Original citation:
Hong, Yi and Jin, Xing (2018) Semi-analytical solutions for dynamic portfolio choice in jump- diffusion models and the optimal bond-stock mix. European Journal of Operational
Research. 265 (1). pp. 389-398. doi:10.1016/j.ejor.2017.08.010.
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Semi-Analyt ical Solut ions for Dynamic Port folio Choice in
J ump-Di
ff
usion Models and t he Opt imal Bond-St ock Mix
Yi Hong
†International Business School Suzhou, X i'an Jiaotong-Liverpool University
Xing J in
‡W arwick Business School, University of W arwick
Abst ract
T his paper st udies t he opt imal port folio select ion problem in jump-diffusion models where
an invest or has a HARA ut ility funct ion, and t here are pot ent ially a large number of asset s
and st at e variables. More specifically, we incorporat e jumps int o bot h st ock ret urns and
st at e variables, and t hen derive semi-analyt ical solut ions for t he opt imal port folio policy
up t o solving a set of ordinary different ial equat ions t o great ly facilit at e economic insight s
and empirical applicat ions of jump-diffusion models. To examine t he effect of jump risk on
invest ors’ behavior, we apply our result s t o t he bond-st ock mix problem and part icularly
revisit t he bond/ st ock rat io puzzle in jump-diffusion models. Our result s cast new light
on t his puzzle t hat unlike pure-diffusion models, it cannot be rat ionalized by t he hedging demand assumpt ion due t o t he presence of jumps in st ock ret urns.
J EL Classificat ion: G11
Keywords: F inance, opt imal port folio select ion, jump-diffusion models, HARA ut ility
funct ions, bond-st ock mix
We are very grat eful t o Emanuele Borgonovo (t he Edit or) and four anonymous referees for t heir insight ful and const ruct ive comment s t hat subst ant ially improve t he pap er. We grat efully acknowledge t he financial supp ort from Xi’an J iaot ong-Liverp ool University (No. RDF -14-02-51).
† Int ernat ional Business School Suzhou (IBSS), Xi’an J iaot ong Liverp ool University, China; Telephone:
+ 86 51288161729; Email: yi.hong@xjt lu.edu.cn.
‡
1
Int roduct ion
As prompt ed by t he seminal work of Mert on (1969), t here is a large lit erat ure on t he dy-
namic port folio choice problem t hat has typically been st udied in cont inuous-t ime models
primarily due t o t heir analyt ical t ract ability. T here are two popular met hods t hat are
widely employed t o solve t his problem. T he first one is t he HJ B-based approach pro-
posed by Mert on (1969), and t he ot her is t he mart ingale approach advanced by Karat zas,
Lehoczky and Shreve (1987) and Cox and Huang (1989). In bot h approaches, t he in-
vest or’s ut ility funct ion plays a fundament al role in seeking t he opt imal port folio policy.1
Unfort unat ely, it is well known t hat semi-analyt ical solut ions t o t he dynamic port folio
choice problem are generally unavailable, alt hough t hey are vit ally import ant t o facilit at e
economic insight s and empirical applicat ions. In t his paper, we solve t he opt imal asset
allocat ion problem in closed form for mult i-asset jump-diffusion models in t he way t hat
t he solut ions provide a new inst rument t o analyze t he behavior of invest ors wit h general
HARA preferences t owards dist inct risk fact ors.
In a growing lit erat ure, numerous effort s have been made t o solve t he port folio choice
problem in closed form. Specifically, Ba jeux-Besnainou and P ort ait (1998) ext end t he
st at ic set up in Markowit z (1952) t o a much more challenging dynamic version and ex-
plicit ly solve t he dynamic mean-variance problem in a complet e pure-diffusion model. Recent ly, by using t he mart ingale approach, Lioui and P oncet (2016) provide closed-form
solut ions t o t he dynamic mean-variance problem in a complet e affne diffusion model.As remarked by t he aut hors, t he dynamic mean-variance model in Sect ion 2.3 of Lioui and
P oncet (2016) may result in t ime-inconsist ent port folio st rat egies, showing t hat t he in-
vest or may find it opt imal t o deviat e from her init ial policy. In cont rast , Basak and
Chabakauri (2010)2 explicit ly solve t he t ime-consist ent dynamic mean-variance policy
based on a recursive represent at ion. In a cont inuous-t ime mean variance model wit h
const raint s on port folio policy, Wang and Forsyt h (2011) develop a numerical scheme t o
det ermine t he opt imal t ime-consist ent asset allocat ion st rat egy3. For a von
Neumann-1T he widely used ut ilit y funct ions b elong t o t he so-called hyp erb olic absolut e risk aversion (HARA)
family, including quadrat ic (wit h rest rict ions on paramet ers), exp onent ial, logarit hmic, and p ower forms.
2
We t hank an anonymous referee for p oint ing t his out t o us.
3
Morgenst ern ut ility, Det emple, Garcia and Rindisbacher (2003) also use t he mart ingale
approach t o solve t he port folio choice problem in a complet e pure-diffusion model which may include a large number of asset s and st at e variables wit h non-affne st ruct ures. T hey
obt ain t he opt imal port folio st rat egy using t he Mont e Carlo simulat ion, yet which may
be t ime-consuming in t he presence of a large number of asset s and st at e variables.
As discussed in Bardhanand and Chao (1996), a jump-diffusion model wit h random jump sizes is inherent ly incomplet e. One of t he key assumpt ions in t he aforement ioned
papers is t he complet eness of t he market . In general, it is a daunt ing t ask t o explicit ly
solve t he opt imal port folio choice problem in an incomplet e market . One usually resort s
t o eit her t he HJ B equat ion or t he mart ingale met hod. As is well known, it is diffcult t o apply t he HJ B equat ion t o a high-dimensional problem in bot h complet e and incomplet e
market s. Furt hermore, it is very challenging t o use t he mart ingale met hod in an incom-
plet e market since t here are infinit ely many mart ingale measures. To solve t he opt imal
port folio problem in incomplet e pure-diffusion models, approximat ion met hods are pro- posed in Bick, Kraft and Munk (2013) and Haugh, Kogan and Wang (2006), respect ively.
Yet , t heir solut ions are numerically approximat ed and t hus may suffer inaccuracy.
In cont rast , by assuming quadrat ic condit ions in pure-diffusion models, Liu (2007)
explicit ly solves t he opt imal dynamic port folio choice problem in bot h complet e and in-
complet e market s, up t o t he solut ions t o a set of ordinary different ial equat ions (ODEs).
Specifically, he solves a set of ODEs by guessing t he exponent ial linear form of t he indi-
rect value funct ion wit hout simulat ion. T his met hod is widely used in t he asset allocat ion
lit erat ure of pure-diffusion models nowadays. However, much less is known about t he con-
dit ions t hat can lead t o t he ODE-based analyt ic solut ion t o t he opt imal port folio choice
problem in jump-diffusion models especially when bot h st ock prices and st at e variables are
allowed t o jump.4 T he ob ject ive of t he present paper is t hen t o generalize t he
afore-4
ment ioned ODE-based approach in pure-diffusion models t o jump-diffusion models which
nest t he former (e.g., Liu (2007)) as special cases.
