Optimal Investment with Transaction Costs and Stochastic Volatility

0.42 MB·

by  Maxim Bichuch

Optimal Investment with Transaction Costs and Stochastic
Volatility
Maxim Bichuch ∗ Ronnie Sircar †
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1 January 2014, revised Thursday 28th August, 2014
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g
u Abstract
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Twomajorfinancialmarketcomplexitiesaretransactioncostsanduncertainvolatility,andweanalyze

7 theirjoint impact on theproblem ofportfolio optimization. Whenvolatility isconstant,thetransaction
2 costsoptimal investmentproblem hasalong history,especially intheuseof asymptoticapproximations
when the cost is small. Under stochastic volatility, but with no transaction costs, the Merton problem

] under general utility functions can also be analyzed with asymptotic methods. Here, we look at the
M
long-run growth rate problem when both complexities are present, using separation of time scales ap-
P proximations. This leads to perturbation analysis of an eigenvalue problem. We find the first term in
. theasymptotic expansion in thetime scale parameter, of theoptimal long-term growth rate, and of the
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optimal strategy, for fixedsmall transaction costs.
i
f
- AMS subject classification 91G80, 60H30.
q
[
JEL subject classification G11.

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v
Keywords Transactioncosts, optimal investment, asymptotic analysis, utility maximization, stochastic
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volatility.
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.
1 1 Introduction
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1 The portfolio optimization problem, first analyzed within a continuous time model in Merton [1969], ig-
: nores two key features that are important for investment decisions, namely transaction costs and uncertain
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volatility. Both these issues complicate the analysis of the expected utility maximization stochastic control
i
X problem,andobtainingclosed-formoptimalpolicies,orevennumericalapproximations,ischallengingdueto
r the increase in dimension by incorporating a stochastic volatility variable, and the singular control problem
a
that arises by considering proportional transaction costs. Here, we develop asymptotic approximations for
a particular long-run investment goal in a model with transaction costs and stochastic volatility.
The typicalproblemhas aninvestor who can investin a marketwith one riskless asset(a money market
account),andoneriskyasset(astock),andwhohastopayatransactioncostforsellingthestock. Thecosts
are proportionalto the dollar amount of the sale, with proportionality constantλ>0. The investmentgoal
is to maximize the long-term growth rate. The original works all assumed stocks with constant volatility.
Transaction costs were first introduced into the Merton portfolio problem by Magill and Constantinides
[1976] and later further investigated by Dumas and Luciano [1991]. Their analysis of the infinite time
horizoninvestmentandconsumptionproblemgivesaninsightinto the optimalstrategyandthe existence of
a “no-trade” (NT) region. Under certain assumptions, Davis and Norman [1990] provided the first rigorous
∗Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA. [email protected].
PartiallysupportedbyNSFgrantDMS-0739195.
†Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA.
[email protected]. PartiallysupportedbyNSFgrantDMS-1211906.
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analysis of the same infinite time horizon problem. These assumptions were weakened by Shreve and Soner
[1994], who used viscosity solutions to also show the smoothness of the value function.
When λ > 0, and the volatility is constant, the optimal policy is to trade as soon as the position is
sufficientlyfarawayfromtheMertonproportion. Morespecifically,theagent’soptimalpolicyistomaintain
her position inside a NT region. If the investor’s position is initially outside the NT region, she should
immediately sell or buy stock in order to move to its boundary. She will trade only when her position is on
the boundary of the NT region, and only as much as necessary to keep it from exiting the NT region, while
no trading occurs in the interior of the region; see Davis et al. [1993].
There is a trade-off between the amount of transaction costs paid due to portfolio rebalancing and the
width of the NT region. A smaller NT region generally increases the amount spent paying transaction
costs in maintaining the optimal portfolio. Not surprisingly, the same behavior persists when volatility is
stochastic,but in this case,the boundaries of NT regionin generalwill no longerbe straightlines as before.
Hence, the approach of this paper, is to find a simple strategy that will be asymptoticaly optimal in both
the volatility scaling and transaction costs parameters.
