OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART
VOLATILITY
NICOLE BÄUERLE∗ AND ZEJING LI‡
Abstract. We consider a multi asset financial market with stochastic volatility modeled by
a Wishart process. This is an extension of the one-dimensional Heston model. Within this
framework we study the problem of maximizing the expected utility of terminal wealth for
power and logarithmic utility. We apply the usual stochastic control approach and obtain
explicitly the optimal portfolio strategy and the value function in some parameter settings. In
particular when the drift of the assets is a linear function of the volatility matrix. In this case
the affine structure of the model can be exploited. In some cases we obtain a Feynman-Kac
representation of the candidate value function. Though the approach we use is quite standard,
the hard part is indeed to identify when the solution of the HJB equation is finite. This involves
a couple of matrix analytic arguments. A two-dimensional example is given to illustrate the
results.
Key words: Wishart Process, Portfolio Problem, CRRA Utility, Stochastic Control,
HJB Equation, Matrix Exponential.
AMS subject classifications: 93E20, 91G80, 91G10.
1. Introduction
Asset price models need a stochastic volatility in order to produce the ’smile effect’ between
the implied volatility and the strike price of a call option or to reproduce the ’leverage effect’,
i.e. that past returns are negatively correlated with future volatilities. To cover these aspects, a
lot of one-factor stochastic volatility models have been introduced in the course of time, among
them the popular Heston model (see ?). However, recent empirical studies show that even
in a single asset model it might be reasonable to model the volatility by at least two factors
(see e.g. ?). Moreover it is often necessary to model a complete portfolio of assets. Hence in
recent times multivariate extensions of the Heston model have been proposed. They all build
on the matrix Wishart process for the volatility which has been introduced in ?. It is a direct
multivariate extension of the Cox-Ingersoll-Ross model and has been extended and used for
financial applications by e.g. ??????. While these papers consider option pricing, hedging,
credit risk and term structure models, we will investigate portfolio optimization problems. More
precisely we consider the classical problem of maximizing the expected utility of terminal wealth
in a multi asset Wishart volatility market for power and logarithmic utility. During the work on
this project we became aware that ? also consider portfolio optimization problems in a similar
setup with the risk-sensitive criterion. However their methods are different and they also focus
on different aspects as we will explain below. Another recent work is ? where solutions of BSDEs
in such a setting are considered and applied to indifference pricing. There also the exponential
utility is dealt with. In ? a model with two assets and two-dimensional Wishart process is
considered and the hedging demand for a power utility problem is treated. Besides these works,
to the best of our knowledge, there are no others yet which deal with optimization problems
in this financial market. Moreover, all these papers consider only financial markets where the
process of volatility and stochastic logarithm of the asset has an affine differential w.r.t. the
volatility. We call this the affine case. In ? the authors consider a general semimartingale
market which is again affine. They consider portfolio problems with power utility but only in
the single asset framework.
c
0000
(copyright holder)
1
2
N. BÄUERLE AND Z. LI
In this paper we use the classical stochastic control approach and solve the portfolio optimization problem with the help of the Hamilton-Jacobi-Bellman (HJB) equation. However it turns
out that this is here indeed a non-trivial task due to rather complicated computations. Moreover
some interesting aspects occur in the sense that in some parameter settings the problem is rather
easy and in some not. We already know from ? (see also ? and ?) that in the one-dimensional
Heston model which is included as a special case, there are parameter setings where the value
function is not finite and thus the stochastic control approach breaks down. See also ? for a
warning word in this direction. It will turn out that in the affine case (i.e. when the drift of the
stocks is a linear function of the Wishart process, which is assumed in most models and also
in the Heston model), there are parameter settings where the value function is finite and can
be computed explicitly. In this case fortunately we can show that the optimal value function is
a Laplace transform of the Wishart volatility process and we can use results in ? where such
expressions have been computed. The affine structure of the process is here exploited. Still the
impression remains that this is a very special case because the corresponding optimal portfolio
strategy is completely deterministic and hence measurable w.r.t. the initial information. We
also identify another situation with special correlation between the Brownian motions which
drive the asset dynamics and those which drive the volatility dynamics and a special Q matrix,
however general drift, where the HJB equation boils down to a linear partial differential equation
and where we have at least a candidate for the value function via a Feynman-Kac representation
formula. In particular the case of uncorrelated Brownian motions belongs to this category.
In ? the authors solve a risk-sensitive portfolio problem in an affine Wishart-volatility model.
While doing this, they also solve the power utility problem. However they use the risk sensitive
approach and do a change of measure first, before they set up the HJB equation. Moreover they
do not get a closed form solution like we do and also do not discuss the cases where the value
function is infinite. On the other side they also tackle infinite horizon problems.
The outline of our paper is as follows: In section 2 we introduce the multi asset financial
market where volatility is modeled by a Wishart process and state the optimization problem. In
section 3 we derive the associated HJB equation and consider some transformations of it. The
next section is then dedicated to finding solutions of this HJB equation. We divide this section
into two parts, where in the first part we consider a general asset drift, but specific correlations
and in the second part we treat the affine model where we derive an explicit solution. In
section 5 we verify that this solution is indeed the value function of our portfolio problem. In
section 6 we briefly discuss the case of logarithmic utility function and in section 7 we consider
a two-dimensional example. The appendix contains some of the proofs and auxiliary results.