More specifically, we first consider const ant relat ive risk aversion (CRRA) ut ility
func-t ions and provide func-t he condifunc-t ions under which func-t he indirecfunc-t value funcfunc-t ion in jump-diffusion models has an exponent ial linear form. T he indirect value funct ion and t he opt imal port -
folio st rat egy can t hen be obt ained by solving a set of ODEs. By providing an effcient two-st ep approach, we furt her ext end our ODE-based met hod t o more general HARA
ut ility funct ions given t heir popularity in financial economics.5 Our result s show t hat t he
indirect ut ility funct ion for a HARA ut ility t akes a form significant ly different from t he
exponent ial linear one for a CRRA ut ility. To t he best of our knowledge6, we are not
aware of any semi-analyt ical solut ion t o t he dynamic asset allocat ion problem in jump-
diffusion models where risk-averse invest ors face jumps in mult iple risky asset s and st at e
variables. More import ant ly, t he semi-analyt ical solut ions may great ly facilit at e economic
insight s and enhance our underst anding of invest ors’ behavior t owards jump risks.
Our paper is closely relat ed t o t he work of J in and Zhang (2012) in t hat t hey use a
decomposit ion approach based on an HJ B equat ion t o solve a port folio select ion problem
t hat includes a large number of risky asset s and st at e variables. But t heir st at e variables
are pure-diffusion processes and t he indirect value funct ion is evaluat ed by t he Mont e Carlo
simulat ion. Our paper also relat es t o t he work of Das and Uppal (2004) and
A¨ıt -Sahalia, Cacho-Diaz and Hurd (2009). T hese st udies solve t he port folio select ion problem in jump-diffusion models, but wit hout st at e variables. In cont rast , we obt ain semi-analyt ical solut ions t o t he opt imal port folio st rat egy under jump-diffusion models t hat
include a large number of asset s and st at e variables. T hese solut ions t herefore allow
select ion, see, for example, Liu, Longst affand P an (2003) and Das and Uppal (2004).
5More imp ort ant ly, P eret s and Yashiv (2016) show t hat t he HARA ut ilit y is more fundament al t o
economic analysis. T his funct ional form is t he unique one which sat isfies basic economic principles in an opt imizat ion cont ext . T herefore, t he use of HARA ut ilit y funct ions is not just a mat t er of convenience or t ract abilit y, but rat her emerges from economic reasoning, i.e., it is inherent in t he economic opt imizat ion problem.
6It should b e not ed t hat for t he logarit hmic ut ilit y maximizat ion under jump diffusions, semi-analyt ical
us t o solve in a comput at ionally effcient way t he dynamic port folio select ion problem in
jump-diffusion models where bot h st ock ret urns and st at e variables can jump.
By using t he t heoret ical framework developed in t his paper, we st udy t he problem of
how jumps in st ock ret urns affect t he opt imal cash-bond-st ock port folio in a dynamic asset allocat ion model where an invest or can t rade one st ock, two bonds, and cash. Especially,
we revisit t he asset allocat ion puzzle raised in Canner, Markiw and Weil (1997). T hey
document t he empirical evidence t hat st rat egic asset allocat ion advices t end t o recommend
a higher bond/ st ock rat io for a more risk-averse invest or. Several st udies have at t empt ed
t o explain t he rat ionality of t his puzzle. For inst ance, Brennan and Xia (2000) and
Ba jeux-Besnainou and P ort ait (2001) relat e t he puzzle t o a hedging component in t he
st ochast ic int erest rat e and provide elegant solut ions t o t he asset allocat ion puzzle. All
of t hese st udies assume t hat bot h t he short -t erm int erest rat e and st ock ret urns follow
pure diffusion processes. Our framework generalizes t hese st udies by incorporat ing jumps
int o st ock ret urns and examining t he role of risk aversion in det ermining t he opt imal
cash-bond-st ock port folio. In part icular, we show bot h t heoret ically and numerically
t hat unlike t he pure-diffusion models in Brennan and Xia (2000), Ba jeux-Besnainou and
Port ait (2001) and Lioui (2007), t here is no clear-cut answer t o t he bond/ st ock rat io
puzzle in jump-diffusion models even despit e t he aforement ioned hedging assumpt ion. In
ot her words, t he puzzle it self cannot be rat ionalized by t he hedging assumpt ion in t he
presence of jumps in st ock ret urns. T he underlying reason for t his is t hat an invest or
responds dist inct ly t o diffusion risk premium and jump risk premium when t here is an increase in t he invest ors’s relat ive risk aversion coeffcient .
In summary, our paper makes t hree cont ribut ions t o t he lit erat ure on port folio choice.
First , our work generalizes t he popular ODE-based approach used in pure-diffusion mod-
els t o jump-diffusion models for CRRA ut ility funct ions, which may great ly alleviat e comput at ional effort s in seeking t he opt imal port folio st rat egy. Second, we provide an
effcient two-st ep met hod for solving HARA preference-based ODEs. T his t hen ext ends t he applicability of our approach wit hin a family of general ut ility funct ions. F inally, we
t o underst and t he nat ure of t his well-known puzzle.
T he rest of t he paper is organized as follows. In Sect ion 2, we present t he framework for
Mert on’s dynamic port folio select ion problem in jump-diffusion models and t hen present
affne condit ions in t he jump-diffusion models. In Sect ion 3, we use t he affne condit ions t o explicit ly solve t he indirect value funct ion and t he opt imal port folio st rat egy in t erms
of t he solut ions t o a set of ODEs for general HARA preferences. In Sect ion 4, we derive
semi-analyt ical solut ions t o t he opt imal bond-st ock mix and especially invest igat e how
jump risk in st ock ret urns affect s bond/ st ock rat ios. Sect ion 5 is devot ed t o a calibrat ion exercise in order t o illust rat e numerically t he t heoret ical result s in Sect ion 4. We conclude
in Sect ion 6. All proofs are collect ed in Appendices.
2
T he Economy
In t his sect ion, we formulat e a model of incomplet e financial market s in a cont inuous t ime
economy where asset prices and st at e variables follow a mult idimensional jump-diffusion process on t he fixed t ime horizon [0; T ] (0 < T < ∞). We consider a complet e probability
space (Ω; F ; P ), where Ω is t he set of st at es of nat ure wit h generic element s ! s, and F is t he -algebra of observable event s, while P is a probability measure on (Ω; F ).
We use an l-dimensional vect or X t = (X 1t ; :::; X lt ) t o denot e t he st at e variables of
t he economy where t he convent ion st ands for t he t ranspose of a vect or or a mat rix.