Smalltransactioncostasymptoticexpansions(inpowersofλ1/3)wereusedinJanecek and Shreve[2004]
for an infinite horizon investment and consumption problem. This approach allows them to find approxi-
mations to the optimal policy and the optimal long-term growthrate, and is also used in Bichuch [2012] for
a finite horizon optimal investment problem. The survey article Guasoni and Muhle-Karbe [2013] describes
recentresultsusingso-calledshadowpricetoobtainsmalltransactioncostasymptoticsfortheoptimalinvest-
ment policy, its implied welfare, liquidity premium, and trading volume. All of the above mentioned litera-
tureontransactioncostsassumesconstantvolatility. SomerecentsexceptionsareKallsen and Muhle-Karbe
[2013a,b],where the stockis ageneralItoˆdiffusion, andSoner and Touzi[2013]wherethe stock isdescribed
byacomplete(localvolatility)model. Wesummarizesomeofthisliteratureandtheindividualoptimization
problems and models that they study in Table 1.
Paper Model Utility Objective Solution
DumasandLuciano[1991] B-S Power LTGR Explicit
DavisandNorman[1990] B-S Power -consumption Numerical
∞
ShreveandSoner[1994] B-S Power -consumption Viscosity
∞
Davisetal.[1993] B-S Exponential Optionpricing Viscosity
WhalleyandWilmott[1997] B-S Exponential Optionpricing λ-expansion
JanecekandShreve[2004] B-S Power -consumption λ-expansion
∞
Bichuch[2012] B-S Power T < λ-expansion
∞
Daietal.[2009] B-S Power T < ODEsFree-Bdy
∞
Gerholdetal.[2014] B-S Power LTGR λ-expansion
GoodmanandOstrov[2010] B-S General T < λ-expansion
∞
Choietal.[2013] B-S Power -consumption ODEsFree-Bdy
∞
KallsenandMuhle-Karbe[2013a] Itˆo General T <= ,consumption λ-expansion
∞
KallsenandMuhle-Karbe[2013b] Itˆo Exponential Optionpricing λ-expansion
SonerandTouzi[2013] LocalVol GeneralonR+ T < λ-expansion
∞
Caflischetal.[2012] Stochvol Exponential Optionpricing λ-SVexpansion
Thispaper Stochvol Power LTGR SVexpansion
Table 1: Problems, models and solution approaches. The acronyms used are: B-S = Black-Scholes, LTGR
= Long-Term Growth Rate, SV=Stochastic Volatility, Free-Bdy=Free Boundary.
Our approachexploits the fast mean-reversionof volatility (particularly when viewed overa long invest-
ment horizon)leading to a singular perturbation analysis of an impulse controlproblem. We treat the case,
when the volatility is slow mean reverting separately. This complements multiscale approximations devel-
oped for derivatives pricing problems described in Fouque et al. [2011] and for optimal hedging and invest-
ment problems in Jonsson and Sircar [2002] and Fouque et al. [2013] respectively. Recently, Caflisch et al.
[2012]studyindifference pricingofEuropeanoptionswithexponentialutility, fastmean-revertingstochastic
volatilityandsmalltransactioncostswhichscalewiththe volatilitytime scale. The currenttransactioncost
problemcanbecharacterizedasafree-boundaryproblem. Thefastmean-reversionasymptoticsforthefinite
horizonfree boundaryproblemarisingfromAmericanoptions pricingwasdevelopedinFouque et al.[2001],
andrecentlytherehasbeeninterestinsimilaranalysisforperpetual(infinitely-lived)Americanoptions(used
as part of a real options model) in Ting et al. [2013], and for a structural credit risk model in McQuade
[2013]. Here, we also have an infinite horizon free-boundary problem, but it is, in addition, an eigenvalue
2
problem.
In Section 2 of this paper, we introduce our model and objective function and give the associated
Hamilton-Jacobi-Bellman (HJB) equation. In Section 3 we perform the asymptotic analysis. We first
consider the fast-scale stochastic volatility in Section 4, where we find the first correctionterm in the power
expansion of the value function, and as a result also find the corresponding term in the power expansion
of the optimal boundary. We perform similar analysis in the case of slow-scale stochastic volatility in Sec-
tion 5. In Section 6 we show numerical calculations based on our results, and give an alternative intuitive
explanation to the findings. We summarize the results obtained in the paper in Section 7, and leave some
technical computations to the Appendix.
