Asset Allocation Models in a Generalized Heston Framework … · 2015. 9. 24. · Asset allocation models in a generalized Heston framework 5889 of the asset prices instead of the

Applied Mathematical Sciences, Vol. 9, 2015, no. 118, 5887 - 5905HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2015.57472

Asset Allocation Models in a Generalized

Heston Framework Using a Gradient Like

Vector Optimization Algorithm

Maria Cristina Recchioni

Dipartimento di ManagementUniversita Politecnica delle Marche

Piazzale Martelli 8, 60121 Ancona, Italy

Adina Scoccia

Dipartimento di ManagementUniversita Politecnica delle Marche

Piazzale Martelli 8, 60121 Ancona, Italy

Copyright c© 2015 Maria Cristina Recchioni and Adina Scoccia. This article is dis-

tributed under the Creative Commons Attribution License, which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper deals with the problem of determining “stable” portfolioallocations over a given time interval. Two portfolio optimization prob-lems are formulated in the multi-asset Heston framework. The portfolioreturn is modeled by using a generalization of the Heston stochasticvolatility model and explicit formuls for the conditional expected mean,the variance and the transition probability density function of the port-folio return process are deduced. These formuls are used to define theobjective functions of the portfolio optimization problems. The opti-mal Pareto sets of these problems are explored to determine allocationswhich belong to the optimal Pareto sets of both optimization problemsand that belong to these sets over a time interval. An experiment onreal data is proposed to illustrate the implications of the “stable” assetallocation.

Keywords: Stochastic volatility model, Kolmogorov backward equation,maximum likelihood function, calibration procedure

5888 Maria Cristina Recchioni and Adina Scoccia

1 Introduction

This paper deals with the problem of determining a long-term optimal portfo-lio when the returns of the risky assets are modeled with a stochastic volatilitymodel. The main result is the formulation of two long-term portfolio opti-mization problems and their solution through the use of new computationalmethods. These optimization problems are formulated assuming that the re-turns of the risky assets in the portfolio are modeled with a generalization ofthe Heston stochastic volatility model (GH model for short) [20].

The GH model is a stochastic volatility model that describes the behaviorof n assets having stochastic variances prescribed by a mean reverting process.This model is obtained by the Heston stochastic volatility model adding to thestochastic differential equation satisfied by each asset price an extra term (see[20] for further details). A semi-explicit formula for the transition probabilitydensity function associated with the stochastic process is derived in [20] andthis formula is used to calibrate the GH model parameters with the maximumlikelihood approach.

Under the assumption that the asset returns are prescribed by the GHmodel in this paper we derive a semi explicit formula for the transition proba-bility density function associated with the stochastic process that describes theportfolio return. This formula is expressed by a one dimensional integral of anexplicitly known elementary integrand function and it can be evaluated usingsimple quadrature rules such as the midpoint quadrature rule. Moreover, usingthis formula we derive explicit formulas of the conditional expected values ofthe portfolio return and of its variance (see formulas (44) and (45)).

These elementary formulas of the GH model make it easy to formulateportfolio optimization problems and to solve them using specific computationalmethods. We study two portfolio optimization problems that are formulatedas bi-criteria optimization problems. The first optimization problem minimizesthe conditional variance and maximizes the conditional expected value of port-folio return. The second one minimizes the conditional variance and maximizesthe probability that the portfolio return will not fall below a given threshold.The first optimization problem reduces to Markowitz’s approach [12] undersuitable choices of the parameters that define the optimization problem itself.

These vector optimization problems are solved using a suitable path follow-ing algorithm that determines the whole optimal Pareto front of the problems.The optimal Pareto sets of these two optimization problems are explored inorder to define a long term allocation procedure. This procedure selects, ifany, points that are optimal for both problems and optimal for several futurehorizons in order to reduce the re-allocation cost.The portfolio is defined by using the risky asset log-returns instead of the as-set prices. Roughly speaking, this corresponds to consider the geometric mean

Asset allocation models in a generalized Heston framework 5889

of the asset prices instead of the arithmetic mean. This choice is not new(see [4], [16], [14], [7] and the references therein) and is motivated by the factthat the geometric mean is less affected by extreme values than the arithmeticmean and that it is useful as a measure of central tendency for some positivelyskewed distributions.

The portfolio optimization problems proposed in this paper are inspiredby recent studies of Herzog et al. [8], Mwambi and Mwamba [14], Stuer etal. [22], Cremers et al. [3], Estrada [5], Gotoh and Takeda [7]. These studiesshow that the changes in the financial markets suggest the need to introducenew objectives in portfolio optimization in order to define efficient hedgingstrategies.

