optimal investments

Motivation Optimal portfolio – dynamic programming General market Random coefficients – Results Derivation

Optimal portfolio under proportional transactioncosts

Petr Zahradnık

KPMS MFF UK; UTIA CAS

14th, 17th April 2014

Petr Zahradnık KPMS MFF UK; UTIA CAS

Optimal portfolio under proportional transaction costs


1 Recapitulation why the size of investments matters

2 Optimal portfolio – dynamic programming

3 General market

4 Random coefficients – Results

5 Derivation




Investments“alpha” considered crucial, the size of investmentsmuch less worried about,

Since Markowitz portfolio selection evolved, varianceconsidered as a measure of risk,

Since Merton, there is a consistent theory of how “optimal”investments in terms of investor’s utility of money orconsumption look like.

One should get investor’s utility function and then simply tryto maximize an expectation of utility at some point in future(or over a time epoch). We are obviously not maximizing theexpectation of our payoff!

Utility functions are not arbitrary, it turns out that they areincreasing and concave. In many aspects a good class of utilityfunctions is the so–called CRRA (a subclass of HARA) classof functions, especially power utility functions: U(x) = xp

p .




Controlling a dynamic model

Suppose (Ω,Ft ,F ,P) and a random (in our case – “wealth”)process X x ,s,u

t 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 andut is generally an Ft–progressively measurable real valued processwhich “steers” the system. Typically, and from now on always, weaim at Markovian control, which means ut(ω) = u(t,Xt(ω)) whereu(., .) is a real valued function of two variables.




Our goal is to maximize

J(s, x , u) = E

(∫ T

sf (t,X x ,s,u

t , ut)dt + g(X s,x ,uT )

)which means we search for an optimal value function

v(s, x) = suput∈AJ(s, x , u)

and the maximizing ut called optimal control. We assume f , greasonable enough and a set A of admissible controls such thatthere exists a strong solution to the above mentioned equation

dXt = µ(t,Xt , ut)dt + σ(t,Xt , ut)dWt

Xs = x .




We cope with such an optimization problem via classical Bellmandynamic programming approach. We assume that there exists anoptimal control and that the optimal value function v(s, x) ∈ C1,2.Fix (s, x) ∈ (0,T )× R and h > 0 such that s + h < T . Fix anarbitrary 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, y) ∀t ∈ (s + h,T ]. (2)

The intuition behind this definition is clear: do whatever you comeup with for a small fraction of time, than switch to an optimalcontrol.




Now compare the two controls ut and ut :

the optimal control ut expected utility on [s,T ] is v(s, x)

ut : The expected utility is given by:

E

[∫ s+h

sf (t,X x ,s,u

t , ut)dt + v(s + h,X s,x ,us+h )

]Clearly, because ut is optimal, it holds that

E

[∫ s+h

sf (t,X x ,s,u

t , ut)dt + v(s + h,X s,x ,us+h )

]≤ v(s, x)

f (s, x , u) + E limh→0

v(s + h,X s,x ,us+h )− v(s, x)

h≤ 0

where we subtracted v(s, x), divided by h and sent h→ 0.




By Ito formula:

dv(t,X ) =vtdt + vxdX +1

2vxxd 〈X 〉

=vtdt + vx(µdt + σdW ) +1

2vxxσ

2dt,

v(s + h,X s,x ,us+h ) =v(s, x) +

∫ s+h

svt(t,X

x ,s,ut )dt

+

∫ s+h

svx(t,X x ,s,u

t )µ(t,X x ,s,ut , ut)dt

+

∫ s+h

svx(t,X x ,s,u

t )σ(t,X x ,s,ut , ut)dWt

+1

2

∫ s+h

svxx(t,X x ,s,u

t )σ2(t,X x ,s,ut , ut)dt.




These results yield

f (s, x , u) + E limh→0

(

∫ s+hs vt(t,X

x ,s,ut )dt

h

+

∫ s+hs vx(t,X x ,s,u

t )µ(t,X x ,s,ut , ut)dt

h

+

∫ s+hs vx(t,X x ,s,u

t )σ(t,X x ,s,ut , ut)dWt

h

+12

∫ s+hs vxx(t,X x ,s,u

t )σ2(t,X x ,s,ut , ut)dt

h)

≤ 0

if v(., .) is nicely behaved, we have a true martingale above andafter switching limit and expectation one gets:

sup(f (s, x , u) + vt + vxµ+1

2vxxσ

2) = 0.