2 A Class of Stochastic Volatility Models with Transaction Costs
An investor can allocate capital between two assets – a risk-free money market account with constant rate
of interest r, and risky stock S that evolves according to the following stochastic volatility model:
dS
t = (µ+r)dt+f(Z )dB1,
S t t
t
1 1
dZ = α(Z )dt+ β(Z )dB2,
t ε t √ε t t
where B1 and B2 are Brownian motions, defined on a filtered probability space (Ω, , (t) ,P), with
t 0
constant correlation coefficient ρ ( 1,1): d B1,B2 = ρdt. We assume that f(z)Fis a{Fsmo}ot≥h, bounded
∈ − t
and strictly positive function, and that the stochastic volatility factor Z is a fast mean-reverting process,
t
(cid:10) (cid:11)
meaning that the parameter ε > 0 is small, and that Z is an ergodic process with a unique invariant
distribution Φ that is independent of ε. We refer to [Fouque et al., 2011, Chapter 3] for further technical
detailsanddiscussion. Additionallyr,µarepositiveconstants,andα,β aresmoothfunctions: exampleswill
be specified later for computations.
2.1 Investment Problem
The investormust choose a policy consisting of two adapted processesL and M that are nondecreasingand
right-continuouswithleftlimits,andL =M =0. ThecontrolL representsthecumulativedollarvalue
0 0 t
− −
of stock purchased up to time t, while M is the cumulative dollar value of stock sold. Then, the wealth X
t
invested in the money market account and the wealth Y invested in the stock follow
dX = rX dt dL +(1 λ)dM ,
t t t t
− −
dY = (µ+r)Y dt+f(Z )Y dB1+dL dM .
t t t t t t− t
The constant λ (0,1) represents the proportional transaction costs for selling the stock.
∈
Next, we define the solvency region
, (x,y); x+ y >0, x+(1 λ)y >0 , (2.1)
S { − }
which is the set of all positions, such that if the investor were forced to liquidate immediately, she would
not be bankrupt. This leads to a definition that a policy (L ,M ) is admissible for the initial position
s s s t
≥
Z = z and (X ,Y ) = (x,y) starting at time t , if (X ,Y ) is in the closure of solvency region, , for
t t t − s s (cid:12)
all s t. (Since−the i−nvestor may choose to immediately rebalance(cid:12)his position, we have denoted the Sinitial
≥
time t ). Let (t,x,y,z) the set of all such policies. Clearly, if (x,y) then we can always liquidate
− A ∈ S
the position, and then hold the resulting cash position in the risk-free money market account. It is easy to
adapt the proof in Shreve and Soner [1994] (for the constant volatility case) to show that (t,x,y,z)= if
and only if (t,x,y,z) [0, ) R. A 6 ∅
We work with CR∈RA o∞r po×wSer×utility functions (w) defined on R :
+
U
w1 γ
−
(w):= , γ >0, γ =1,
U 1 γ 6
−
3
where γ is the constant relative risk aversionparameter. We are interested in maximizing:
1
sup liminf log 1 Ex,y,z X +Y λY+ , (x,y,z) , R,
(L,M)∈A(0,x,y) T→∞ T U− (cid:0) 0 (cid:2)U(cid:0) T T − T (cid:1)(cid:3)(cid:1) ∈S ×
where Ex,y,z[] := E[ X = x,Y = y,Z = z]. This is a problem in optimizing long term growth.
To see thte ec·onomic ·intetr−pretationt−note thatt the quantity U 1 Ex,y,z U X +Y λY+ is the cer-
(cid:12) − 0 T T − T
tainty equivalent of th(cid:12)e terminal wealth X + Y λY+. Hence if we can match this certainty equiv-
alent with (x + y λy+)e(r+δε)T – the invTestor’Ts−initiaTl capita(cid:0)l comp(cid:2)oun(cid:0)ded at some rate(cid:1)(cid:3)r(cid:1)+ δε, then
1 logU 1 Ex,y,z U− X +Y λY+ = r+δε. For a survey and literature on this choice of objective
T − 0 T T − T
function we refer to Guasoni and Muhle-Karbe [2013]. This choice of optimization problem ensures the
(cid:0) (cid:2) (cid:0) (cid:1)(cid:3)(cid:1)
simplest HJB equation, which in this case turns out to be linear and time independent.