Specifically, the work of Mwambi and Mwamba proposes an investmentstrategy in the framework of the mean-variance portfolio selection based onthe assumption that the asset prices are log-normally distributed. Underthis assumption they formulate and solve portfolio optimization problems.In Estrada’s paper [5] a comparative analysis between the Growth OptimalPortfolio (GOP) and maximization of the Sharpe ratio is proposed. Eltonand Gruber [4] study the practicality of the use of the geometric mean in theportfolio selection. Specifically, they investigate an algorithm to maximizethe geometric mean with assumption log-normally distributed asset prices andthey prove that the maximum geometric mean lies on the efficient frontier inthe mean variance space. The work of Platen [17] shows that the optimalportfolio selection plays a fundamental role in financial market modeling andthe recent studies are devoted to select portfolios that maximize investors’expected utilities such as those proposed by Cremers et al [3], and Tim andKritzman [23].

In addition, Gotoh and Takeda [7] illustrate a portfolio optimization modelwhich minimizes the upper and lower bounds of loss probability of the under-lying return distribution. This is done by reducing the portfolio optimizationproblem to two fractional programming models based on ratios which includethe value at risk in the numerator and some norm of the portfolio in the de-nominator.Finally, Kim et al. [10] show that building robust portfolios reduces the uncer-tainty in their performance. To do this they formulate portfolio optimizationmodels based on a suitable modification of the Markowitz model.Our work is in line with this recent literature. In fact, we determine ro-bust portfolios using a suitable analytical treatment of the stochastic volatilitymodel which describes the underlying returns.

The paper is organized as follows. In Section 2 we derive the stochasticprocess that prescribes the dynamic of the portfolio when the asset prices aredescribed by the generalization of the Heston model introduced in [20]. InSection 3 we derive a formula for the probability density function associated


with the stochastic variable that describes the portfolio dynamics. In Section 4we formulate two asset allocation problems as bi-criteria optimization problemsand we propose some experiments on real data to show how the asset allocationmodels work.

2 A stochastic model for the asset log-return

portfolio dynamics

Let ci, i=1,2,. . . ,n, be real nonnegative constants and c = (c1, c2, . . . , cn)T ∈ Rn

such that c1 + c2 + . . . + cn = 1. We consider the following asset log-returnportfolio:

Rt =n∑i=1

cixi,t, t > 0, (1)

where xi,t, t > 0, is the log-returns relative to the asset price Si,t, i = 1, 2, . . . , n,that is

xi,t = lnSi,tSi,0

. (2)

The asset dynamics are described by equations

dSi,t = Si,tµidt+ Si,t√vi,tdW

0i,t + Si,taidQ

0i,t, (3)

dvi,t = χi(θi − vi,t)dt+ εi√vi,tdW

1i,t, (4)

where µi, ai, χi, θi, εi, for i = 1, 2, . . . , n, are suitable real constants, satisfyingthe following conditions:

ai, χi, θi, εi > 0, (5)

2χiθiε2i

> 1, (6)

and W 0i,t,Q

0i,t,W

1i,t are standard Wiener processes such that W 0

i,0 = W 1i,0 =

Q0i,0 = 0, i = 1, 2, . . . , n. We assume that the following relations hold:

E(dW 0i,tdQ

0j,t) = 0 ∀ i, j, E(dW 1

i,tdQ0j,t) = 0 ∀ i, j, (7)

E(dW 0i,tdW

1j,t) = 0 i 6= j, E(dW 0

i,tdW1i,t) = ρidt, (8)

E(dW 0i,tdW

0j,t) = 0 i 6= j, E(dW 0

i,tdW0i,t) = dt, (9)

E(dW 1i,tdW

1j,t) = 0 i 6= j, E(dW 1

i,tdW1i,t) = dt, (10)

E(dQ0i,tdQ

0j,t) = ρi,jdt i 6= j, E(dQ0

i,tdQ0i,t) = dt, (11)

where dW 0i,t, dQ

0i,t, dW

1i,t are the stochastic differentials of the Wiener processes,

E(·) denotes the expected value of · and ρi, ρi,j ∈ [−1, 1] are constants known


as correlation coefficients. It follows that

dxi,t = (µi −1

2a2i −

1

2vi)dt+

√vi,tdW

0i,t + aidQ

0i,t, t > 0, i = 1, 2, . . . , n, (12)