2. The Wishart Volatility Market
It is well-known that the classical standard Black-Scholes model is not flexible enough to create
the simile effect, nor does it satisfy the leverage effect either.
To cover these shortages of the standard Black-Scholes model, ? have presented a multivariate
Wishart stochastic volatility. The model introduced below possesses a generalized drift compared
with ?. It is an extension of the one-dimensional Heston model.
In our model the market consists of one riskfree asset with price process St0 t≥0 and d risky
assets. The constant riskfree rate is r ≥ 0 and the dynamic of the riskfree asset is
dSt0 = St0 rdt, St0 = 1.
We denote by (St,i )t≥0 , 1 ≤ i ≤ d the price processes of the d risky assets and by (St )t≥0 =
(St,1 , . . . , St,d )t≥0 the vector process. The return of (St )t≥0 owns a Wishart stochastic volatility
(Σt )t≥0 . The joint dynamics of (St )t≥0 and (Σt )t≥0 are given by the following (vector-matrix-)
stochastic differential system:
1/2
dSt = diag(St ) B(Σt )dt + Σt dWtS ,
(2.1)
1/2
1/2
dΣt = N N T + M Σt + Σt M T dt + Σt dWtσ Q + QT (dWtσ )T Σt ,
(2.2)
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
3
where WtS t≥0 , is a d dimensional Brownian motion vector and (Wtσ )t≥0 is a d × d Brownian
motion matrix respectively. All processes are defined on a common probability space (Ω, F, P).
In what follows, (Ft )t≥0 denotes the corresponding Brownian filtration. The entries between
S ,Wσ i = ρ
WtS t≥0 and (Wtσ )t≥0 can be correlated. We assume that dhWt,k
k,ij dt for 1 ≤
t,ij
k, i, j ≤ d. The matrix diag(St ) is a diagonal matrix with entries St,1 , . . . , St,d on the diagonal.
Further N , M, Q are d × d matrices with N ∈ GLd (R) the set of real invertible matrices of
dimension d × d. We also assume that N N T (d + 1) QT Q (where A B means that A − B is
positive semidefinite) which according to Theorem 2.2 in ? implies that (2.2) has a unique global
strong solution on Sd+ (R) which is the set of symmetric positive definite matrices of dimension
d × d. As usual we denote for Σ ∈ Sd+ (R) by Σ1/2 the unique matrix A ∈ Sd+ (R) such that
A2 = Σ. The function B : Sd+ (R) → Rd is measurable and will be specified later.
In what follows we will assume for the correlation coefficients ρk,ij , 1 ≤ k, i, j ≤ d between
the Brownian motions that ρk,ij = 0 for k 6= i and ρk,kj =: ρj is independent of k. In particular
we denote ρ = (ρ1 , . . . , ρd )T .
Example 2.1. The Wishart stochastic volatility model can be regarded as an extension of the
Heston model to the multidimensional case. Recall that the one-dimensional asset return process
(St )t≥0 in the Heston model is determined by the stochastic process:
p
dSt = St (µ + λZt )dt + Zt dWtS ,
whereas the volatility process (Zt )t≥0 follows a Cox-Ingersoll-Ross process:
p
dZt = κ (θ − Zt ) dt + ξ Zt dWtZ ,
where (WtS )t≥0 , (WtZ )t≥0 are Brownian motions with correlation ρ and µ, κ, θ and ξ are suitable
constants in R. One can easily get that the dynamics above are specifications of (2.1) and (2.2)
in the one-dimensional case.
We assume now that an agent can invest into this financial market and define the portfolio
strategy process (πt )t≥0 as an Rd -valued progressively measurable process with respect to (Ft )t≥0
where πt = (πt,1 , . . . , πt,d )T and πt,k represents the proportion of wealth invested into stock k
at time t. Obviously πt0 := 1 − πtT 1 is the proportion of wealth invested in the bond where
1 = (1, . . . , 1)T ∈ Rd . Under a fixed portfolio strategy (πt )t≥0 , the portfolio wealth process
(Xtπ )t≥0 owns the following dynamic:
dXtπ = Xtπ πtT
dS 0
dSt
+ X π πt0 0t .
St
St
Applying the dynamic (2.1), it yields
dXtπ
1/2
= πtT (B(Σt ) − r) + r dt + πtT Σt dWtS ,
π
Xt
(2.3)
(2.4)
with X0π = x0 and r = (r, . . . , r)T ∈ Rd . We consider only portfolio strategies where (2.4) has
a unique strong solution and call them admissible. The solution of the portfolio wealth process
(Xtπ ) is given as follows:
Z T Z T
1 T 1/2 2
T
π
T 1/2
S
XT = x0 exp
πs (B(Σs ) − r) + r − kπs Σs k2 ds +
πs Σs dWs .