2.2 HJB Equation
Consider first the value function for utility maximization at a finite time horizon T:
V(t,x,y,z)= sup Ex,y,z X +Y λY+ .
t U T T − T
(L,M) (t,x,y,z)
∈A (cid:2) (cid:0) (cid:1)(cid:3)
b
From Itoˆ’s formula it follows that
dV(t,X ,Y ,Z )
t t t
1 1
=b Vt+rXtVx+(µ+r)YtVy + 2f2(Zt)Yt2Vyy+ √ερf(Zt)β(Zt)YtVyz dt
(cid:18) (cid:19)
1 1 1
+ bα(Z )Vb+ β2(Z )V2b dt+f(Z )Y VbdB1+ β(Z )V dB2b
ε t z 2 t zz t t y t √ε t z t
(cid:18) (cid:19)
+ V V bdL + (1 bλ)V V dM . b
y x t x y t
− − −
(cid:16) (cid:17) (cid:16) (cid:17)
SinceV mustbeasubpemabrtingale,thedt,dLbandbdM termsmustnotbepositive. ItfollowsthatV V 0
t t y x
− ≤
and (1 λ)V V 0. Alternatively,
x y
− − ≤
b V 1 b b
x
1 . (2.2)
b b ≤ V ≤ 1 λ
y −
b
Wewilldefinetheno-trade(NT)region,associatedwithV,tobethe regionwherebothoftheseinequalities
b
are strict. Moreover, for the optimal strategy, V is a martingale, and so the dt term above must be zero
inside the NT region. Thusdit will then satisfy the HJBbequation
b
mdax (∂ + ε)V,(∂ ∂ )V,((1 λ)∂ ∂ )V =0, V(T,x,y,z)= (x+y λy+), (2.3)
t y x x y
D − − − U −
n o
where b b b b
1 1
ε =rx∂ +(µ+r)y∂ + f2(z)y2∂2 + ρf(z)β(z)y∂2
D x y 2 yy √ε yz
1 1
+ α(z)∂ + β2(z)∂2 .
ε z 2 zz
(cid:18) (cid:19)
The fact that V is a viscosity solutionof (2.3) is standard, and a similar proofcan be found for example
in Shreve and Soner [1994], and thus will be omitted here. We will furthermore assume that the viscosity
solution V of (2.3)bis in fact a classical solution, that is we will assume that it is sufficiently smooth. It can
be shownthat V is smoothinside eachofthree regions: the NT, andthe regionswhere (∂ ∂ )V =0, and
y x
−
((1 λ)∂b ∂ )V =0. The assumption that it is also smooth on the boundary of the NT is the smooth fit
x y
− −
assumption, whbich is very common; see, for instance, Goodmdan and Ostrov [2010]. b
b d
4
Next, we look for a solution of the HJB equation (2.3) of the form
V(t,x,y,z)=x1 γvλ,ε(ζ,z)e(1 γ)(r+δε)(T t), ζ = y, (2.4)
− − −
x
where δε is a constant, and the function vλ,ε is to be found. However, we will not impose the final time
conditiononV. Fornow,wewillonlyassumethatitissmoothand vλ,ε isboundedawayfromzero. Wewill
| |
define the NT region(associatedwith V) as the regionwhere (∂ + ε)V =0. Additionally, we will assume
t
D
that for any point (t,x,y,z) in the NT region, the ratio y/x is bounded. We note that V is not equivalent
to the value function V, since we have not imposed the final time condition V(T,x,y,z)= (x+y λy+).
U −
In fact there is no reason to believe that the final time condition can be satisfied if V is given by (2.4).
However, if we finbd a constant C such that V C V , then it would follow that δε is the optimal
| | ≤ | |
growthrateandthe NTregionforthelong-termoptimalgrowthproblemcanbedefinedastheregionwhere
(∂ + ε)V =0. In other words, b
t
D
1 1
liminf log 1 V(0,x,y,z) = liminf log 1(V(0,x,y,z))
− −
T T U T T U
→∞ (cid:16) (cid:17) →∞
1 logV(0,x,y,z)
b = liminf
T T 1 γ
→∞ −
= r+δε.
We will now show that there exists a constant C such that V C V . Indeed, note that the utility
| | ≤ | |
function is homogeneous of degree 1 γ, that is (w)=w1 γ (1), it follows that
−
U − U U
b
+
Y Y
U XT +YT −λYT+ =XT1−γU 1+ XTT −λ(cid:18)XTT(cid:19) !.
(cid:0) (cid:1)
By our assumption, Y /X is bounded, being inside the NT region. Hence, there exists a constant C such
T T
that
y
V(T,x,y,z) = x+y λy+ C V(T,x,y,z) =x1−γCvλ,ε ,z . (2.5)
| | |U − |≤ | | x
(cid:0) (cid:1) (cid:16) (cid:17)
b
Since both V and V solve the HJB equation (2.3), it follows by a comparison theorem that V C V
| | ≤ | |
everywhere. For the reader convenience, we have sketched the proof of it in Appendix A.