1i,t, t > 0, i = 1, 2, . . . , n. (13)

with the initial conditions:

xi,0 = xi,0 = 0, i = 1, 2, . . . , n, (14)

vi,0 = vi,0, i = 1, 2, . . . , n, (15)

where xi,0, vi,0, i = 1, 2, . . . , n, are random variables that we assume to beconcentrated in a point with probability one. Eqs. (1), (12) and (13) implythat the portfolio return Rt, t > 0, is the solution of the following system ofstochastic differential equations:

dRt =n∑i=1

(ciµi −ci2a2i −

ci2vi)dt+

n∑i=1

ci√vi,tdW

0i,t +

n∑i=1

ciaidQ0i,t,

t > 0, (16)


1i,t, t > 0, i = 1, 2, . . . , n. (17)

The equations (16)–(17) must be equipped with the initial condition:

R0 = R0 =n∑i=1

cixi,0 , (18)

vi,0 = vi,0, i = 1, 2, . . . , n, (19)

where R0, vi,0, i = 1, 2, . . . , n, are random variables that we assume to beconcentrated in a point with probability one. For simplicity, we identify therandom variables R0, vi,0, i = 1, 2, . . . , n, with the points where they are con-centrated. Eqs. (1) and (2) imply R0 = 0. Let pf (R, v1, .., vn, t, R

′, v′1, .., v′n, t′),

R,R′ ∈ R, t, t′, vi, v′i ∈ R+, i = 1, 2, . . . , n, t′ > t, be the transition probabil-

ity density function associated with the stochastic process implicitly definedby (16)–(17), that is, the probability density function of having Rt′ = R′,vi,t′ = v′i, given that Rt = R, vi,t = vi, i = 1, 2, . . . , n, when t′ > t. To de-rive a formula for pf in analogy with Lipton [11] (pages 602–605), we considerthe backward equation satisfied by pf as a function of the “past” variables(R, v1, .., vn, t) ∈ R× R(n+1)+ , that is:

−∂pf∂t

=1

2

∂2pf∂R2

n∑i=1

c2i (vi + a2

i ) +1

2

n∑i=1

∂2pf∂v2

i

ε2i vi

+n∑i=1

∂2pf∂R∂vi

ciεiviρi +1

2

∂2pf∂xi∂xj

n∑i=1

n∑j=1,j 6=i

ρi,jcicjaiaj

+n∑i=1

∂pf∂Ri

ci

(µi −

a2i

2− vi

2

)+

n∑i=1

∂pf∂vi

χi(θi − vi) (20)


with the final condition:

pf (R, v1, .., vn, t′, R′, v′1, .., v

′n, t′) = δ(R−R′)

n∏j=1

δ(vj − v′j) (21)

where, as mentioned above, δ(·) denotes Dirac’s delta. As shown in the nextsection we have:

pf (R, v1, .., vn, t, R′, v′1, .., v

′n, t′) =

1

2π

∫Rdk e

ık[

(R′−R)−τn∑j=1

cj

(µj−

a2j2

)]e−

n∑j=1

2χjθj

ε2j

(νRj +ζRj )τ

·

e−k2

2cTDΓDc

{n∏j=1

4sβ,jε2jsRγ,j

e−2χjθj/ε2j ln(sRj,β/(2ζ

Rj )) e−2vj((ζ

Rj )2−(νRj )2)sRj,γ/(ε

2jsRj,β) ·

e−MRj (vRj +v′j)

(v′jvRj

)(χjθj/ε2j )−1/2

I2χjθj/ε2j−1

(2MR

j (vRj vj)1/2)}

(22)

where τ = t′ − t, νRj , ζRj for j = 1, 2, . . . , n are given by:

νRj = −1

2(χj + ı kcjεjρj) (23)

ζRj =1

2

(4(νRj )2 + ε2

j(k2c2j − ı kj)

)1/2, (24)

and the quantities sRj,β, sRj,γ and vRj , MRj , j = 1, 2, . . . , n are given by:

sRj,γ = 1− e−2ζRj τ , sRj,β = (ζRj − νRj ) + (ζRj + νRj )e−2ζRj τ , τ > 0, (25)

vRj =4vj(ζ

Rj )2e−2ζRj τ

(sRj,β)2, MR

j =2sRj,β

εRj2sRj,γ

, τ > 0. (26)

Integrating equation (22) with respect to the “future” variances (v′1, v′2, . . . , v

′n)

we obtain:

pM(R, v1, . . . , vn, t, R′, t′) =

1

2π

∫Rdk e

ık[

(R′−R)−(t′−t)n∑j=1

cj

(µi−

a2j2

)]e−(t′−t)

n∑j=1

2χjθj

ε2j

(νRj +ζRj )