(2.5)
2
0
0
Now denote by U : R+ → R a (strictly increasing, strictly concave) utility function. We want
to solve the classical problem of maximizing expected utility of terminal wealth for power and
logarithmic utility. The value function of the optimization problem reads as
V (t, x, Σ) = sup Et,x,Σ [U (XTπ )] ,
π
x > 0, Σ ∈ Sd+ (R) , t ∈ [0, T ]
(2.6)
where Et,x,Σ is the expectation w.r.t. the conditional distribution Xt = x, Σt = Σ and where
the supremum is taken over all admissible portfolio strategies. We will follow the usual way
4
N. BÄUERLE AND Z. LI
using stochastic control, i.e. we will first derive the HJB equation, then find a solution and
finally verify it. However, the challenge here is to identify parameter cases where indeed the
value function is finite. As pointed out in ? and ? this is an important issue and in this case a
non-trivial task.
3. The HJB Equation and its Transformations
In what follows we assume that U (x) = γ1 xγ for γ < 1, γ 6= 0. The formal derivation of the
HJB equation is as follows: Since the process V (t, Xtπ , Σt ) t≥0 is a supermartingale under any
admissible portfolio strategy π and a martingale under the optimal one, the drift of the process
which is derived using the Itô-Doeblin formula has to be zero when maximized over all portfolio
allocations. In order to apply the Itô-Doeblin formula we have to compute
h
i
dhΣlk , Σpq it = Σlp (t) QT Q kq + Σpk (t) QT Q ql + Σlq (t) QT Q kp + Σkq (t) QT Q lp dt
h
i
dhΣlk , X π it = Xtπ (πtT Σt )l (QT ρ)k + (πtT Σt )k (QT ρ)l dt
dhX π it = (Xtπ )2 πtT Σt πt dt.
We will also denote by Gt and Gx the partial derivative w.r.t. t and x and we denote the operator
matrix
∂
.
∇ :=
∂Σij 1≤i,j≤d
Thus, a candidate G(t, x, Σ) ∈ C 1,2,2 [0, T ] × R+ × Sd+ (R) for the value function should satisfy
the HJB equation:
1
0 = Gt + T r N N T + M Σ + ΣM T ∇G + rxGx + T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) G+
2
n
o
1
+ sup xuT (B − r) Gx + x2 uT ΣuGxx + xuT Σ(∇ + ∇T )Gx QT ρ ,
(3.1)
2
u∈Rd
γ
with terminal condition G (T, x, Σ) = γ1 xγ . When we use the usual Ansatz G(t, x, Σ) = xγ g(t, Σ)
with g > 0 and plug in the expressions above we end up with the HJB equation
1
1
gt + T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) g + T r (N N T + M Σ + ΣM T )∇g + rg
γ
2
n
o
γ−1 T
+ sup uT (B − r)g +
u Σug + uT Σ(∇ + ∇T )gQT ρ = 0.
(3.2)
2
u∈Rd
Obviously a maximizer of this HJB equation is given by
(B (Σ) − r)g(t, Σ) + Σ(∇ + ∇T )g(t, Σ)QT ρ
π ∗ (t, Σ) = Σ−1
,
(1 − γ) g (t, Σ)
Σ ∈ Sd+ (R), 0 ≤ t ≤ T.
(3.3)
Plugging the maximum point into the HJB equation we arrive at
1
gt + T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) g + +T r N N T + M Σ + ΣM T ∇g + γrg
2
γ
+
(B − r)T Σ−1 (B − r) g 2 + 2ρT Q(∇ + ∇T )g (B − r) g
2 (1 − γ) g
+ ρT Q(∇ + ∇T )gΣ(∇ + ∇T )gQT ρ = 0.
(3.4)
Next we try to simplify this partial differential equation further. In particular we want to use
the transformation
g(t, Σ) = h(t, Σ)δ
(3.5)
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
5
for a suitable δ to obtain a linear partial differential equation. This is possible under the further
assumption
ρ = ρ̂1
and Qij = ci Q1j
ci ∈ R,
2 ≤ i ≤ d, c1 := 1.
(3.6)
In this case we obtain
P
d
i=1 ci
QT ρρT Q = ρ̂2 QT 11T Q = ρ2 QT Q with ρ2 = ρ̂2 Pd
2
2
i=1 ci
.
Thus, when we use the transformation (3.5) with
δ :=
(1 − γ)
,
(1 − γ) + γρ2
(3.7)
equation (3.4) reduces to a linear partial differential equation
1
ht + T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) h+
2
γr
γ
T r N N T + M Σ + ΣM T + H ∇h +
+
(B − r)T Σ−1 (B − r) h = 0 (3.8)
δ
2 (1 − γ) δ
with terminal condition h(T, Σ) = 1 and matrix
H :=
γ
QT ρ(B − r)T + (B − r)ρT Q .
(1 − γ)
This specific transformation has been used before by ? and in particular by ? in the onedimensional Heston model and in ? in a model with partial observation. Here we get exactly
the same δ as in ?, p. 305.
4. Solutions of the HJB Equation
4.1. A Candidate for general Drift. It is now possible to formally derive a solution via a
Feynman-Kac formula. So far we worked under the physical measure P. We now denote a new
measure by P̃ associated with the following Radon-Nikodym derivative:
Z T
Z
dP̃ 1 T
2
T
σ
kθ(Σs )k ds
(4.1)
Zt :=
T r(θ(Σs ) dWs ) −
= exp
dP 2 t
t
Ft
with
γ 1/2 −1
Σ
(B(Σ) − r) ρT , Σ ∈ Sd+ (R).