T he st at e variables X t may include st ochast ic volat ility and st ochast ic int erest rat e as it s
component s. We assume t hat st at e variables X t follow a jump-diffusion process
dX t = bx (X t )dt + x (X t )dB X (t) + x (X t )(Y x • dN (t)) (1)
where bx (X t ) is an l-dimensional vect or funct ion, x (X t ) is an l × l mat rix funct ion of X t ,
and x (X t ) is an l × m mat rix funct ion of X t , respect ively. B X (t) = (B X (t); :::; B X (t))
is an l-dimensional st andard Brownian mot ion; N (t) = (N1(t); :::; Nm (t)) is an m -
dimensional mult ivariat e Poisson process wit h Nk (t) denot ing t he number of type k jumps
up t o t ime t, while Y x = (Y x ; :::; Y x ) represent s an m -dimensional jump size process J
J 1 l
wit h Y x denot ing t he amplit ude of t he type k jump condit ional on t he occurrence of t he k
-t h jump. For any -two n-dimensional vect ors x = (x1; :::; xn ) and y = (y1; :::; yn ) , we denot e
t he component -wise mult iplicat ion as x • y = (x1y1; :::; xn yn ) . Not e t hat unlike Liu
(2007) and J in and Zhang (2012), t he above specificat ion of X t includes jumps in st at e
variables. For inst ance, we can incorporat e jumps int o a volat ility process. By let t ing
Y x = 0, our jump-diffusion model reduces t o it s pure-diffusion count erpart for t he st at e
variables X t .
T he uncert ainty of t he economy is also generat ed by a d-dimensional st andard
Brow-nian mot ion B S (t) = (B S (t); :::; B S (t)) which drives st ock prices defined below. Assume
B S (t) and B X (t) are correlat ed and E [dB X (t)d(B S (t)) ] = t dt, for some l × d mat rix
t . T he informat ion flow in t he economy is given by t he nat ural filt rat ion, i.e., t he right
-cont inuous and augment ed filt rat ion { Ft }t∈[0;T ] = { F S ∨F X ∨F N ; t ∈[0; T ]}, where
F S = (B S (s); 0 ≤ s ≤ t), F X = (B X (s); 0 ≤ s ≤ t) and F N = (N (s); 0 ≤ s ≤ t).
We suppose t hat observable event s are event ually known, i.e., F = FT . For illust rat ive
purposes, we assume t hat Nk admit s st ochast ic int ensity k (X t ) t hat represent s t he rat e
of t he jump process at t ime t.
T he market includes n + 1 asset s t raded cont inuously on t he t ime horizon [0; T ]. One of
t hese asset s, risk-free, has a price S0(t) which evolves according t o t he different ial equat ion
dS0(t) = S0(t)r(X t )dt; S0(0) = 1:
T he remaining n asset s, called st ocks, are risky, and t heir prices are modeled by t he linear
st ochast ic different ial equat ion
dSi (t)
Si (t−)
= bi (X t )dt + b(X t )dB S (t) + q(X t )(Y s • dN S (t))
where i = 1; :::; n, N S (t) = (N1(t); :::; Nn − d(t)) , and Y s = (Y s ; :::; Y s− d) , wit h Y s
denot ing t he amplit ude of t he type k jump condit ional on t he occurrence of t he k-t h
jump. Here b(X t ) is t he d-dimensional diffusion coeffcient row vect or and q(X t ) is
t he (n − d)-dimensional jump coeffcient row vect or. In part icular, t he Brownian mot ions k
1 d
t t t
t t t
i i
1 n k
1−γ, ∀x >0;
−∞, ∀x≤0,
(2)
where γ(̸= 1) is the relative risk aversion (RRA) coefficient. We will solve the
op-timal portfolio choice problem for more general HARA utility functions in the next
section. Specifically, we consider an investor with the utility function U(x), endowed
with some initial wealth w0 that is invested in the above-mentioned n+ 1 assets. Let
π(t) = (π1(t), ..., πn(t))⊤ denote a trading strategy, where the Ft-predictableπi(t) is the
proportion of the total wealth invested in thei-th risky asset held at timet. Furthermore,
π(t) satisfies the standard square-integrability condition discussed in Bremaud (1981).
Moreover, the portfolio policyπ(t) has an associated wealth process Wtthat evolves as
Wt = W0+
∫ t
0
r(s)Wsds+ ∫ t
0
Wsπ⊤(s)(b(s)−r(s)1n)ds
+ ∫ t
0
Wsπ⊤(s)Σb(Xs)dBS(s) + ∫ t
0
Ws−π⊤(s−)Σq(Xs)(Ys•dNS(s)) (3)
where b(t) = (b1(Xt), ..., bn(Xt))⊤, Σb(Xt) is an n×dmatrix with σbi being its i-th row,
Σq(Xt) is then×(n−d) matrix withσiq being itsi-th row. Here we use1nto denote the
n-dimensional column vector of ones. The portfolio policyπ(t) is said to be admissible if
the corresponding wealth process satisfiesWt≥0 almost surely. We useA(w0) to denote
the set of all admissible trading strategies. Then, Merton’s portfolio choice problem states represent frequent small movement s in st ock prices, while t he jump processes represent
infrequent large shocks t o t he market . Assuming n − d ≤ m , t he jumps N S (t) can be
regarded as common jumps in st ock ret urns and st at e variables.
To obt ain t he semi-analyt ical solut ions t o t he opt imal port folio choice problem, we now
t urn t o t he assumpt ion for affne models. In t his paper, we focus on Mert on’s problem of
maximizing t he expect ed ut ility from t he t erminal wealt h.7 In t his sect ion, for illust rat ive
purposes, we follow t he lit erat ure t o consider t he CRRA ut ility funct ion given by
U(x) = ; ∀x > 0;
−∞ ; ∀x ≤ 0;
(2)
where (= 1) is t he relat ive risk aversion (RRA) coeffcient . We will solve t he op-t imal porop-t folio choice problem for more general HARA uop-t iliop-ty funcop-t ions in op-t he nexop-t
sect ion. Specifically, we consider an invest or wit h t he ut ility funct ion U(x), endowed wit h some init ial wealt h w0 t hat is invest ed in t he above-ment ioned n + 1 asset s. Let
(t) = ( 1(t); :::; n (t)) denot e a t rading st rat egy, where t he Ft -predict able i (t) is t he
proport ion of t he t ot al wealt h invest ed in t he i-t h risky asset held at t ime t. Furt hermore,
(t) sat isfies t he st andard square-int egrability condit ion discussed in Bremaud (1981).
Moreover, t he port folio policy (t) has an associat ed wealt h process Wt t hat evolves as
Wt = W0 +
∫ t
r(s)Ws ds + ∫ t
Ws (s)(b(s) − r(s)1n )ds ∫ t
0 ∫
t + Ws (s)Σb(X s )dB S (s) +
0
Ws− (s−)Σq(X s )(Y s • dN S (s)) (3)
where b(t) = (b1(X t ); :::; bn (X t )) , Σb(X t ) is an n × d mat rix wit h b being it s i-t h row, Σq(X t ) is t he n × (n − d) mat rix wit h q being it s i-t h row. Here we use 1n t o denot e t he n -dimensional column vect or of ones. T he port folio policy (t) is said t o be admissible if t he
corresponding wealt h process sat isfies Wt ≥ 0 almost surely. We use A (w0) t o denot e t he
set of all admissible t rading st rat egies. T hen, Mert on’s port folio choice problem st at es
7
A semi-analyt ical solut ion can b e obt ained for t he opt imal p ort folio choice problem wit h t he ut ilit y funct ion defined by (2) in Liu (2007) when t he Brownian mot ions in prices and st at e variables are t he same, namely, B X(t) = B S(t). T his condit ion is sat isfied in t he applicat ions in Sect ion 4.
x 1−
1−
i
t hat t he invest or at t empt s t o maximize t he following quant ity
u(w0; X 0) = max J (w0; X 0) = E [U(WT )]:
∈A (w0 )
We consider t he general case: n − d < m because, by let t ing Y x = 0; k = m 0 + 1; :::; m ,
we can get t he model where t here are only m 0 (≤ n − d) types of jumps in st at e variables.