The nonlinear PDE

supu

[f (s, x , u) + vt + vx(s, x)µ(s, x , u) +

1

2vxx(s, x)σ2(s, x , u)

]= 0

is called the Hamilton–Jacobi–Bellman equation. In its mostnotorious form in the classical frictionless BS market, such anequation becomes: (zero interest rates: dXt = γdSt)

supθ

[f (s, x , u) + vt + vxαxθ +

1

2vxxσ

2x2θ2

]= 0.

With one boundary condition v(x ,T ) = U(x). In the classicalMerton problem maximizing utility at T , f = 0 and it is easy tocompute the maximizing θ = −αvx (x ,t)

σ2xvxx (x ,t). The PDE can be solved

analytically by guess and verify method to get:

v(t, x) =x1−p

1− pexp

α2

2σ2

1− p

p(T − t)

→ θ =

α

pσ2




When considering maximization of utility at time T from theprobabilistic side of view, we arrive at a need to satisfy averification lemma:Let v(y , t) be such that

v(y ,T ) = U(y)

v(Xt , t) is a super–martingale for any controlled self–financingportfolio process, and there exists a process Xt such thatv(Xt , t) is a martingale.

Thenv(x , s) = sup

uEU(X x ,s,u

T ).

Remark: The above developed optimal control posseses unrealisticbehaviour – rebalancing infinitely often.




I concentrate on random or “stochastic” coefficients of a linearBrownian motion, in a one–dimensional case only. StandardBrownian motion rather than Geometric is not an original approachand appears in Janecek, Shreve (2011) for example, or in yetunpublished works by Petr Dostal, who reason that arithmeticBrownian motion is even a more realistic model for futures forexample. I stick to “futures” onwards.I postulate the following easiest possible setup. Let Xt be theinvestors capital, Lt ,St the cumulative number of bought, soldfutures respectively:

dFt = αtdt + σtdWt ;

dαt = a0tdt + atdW

1t

dσt = b0t dt + btdW

2t

Yt = Lt − St ;dYt = dLt − dSt = (lt − st)dt;

dXt = γtdFt − δltdt − δstdt.Petr Zahradnık KPMS MFF UK; UTIA CAS



It should be noted that Lt , St have finite variation (they arepositive nondecreasing) and hence must have zero quadraticvariation. We have a sufficiently rich filtration so that threemutually independent Brownian motions exist, above–mentionedW , W 1 and W 2 are however generally correlated.Inspired by the verification lemma in the frictionless case, it isstraightforward to come up with an analogous verification lemmafor transaction costs. To know the optimal control and optimalvalue 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) Yt

such that it is a martingale.




This lemma results in a theorem which to be proved needs onlysome more Ito lemma application. Here I stick to random volatilityonly 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 − vxδ, vy + vxδ) = 0, with boundary conditionv(x , y , a, s,T ) = U(x).

Lv = vt + vxyα + 12vxxs

2y2 + 12vssa

2s2 + vxsys2aρs2




Shreve, Soner (1994) have a similar HJB and thoroughly arguewhy a solution in a general viscosity sense exists. I conjecture thatsuch a solution exists here too. Being interested in the optimalpolicy, hence in the optimal wedge, Janecek, Shreve (2011) have ina similar setup proved that once coefficients are constant, viadecomposing the loss in utility caused by transaction costs anddisplacement, there is an optimal wedge too. Denote πt = Yt

Xt,

θ = αpσ2 , the wedge is of the following form:

π ∈ (θ − xt , θt + xt)

where

xt =

(3δ

2pθ4

)1/3

.




Now we get back to our verification lemma and proceed to solving:

maxy (Lv , vy − vxδ, vy + vxδ) = 0, with boundary conditionv(x , y , s,T ) = U(x).

Lv = vt + vxyα + 12vxxs

2y2 + 12vssa

2σ2 + vxsys2aρσ2

This can be tried via asymptotic expansion - a method known influid mechanics etc. where it is used extensively to conjecture thequalities of generally unknown solutions. Other approaches includevarious probabilistic approaches in which a shape of the valuefunction is “guessed” and one tries to make the “martingaleproperty” hold almost – Landau symbols are used here too.