1−γ
P
For a d × d matrix A we define by kAk2 = i,j a2ij the Frobenius norm. Then, we obtain the
following theorem which is proved in the appendix:
θ (Σ) :=
Theorem 4.1. If the Radon-Nikodym derivative (Zt ) is an (Ft )-martingale and if
Z T γ
γr
Σ,t
+
(B (Σs ) − r)T Σ−1
(B
(Σ
)
−
r)
ds
∈ C 1,2 (O)
h̃(t, Σ) := Ẽ
exp
s
s
δ
2
(1
−
γ)
δ
t
(4.2)
d
with O = [0, T ] × S (R), then the solution h of (3.8) if it exists, has the Feynman-Kac representation h̃, where Ẽ denotes the expectation under P̃.
Hence we have a candidate solution for the stochastic control problem. However for a general
function B it is difficult to compute the expectation or verify the solution. In the next subsection
we consider the special case of a linear function B where we get an explicit solution.
6
N. BÄUERLE AND Z. LI
4.2. The Linear Drift Case. In this subsection, we consider a special case of (St )t≥0 with
drift coefficient B(Σ) satisfying B(Σ) − r = Σv for a v ∈ Rd . We drop the former restrictions
on Q and ρq,ij in (3.6). The asset dynamic (St )t≥0 can now be written as
1/2
dSt = diag (St ) (r + Σt v) dt + Σt dWtS .
(4.3)
The key property that makes this case solvable is the fact that the Wishart process is an affine
process and its Laplace transform can be computed (see e.g. ?). We will later see that under the
optimal strategy the expected utility of terminal wealth exactly reduces to a Laplace transform
of the Wishart process. The partial differential equation in (3.4) now reads
1
gt + T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) g + T r N N T + M Σ + ΣM T ∇g + γrg
2
γ
+
vT Σvg 2 + 2ρT Q(∇T + ∇)gΣvg + ρT Q(∇T + ∇)gΣ(∇ + ∇T )gQT ρ = 0 (4.4)
2 (1 − γ) g
with g(T, Σ) = 1.
In the next theorem Sd (R) denotes the set of symmetric real matrices of dimension d. The
proof can again be found in the appendix.
Theorem 4.2. The partial differential equation (4.4) with boundary condition g(T, Σ) = 1
possesses the following solution in case the expressions are finite:
g(t, Σ) = exp (φ(T − t) + T r[ψ(T − t)Σ]) ,
(4.5)
where φ(t) ∈ R and ψ(t) ∈ Sd (R) for t ∈ [0, T ] are solutions of the following Riccati equations
system:
ψt (t) = ψ(t)M̃ + M̃ T ψ(t) + 2ψ(t)Q̃T Q̃ψ(t) + Γ̃,
(4.6)
φt (t) = T r[ψ(t)N N T ] + γr
(4.7)
with
M̃ = M +
γ
QT ρvT ,
(1 − γ)
Q̃T Q̃ = QT Q +
γ
QT ρρT Q,
(1 − γ)
Γ̃ =
γ
vvT
2(1 − γ)
and the initial conditions: ψ(0) = 0 ∈ Sd (R), φ(0) = 0 ∈ R.
The Riccati equations have a finite solution in some parameter settings. In the next propoA
−A
where eA is the usual matrix
sition sinh(A) for A ∈ Sd+ (R) is defined as sinh(A) = e −e
2
exponential. Moreover, the log which appears there is the matrix logarithm. The proof can be
found in the appendix.
Proposition 4.3. For t ∈ [0, T ] define
p
p
p
p
p
−1 p
κ(t) := − C2 cosh( C2 t) + C1 sinh( C2 t)
C2 sinh( C2 t) + C1 cosh( C2 t) ,
C2 := Q̃ − 2Γ̃ + M̃ T (Q̃T Q̃)−1 M̃ Q̃T ∈ Sd (R),
1
C1 := − Q̃ M̃ T (Q̃T Q̃)−1 + (Q̃T Q̃)−1 M̃ Q̃T ∈ Sd (R)
2
with M̃ , Q̃, Γ̃ given in Theorem 4.2. If
p
C2 + C1 0 and Q̃T Q̃ ∈ GLd (R)
(4.8)
−2Γ̃ + M̃ T Q̃−1 Q̃−T M̃ 0,
is satisfied, then the partial differential equation (3.8) possesses on [0, T ] the solution (4.5), with
p
1
1
(4.9)
ψ(t) = Q̃−1 C2 κ(t)Q̃−T − M̃ T (Q̃T Q̃)−1 + (Q̃T Q̃)−1 M̃
2
4
1 φ (t) = − T r N N T M̃ T (Q̃T Q̃)−1 + (Q̃T Q̃)−1 M̃ t
4
h p p
p
p
i
1 −T
−1
− T r Q̃ N N T Q̃−1 · log
C2
C2 cosh
C2 t + C1 sinh
C2 t
.