·

e−(t′−t) k2

2cTDΓDc

{n∏i=1

e−2χjθj/ε2j ln(sRj,β/(2ζ

Rj )) e−2vj((ζ

Rj )2−(νRj )2)sRj,γ/(ε

2jsRj,β)

}(27)

Note that the function pM depends on the stochastic variances at time t (inthe past). The quantities v1, v2, . . . , vn are not really observable in the marketso that there are at least three ways of acting:


i) take an average value on Rn+;

ii) assign them estimating their values from data using for example high-low volatility estimator, such as for example, σ2

t = (lnHt − lnLt)2/(4 ln 2) or

σ2t = 0.5(lnHt− lnLt)

2−0.39(ln pt− ln pt−1)2 where Ht, Lt and pt, pt−1 are thehighest, the lowest, the closing and the opening prices of the asset considered(see [18] and the references therein for further details);

iii) consider the initial variances as model parameters in the calibrationprocedure.

In this paper we choose the third approach.

3 Transition probability density function for

the log-return portfolio

Let us derive the integral representation formula for pf,R (see formula (22)).Letting τ = t′ − t, we can introduce the function pb defined as follows:

pb(τ, R, v1, .., vn, R′, v′1, .., v

′n) =

pf (R, v1, .., vn, t, R′, v′1, .., v

′n, t′), t′ = t+ τ, τ > 0

(28)

In fact, it is easy to see that the function pf is only a function of τ = t′ − tinstead of being a function of t and t′ separately. Using the fact that τ = t′− tand (28), equation (20) can be rewritten as an equation for pb, that is:

∂pb∂τ

=1

2

∂2pb∂R2

n∑i=1

c2i (vi + a2

i ) +1

2

n∑i=1

∂2pb∂v2

i

ε2i vi

+n∑i=1

∂2pb∂R∂vi

ciεiviρi +1

2

∂2pb∂xi∂xj

n∑i=1

n∑j=1,j 6=i

ρi,jcicjaiaj

+n∑i=1

∂pb∂Ri

ci

(µi −

a2i

2− vi

2

)+

n∑i=1

∂pb∂vi

χi(θi − vi) (29)

with initial condition:

pb(0, R, v1, .., vn, R′, v′1, .., v

′n) = δ(R−R′)

n∏i=1

δ(vi − v′i) (30)

and appropriate boundary conditions. Now we consider the following repre-sentation formula for pb:

pb(τ, R, v1, .., vn, R′, v′1, .., v

′n) =

1

(2π)n+1

∫Rdk eıkR

∫Rdl1 e

ıl1v′1 . . .

∫Rdln e

ılnv′n f(τ, R, v1, .., vn, k, l1, .., ln) (31)


where f is the Fourier transform with respect to the “future” variables (R′, v′1,v′2, . . . , v

′n) and k, li, i = 1, 2, . . . , n, are the conjugate variables in the Fourier

transform of R′ and v′i, i = 1, 2, . . . , n. Letting l′i = 2ε2ili, i = 1, 2, . . . , n, from

formula (31) we obtain:

pb(τ, R, v1, .., vn, R′, v′1, .., v

′n) =

1

(2π)n+1

(2

ε21

)n·∫

Rdk eıkR

′∫Rdl1 e

2

ε21ıl1v1

. . .

∫Rdlne

2

ε2nılnvn

f(τ, R, v1, .., vn, k, l1, .., ln) (32)

where

f(τ, R, v1, . . . , vn, k, l1, . . . , ln) =

f(τ, R, v1, . . . , vn, k, l′1(l1), . . . , l′n(ln)). (33)

Since the Fourier transform has been taken with respect to the “future” vari-ables (R′, v′1, . . . , v

′n), it is easy to see that f satisfies equation (29), so problem

(29) is translated into the problem of solving the following partial differen-tial equation satisfied by the function f as a function of the “past” variables(R, v1, . . . , vn):

∂f

∂τ=

1

2

∂2f

∂R2

n∑i=1

c2i (vi + a2

i ) +1

2

n∑i=1

∂2f

∂v2i

ε2i vi

+n∑i=1

∂2f

∂R∂viciεiviρi +

1

2

∂2f

∂xi∂xj

n∑i=1

n∑j=1,j 6=i

ρi,jcicjaiaj

+n∑i=1

∂f

∂Ri

ci

(µi −

a2i

2− vi

2

)+

n∑i=1

∂f

∂viχi(θi − vi) (34)


f(0, R, v1, . . . , vn, k, l1, .., ln) = e−ı kRn∏i=1

e−ı 2

ε2i

livi. (35)