Using t he st andard approach t o st ochast ic cont rol and an appropriat e It o’s lemma for
jump-diffusion processes, t he opt imal port folio policy and t he corresponding indirect value funct ion J of t he invest or’s problem t hen follow t he HJ B equat ion:
{ 0 = max Jt +
1 2W
2 ΣbΣ
b JW W + Wt [ (b(t) − r1n ) + r]JW
+ bx (t)JX + Wt Σb t x (t)JW X + 1 2T r(
x (t) x (t)J
X X ) (4)
+ n − d
k= 1
k E [J (Wt + Wt qk Y s ; X t + Y x ; t) − J (Wt ; X t ; t)]
+ ∑
m
k= n − d+ 1
k E [J (Wt ; X t + Y x ; t) − J (Wt ; X t ; t)]
where qk denot es t he k-t h column of q. T he above HJ B equat ion nest s t he HJ B
equat ion (3) for t he pure-diffusion model in Liu (2007) as a special case by let t ing n−d = 0. In ot her words, we generalize t he models in Liu (2007) by incorporat ing jumps int o st ock
ret urns and st at e variables. It is well-known t hat in t he pure-diffusion model in Liu (2007), t he indirect value funct ion J (Wt ; X t ; t) is conject ured t o have t he form: J (Wt ; X t ; t) =
W 1−
1−
[
eA (t)+ B (t) X t
]
, where A(t) is a scalar and B (t) is an l × 1 vect or. T hen, under
t he quadrat ic condit ions, a set of ODEs for t he funct ions A(t) and B (t) are obt ained by
subst it ut ing t he funct ion J and t he opt imal port folio st rat egy int o t he HJ B equat ion
(4). As shown below, t he argument in Liu (2007) does not t rivially apply t o jump-diffusion models because t he port folio policy may depend on t he st at e variables X t .
We now illust rat e t he diffculty caused by jumps. More specifically, compared wit h t he HJ B equat ion (3) in Liu (2007) for pure-diffusion models, t he jump t erms in t he above
HJ B equat ion creat e new diffcult ies for semi-analyt ical solut ions t o t he opt imal port folio k
t
∑
k k
k
}
choice problem in jump-diffusion models. We now consider a simple case where t here are
no jumps in t he st at e variables X t by let t ing Y x = 0; k = 1; :::; n − d. As in Liu (2007), we
subst it ut e t he indirect value funct ion J (Wt ; X t ; t) = W 1−
(f (t; X t )) int o (4) and obt ain
t he following form for t he last t erm:
n − d
E [J (Wt + Wt qk Y s ; X t ; t) − J (Wt ; X t ; t)] k= 1
= W
1−
(f (t; X t )) n − d
k= 1
k (X t )E [(1 + qk Y s )1− − 1]:
As is well-underst ood from, for inst ance, Liu (2007), in order t o gain an explicit solut ion for
t he indirect value funct ion J (Wt ; X t ; t) of t he form J (Wt ; X t ; t) = W 1−
eA (t)+ B (t) X t ,
t he t erm E [(1 + qk Y s )1− ] should be an affne funct ion of t he st at e variables X t . T his
t erm, however, is hard t o be an affne funct ion of t he st at e variables X t unless t he opt imal
jump exposure qk is a det erminist ic funct ion of t ime t, because t he funct ion x1− is
generally not an affne funct ion. Based on t his observat ion and inspired by t he result s in Liu (2007) and t he result of decomposit ion of opt imal port folio weight s in J in and Zhang (2012),
we are able t o specify an affne model8 which leads t o ODEs for A(t) and B (t) given in P roposit ion 1 in Sect ion 3.
More specifically, by set t ing ak = E (Y s ); k = 1; :::; n − d, we assume t hat t he mat rix
Σ = [Σ b;Σ q] is invert ible. T he market price of risk is t hen represent ed by
b
= Σ−1(b(t) − r1
n + Σq( • a)); (5)
where • a = ( 1a1; :::; n − dan − d) ; b = ( b; :::; b) and q = ( q; :::; q− d) . As shown
in Sect ion 4, b denot es t he risk premium for t he Brownian mot ion B S ; i = 1; :::; d, while
q
represent s t he risk premium for t he jump N S ; k = 1; :::; n − d, in t he st ock ret urns. We
furt her make t he following assumpt ions:
8
Here, for exp osit ional purp oses, we consider affne models only as it is st raight forward t o generalize our result s t o quadrat ic processes defined in Liu (2007).
k
t
1−
∑
k
t 1 −
∑
k
1−
[ ]
k
k
q
1 d 1 n
i i
A ssum pt ion 1
bx (X ) = k − K X ; x x = h0 + h1 · X ;
r = 0 + X ; b b = H0 + H X ; (6)
x
t b = g0 + g1X ; x t t x − x x = l0 + l1 · X ;
= 0 + 1X ; q = 0 k ; k = 1; :::; n − d;
where k ; 1; H1 and g0 are l × 1 const ant vect ors; K ; h0; g1 and l0 are l × l const ant mat rices;
0; H0 and 0 are const ant s; 0 is an (n − d)× 1 const ant vect or; 1 is an (n − d)× l const ant
mat rix; h1 = hi1j k ; i; j ; k = 1; :::; l and l1 = li j k ; i; j ; k = 1; :::; l are const ant t ensors wit h
t hree indices (one upper index and two lower indices). In part icular, h1 · X is an l × l
mat rix whose (j ; k) element is given as follows:
∑l
(h1 · X )j k = hi1j k X it : i= 1
T he l × l mat rix l1 · X is defined exact ly in t he same manner. T he above assumpt ions
except t he last two are similar t o t hose made in Liu (2007), while t he last two assumpt ions
on jump int ensity and jump risk premium are also st andard in lit erat ure, and t he last
assumpt ion st at es t hat t he risk premium for t he k-t h jump is proport ional t o it s int ensity.
3
T he P ort folio Choice P roblem
Given t he affne models in t he proceeding sect ion, we now explicit ly solve t he opt imal port folio choice problem for hyperbolic absolut e risk aversion (HARA) ut ility funct ions
up t o solving a set of ODEs. T he most popular ut ility funct ions used in almost all
applied t heories and empirical st udies in finance belong t o t he class of linear risk t olerance
(LRT ) or HARA ut ility funct ions, including t he quadrat ic funct ion (wit h rest rict ions on
paramet ers), t he CRRA ut ility, t he exponent ial ut ility and t he logarit hmic ut ility as
special cases. T herefore, t he explicit solut ions t o t he port folio choice problem for HARA
preferences may cast new light on invest ors’ behavior t owards dist inct risk fact ors in a
1 1
k k
k
st ochast ic invest ment environment . More specifically, a HARA ut ility funct ion is given
by
U(x) =
0
1− (x − )
1− ; ∀x >
: (7)
For = 0; U(x) reduces t o a CRRA ut ility funct ion (2). Here we consider a realist ic
case wit h > 0,9 t hat is, t he relat ive risk aversion is decreasing wit h wealt h. In Ba jeux-
Besnainou and P ort ait (2001), t hey int erpret t he const ant as a “subsist ence level”.