(4.10)
2
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
7
Remark 4.4.
a) The first two conditions in (4.8) for the existence of the explicit solution
of the Riccati equations reduce to the single condition
Γ̃ ≺ M̃ T (2Q̃T Q̃)−1 M̃ ,
(4.11)
if M̃ T (Q̃T Q̃)−1 + (Q̃T Q̃)−1 M̃ is negative semidefinite. Under this assumption,
C1 is
√
positive semidefinite, thus the condition (4.11) implies the condition C2 + C1 0. If
γ > 0 and Q ∈ GLd (R) then Q̃T Q̃ ∈ GLd (R) is always satisfied. Note that for γ < 0 the
condition Γ̃ M̃ T (2Q̃T Q̃)−1 M̃ is always fulfilled.
b) In the special case d = 1 the conditions (4.8) coincide with the results in the Heston
model in Proposition 5.2 of ?. Let us use the terminologies in ? and denote for d = 1
γ σ
κ
σ
κ
κ̃
ρλ̄ =: − .
M := − < 0, v := λ̄, ρ := ρ, Q := , M̃ = − +
2
2
2 1−γ 2
2
2
γ
ρ2 ) =: σ̃ 2 and M̃ T (Q̃T Q̃)−1 +
In ? it is assumed that κ̃ > 0. Then Q̃T Q̃ = σ4 1 + 1−γ
(Q̃T Q̃)−1 M̃ = − σ̃κ̃2 is negative semidefinite. The condition in (4.11) can be written as
1
γ
2γ λ̄ρσκ γ 2 λ̄2 ρ2 σ 2
2
2
·
λ̄ < κ −
+
.
γ
2
2
2 (1 − γ)
(1 − γ)
(1 − γ)
2(σ + (1−γ)
ρ2 σ 2 )
γ
ρ2 σ 2 is always positive, thus, multiplying both sides with
Note that the term σ 2 + (1−γ)
this expression, the inequality above can be simplified to
γ λ̄
λ̄ ρκ
κ2
+
< 2
1−γ 2
σ
2σ
which is condition (26) in ?.
c) The one-dimensional Heston model with power utility has also been solved in ? using
martingale methods. They deal with parameter settings where the value function is finite
for certain time horizons up to a critical one. The conditions in (4.8) ensure that the
value function exists for all T ≥ 0. Inspecting
4.3 one may also
√ of Proposition √
√ the proof
be able to identify cases where the matrix C2 cosh( C2 t) + C1 sinh( C2 t) ∈ GLd (R)
for some t up to a critical one.
Hence we finally found a solution of our HJB equation which is indeed finite.
5. Verification
In this section we verify that
xγ
exp (φ(T − t) + T r[ψ(T − t)Σ])
(5.1)
γ
with ψ, φ in (4.9),(4.10) is indeed the value function of our portfolio optimization problem for
the special case B (Σ) − r = Σv under condition (4.8).
First note that our candidate for the optimal portfolio strategy π ∗ is given through (3.3) by
∗
πt = π ∗ (t, Σt ) with
G (t, x, Σ) =
v
2ψ(T − t)QT ρ
+
.
(5.2)
1−γ
1−γ
Note that π ∗ (t, Σ) = πt∗ , i.e. the optimal strategy is purely deterministic and does not depend
on Σ.
We will show directly that the corresponding value attained by this portfolio strategy is
V (t, x, Σ) and every other admissible portfolio strategy will not yield a larger value.
π ∗ (t, Σ) =
Theorem 5.1 (Verification). Suppose (4.8) holds. Given (πt∗ ) as in (5.2), there is
h π ∗ γ i xγ
t,x,Σ (XT )
E
=
exp (φ(T − t) + T r[ψ(T − t)Σ]) , t ∈ [0, T ], x > 0, Σ ∈ Sd+ (R)
γ
γ
8
N. BÄUERLE AND Z. LI
and for every other admissible portfolio strategy π we obtain
h (X π )γ i xγ
T
Et,x,Σ
≤
exp (φ(T − t) + T r[ψ(T − t)Σ]) , t ∈ [0, T ], x > 0, Σ ∈ Sd+ (R).
γ
γ
Thus, (πt∗ ) in (5.2) is the optimal portfolio strategy.
Proof. The inequality for every admissible portfolio strategy is standard and follows e.g. like in
? Proposition 4.3. For the equation recall that we have
Z T
Z T 1 T 1/2 2
T
π
π
πs Σs v + r − kπs Σs k2 ds +
πsT Σs1/2 dWsS
XT = Xt exp
2
t
t
with Xtπ = x. Let us denote
Z
Z T
γ2 T
dQ ∗ T 1/2
S
∗ T 1/2 2
(π
)
Σ
dW
−
=
exp
γ
k(π
)
Σ
k
ds
,
Zt :=
s
s
s
s
s
dP Ft
2 t
t
(5.3)
which is a martingale by Proposition 8.2 in the appendix. Using Girsanov’s Theorem we obtain
∗
x−γ Et,x,Σ (XTπ )γ
Z T
Z T
1
∗ T 1/2 2
∗ T 1/2
S
t,x,Σ
∗ T
=E
exp γ
(πs ) Σs v + r − k(πs ) Σs k2 ds + γ
(πs ) Σs dWs
2
t
t
Z T
1
γ
t,x,Σ
∗ T
∗ T
1/2 2
∗ T
1/2 2
= EQ
exp γ
(πs ) Σs v + r − k(πs ) (Σs ) k2 + k(πs ) (Σs ) k2 ds
2
2
t
Z T
γ−1 ∗ T
(πs ) Σs πs∗ ds
= Et,x,Σ
exp γ
(πs∗ )T Σs v + r +
Q
2
t
Z T
γ(γ − 1) ∗ ∗ T
∗ T
= Et,x,Σ
exp
γr(T
−
t)
+
T
r
γv(π
)
+
π
(π
)
Σs ds .