The coefficients of the partial differential operator appearing on the right handside of (34) are first degree polynomials in vi, i = 1, 2, . . . , n, we seek a solutionof problem (34), (35) of the following form (see Lipton, 2001 [11]):

f(τ, R, v1, . . . , vn, k, l1, . . . , ln) =

e−ı k R

(n∏i=1

e− 2

ε22viBi(τ,k,l1,...,ln)

)eA(τ,k,l1,...,ln) (36)

where A and Bi, i = 1, 2, . . . , n, are functions to be determined. Substitutingformula (36) into equation (34), after some elementary algebra one notices that,


in order to satisfy (34), it is sufficient for the functions A and Bi, i = 1, 2, . . . , nto satisfy the following ordinary differential equations:

dA

dτ(τ, k, l1, . . . , ln) = −k

2

2cTDΓDc− ı k

n∑i=1

ci

(µi −

a2i

2

)−

n∑i=1

Biχiθi (37)

−dBi

dτ(τ, k, l1, . . . , ln) =

ε2i

2B2i + (ıkciεiρi + χi)Bi −

1

2(k2c2

i − ıkci) (38)


A(0, k, l1, . . . , ln) = 0, Bi(0, k, l1, . . . , ln) = ı li, i = 1, . . . , n . (39)

Note that the initial condition (39) has been obtained by imposing (35) onthe function f given by (36). Equation (38) is a Riccati equation that can besolved elementarily, then integrating (37) with respect to τ we obtain:

A(τ, k, l1, . . . , ln) = −k2

2cTDΓDcτ − ıτ k

n∑i=1

ci

(µi −

a2i

2

)−

n∑i=1

2χiθiε2i

ln((νRi + ζRi − ı li)e−2ζRi τ + (ı li − νRi + ζRi )

2ζRi

)−

n∑i=1

2

ε2i

χiθi(νRi + ζRi )τ (40)

Bi(τ, k, li) =

(νRi − ζRi )(νRi + ζRi − ı li)e−2ζRi τ + (νRi + ζRi )(ı li − νRi + ζRi )

(νRi + ζRi − ı li)e−2ζRi τ + (ı li − νRi + ζRi )

(41)

where

νRi = −1

2(χi + ı kciεiρi) , (42)

ζRi =1

2

(4(νRi )2 + ε2

i (k2c2i − ı ki)

)1/2. (43)

Substituting equations (40) and (41) into equation (32) and integrating withrespect to the variables li, i = 1, 2, . . . , n (see, for example, Oberhettinger [15]),an elementary computation gives formula (22).


4 Asset allocation problems: an application

on real data

In this section we compare two asset allocation problems which are formulatedas bi-criteria optimization problems.

The two allocation problems are based on the assumption that the assetdynamics are described by the stochastic model (3), (4) and, as a consequence,the portfolio dynamics is given by (16) and (17). Furthermore, the asset allo-cation weights should be positive and no short-selling is allowed.

The first asset allocation problem, named P1, is based on a mean-varianceapproach and the second one, named P2, is based on the minimization ofthe variance and on the maximization of the probability that the value of theportfolio will not fall below a given threshold. Let us go into detail. Firstly, wederive formulas for the mean value and the variance of the portfolio conditionedto the observations made at a given time. Using formulas (1), (46), (47), (49)we obtain for t′ > t:

Rt,t′(c) = E(Rt′ |Rt = R, vi,t = vi, i = 1, 2, . . . , n) =n∑i=1

cixi,t,t′ (44)

vR,t,t′(c) = E((Rt′ − Rt,t′)2 |Rt = R, vi,t = vi, i = 1, 2, . . . , n) =

n∑i=1

c2i vi,t,t′ + 2(t′ − t)

n−1∑i=1

n∑j=i+1

cicj ρi,j,t,t′ (45)

where R =∑n

i=1 cixi while xi and vi, i = 1, 2, . . . , n, are the observation madeat time t. The quantities xi,t,t′ , vi,t,t′ , and ρi,j,t,t′ , i 6= j, i, j = 1, 2, . . . , n, t′ > tare the expected value and the variance of the variable xi,t′ conditioned to theobservation made at t (see [20] for further details):

xi,t,t′ = E(xi,t′ |xi,t = xi, vi,t = vi) =

xi + (t′ − t)(µi −θi2− 1

2a2i ) +

(θi − vi)2χi

(1− e−χi(t′−t)

),

(46)


vi,t,t′ = E((xi,t′ − xi,t′)2 |xi,t = xi, vi,t = vi) =

(t′ − t)a2i +

θiεi8χ3

i

(1− e−χi(t′−t)