Inserting the trabnsformation (2.4) into (2.3) leads to the following equation for (vλ,ε,δε): b
1 1
max + +( (1 γ)δε ) , , vλ,ε =0, (2.6)
0 1 2
εL √εL L − − · B S
(cid:26) (cid:27)
where we define the operators in the NT region by
1 1
= β2(z)∂2 +α(z)∂ , =ρf(z)β(z)ζ∂2 , = f2(z)ζ2∂ +µζ∂ , (2.7)
L0 2 zz z L1 ζz L2 2 ζζ ζ
and the buy and sell operators by
=(1+ζ)∂ (1 γ) , (2.8)
ζ
B − − ·
1
= +ζ ∂ (1 γ) , (2.9)
ζ
S 1 λ − − ·
(cid:18) − (cid:19)
respectively. For future reference, we also define their derivatives
=∂ =(1+ζ)∂ +γ∂ ,
′ ζ ζζ ζ
B B
1
′ =∂ζ = +ζ ∂ζζ +γ∂ζ.
S S 1 λ
(cid:18) − (cid:19)
5
2.3 Free Boundary Formulation & Eigenvalue Problem
We will look for a solution to the variational inequality (2.6) in the following free-boundary form. The NT
region is defined by (2.2), but for the function V. Using the transformation (2.4), this translates to
vλ,ε 1
1+ζ <(1 γ) < +ζ
− vλ,ε! 1 λ
ζ −
for vλ,ε(ζ,z). Similar to the case with constant volatility, we assume that there exists a no-trade region,
within which 1 + 1 +( (1 γ)δε ) vλ,ε = 0, with boundaries ℓε(z) and uε(z). We write this
εL0 √εL1 L2− − ·
region as (cid:16) (cid:17)
min ℓε(z),uε(z) <ζ <max ℓε(z),uε(z) ,
{ } { }
where ℓε(z) and uε(z) are free boundaries to be found. In typical parameter regimes, we will have 0 <
ℓε(z) < uε(z), so we can think of them as lower and upper boundaries respectively, with ℓε being the buy
boundary, and uε the sell boundary. (The other two possibilities are that ℓε <uε <0 with ℓε being the buy
boundary, and uε the sell boundary, or that ℓε < uε < 0 with ℓε being the sell boundary, and uε the buy
boundary. Under a constant volatility model these cases can be categorized explicitly in term of the model
parameters: see Remark 1).
Inside this region we have from the HJB equation (2.6) that
1 1
+ +( (1 γ)δε) vλ,ε =0, ζ (ℓε(z),uε(z)). (2.10)
0 1 2
εL √εL L − − ∈
(cid:18) (cid:19)
The free boundaries ℓε and uε are determined by continuity of the first and second derivatives of vλ,ε with
respect to ζ, that is looking for a C2 solution. In the buy region,
vλ,ε =0 in ζ <ℓε(z), (2.11)
B
and so the smooth pasting conditions at the lower boundary are
Bvλ,ε |ℓε(z) =(1+ℓε(z))vζλ,ε(ℓε(z))−(1−γ)vλ,ε(ℓε(z))=0, (2.12)
B′vλ,ε |ℓε(z) :=(1+ℓε(z))vζλζ,ε(ℓε(z))+γvζλ,ε(ℓε(z))=0. (2.13)
In the sell region, the transaction cost enters and we have:
vλ,ε =0 in ζ >uε(z). (2.14)
S
Therefore the sell boundary conditions are:
1
Svλ,ε |uε(z) = 1 λ +uε(z) vζλ,ε(uε(z))−(1−γ)vλ,ε(uε(z))=0, (2.15)
(cid:18) − (cid:19)
1
S′vλ,ε |uε(z) := 1 λ +uε(z) vζλζ,ε(uε(z))+γvζλ,ε(uε(z))=0. (2.16)
(cid:18) − (cid:19)
Wenotethat(2.10),(2.11)and(2.14)arehomogeneousequationswithhomogeneousboundaryconditions
(2.12), (2.13), (2.15) and (2.16), andso zerois a solution. Howeverthe constantδε is also to be determined,
and in fact it is an eigenvalue found to exclude the trivial solution and give the optimal long-term growth
rate. In the next section, we constructan asymptotic expansionin ε for this eigenvalue problemusing these
equations.