(5.4)
s
s s
Q
2
t
In what follows let us introduce the deterministic matrix-valued process
γ(γ − 1) ∗ ∗ T
πs (πs )
2
which appears in (5.4). By plugging in the optimal strategy π ∗ and using (4.6) we obtain:
o
γ n1 T
Fs =
vv + vρT Qψ(T − s) − ψ(T − s)QT ρvT − 2ψ(T − s)QT ρρT Qψ(T − s)
1−γ 2
=ψt (T − s) − ψ(T − s)M − M T ψ(T − s) − 2ψ(T − s)QT Qψ(T − s)
o
γ n
+
2ψ(T − s)QT ρvT + 4ψ(T − s)QT ρρT Qψ(T − s) .
γ−1
Fs = γv(πs∗ )T +
1/2
Now note that under Q defined in (5.3) the process Ŵtσ := Wtσ − γΣt πt∗ ρT is also a standard
Brownian motion and the dynamics of (Σt ) under Q is given by
1/2
1/2
dΣt = N N T + M Σt + Σt M T dt + Σt dWtσ Q + QT (dWtσ )T Σt
1/2
1/2
= N N T + M Σt + Σt M T + γΣt π ∗ ρTQ + γQTρ(π ∗ )TΣt dt + Σt dŴtσ Q + QT(dŴtσ )TΣt .
Hence under Q the process (Σt ) is again a Wishart process with drift
N N T + M Σt + Σt M T + γΣt π ∗ ρT Q + γQT ρ(π ∗ )T Σt
γ
2γ
=N N T + M Σt + Σt M T +
Σt vρT Q +
Σt ψ(T − t)QT ρρT Q
1−γ
1−γ
γ
2γ
+
QT ρvT Σt +
QT ρρT Qψ(T − t)Σt .
1−γ
1−γ
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
9
RT
Next we compute t T r(Fs Σs )ds under Q. For this instance note that due to the product rule
and since ψ(0) = 0, Σt = Σ we obtain
Z
T
Z
T r(ψt (T − s)Σs )ds = T r Σψ(T − t) +
t
T
ψ(T − s)dΣs .
(5.5)
t
Plugging in the dynamics of (Σt ) under Q we obtain:
Z
T
T r(Fs Σs )ds
t
Z
T
T r ψ(T − s)N N
=T r (Σψ(T − t)) +
T
Z
+ Tr
T r 2ψ(T − s)QT Qψ(T − s)Σs ds
t
t
Z
T
ds −
T
σ
ψ(T − s)Σ1/2
s dŴs Q +
Z
T
ψ(T − s)QT (dŴsσ )T Σs1/2 .
t
t
Note that the differential equation (4.7) can be written as
Z
φ(T − t) =
T
T r ψ(T − s)N N T ds + γr(T − t).
t
Hence we obtain:
∗
x−γ Et,x,Σ (XTπ )γ
Z
h
t,x,Σ
exp γr(T − t) +
= EQ
T
i
T r(Fs Σs )ds
t
= Et,x,Σ
Q
h
exp T r Σψ(T − t) + φ(T − t) −
Z
Z
T
T r 2ψ(T − s)QT Qψ(T − s)Σs ds
t
T
Z
T
i
σ
+ Tr
ψ(T − s)Σ1/2
ψ(T − s)QT (dŴsσ )T Σs1/2
s dŴs Q +
t
t
= exp T r Σψ(T − t) + φ(T − t)
h
Z T
Z T
i
T
1/2
σ
Et,x,Σ
exp
−
T
r
2Qψ(T
−
s)Σ
ψ(T
−
s)Q
ds
+
2T
r
Qψ(T
−
s)Σ
d
Ŵ
.
s
s
s
Q
t
t
With the help of Proposition 8.1 it can be shown that the expression within the expectation is
a Q-martingale with expectation 1 which yields:
π∗ γ xγ
t,x,Σ (XT )
E
= exp T r (Σψ(T − t)) + φ(T − t)
γ
γ
and the statement is shown.
Remark 5.2. The optimal portfolio strategy (πt∗ ) in (5.2) can be decomposed into the Merton
v
ratio 1−γ
and the hedging demand given by
2ψ(T − t)QT ρ
.
1−γ
In case there is no correlation between the Brownian motions that drive the assets and those
which drive the volatility process, i.e. ρ = 0, the optimal portfolio strategy reduces to the
Merton ratio and does not depend on time. In any case note that the optimal portfolio strategy
does not depend on N N T .
10
N. BÄUERLE AND Z. LI
6. Logarithmic Utility Case
In case the utility function in problem (2.6) is the logarithmic utility U (x) = log x, the
problem can be solved by pointwise maximization. Indeed we obtain in the general drift case
for an admissible portfolio strategy:
Et,x,Σ log(XTπ )
i
hZ T
hZ T
1 T 1/2 2 i
T
t,x,Σ
t,x,Σ
πsT Σs1/2 dWsS
(6.1)
πs (B (Σs ) − r) + r − kπs Σs k2 ds + E
= log x + E
2
t
t
i
hZ T
1 T 1/2 2
T
t,x,Σ
= log x + E
(6.2)
πs (B (Σs ) − r) + r − kπs Σs k2 ds
max
2
t πs ,s∈[t,T ]
RT
1/2
when we assume that t πsT Σs dWsS is a true martingale. Obviously the maximizer is here
given by
π ∗ (Σ) = Σ−1 (B(Σ) − r) .