Canakoglu and Ozekici (2012) consider t he opt imal port folio select ion problem in
a cont inuous-t ime pure-diffusion set t ing where t he market st at es follow Markov pro- cesses. T hey ut ilize t he HJ B-based approach t o obt ain semi-analyt ical solut ions for t he CRRA
ut ility, t he exponent ial ut ility and t he logarit hmic ut ility, respect ively. In Ba jeux-
Besnainou and P ort ait (2001), t hey obt ain closed-form solut ions t o t he opt imal dynamic
port folios for t he HARA ut ility in pure-diffusion models. Specifically, t hey employ t he duality result s developed by Karat zas, Lehoczky and Shreve (1987), subst ant ially root ed
in t he key assumpt ion of t he exist ence of a unique equivalent mart ingale measure in a
complet e market . In cont rast , t he market s in t his paper are incomplet e due t o random
jump sizes and t hus t here exist infinit ely many equivalent mart ingale measures. As in J in,
Luo and Zeng (2016), t o solve an opt imal dynamic port folio problem for t he HARA un-
t ility, we resort t o t he duality result s for incomplet e market s developed by Kramkov and
Schachermayer (1999) in combinat ion wit h t he result s developed for t he CRRA ut ility.
But our result s differ from J in, Luo and Zeng (2016) in t hat we incorporat e jumps int o
st at e variables and solve t he opt imal port folio problem based on a set of ODEs inst ead
of a simulat ion-based approach used in t heir paper. Our main result s are summarized in
t he following two proposit ions.
9
For t he case < 0, similar t o t he result s in Sect ion 6.3 of Mert on (1990), t he unconst rained p olicies derived by t he met hod in t he present pap er may violat e t he nonnegat ivit y condit ion on wealt h. T hus, we need t o solve t he const rained problem wit h a p osit ive wealt h process. T his is b eyond t he scop e of t he present pap er and we leave it as a fut ure research.
P rop osit ion 1 Under A ssum ption 1, the indirect value function is represented as
J (Wt ; X t ; t) = (
Wt − e (t)− A (t)+ ( (t)− B (t)) X t
1 −
) 1− [
eA (t)+ B (t) X t
]
(8)
where A(t), B (t), (t) and (t) are obtained by ODEs in A ppendix A .
P ro of. See Appendix A.
T he result in (8) suggest s t hat unlike t he indirect ut ility funct ion for a CRRA ut ility
by set t ing = 0, t he one for a HARA preference cannot be separat ed int o a product of
two funct ions, one depending on t he wealt h W and t he ot her on t he st at e variables X t and
t ime t. T his result ext ends t he lit erat ure on t he opt imal port folio choice wit h a HARA
ut ility. For det ailed discussions, for example, Mert on (1990) and P eret s and Yashiv (2016)
suggest t hat t he above decomposit ion holds t rue due t o const ant invest ment opport unit ies.
P rop osit ion 2 Under A ssum ption 1, the optim al portfolio weight = ( ; :::; ) is
given by
= (
e 1; :::;e d;e 1; :::;e (n − d) )
Σ−1 (9)
where the optim al e is given by
e = W − g(t; X t ) [
eb
+ t x B (t) ]
+ Σb t
x ( (t) − B (t))g(t; X t )
W (10)
and e k solves the following optim ization problem :
( max
eq k ∈F k
eqk W (W − g(t; X t ))− ( eq − k ak )
+ k
1 − E W (1 + qk Y
s ) − g(t; X t )e (t) x k Y x
) 1−
e B (t) x k Y x (11)
for k = 1; :::; n − d, where Fk is the set of feasible k -th jum p exposures satisfying the jum p
induced no-bankruptcy condition, nam ely, Fk = { x |x · y > −1;∀y ∈A k } , with A k denoting
the support of the k -th jum p size Y s , and g(t; X t ) = e (t)− A (t)+ ( (t)− B (t)) X t .
P ro of. See Appendix B.
1 n
b b q q
b
b W
q
k
k
[ ( ] )
J k J k
The second term in (10) indicates that as opposed to a CRRA utility (η = 0), a HARA
utility (η̸= 0) has a separate hedging demand for the interest rate related risk. This term
will disappear if the interest rate is a constant since in this case,β(t) =γB(t) as can seen
in the proof of Appendix A. Furthermore, letting η = 0 in (11) and using Assumption 1
gives the optimal jump exposure problem for a CRRA utility:
max
e
πqk∈Fk
( e πqk
(
θ0k−ak )
+ 1
T he second t erm in (10) indicat es t hat as opposed t o a CRRA ut ility ( = 0), a HARA
ut ility ( = 0) has a separat e hedging demand for t he int erest rat e relat ed risk. T his t erm
will disappear if t he int erest rat e is a const ant since in t his case, (t) = B (t) as can seen
in t he proof of Appendix A. Furt hermore, let t ing = 0 in (11) and using Assumpt ion 1
gives t he opt imal jump exposure problem for a CRRA ut ility:
max
eq k ∈F k (
eqk ( 0
− ak
) + 1 1 − E [
(W (1 + qk Y s ))1− e B (t) x
k Y x] )
(12)
T he ob ject ive funct ion in t he opt imizat ion problem in (12) does not include t he st at e
variables X t and t hus, for each k, t he opt imal jump exposure e k is det erminist ic.10
T his just ifies t he conject ured exponent ial linear form of t he indirect value funct ion for a
CRRA ut ility. It is wort h ment ioning t hat despit e t he det erminist ic jump exposure e k ,
t he opt imal port folio policy is st ill dependent on t he st at e variables X t t hrough t he
opt imal diffusion exposures (e 1; :::;e d) and t he mat rix Σ. T his st at e-dependent port folio st rat egy reflect s t he invest or’s market t iming behavior.
As we discuss in Appendix B, t he conject ure-based approach used in Liu (2007) is very
likely inapplicable t o a HARA ut ility in jump-diffusion models as it is hard t o subst it ut e t he opt imal jump exposure in (11) int o t he HJ B equat ion. T wo reasons account for t his
diffculty. On t he one hand, as shown in t he first -order condit ion for e k in Appendix A, it is generally impossible t o solve t he opt imal e k in closed form unless all jumps are
const ant s. On t he ot her hand, t he opt imizat ion problem in (11) shows t hat t he jump
exposure e k depends on bot h t he wealt h W and t he st at e variables X t and t hus is not
det erminist ic, making it hard t o use t he conject ure-based met hod. As a result , we propose
a two-st ep approach t o solving t he opt imal asset allocat ion problem for t he HARA ut ility
funct ion specified in (7) summarized as follows:
(i) In t he first st ep, t he funct ions (t); (t); A(t) and B (t) are det ermined by solving t he
opt imal asset allocat ion problem for a CRRA ut ility funct ion in (2);
1 0It will b e shown in App endix A t hat t he result of t he det erminist ic jump exp osure e
k of t he CRRA
ut ilit y funct ion is part icularly useful when we solve t he opt imal p ort folio choice problem in closed form wit h a more general HARA ut ilit y funct ion.
k k J k
q
q
b b
q
q
q
(ii) In t he second st ep, t he indirect ut ility funct ion J (Wt ; X t ; t) of t he HARA ut ility
funct ion is evaluat ed by (8) and t hen t he opt imal port folio weight s are det ermined
t hrough (9), (10) and (11).