An interesting possibility is proposed by Petr Dostal, who opts formaximizing certainty equivalent. The goal of such an approach is

to find a process νt and an optimal strategy such that eγUt is amartingale and for any other strategy eγUt is a super–martingale,where:

Ut = logXt −∫ t

0νsds.

Intuition:

assume logXn −∑

n νk a martingale

assume eγ(log Xn−∑

n νk ) a martingale

Interestingly enough, several formulations, as much as they differin the value of the task, lead to the same optimal behaviour:WHICH IS WHAT WE WOULD HOPE FOR IN REAL WORLDPROBLEMS!




General Result:

π ∈ (θ − xt , θt + xt)

xt =

(3δ

2p(θ4 + 2θ2d[π,F ]t

d[F ]t+

d[π]td[F ]t

)

) 13

In other words, if we make a (reality driven) assumptiondθt = φdt + ΦdWt + Ψ1

tdW1t + Ψ2

tdW2t :

xt =

(3δ

2pK (θ)

) 13

, where:

K (θ) =

(σtθ

2 + Φt

)2+(Ψ1)2

+(Ψ2)2

σ2t

.




Some sanity checks:

for nonrandom coefficients: K (θ) =(σtθ2)

2

σ2 .

Φ < 0 leads to smaller wedges.

Ψ1 6= 0,Ψ2 6= 0 lead to greater wedges.

Ψ1 →∞,Ψ2 →∞ lead to no trading.

. . .




Remarks on the structure:

transaction costs lead to reduced wealth, so the wedge is notcentered around the no transaction-costs position ceterisparibus. However, it is centered around the same relativeposition.

transaction costs increase the width of the interval, highercosts lead to less activity.

risk tolerance plays a role in the width of the wedge asexpected - 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.

. . .




First step - homotheticity of the value function

Proposition (Homotheticity of type I)

v(t, kx , ky , s) = k1−pv(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 utilityfunction. The second is independent of the choice, comes from thearithmetic Brownian motion setup.




Dimensions reduction

From the type I homotheticity:

v(t, x , y , s) = x1−pv(t, 1, y/x , s) = x1−pu(t, z , s)

Hence it might be useful to drop one dimension and consider thevariable y/x instead. This is motivated not only by technicaladvantages but by resemblance to the no transaction case too. Weare onwards using the function u, often dropping variables whichremain fixed in the computation. Subscripts onwards denotederivatives with respect to the respective variable.




Solutions in buy and sell regions

v(x , y , s, t) = x1−pu(z , s, t)

vx(x , y) = x−p((1− p)u(z)− uz(z)z)

vy (x , y) = x−puz(z)

In the buy resp. sell regions, with such a dimension reduction thePDEs 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 riskaverse investors (p > 1) it is worse to oversize.)




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 asymptoticexpansion. We can employ continuity arguments of the valuefunction and of its derivatives (we used Itos formula!) and theknowledge of the solutions in the buy and sell regions.

An interesting question arises, in what powers is it reasonableto perform the expansion? The powers of δ1/3 are to be foundJanecek, Shreve(2004) and Rogers(2004) argues, why it mustbe 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) + δ1/3Z .




The expansion may be tried as follows. Notice how Z appears. Thisis because of the continutity arguments from the buy sell regions.

u(t,Z , s; δ) = [H0(t, s) + δ1/3ZH1(t, s) + δ2/3H2(t, s) + δH3(t, s) . . .

+ δ4/3H4(t,Z , s) + δ5/3H5(t,Z , s) . . .]

Now we must compute derivatives and do some miracles with thePDEs connected with respective powers. I haven’t succeeded inderiving the optimal wedge bounds for random volatility yet. I havehowever succeeded in deliberately deriving the optimal wedgebounds for constant volatility via such an expansion and theycoincide with the result of Janecek, Shreve (2011)and others:

zb, zs = z∗ ±(

3δ

2p(z∗)4

)+ O(δ2/3).




I am however quite confident that this approach should lead to thedesired 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 meansof derivation, so we probably have a rule for investors. Now weneed to find a good enough formal language to make it publishablein a scientific paper.

Thank you for attention!



optimal investments

Documents