RT
1/2
For the case B(Σ) = r + Σv with a v ∈ Rd , it is evident that πt∗ ≡ v and t vT Σs dWsS is a
true martingale.
Plugging π ∗ into (6.2) yields
Z
1 T t,x,Σ (B (Σs ) − r)T Σ−1
(B
(Σ
)
−
r)
ds.
E
sup Et,x,Σ log(XTπ ) = log x + r (T − t) +
s
s
2 t
(πs )
For the further computation we need to calculate the conditional expectation of the function
f (Σt ) := (B (Σt ) − r)T Σ−1
t (B (Σt ) − r) .
Note that Σt has a Wishart distribution for fixed t. We refer to ? for the conditional expectation
of the moments of real inverse Wishart distributed matrices and ? for the moments of central
and noncentral Wishart distributions.
7. An Example
In this subsection we give an example of the value function V (t, x, Σ) for the case of B(Σ)−r =
Σv. By Theorem 4.2 we have that
V (t, x, Σ) =
xγ
exp (φ(T − t) + T r[ψ(T − t)Σ]) ,
γ
where φ(T − t) and ψ(T − t) are determined by (4.9) and (4.10). In our example, we consider the
financial market with one riskfree asset and d = 2 risky assets. The parameters of the volatility
process (Σt ) are given by
M1 0
Q1 0
ρ
v1
M=
,Q =
,ρ =
,v =
0 M2
0 Q2
0
0
for M1 , M2 , Q1 , Q2 , v1 ∈ R. Then we obtain
γv12
a 0
2(1−γ)
M̃ =
, Γ̃ =
0 M2
0
0
0
!
T
,
Q̃ Q̃ =
Q21 c 0
0 Q22
γ
γ
with a = M1 + 1−γ
v1 Q1 ρ and c = 1 + 1−γ
ρ2 . In what follows we assume that Q1 , Q2 6= 0, c > 0.
Then we obtain
√
Q1 c 0
Q̃ =
0
Q2
and
C1 =
−a
0
0 −M2
,
C2 =
b 0
0 M22
,
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
with b = a2 −
11
γ
2 2
(1−γ) v1 Q1 c.
By Proposition 4.3 we have to assume
p
−2Γ̃ + M̃ T Q̃−1 Q̃−T M̃ 0,
C2 + C1 0 and Q̃T Q̃ ∈ GLd (R).
√
which is satisfied if M2 6= 0, b − a ≥ 0 and b > 0. We obtain
 √

√
√
√
b sinh( bt)−a cosh( bt)
p
− √b cosh(√bt)−a sinh(√bt)
0
b
0
.
C2 =
, κ(t) = 
|M
2|
0 |M2 |
0
(7.1)
M2
Then ψ(t) reduces to
ψ(t) =
γv12
2(1−γ)
√
√
!
−1
b coth( bt) − a
0
.
0
0
The solution of φ (t) can be deduced from (4.7) as
Z t
Z
T
2
2
φ (t) =T r
ψ (u) duN N
= N11 + N12
0
t
0
√
√
−1
γv12
b coth( bu) − a du.
2(1 − γ)
The optimal portfolio strategy is here given by
√
√
−1 ργv1
1
v1
b coth( bt) − a
∗
+
πt =
0
0
1−γ
(1 − γ)2
√
i.e. in this√example one would never invest into the second asset. Also note that due to b−a ≥ 0
and coth( bt) ≥ 1, the hedging demand in the first asset is positive if ργv1 > 0 and negative if
ργv1 < 0.
8. Appendix
This section contains some proofs and auxiliary results.
Proof of Theorem
4.1: Observe first that under the new measure P̃, the process defined by
Rt
σ
:= Wt − 0 θ (Σs ) ds is a d × d matrix Brownian motion on [0, T ] by the Girsanov theorem.
Thus, under P̃ the process (Σt ) has dynamics
1/2
1/2
dΣt = N N T + M Σt + Σt M T + H dt + Σt dW̃tσ Q + QT (dW̃tσ )T Σt ,
W̃tσ
where we have used that Σ1/2 θQ + QT θT Σ1/2 = H. Hence the process (Σt ) is again a Wishart
process under P̃ however with different drift. The characteristic operator of this process is for
f ∈ C 2 Sd+ (R) given by
1
(Af )(Σ) = T r Σ(∇ + ∇T )(QT Q)(∇T + ∇) f + T r
2
N N T + M Σ + ΣM T + H ∇f
under P̃. Then the partial differential equation (3.8) can be written as
γr
γ
+
(B − r)T Σ−1 (B − r) h,
Ah = −ht −
δ
2 (1 − γ) δ
h(T, Σ) = 1.
Applying the theorem of Feynman-Kac, we conclude that the representation in (4.2) is the solution of (3.8) under proper conditions.