Our two-st ep approach t herefore cont ribut es t o t he lit erat ure in solving t he opt imal port -
folio choice problem for HARA preferences effcient ly in jump-diffusion models.
4
D ynam ic A sset A llocat ion for St ocks, B onds and Cash
We now apply t he result s in Sect ion 3 t o examine t he impact of jumps in st ock ret urns
on t he opt imal cash-bond-st ock mix in a dynamic model where an invest or can t rade
one st ock, two bonds, and cash (or t he called money market account ). A closely relat ed
problem is t he asset allocat ion puzzle raised in Canner, Markiw and Weil (1997). T hey
empirically document t hat t he st rat egic asset allocat ion advice t ends t o recommend a
higher bond/ st ock rat io for an invest or wit h more risk aversion. T his finding, however, is inconsist ent wit h Tobin (1958)’s Separat ion T heorem t hat t he rat io of bonds t o st ocks in
t he opt imal port folio is t he same for all invest ors regardless of t heir risk aversion.
Brennan and Xia (2000) and Ba jeux-Besnainou and P ort ait (2001) relat e t his puzzle
t o a hedging component in t he st ochast ic int erest rat e and provide elegant solut ions t o
t he asset allocat ion puzzle. More specifically, as point ed out by Lioui (2007), t he puzzle
can be resolved under t he assumpt ion t hat one or several bonds can perfect ly hedge t he
risk from t he int erest rat e and t he market price of risk. Yet , Lioui (2007) argues t hat
t here is no clear-cut answer t o t he puzzle if t he hedging assumpt ion is invalid. All of t hese
st udies assume t hat t he short -t erm int erest rat e and st ock ret urns follow pure-diffusion
processes. T his sect ion at t empt s t o generalize t hese st udies by incorporat ing jumps int o
st ock ret urns11 and examining t he role of risk aversion in det ermining t he opt imal cash-
bond-st ock mix in t he presence of jump risk. Int erest ingly, we will show t hat unlike t he
pure-diffusion model in Lioui (2007), t here is no clear-cut answer t o t he bond/ st ock rat io
1 1
puzzle in a jump-diffusion model even despit e t he aforement ioned hedging assumpt ion.
T his finding demonst rat es t hat t he puzzle cannot be rat ionalized by t he hedging assump- t ion in t he presence of jumps and t hus st rengt hens t he claim made by Lioui (2007) t hat
t he asset allocat ion puzzle is st ill a puzzle.
Like Lioui (2007), we adopt a two-fact or t erm st ruct ure model t hat is a simplified
version of t he mult i-fact or models in Sangvinat sos and Wacht er (2005). We ext end it by
adding a jump component in t he st ock price. T he model assumes t he following dynamics
under t he physical measure P :
r(X (t); t) = 0 + X (t);
dX (t) = K ( − X (t))dt + X dZ (t); (13)
where r(t) is t he short -t erm int erest rat e; X (t) is a 2 × 1 vect or of st at e variables; Z (t) =
(Z1(t); Z2(t)) is a st andard 2-dimensional Brownian mot ion; 0 ∈R ; ∈R 2× 1; K ∈
R 2× 2; ∈ R 2× 1;
X = ( X i j )1≤ i;j ≤2 is a 2 × 2 non-singular mat rix, and all of t hese
paramet ers are assumed t o be const ant s.
For simplicity, we incorporat e only one type of jump int o t he st ock ret urns. We specify
t he Radon-Nikodym derivat ive as dQP = t = Z N as follows:
Z = Z exp (
−Λ¯ (t) dZ (t) − 1
∫
0
t
¯ (t) ¯ (t) dt
)
N (t)
N = N #(t
i ) (ti ; zi ) exp i= 1
( ∫
0
t∫ A
(1 − #(s) (s; z)) (X s )Φ(s; dz)ds )
where Λ¯ (t) = ¯1 + ¯2X (t), ¯1 ∈R 2× 1 is a const ant vect or; ¯2 ∈R 2× 2 is a const ant mat rix;
ti is t he i−t h jump t ime up t o t; zi is t he corresponding jump size; #(s) and (s; z) are
posit ive st ochast ic processes, and (s; z) sat isfies t he relat ionship of ∫A (t; z)Φ(t; dz) = 1,where A and Φ(t; dz) are t he support and dist ribut ion of t he jump size, respect ively. By T heorem T 10 of Bremaud (1981), under t he probability measure Q, t he int ensity Q
is # and t he density funct ion ΦQ (t; dz) is (z)Φ(t; dz).
Due t o no jumps in t he int erest rat e, a zero-coupon can be priced by using Radon-d t t
t 0 2 Λ Λ
Pi(t)
= (A2(τi)σXΛ(¯ t) +r(t))dt+A2(τi)σXdZ(t), i= 1,2, (14)
where τi = Ti −t and Ti denotes the maturity date of bond i with τ1 ̸= τ2, while
A2(τi) = (A21(τi), A22(τi)) is a 1×2 row vector fori= 1,2. And moreover, from Appendix
A in Sangvinatsos and Wachter (2005),A2(τ) solves the following ODE
dA2(τ)
Nikodym derivat ive Z . As shown in Sangvinat sos and Wacht er (2005), t he nominal bond
price evolves as follows:
dPi (t)
Pi (t)
= (A2( i ) X Λ¯ (t) + r(t))dt + A2( i ) X dZ (t); i = 1;2; (14)
where i = Ti − t and Ti denot es t he mat urity dat e of bond i wit h 1 = 2, while A2
( i ) = (A21( i ); A22( i )) is a 1× 2 row vect or for i = 1;2. And moreover, from Appendix A
in Sangvinat sos and Wacht er (2005), A2( ) solves t he following ODE
dA2( )
d = −A2( )(K + X ¯2) − ; (15)
wit h t he boundary condit ion A2(0) = 01× 2.
To explain t he asset allocat ion puzzle, Lioui (2007) assumes t hat only t he short rat e
is st ochast ic while t he market prices are det erminist ic. For comparison, we follow Lioui
(2007) t o assume t hat t he price of risk Λ¯ (t) is a const ant vect or by set t ing ¯2 = 02× 2, and
t hen solve t he equat ion in(15) t o obt ain t he following
A2( ) = (e− K −1)K −1: (16)
Denot e t he vect ors of volat ility and risk premia of t he two bonds by
P =
A2( 1) X
A2( 2) X
= A2( 1)
A2( 2)
X = A2 X ;
and P = P Λ¯ (t), respect ively.