3 Fast-scale Asymptotic Analysis
We look for an expansion for the value function
vλ,ε =vλ,0+√εvλ,1+εvλ,2+ , (3.1)
···
6
as well as for the free boundaries
ℓε =ℓ +√εℓ +εℓ + , uε =u +√εu +εu + , (3.2)
0 1 2 0 1 2
··· ···
and the optimal long-term growth rate
δε =δ +√εδ + , (3.3)
0 1
···
which are asymptotic as ε 0.
↓
Crucial to this analysis is the Fredholm alternative (or centering condition) as detailed in Fouque et al.
[2011]. In preparation, we will use the notation to denote the expectation with respect to the invariant
h·i
distribution Φ of the process Z, namely
g := g(z)Φ(dz).
h i
Z
The Fredholm alternative tells us that a Poissonequation of the form
v+χ=0
0
L
hasasolutionvonlyifthesolvabilitycondition χ =0issatisfied,andwereferforinstanceto[Fouque et al.,
h i
2011, Section 3.2] for technical details.
It is also convenient to introduce the differential operators
∂k
D =ζk , k =1,2, , (3.4)
k ∂ζk ···
in terms of which the operators and in (2.7) are
1 2
L L
1
=ρf(z)β(z)∂ D , = f(z)2D +µD .
1 z 1 2 2 1
L L 2
In the following, a key role will be played by the squared-averagedvolatility σ¯ defined by
σ¯2 = f2 . (3.5)
The principal terms in the expansions will be related(cid:10)to(cid:11)the constant volatility transaction costs problem,
and we define the operator (σ;δ) that acts in the no trade region by
LNT
1
(σ;δ)= σ2D +µD (1 γ)δ , (3.6)
LNT 2 2 1− − ·
and it is written as a function of the parameters σ and δ.
The zero-order terms in each of the asymptotic expansions (3.1), (3.2) and (3.3) are known and will be
re-derived in Section 4. In the rest of this section, we calculate the next terms in the above asymptotic
expansion in the case of fast-scale stochastic volatility.
3.1 Power expansion inside the NT region
In this subsection we will concentrate on constructing the expansion inside the NT regionζ (lε(z),uε(z)),
∈
where (2.6) holds. We now insert the expansion (3.1) and match powers of ε.
Thetermsoforderε 1 leadto vλ,0 =0. Sincethe operatortakesderivativesinz,weseekasolution
− 0 0
L L
of the form vλ,0 =vλ,0(ζ), independent of z.
At order ε 1/2, we have vλ,0+ vλ,1 = 0. But since takes a derivative in z, vλ,0 = 0, and so
− 1 0 1 1
L L L L
vλ,1 =0. Again, we seek a solution of the form vλ,1 =vλ,1(ζ) that is independent of z.
0
L
The terms of order one give
( (1 γ)δ )vλ,0+ vλ,1+ vλ,2 =0.
2 0 1 0
L − − L L
7
Since we have that takes derivatives in z, and vλ,1 is independent of z, we have that
1
L
( (1 γ)δ )vλ,0+ vλ,2 =0. (3.7)
2 0 0
L − − L
This is a Poissonequation for vλ,2 with ( (1 γ)δ ) vλ,0 =0 as the solvability condition. We observe
2 0
h L − − i
that
( (1 γ)δ ) = (σ¯;δ ),
h L2− − 0· i LNT 0
whereσ¯ isthesquare-averagedvolatilitydefinedin(3.5),and istheconstantvolatilitynotradeoperator
LNT
defined in (3.6). Then we have
(σ¯;δ )vλ,0 =0, (3.8)
LNT 0
which, along with boundary conditions we will find in the next subsection, will determine vλ,0.
To find the equation for the next term vλ,1 in the approximation, we proceed as follows. We write the
first term of (3.7) as
1
( (1 γ)δ )vλ,0 =(( (1 γ)δ ) (σ¯;δ ))vλ,0 = f2(z) σ¯2 D vλ,0.
L2− − 0 L2− − 0 −LNT 0 2 − 2
(cid:0) (cid:1)
The solutions of (3.7) are given by
1 1
vλ,2 =−L−01L2vλ,0 =−2L−01 f2(z)−σ¯2 D2vλ,0 =−2 (φ(z)+c(ζ))D2vλ,0, (3.9)
(cid:0) (cid:1)
where c(ζ) is independent of z, and φ(z) is a solution to Poissonequation
φ(z)=f2(z) σ¯2,
0
L −
Continuing to the order √ε terms, we obtain
( (1 γ)δ )vλ,1+ vλ,2+ vλ,3 (1 γ)δ vλ,0 =0.
