x.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
We cope with such an optimization problem via classical Bellman
dynamic programming approach. We assume that there exists an
optimal control and that the optimal value function v (s, x) ∈ C 1,2 .
Fix (s, x) ∈ (0, T ) × R and h > 0 such that s + h < T . Fix an
arbitrary control u(t, y ) and define a new control u˜
u(t,
˜ y ) = u(t, y ) ∀t ∈ [s, s + h]
(1)
u(t,
˜ y ) = u(t,
ˆ y ) ∀t ∈ (s + h, T ].
(2)
The intuition behind this definition is clear: do whatever you come
up with for a small fraction of time, than switch to an optimal
control.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Now compare the two controls u˜t and uˆt :
the optimal control uˆt expected utility on [s, T ] is v (s, x)
u˜t : The expected utility is given by:
Z s+h
x,s,u
s,x,u
E
f (t, Xt
, ut )dt + v (s + h, Xs+h )
s
Clearly, because uˆt is optimal, it holds that
Z s+h
x,s,u
s,x,u
E
f (t, Xt
, ut )dt + v (s + h, Xs+h ) ≤ v (s, x)
s
s,x,u
v (s + h, Xs+h
) − v (s, x)
≤0
h→0
h
f (s, x, u) + E lim
where we subtracted v (s, x), divided by h and sent h → 0.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
By Ito formula:
1
dv (t, X ) =vt dt + vx dX + vxx d hX i
2
1
=vt dt + vx (µdt + σdW ) + vxx σ 2 dt,
2
Z s+h
s,x,u
v (s + h, Xs+h
) =v (s, x) +
vt (t, Xtx,s,u )dt
s
Z s+h
+
vx (t, Xtx,s,u )µ(t, Xtx,s,u , ut )dt
s
Z s+h
+
vx (t, Xtx,s,u )σ(t, Xtx,s,u , ut )dWt
s
Z
1 s+h
vxx (t, Xtx,s,u )σ 2 (t, Xtx,s,u , ut )dt.
+
2 s
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
These results yield
R s+h
vt (t, Xtx,s,u )dt
f (s, x, u) + E lim ( s
h→0
h
R s+h
x,s,u
vx (t, Xt
)µ(t, Xtx,s,u , ut )dt
+ s
h
R s+h
x,s,u
vx (t, Xt
)σ(t, Xtx,s,u , ut )dWt
+ s
h
R
x,s,u 2
1 s+h
)σ (t, Xtx,s,u , ut )dt
vxx (t, Xt
2 s
)
+
h
≤0
if v (., .) is nicely behaved, we have a true martingale above and
after switching limit and expectation one gets:
1
sup(f (s, x, u) + vt + vx µ + vxx σ 2 ) = 0.
2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
The nonlinear PDE
1
2
supu f (s, x, u) + vt + vx (s, x)µ(s, x, u) + vxx (s, x)σ (s, x, u) = 0
2
is called the Hamilton–Jacobi–Bellman equation. In its most
notorious form in the classical frictionless BS market, such an
equation becomes: (zero interest rates: dXt = γdSt )
1
2 2 2
supθ f (s, x, u) + vt + vx αxθ + vxx σ x θ = 0.
2
With one boundary condition v (x, T ) = U(x). In the classical
Merton problem maximizing utility at T , f = 0 and it is easy to
x (x,t)
compute the maximizing θˆ = σ−αv
2 xv (x,t) . The PDE can be solved
xx
analytically by guess and verify method to get:
2
x 1−p
α 1−p
α
v (t, x) =
exp
(T
−
t)
→θ=
1−p
2σ 2 p
pσ 2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
When considering maximization of utility at time T from the
probabilistic side of view, we arrive at a need to satisfy a
verification lemma:
Let v (y , t) be such that
v (y , T ) = U(y )
v (Xt , t) is a super–martingale for any controlled self–financing
portfolio process, and there exists a process Xˆt such that
v (Xˆt , t) is a martingale.
Then
v (x, s) = sup EU(XTx,s,u ).
u
Remark: The above developed optimal control posseses unrealistic
behaviour – rebalancing infinitely often.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
I concentrate on random or “stochastic” coefficients of a linear
Brownian motion, in a one–dimensional case only. Standard
Brownian motion rather than Geometric is not an original approach
and appears in Janecek, Shreve (2011) for example, or in yet
unpublished works by Petr Dostal, who reason that arithmetic
Brownian motion is even a more realistic model for futures for
example. I stick to “futures” onwards.