)2

+θi

((t′ − t)−

(1− e−χi(t′−t)

)χi

)[(ρi −

εi2χi

)(ρi −

3εi2χi

)+ (1− ρ2

i )

]

+viχi

(1− e−χi(t′−t)

)[(ρi −

εi2χi

)2

+ (1− ρ2i )

](47)

+viχi

(t′ − t)εie−χi(t′−t)(−ρi +

εi2χi

)(48)

and finally:

ρi,j,t,t′ = E ((xi,t′ − xi,t′)(xj,t′ − xj,t′) |xi,t = xi, vi,t = vi, xj,t = xj, vj,t = vj)

= aiajρi,j(t′ − t), (49)

Let us now formulate the two optimization problems that model the assetallocation problems P1 and P2:Problem P1:

minc∈Rn

fP1,t,t′

(c), t < t′, (50)

subject to the constraints

c1 + c2 + . . .+ cn = 1, ci ≥ 0, i = 1, 2, . . . , n, (51)

where fP1,t,t′

= (f1,P1,t,t′ , f2,P1,t,t′) is the vector function defined by:

f1,P1,t,t′(c) = −Rt,t′(c), t < t′ (52)

f2,P1,t,t′(c) = vR,t,t′(c), t < t′. (53)

Problem P2:

minc∈Rn

fP2,t,t′

(c), 0 < t < t′, (54)

subject to the constraints

c1 + c2 + . . .+ cn = 1, ci ≥ 0, i = 1, 2, . . . , n, (55)

where fP2,t,t′

= (f1,P2,t,t′ , f2,P2,t,t′) is the vector function defined by:

f1,P2,t,t′(c) = −∫ +∞

R∗dR′pM,R(R(c), v1, . . . , vn, t, R

′, t′), t < t′, (56)

f2,P2,t,t′(c) = vR,t,t′(c), t < t′, (57)


where R∗ is a positive number that is the threshold for portfolio value andpM,R is given by (27). Note that we have highlighted the dependence on c ofR in formula (56).

Problems P1 and P2 depend on the observation made at time t of theasset log-returns and on a future time t′ (the horizon), t′ > t. Both the assetallocation strategies try to minimize the portfolio variance at time t′ in thefuture. One strategy (Problem P1) tries to maximize the mean value of theportfolio at a time t′ in the future given the fact that at time t the values ofthe asset log-returns in the portfolio are xi, i = 1, 2, . . . , n. The other strategy(Problem P2) tries to maximize the probability that the value of the portfolio isgreater than a fixed threshold R∗ at time t′ given the values xi, i = 1, 2, . . . , nof the asset log-returns at time t, t < t′.

The proposed models, using a kind of tracking procedure, try to get the“best” allocation in a future time.

That is, calibrating today, at time t, the stochastic model (16), (17), as-signing a maximum time horizon T > 0 we can solve (today) Problem P1 andProblem P2 for several values of the time horizon Tj ≤ T for j = 1, 2, . . . , nT .This corresponds to choose t′ = Tj, j = 1, 2, . . . , k, in formulas (44), (45). Weobtain the optimal Pareto fronts F (Tj) and the corresponding optimal Paretosets A(Tj) ⊂ Rn for any j = 1, 2, . . . , nT .

This knowledge of the Pareto set A(Tj), for j = 1, 2, . . . , nT , can be used inthe decisional process to achieve some goals. In the following we try to pursuethe following three goals:

Goal 1 choose an asset allocation that is as near as possible to a point of theoptimal Pareto set for any Tj (j = 1, 2, . . . , nT );

Goal 2 choose an asset allocation such that for any Tj (j = 1, 2, . . . , nT )guarantees a portfolio variance below a given target;

Goal 3 choose an asset allocation such that for any Tj (j = 1, 2, . . . , nT )guarantees a portfolio return above a given target.

Goals 2 and 3 are classical ones and have been studied in the literature(see, for example, [12], [23], [14], [8]).

In pursuing Goal 1 we have the advantage of reducing the costs impliedby the asset reallocation. In fact, roughly speaking, if there exists an assetallocation that belongs to the optimal Pareto set A(Tj) for any j = 1, 2, . . . , nT ,the reallocation cost is zero on average. However, if we cannot find an assetallocation that is optimal for every Tj we choose an asset allocation todaythat is optimal at least for the first horizon T1 and whose sum of Euclideandistances from the other optimal sets A(Tj) is the smallest.