2 0 1 0 1
L − − L L − −
Once again, this is a Poisson equation for vλ,3 whose centering condition implies that
( (1 γ)δ ) vλ,1+ vλ,2 (1 γ)δ vλ,0 =0.
2 0 1 1
h L − − i L − −
From (3.9), it follows that (cid:10) (cid:11)
1 1
(σ¯;δ )vλ,1 (1 γ)δ vλ,0 = vλ,2 = φ D vλ,0 = ρ βfφ D D vλ,0. (3.10)
LNT 0 − − 1 − L1 2hL1 i 2 2 h ′i 1 2
(cid:10) (cid:11)
We define
1
V = ρ βfφ . (3.11)
3 ′
−2 h i
Then we write the equation (3.10) as
(σ¯;δ )vλ,1 = V D D vλ,0+(1 γ)δ vλ,0. (3.12)
LNT 0 − 3 1 2 − 1
3.2 Boundary Conditions
So far we have concentrated on the PDE (2.6) in the NT region. We now insert the expansions (3.1) and
(3.2) into the boundary conditions (2.12)–(2.16). The terms of order one from (2.12) and (2.13) give
vλ,0 =0, and vλ,0 =0, (3.13)
B |ℓ0 B′ |ℓ0
while the terms of order one from (2.12) and (2.13) give
vλ,0 =0, and vλ,0 =0, (3.14)
S |u0 S′ |u0
Since vλ,0 is independent of z, these equations imply that ℓ and u are also independent of z (they are
0 0
constants).
8
Taking the order √ε terms in (2.12) gives
(1+ℓ ) vλ,1(ℓ )+ℓ vλ,0(ℓ ) +ℓ vλ,0(ℓ ) (1 γ) vλ,1(ℓ )+ℓ vλ,0(ℓ ) =0.
0 ζ 0 1 ζζ 0 1 ζ 0 − − 0 1 ζ 0
(cid:16) (cid:17) (cid:16) (cid:17)
Using the fact that vλ,0 =0, we see the terms in ℓ cancel, and we obtain
B |ℓ0 1
vλ,1 =0, (3.15)
B |ℓ0
which is a mixed-type boundary condition for vλ,1 at the boundary ℓ .
0
From the order √ε terms in (2.13), we obtain
ℓ vλ,0(ℓ )+(1+ℓ )vλ,0(ℓ )+γvλ,0(ℓ ) + (1+ℓ )vλ,1(ℓ )+γvλ,1(ℓ ) =0,
1 ζζ 0 0 ζζζ 0 ζζ 0 0 ζζ 0 ζ 0
(cid:16) (cid:17) h i
and so, as vλ,1 does not depend on z, ℓ is also a constant (independent of z) given by
1
vλ,1
ℓ = B′ |ℓ0 . (3.16)
1 − (1+ℓ0)vζλζ,0ζ(ℓ0)+(1+γ)vζλζ,0(ℓ0)!
Similar calculations can be performed on the (right) sell boundary uε u +√εu , where vλ,ε = 0.
0 1
≈ S
The analogous equations to (3.15) and (3.16) are
vλ,1 =0. (3.17)
S |u0
vλ,1
u = S′ |u0 . (3.18)
1
− 1 +u vλ,0(u )+(1+γ)vλ,0(u )
1 λ 0 ζζζ 0 ζζ 0
−
(cid:16) (cid:17) 
Note that (3.17) is a mixed-type boundary condition for vλ,1 at the boundary u , and (3.18) determines the
0
constant correction term u to the sell boundary.
1
3.3 Determination of δ
1
Thenexttermvλ,1 intheasymptoticexpansionsolvestheODE(3.12),withboundaryconditions(3.15)and
(3.17),butwealsoneedtofindδ whichappearsinthe equation. Infact,the Fredholmsolvabilitycondition
1
for this equation determines δ , and so we look for the solution w of the homogeneous adjoint problem.