I postulate the following easiest possible setup. Let Xt be the
investors capital, Lt , St the cumulative number of bought, sold
futures respectively:
dFt = αt dt + σt dWt ;
dαt = at0 dt + at dWt1
dσt = bt0 dt + bt dWt2
Yt = Lt − St ; dYt = dLt − dSt = (lt − st )dt;
dXt = γt dFt − δlt dt − δst dt.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
It should be noted that Lt , St have finite variation (they are
positive nondecreasing) and hence must have zero quadratic
variation. We have a sufficiently rich filtration so that three
mutually independent Brownian motions exist, above–mentioned
W , W 1 and W 2 are however generally correlated.
Inspired by the verification lemma in the frictionless case, it is
straightforward to come up with an analogous verification lemma
for transaction costs. To know the optimal control and optimal
value and utility, it suffices to find a (value) function v (x, y , a, s, t)
such that
v (x, y , a, s, T ) = U(x)
v (Xt , Yt , αt , σt , t) is a super–martingale for any
(Xt , Yt , αt , σt , t) and there exists (an optimal control) Yˆt
such that it is a martingale.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
This lemma results in a theorem which to be proved needs only
some more Ito lemma application. Here I stick to random volatility
only to save notation, i.e. only two Brownian motions W and W 2
with instantaneous correlation ρ. The function v should then solve:
maxθ (Lv , vy − vx δ, vy + vx δ) = 0, with boundary condition
v (x, y , a, s, T ) = U(x).
Lv = vt + vxy α + 12 vxx s 2 y 2 + 21 vss a2 s 2 + vxs ys 2 aρs 2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Shreve, Soner (1994) have a similar HJB and thoroughly argue
why a solution in a general viscosity sense exists. I conjecture that
such a solution exists here too. Being interested in the optimal
policy, hence in the optimal wedge, Janecek, Shreve (2011) have in
a similar setup proved that once coefficients are constant, via
decomposing the loss in utility caused by transaction costs and
t
displacement, there is an optimal wedge too. Denote πt = Y
Xt ,
α
θˆ = pσ
2 , the wedge is of the following form:
π ∈ (θˆ − xˆt , θˆt + xˆt )
where
xˆt =
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Optimal portfolio under proportional transaction costs
3δ ˆ4
θ
2p
1/3
.
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Now we get back to our verification lemma and proceed to solving:
maxy (Lv , vy − vx δ, vy + vx δ) = 0, with boundary condition
v (x, y , s, T ) = U(x).
Lv = vt + vxy α + 12 vxx s 2 y 2 + 21 vss a2 σ 2 + vxs ys 2 aρσ 2
This can be tried via asymptotic expansion - a method known in
fluid mechanics etc. where it is used extensively to conjecture the
qualities of generally unknown solutions. Other approaches include
various probabilistic approaches in which a shape of the value
function is “guessed” and one tries to make the “martingale
property” hold almost – Landau symbols are used here too.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
An interesting possibility is proposed by Petr Dostal, who opts for
maximizing certainty equivalent. The goal of such an approach is
ˆ
to find a process νt and an optimal strategy such that e γ Ut is a
martingale and for any other strategy e γUt is a super–martingale,
where:
Z t
Ut = log Xt −
νs ds.
0
Intuition:
assume log Xn −
assume
P
e γ(log Xn −
νk
Pn
ν
n k)
a martingale
a martingale
Interestingly enough, several formulations, as much as they differ
in the value of the task, lead to the same optimal behaviour:
WHICH IS WHAT WE WOULD HOPE FOR IN REAL WORLD
PROBLEMS!
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
General Result:
π ∈ (θˆ − xˆt , θˆt + xˆt )
1
3δ ˆ4
d[π]t 3
2 d[π, F ]t
ˆ
xˆt =
(θ + 2 θ
+
)
2p
d[F ]t
d[F ]t
In other words, if we make a (reality driven) assumption
dθt = φdt + ΦdWt + Ψ1t dWt1 + Ψ2t dWt2 :
1
3
3δ
K (θ) , where:
xˆt =
2p
2
2
2
σt θ2 + Φt + Ψ1 + Ψ2
K (θ) =
.
σt2
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Some sanity checks:
2
for nonrandom coefficients: K (θ) =
(σt θ2 )
σ2
.