To compare wit h t he result s of t he bond/ st ock rat io in a pure-diffusion model in
Brennan and Xia (2000), we assume t hat t he invest or who has a CRRA ut ility funct ion is
allowed t o invest in two bonds, one st ock, and cash. In addit ion t o t he above two bonds,
price B (t) and one st ock index wit h t he price S (t) where B (t) and S (t) sat isfy
dB (t)
B (t)
dS (t)
S (t)
= r(t)dt; (17)
= ( S + r(t))dt + S dZ (t) + J dN (t) − gP P dt; (18)
where S = S Λ¯ (t) + gP P − gQ Q ; S = ( S 1; S 2); gP and P are t he expect ed jump
size and jump int ensity under t he physical measure P , respect ively; gQ and Q are t he
expect ed jump size and jump int ensity under t he risk neut ral measure Q, respect ively.
Specifically, S is t he t ot al risk premium for t he st ock wit h t he t erm S Λ¯ (t) compensat ing
for t he diffusion risk, while t he t erm gP P − gQ Q compensat es for t he jump risk.
T his specificat ion implies t hat t he two bonds and cash are relat ively safer t han st ock
during a t urbulent period when jump occurs. As is well underst ood, jumps in st ock ret urns
have significant impact s on t he opt imal port folio choice. For inst ance, Liu, Longst affand Pan (2003) demonst rat e t hat in t he presence of jumps in st ock ret urns invest ors are less
willing t o t ake levered or short posit ions t han in a st andard diffusion model. Furt hermore, even when t he chance of a large jump is remot e, an invest or has st rong incent ives t o
significant ly reduce her exposure t o t he st ock market . T he reason is t hat , if a jump occurs, invest ed wealt h can change significant ly from it s current value, and such changes
cannot be hedged t hrough cont inuous rebalancing, result ing in pot ent ially large losses for
invest ors wit h levered or short posit ions. In st ark cont rast , t he changes in bond prices
can be hedged t hrough cont inuous rebalancing as t hey follow pure-diffusion processes. A nat ural quest ion is: how does a risk-averse invest or choose her bond-st ock mix when
facing uncert ain abrupt changes in st ock ret urns? More concret ely, does a more risk-averse
invest or hold more bonds and/ or cash t han a less risk-averse invest or does? To answer
t hese quest ions, we let 1; 2 and denot e t he fract ions of t he wealt h invest ed in t he
two bonds and t he st ock, respect ively. And hence, t he remainder C = 1− 1 − 2 −
is invest ed in cash. T he following proposit ion present s a semi-analyt ical solut ion t o t he
opt imal st rat egy.
B B S
P rop osit ion 3 T he optim al portfolio weight = ( 1; 2; ) is given by
[ ¯(t)
( 1; 2) = + f X
]
−1 − ˜
q S −1; (19)
= ˜ ; (20)
where the function f (t; X t ) is given in A ppendix A , and e solves the following optim
iza-tion problem :
sup
eq ∈F
eq(−gQ Q ) + P
1 − ∫
A
(1 + eqz)1− Φ(dz); (21)
where F speci es the set of feasible jum p exposures satisfying the jum p induced no-
bankruptcy condition, and A and Φ(dz) are the support and distribution of the jum p size.
P ro of. See Appendix C.
Int erest ingly, Equat ion (20) shows t hat t he demand for t he st ock index has a specu-
lat ive component t o gain t he risk premium only from jumps as suggest ed by t he st at ic
opt imizat ion problem for e , while t he burden of hedging t he int erest rat e risk and t he
market price of risk is borne by t he two bonds. T his result holds t rue regardless of whet her
or not 1 = T , namely, t he mat urity of a bond equal t o t he invest ment horizon. T he reason
underlying t he result s in P roposit ion 3 is t hat t he two bonds span t he risk of t he int erest
rat e and t he market price of risk while only st ock spans t he jump risk. In cont rast , t he
bond port folio weight s have t hree component s. T he first is t he myopic demand for t he
risk premia of two diffusion risks; t he second is t he hedging demand against t he risk st em-
ming from t he two diffusion risks; t he t hird one is anot her myopic demand for t he jump risk premium. More specifically, as shown in Appendix C, t he first two component s are
ident ical t o t he opt imal weight s in t he market where t he st ock is not available for t rading.
And t hus, t he t hird component det ermines more or fewer bonds t he invest or holds when
she can t rade t he st ock. Alt hough t he two bonds are independent of jumps, t he invest or
can gain t he jump risk premium by invest ing more in t he two bonds, as t he two bonds
and t he st ock are correlat ed via diffusion, suggest ed by t he t erm S −1.
To make t he int uit ion behind t he result s as clear as possible, we concent rat e on a B B S
B B Λ X
f p p
S q
q
q
simple case by furt her assuming t hat t he jump sizes J = gP and J = gQ are negat ive
const ant s under bot h t he physical measure P and t he risk-neut ral measure Q. We follow
Sangvinat sos and Wacht er (2005) t o assume t hat t he st at e variables X 1 and X 2 follow
t he equat ions below.
dX 1(t) = K 1( 1 − X 1(t))dt + X 1 1 dZ1(t);
dX 2(t) = K 2( 2 − X 2(t))dt + X 2 2 dZ2(t); (22)
where K 1 and K 2 are posit ive const ant s. In t his case, by (16), we have
A2i ( ) =
e− K i − 1
i ; i = 1;2: i
We furt her assume t hat X 1 is a permanent st at e variable wit h a low value of K 1 while
X 2 is a t ransit ory st at e variable wit h a high one of K 2. Like Table II in Sangvinat sos and
Wacht er (2005), we let X 1 1 > 0; X 2 2 > 0; S 1 < 0; S 2 > 0 and S 1 X 1 1 + S 2 X 2 2 < 0
so t hat t he st ock ret urns are negat ively correlat ed wit h bot h t he st at e variable X 1(t) and
t he int erest rat e r(t). T he negat ive correlat ion between st ock ret urns and int erest rat es
has been document ed in t he lit erat ure (see, for example, Fama (1981) and Sangvinat sos
and Wacht er (2005)). From (14), it is easy t o check t hat t he bond ret urn and t he int erest
rat e are negat ively correlat ed as A21( ) < 0 and A22( ) < 0. Furt hermore, in order t o
invest igat e whet her or not t he explanat ion of Lioui (2007) for t he bond/ st ock rat io puzzle
is st ill valid in our jump-diffusion model, we assume t hat t he mat urity 1 of t he first bond
is equal t o t he invest ment horizon T . T hen, t he opt imal port folio weight s in P roposit ion
3 are given explicit ly in t he following result .
P rop osit ion 4 T he optim al portfolio weight = ( 1; 2; ) is given by
1 ( ¯1(t) ¯2(t) ) ( ˜ ( S 1 S 2 )
|A2| X 1 1 X 2 2
2 =
1
|A2| −
¯1(t)
X 1 1
A22( 1) +
¯2(t)
X 2 2
A21( 1) −
|A2| −
S 1
X 1 1
A22( 1) +
S 2
X 2 2
A21( 1) ;
= ˜ = 1
gP gP P −1 ; (23)
K
B B S
B1 =
Λ Λ q
A22( 2) − A21( 2) + 1 −
1 )
−
|A2| X 1 1
A22( 2) −
X 2 2
A21( 2) ;
( ) ˜ ( )
B
[ ( gQ Q ) −1
Λ Λ
]
q
where |A2| = A21( 1)A22( 2) − A21( 2)A22( 1) < 0.