Note that the approach proposed here is based on the assumption that theasset log-return dynamics is described by the stochastic model (12), (13). This


could be considered a restriction of the proposed asset allocation model. How-ever, our model is a generalization of the well known Heston model [9] that hasbeen shown to describe satisfactorily asset dynamics in several research papers(see, for example, [21], [6], [1]). Furthermore, the generalization of the Hestonmodel used to describe the portfolio dynamics allows for a correlation betweenthe assets without losing semi-explicit formulas for the transition probabilitydensity function associated with the stochastic process implicitly defined bythe model itself.

The asset allocation procedure can be summed up in three main steps:

(a) Given the log-returns xi,tj , i = 1, 2, . . . , n, observed at tj, j = 1, 2, . . . , h,tj ∈ [0, t], with t the current date, estimate the parameters of the stochas-tic model;

(b) Given a predetermined value of the portfolio R∗ and a time horizon T ,T > t solve problem P1 (i.e. maximize the conditional mean and mini-mize the conditional variance) and problem P2 (i.e. maximize the prob-ability of exceeding a predetermined value of the portfolio R∗ and mini-mize the conditional variance) for several time values Tj, j = 1, 2, . . . , nT ,tj ∈ [t, T ];

(c) Select on the optimal Pareto sets A(Tj), tj ∈ [t, T ] of problem P1 and P2

the point c∗ that satisfies a given goal (Goals 1-3).

The portfolio can be re-balanced with a given frequency (i.e.: daily, monthly,quarterly, semiannual or annually) applying steps (a)-(c) with a time horizondepending on the re-balancing frequency.

We apply this procedure on real data. We consider the historical series ofdaily data of four market indexes: DAX, FTSE 100 UK, S& P 500 and FTSEMIB observed in the period going from March 5, 2010 to April 5, 2011. Wecalibrate the model parameters using the daily data from March 5, 2010 toMarch 5, 2011 and we use the data of April 2011 to study the behavior of theproposed strategies for the asset allocation (i.e. Problem P1 and P2).

Step (a) produces a vector Θ∗ containing the parameters, the correlationcoefficients and the initial stochastic variances of the stochastic volatility model(12), (13). Once computed the vector Θ∗, it is possible to solve the two assetallocation problems P1 and P2. Problems P1, P2 are two linearly constrainedmultiobjective optimization problems that we solve using the algorithm devel-oped in [13] and [19].This computational method is an interior point method based on a suitabledirection that plays the role of projected gradient-like direction for the vectorobjective function. More specifically, the algorithm is based on the numericalintegration of a dynamical system defined by a vector field that is a linear com-bination of descent directions for the objective functions. The limit points of


the trajectories solutions of the dynamical system belong to the optimal Paretoset so that integrating the dynamical system starting from a sufficiently largenumber of initial points belonging to the feasible region we are able to recon-struct the whole optimal Pareto set and the corresponding Pareto set. For thesake of the simplicity for technical details on this algorithm we refer to thepapers [13] and [19].

5 Some experiments on real data

In this section we perform some experiments described using real data. Inparticular we consider four market indexes: DAX, FTSE 100 UK, S& P 500and FTSE MIB observed in the period going from March 5, 2010 to April 5,2011. In particular we use daily data of the period March 5, 2010 March 5,2011to calibrate the model parameters with the procedure described in [20] and thedata of April 2011 are used to study the behavior of the optimal portfolios.

The observed log-return values of the indexes considered in the numericalexperiment are shown in Figure 1.

50 100 150 200 250−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

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x1

Day index

DAX index log−return

50 100 150 200 250−0.8

−0.6

−0.4

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x2

Day index

S&P500 index log−return

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Day index

FTSE 100 UK index log−return

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x4

Day index

MIB index log−return

Figure 1: Log-return values as function of the time (1-March, 5, 2010, 2-March6, 2010 and so on).

Now we solve problem P1 and P2 that model the asset allocation problemsusing the algorithm presented in [19]. We calibrate our portfolio using firsttime horizon t′ = T1 (two weeks in the future) and then time horizon t′ = T2

(one month in the future). Our calibration day is March 5, 2011. So doingour time horizons are about April 15, 2011 and April 30, 2011. We set thethreshold R∗ = log(1 + ∆R∗), with ∆R∗ = 5%.