Φ < 0 leads to smaller wedges.
Ψ1 6= 0,Ψ2 6= 0 lead to greater wedges.
Ψ1 → ∞,Ψ2 → ∞ lead to no trading.
...
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Remarks on the structure:
transaction costs lead to reduced wealth, so the wedge is not
centered around the no transaction-costs position ceteris
paribus. However, it is centered around the same relative
position.
transaction costs increase the width of the interval, higher
costs lead to less activity.
risk tolerance plays a role in the width of the wedge as
expected - more risk aversion leads to closer tracking.
risk tolerance plays role in the optimal relative position which decreases with increasing aversion. This is as expected
- certainty equivalent scales in the same way.
quadratic variation of the risky asset is in the denominator –
more volatility leads to closer tracking.
...
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
First step - homotheticity of the value function
Proposition (Homotheticity of type I)
v (t, kx, ky , s) = k 1−p v (t, x, y , s)
Proposition (Homotheticity of type II)
v (t, x, y , s; α, δ) = v (t, x, ky , s/k; α/k, δ/k)
Proofs: The first is deduced from the properties of a power utility
function. The second is independent of the choice, comes from the
arithmetic Brownian motion setup.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Dimensions reduction
From the type I homotheticity:
v (t, x, y , s) = x 1−p v (t, 1, y /x, s) = x 1−p u(t, z, s)
Hence it might be useful to drop one dimension and consider the
variable y /x instead. This is motivated not only by technical
advantages but by resemblance to the no transaction case too. We
are onwards using the function u, often dropping variables which
remain fixed in the computation. Subscripts onwards denote
derivatives with respect to the respective variable.
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
Solutions in buy and sell regions
v (x, y , s, t) = x 1−p u(z, s, t)
vx (x, y ) = x −p ((1 − p)u(z) − uz (z)z)
vy (x, y ) = x −p uz (z)
In the buy resp. sell regions, with such a dimension reduction the
PDEs actually reduce to ODEs, let us solve it for the buy region,
for z < zl :
−δ(1 − p)u(z) = uz (z)(1 − δz)
u(z) = u(zl )(1 + δz)1−p
Similarly for the sell region u(z) = u(zs )(1 − δz)1−p (For risk
averse investors (p > 1) it is worse to oversize.)
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
NT region: asymptotic expansion
These were the exact solutions in the NT region.
In the NT region, the solution can’t be found in a closed form,
one of the possibilities to tackle this is the asymptotic
expansion. We can employ continuity arguments of the value
function and of its derivatives (we used Itos formula!) and the
knowledge of the solutions in the buy and sell regions.
An interesting question arises, in what powers is it reasonable
to perform the expansion? The powers of δ 1/3 are to be found
Janecek, Shreve(2004) and Rogers(2004) argues, why it must
be that way. His intuition is actually included in Janecek,
Shreve(2004) and Janecek, Shreve(2011) partially too.
Instead of z we use a more natural scaled Z from
z = z ∗ (t, s) + δ 1/3 Z .
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
The expansion may be tried as follows. Notice how Z appears. This
is because of the continutity arguments from the buy sell regions.
u(t, Z , s; δ) = [H0 (t, s) + δ 1/3 ZH1 (t, s) + δ 2/3 H2 (t, s) + δH3 (t, s) . . .
+ δ 4/3 H4 (t, Z , s) + δ 5/3 H5 (t, Z , s) . . .]
Now we must compute derivatives and do some miracles with the
PDEs connected with respective powers. I haven’t succeeded in
deriving the optimal wedge bounds for random volatility yet. I have
however succeeded in deliberately deriving the optimal wedge
bounds for constant volatility via such an expansion and they
coincide with the result of Janecek, Shreve (2011)and others:
3δ ∗ 4
∗
(z ) + O(δ 2/3 ).
zb , zs = z ±
2p
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
Motivation
Optimal portfolio – dynamic programming
General market
Random coefficients – Results
Derivation
I am however quite confident that this approach should lead to the
desired outcome. The reformulation of Petr Dostal also works well
– but the proof techniques are not ideal in my opinion.
The message is: we have a result which seems ok by several means
of derivation, so we probably have a rule for investors. Now we
need to find a good enough formal language to make it publishable
in a scientific paper.
Thank you for attention!
Petr Zahradn´ık
Optimal portfolio under proportional transaction costs
KPMS MFF UK; UTIA CAS