P ro of. See Appendix C.
T he above result s suggest t hat Bond 1 perfect ly hedges t he int erest rat e risk, which
is t he same as a pure-diffusion model in Lioui (2007). Using t he fact s t hat A21( ) <
0; A22( ) < 0; |A2| < 0 and S 1 < 0, we can verify t hat t he coeffcient of ˜ in t he first
equat ion in (23) is posit ive while t he one in t he second equat ion in (23) is negat ive. In
ot her words, t o gain jump risk premia, t he invest or holds more short -t erm bonds (Bond
1) and less long-t erm bonds (Bond 2) t o offset t he posit ion in Bond 1. Meanwhile, t he t ot al
demand for t he two bonds due t o jump risk is posit ive, which can be rewrit t en as
˜ [ S 1 ]
−
|A
S 2
2| X 1 1
(A22( 2) − A22( 1)) +
X 2 2
(A21( 1) − A21( 2)) (24)
and t he coeffcient of ˜ is posit ive.
We now t urn t o t he impact of t he risk aversion coeffcient on t he bond/ st ock rat io.
From P roposit ion 4, t he bond/ st ock rat io is separat ed int o t hree t erms t hat correspond
t o t hree part s in t he port folio on t he bonds: mean-variance allocat ion, hedging demand
for int erest risk, and myopic demand for jump risk. T he second t erm is act ually exploit ed
t o explain t he asset allocat ion puzzle in t he lit erat ure (e.g., Brennan and Xia (2000),
Ba jeux-Besnainou and P ort ait (2001) and Lioui (2007)). It is int erest ing t o invest igat e
whet her t he rat io increases wit h t he relat ive risk aversion coeffcient in our model here. For t his purpose, we follow Brennan and Xia (2000) t o rewrit e t he t ot al demand for t he
two bonds in P roposit ion 4 as:
= a + 1 − 1 − b˜ ;
wit h
[
a = 0
1 (t) (A
22( 2) − A22( 1)) + 2 (t) (A21( 1) − A21( 2))
]
;
b = |A 12 | S 1 X 1 1
X 2 2
S 2 (A22( 2) − A22( 1)) + X
2 2 (A21( 1) − A21( 2)) ]
:
q
q
q
B q
And hence, t he bond/ st ock rat io is obt ained as:
f ( ) = = (
a −1 + 1
)
S
1
˜ − b; (25)
implying t hat by using t he t hird equat ion in (23),
) 1 [ 1 ( a −1 ) ( gQ Q) −1 ( gQ Q ) ]
1˜ gP ˜ gP P gP P
As shown below, t he funct ion f ′( ) can be eit her posit ive or negat ive depending on t he
model paramet ers. For inst ance, we show t hat it can be negat ive under cert ain condit ions.
For t his, we consider t he case of a > 1 in which t he invest or t akes highly levered posit ions
in bonds as document ed in Table VI of Sangvinat sos and Wacht er (2005) and in t he
numerical analysis in t he following sect ion (Sect ion 5).
We now rewrit e f ′( ) as
1
1 − a −
1˜
(
a − 1 ) ln
1− (
gQ Q gP P
gQ Q gP P
)
:
Assuming 1 ≤ ≤ 3, we can show t hat f ′( ) < 0 when gQ QP > g(a) = (
a+ 2
a−1
) 3
, t hat is,
t he rat io relat ed t o t he jump risk premium is higher t han g(a) which is a funct ion of t he
diffusion risk premia. T herefore, in t his case, t he rat io is a decreasing funct ion of in q
t he range of [1;3]. T he reason for t his is t hat unlike a pure-diffusion model, t he demand ˜
for t he st ock is not proport ional t o 1= as indicat ed by t he t hird equat ion in (23). In fact ,
˜ decreases slower t han 1= when increases in t hat @~ = 21gP ln (
gQ Q gP P
) ( gQ Q gP P
) −1=
and ggQ QP P increases wit h for gQ Q
> 1. In ot her words, t he invest or wit h more
risk aversion holds relat ively more st ocks t han bonds t o exploit t he jump risk premium
when t he premia compensat ed for bot h t he jump risk and diffusion risks sat isfy t he afore-
ment ioned condit ion. T his is in cont rast wit h t he observat ions in a pure-diffusion model. Specifically, our jump-diffusion model reduces t o a pure-diffusion model by replacing t he
jump component in st ock ret urns wit h a diffusion one Z3(t). T hen, t he result s in P
ropo-B
˜ q
f ′( ) = df ( = 1 − a − + 1 ln :
d q q
q
f ′( ) = + 1 ( ) 1
gP B ~ q q q @ ( ) −1=
sit ion 4 except remain unchanged. Specifically, = Λ3= , where Λ3 > 0 is t he risk
premium for t he diffusion t erm Z3(t). As a result , f ( ) = (a −1 + ) 1 − b, which is an
3
increasing funct ion of . And t hus, as argued in Lioui (2007), t his leads t o t he resolut ion
of t he asset allocat ion puzzle in pure-diffusion models. In short , t he rat ionality of t he
bond/ st ock rat io puzzle cannot be explained by t he int ert emporal hedging demand in t he
presence of jumps in st ock ret urns, and t hus our jump-diffusion model provides anot her
channel t o st rengt hen t he issue addressed by Lioui (2007) t hat t he asset allocat ion puzzle
is st ill a puzzle.
F inally, we conduct a comparat ive st at ic analysis t o invest igat e t he effect of t he jump
paramet ers on t he cash-bond-st ock mix. For simplicity, we just vary t he jump int ensity
P while keeping t he ot her paramet ers fixed. T he t hird equat ion in (23) suggest s t hat :
@˜
@P = 1
gP ( P )
1
−1( gQ
g
Q ) −1 < 0;
implying t hat t he t ot al demand for t he two bonds decreases wit h P from (24) while
t he cash holding increases wit h P . In cont rast , t he bond/ st ock rat io increases wit h P
by (25) if a > 1. T he invest or hence holds less in st ocks when facing more frequent jumps.
Namely, t he invest or reduces her posit ion in st ocks during a t urbulent t ime of t he st ock
market , and also reduces her bond holding based on t he above discussion. Int erest ingly,
t he invest or holds more bonds relat ive t o st ocks as indicat ed by t he increasing bond/ st ock
rat io. As a result , t he invest or holds more cash and relat ively more bonds, reflect ing t he
phenomenon of flight -t o-safety, when facing a high possibility of jump risk.
5
N um erical R esult s
In t his sect ion, we use a numerical example t o illust rat e t he t heoret ical findings in t he
preceding sect ion. Especially, we invest igat e t he effect of t he ext reme negat ive jump risk on t he bond/ st ock rat io. T he recent financial crises have fuelled a renewed int erest
in modeling, est imat ing, and deriving t he implicat ions of ext reme t ail event s. It has
been document ed in t he lit erat ure t hat t he dist ribut ion for ext reme event s can be well
S S
q
P
B