1
To do that we first multiply both sides of (3.12) by w and integrate from ℓ to u :
0 0
u0 u0 u0
w vλ,1dζ = V D D vλ,0wdζ +(1 γ)δ vλ,0wdζ. (3.19)
LNT − 3 1 2 − 1
Zℓ0 Zℓ0 Zℓ0
Integration by parts gives
u0 u0 σ¯2 σ¯2 u0
wLNTvλ,1dζ = vλ,1L∗NTwdζ+ 2 vζλ,1ζ2w− 2 (ζ2w)′vλ,1+µζwvλ,1 , (3.20)
Zℓ0 Zℓ0 (cid:20) (cid:21)ℓ0
where = (σ¯;δ ) is the adjoint operator to :
L∗NT L∗NT 0 LNT
1
L∗NT(σ¯;δ0)(w)= 2σ¯2∂ζζ ζ2w −µ∂ζ(ζw)−(1−γ)δ0w.
(cid:0) (cid:1)
We set w to satisfy
(σ¯;δ )(w)=0, (3.21)
L∗NT 0
and, to cancel the boundary terms in (3.20), the boundary conditions
ℓ w (ℓ ) k w(ℓ )=0, u w (u ) k w(u )=0, (3.22)
0 ′ 0 0 0 ′ 0 + 0
− − −
where we define the constants
µ
k :=(1 γ)π +(k 2), and k := . (3.23)
± − ± − 1σ¯2
2
9
Lemma 3.1. The solution w to the adjoint equation (3.21) with boundary conditions (3.22) is, up to a
multiplicative constant, given by
w(ζ)=ζk 2vλ,0(ζ), (3.24)
−
where k was defined in (3.23).
Proof. Making the substitution (3.24) into (3.21) leads to the equation (3.8) satisfied by vλ,0. Similarly
inserting (3.24) into the boundary conditions (3.22) leads to the boundary conditions (3.13) and (3.14)
satisfied by vλ,0. The conclusion follows.
Now the left hand side of (3.19) is zero, and so we find that δ is given by
1
δ = V3 ℓu00wD1D2vλ,0dζ. (3.25)
1 (1 γ) u0wvλ,0dζ
− R ℓ0
Note that δ is well defined, as the undetermined multRiplicative constant of vλ,0 cancels in the ratio.
1
3.4 Summary of the Asymptotics
To summarize, we have sought the zeroth and first order terms in the expansions (3.1), (3.2) and (3.3) for
(vλ,ε,ℓε,uε,δε). The principal terms are found from the eigenvalue problem described by ODE (3.8), with
boundary and free boundary conditions (3.13)-(3.14):
(σ¯;δ )vλ,0 =0, ℓ ζ u ,
LNT 0 0 ≤ ≤ 0
vλ,0 =0, and vλ,0 =0; vλ,0 =0, and vλ,0 =0.
B |ℓ0 B′ |ℓ0 S |u0 S′ |u0
The next term in the asymptotic expansion of the boundaries of the NT region, and of the optimal
long-term growth rate ℓ ,u , and δ respectively, are given by (3.16), (3.18) and (3.25), and vλ,1 solves the
1 1 1
ODE (3.12), with boundary conditions (3.15) and (3.17):
(σ¯;δ )vλ,1 = V D D vλ,0+(1 γ)δ vλ,0, ℓ <ζ <u ,
LNT 0 − 3 1 2 − 1 0 0
vλ,1 =0, and vλ,1 =0.
B |ℓ0 S |u0
We describe the essentially-explicit solutions to these problems in the next section.
4 Building the Solution
In the previous section we have established that (vλ,0,δ ) solve the constant volatility optimal growth rate
0
with transaction costs problem, which is described in Dumas and Luciano [1991], but using the averaged
volatility σ¯, where σ¯2 = f2 . In this section, we review how to find (vλ,0,δ ), and then use them to build
0
h i
the stochastic volatility corrections (vλ,1,δ ).
1
4.1 Building vλ,0 and δ
0
We denote by (V (ζ;σ),∆ (σ),L (σ),U (σ)) the solution to the constant volatility problem with volatility
0 0 0 0
parameter σ and corresponding eigenvalue ∆ , and so
0
vλ,0(ζ)=V (ζ;σ¯), δ =∆ (σ¯), and (ℓ ,u )=(L (σ¯),U (σ¯)). (4.1)
0 0 0 0 0 0 0
Assumption 4.1. Without loss of generality assume that µ > 0. The case µ < 0 can be handled similarly
to the current case. The case µ = 0 is not interesting, as in this case one would not hold the risky stock at
all. We also assume that the optimal proportion of wealth invested into the risky stock in case of constant
volatility σ and zero transaction costs is less than 1:
µ
π := <1. (4.2)
M γσ2
We will refer to π as the Merton proportion.
M
10