Figure 2 shows the optimal Pareto fronts obtained solving Problem P1 (a)and Problem P2 (b)


0 1 2 3 4 5 6 7 8 9

x 10−3

0.01

0.015

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0.025

0.03

0.035

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0.045

0.05

Portfolio conditioned variance

Con

ditio

ned

Por

tofo

lio m

ean

Pareto front Problem P1 (blu−squares T=two weeks, green−stars T=one month)

(a)0 1 2 3 4 5 6 7 8 9

x 10−3

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1

Portfolio conditioned variance

Pro

babi

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that

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tofo

lio v

alue

is g

reat

er th

an R

*

Pareto front Problem P2 (blu−squares T=two weeks, green−stars T=one month)

(b)

Figure 2: Optimal Pareto front obtained solving Problem P1 (a) and ProblemP2 (b) for two values of the time horizon. Blue line - squares T1 (two weeks),green line - stars T2 (one month).

Let us denote with OP1,Tj and OP2,Tj , for j = 1, 2, the optimal Pareto setsobtained solving Problem P1 and Problem P2 respectively with time horizonsT1 = twoweeks and T2 = onemonth.

0

0.25

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1

0

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c1

Some points of the optimal Pareto Set Problem P1

c2

c 3

(a)0

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0

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0.75

10

0.25

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1

c1


c2

c 3

(b)

Figure 3: Some points of the optimal Pareto set obtained solving Problem P1

(a) and Problem P2 (b) for two values of the time horizon. Green stars T1 (twoweeks), blue stars T2 (one month).

Figures 3 (a), (b) and 4 (a), (b) show the first three components of thevectors ct = (c1, c2, c3, c4) belonging to the optimal sets OPi,Tj i, j = 1, 2. Notethat we represent (c1, c2, c3)T since we have c4 = 1− c1 − c2 − c3 and that thedistribution of the points c = (c1, c2, c3) varies passing from OPi,T1 to OPi,T2 ,i = 1, 2, or passing from OP1,Tj to OP2,Tj , j = 1, 2. In particular we can seethat the sets OP1,T1 ∩OP1,T2 and OP2,T1 ∩OP2,T2 , OP1,T1 ∩OP2,T1 are non emptysets.These optimal points are interesting for two reasons. The first one is thatdistributing the assets according with one of these points we are guaranteed


0

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Some Points of the optimal Pareto Set of Problem P1 (green) and P

2 (blue)

c2

c 3

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c1

Some Points of the optimal Pareto Set of Problem P1 (green) and P

2 (blue)

c2

c 3

(b)

Figure 4: Some points of the optimal Pareto set obtained solving ProblemP1 and Problem P2 for time horizon T1 (two weeks) (a) and for time horizonT2 (one month) (b). Green stars are points of the sets OP1,Tj , j = 1, 2, bluesquares are points of the sets OP2,Tj , j = 1, 2.

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.110

0.2

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1

(blue−∆ R*=10%, green− ∆ R*=5%)

Portfolio conditioned variance − T=one month

Pro

babi

lity

that

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lio v

alue

is g

reat

er th

an R

*

Optimal Pareto front − Problem P2

(a)0

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c1

c2


(blue−∆ R*=10%, green− ∆ R*=5% − T=one month)

c 3

(b)

Figure 5: Optimal Pareto front (a) and Optimal Pareto set (b) obtained solvingProblem P2 for T = T2 (one month). Green stars are points obtained choosing∆R∗ = 5% and blue squares are points obtained choosing ∆R∗ = 10%.

that two goals are satisfied (the maximization of the conditioned mean and ofthe probability that the portfolio value will not fall below a fixed threshold).

The second one is that passing from a time horizon of two week to a timehorizon of a month the portfolio should be slightly reallocated and this factimplies that on average the reallocation costs are reduced.

Finally we show how the Pareto front changes when ∆R∗ increases. Figure5 (a) shows the behavior of the Pareto fronts when ∆R∗ = 5% and ∆R∗ = 10%when the time horizon is a month and Figure 5 (b) shows the optimal Paretosets. We can see that increasing ∆R∗ we get the same probability for highervalues of the conditioned variance that is a reasonable behavior. However theintersection of the two optimal sets is not empty.


6 Conclusions

This paper presents two asset allocation problems where the dynamics of theassets are described by a generalization of the Heston stochastic volatilityproposed in [20]. The aim of the paper is to determine asset allocations thatare “stable” over a time interval. The term “stable” means that the allocationbelongs to the optimal Pareto front for a given time horizon, thus reducing there-allocation costs. To efficiently solve the allocation problems we derive anintegral representation formula for the probability density function associatedwith the portfolio return variable which we use to formulate Problem P2. Theempirical analysis shows that these optimal and “stable” allocations exist.However, and that they deserve further investigation. However, it would beinteresting to study easily verifiable conditions for these stable allocations.

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Received: July 21, 2015; Publsihed: September 23, 2015

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