12
N. BÄUERLE AND Z. LI
Proof of Theorem 4.2: We simply verify that g given in (4.5) satisfies (4.4). To this end note
that
gt = −g φt (T − t) + T r(ψt (T − t)Σ) ,
∇g = ∇T g = gψ(T − t),
gΣlk ,Σij = gψji (T − t)ψkl (T − t).
Plugging these derivatives into (4.4) yields (obviously g cancels out):
0 = − φt + T r(ψt Σ) + 2T r(Σψ(QT Q)ψ)
γ
+ T r ψ(N N T + M Σ + ΣM T ) + γr +
T r vT Σv + 4ρT QψΣv + 4ρT QψΣψQT ρ .
2(1 − γ)
In order to see that the right hand side is really zero, keep in mind that φ satisfies (4.7) and ψ
satisfies (4.6) and that the following relations hold
γ
T r vT Σv = T r Γ̃Σ ,
2(1 − γ)
γ
γ
γ
T r 4ρT QψΣv =
T r vρT QψΣ +
T r ψQT ρvT Σ ,
2(1 − γ)
(1 − γ)
(1 − γ)
T
T
T
T
T r ρ QψΣψQ ρ = T r ψQ ρρ QψΣ .
To ensure g(T, Σ) = 1, we need the initial conditions ψ(0) = 0 and φ(0) = 0.
Proof of Proposition 4.3: The explicit representation of ψ and φ follows directly from Theorem
11 in ?. Given Q̃T Q̃ ∈ GLd (R) these expressions are well-defined and finite when
p
p
p
K(t) := C2 cosh( C2 t) + C1 sinh( C2 t) ∈ GLd (R), t ∈ [0, T ].
Indeed we show now that K(t) ∈ GLd (R), for all t ≥ 0, if and only if
p
−2Γ̃ + M̃ T (Q̃T Q̃)−1 M̃ 0 and
C2 + C1 0.
(8.1)
First of all note that C2 is symmetric and for the well definedness of the matrix square root
of C2 , we need C2 to be nonnegative definite, which √
is equivalent to −2Γ̃ + M̃ T (Q̃T Q̃)−1 M̃ 0.
Consider the situation t = 0. We obtain K(0) = C2 . Then K(0) ∈ GLd (R) if and only if
−2Γ̃ + M̃ T (Q̃T Q̃)−1 M̃ 0.
Subsequently, we show the sufficiency of the conditions for t > 0. For the sake of simplicity,
we write
√
√
1 p
1 p
K(t) =
C2 + C1 e C2 t +
C2 − C1 e− C2 t .
(8.2)
2
2
√
√
√
−1
√
Note that for C2 ∈ Sd+ (R) and t > 0, there is e C2 t ∈ Sd+ (R) and e− C2 t = e C2 t
∈ Sd+ (R).
√
√
Moreover, expanding the matrix exponential functions
as√a series, one gets e C2 t − e− C2 t 0
√
√
for C2 ∈ Sd+ (R), t > 0. Hence, one can write e C2 t = e− C2 t + P (t) with P (t) ∈ Sd+ (R). Then
there is
√
1 p
√
1 p
K(t) =
C2 + C1 e− C2 t + P (t) +
C2 − C1 e− C2 t
2
2
√
p
p
1
= C2 e − C2 t +
C2 + C1 P (t),
2
−1 (t) =
which is always
invertible
if
and
only
if
det(K(t))
=
6
0,
for
all
t
>
0.
Note
that
det
K(t)P
det K(t) det P −1 (t) and det P −1 (t) > 0, for all t > 0, since P (t) ∈ Sd+ (R) for all positive
OPTIMAL PORTFOLIOS FOR FINANCIAL MARKETS WITH WISHART VOLATILITY
13
t, one gets K(t) ∈ GLd if and only if det K(t)P −1 (t) 6= 0, for all t > 0. Because of
√
−1 1 p
√
√
p
K(t)P −1 (t) = C2 e− C2 t e C2 t − e− C2 t
+
C2 + C1
2
√
p −1
p −1 −1 1 p
C2
C2
C2 + C1
= e2 C2 t
−
+
2
−1 1 p
3
p
4t
+
= 2tI + 2t2 C2 +
C2 + C1 ,
C2 + . . .
{z 3
} 2
|
0
√
it follows det K(t)P −1 (t) 6= 0, for all t > 0 from C2 + C1 0.
√
Eventually, we show
√ the necessity of the conditions. We assume C2 + C1 0. Note that
for
√ t = 0, K(0) = √C2 0, which implies that all the eigenvalues of K(0) are positive. If
C2 + C1 0, i.e. √ C2 + C1 possesses at least one negative eigenvalue, one can identify the
√
matrix
C2 + C1 e C2 t owns also at least one negative eigenvalue through a matrix similarity
transformation:
p
√ √ √ 1/2 1/2 p
σ
C2 + C1 e C2 t = σ e C2 t
C2 + C1 e C2 t
,
where σ(A) denotes the spectrum of the matrix A. Then for t large enough, it follows that K(t)
owns at least one negative eigenvalue. Since the spectrum of a matrix is a continuous function
on√the entries of the matrix (see ?), we conclude that there exists a t > 0 with K(t) ∈
/ GLd (R),
if C2 + C1 0.
Proposition 8.1. Let us denote
Z T
1Z T
1/2
σ
1/2 2
Zt := exp
T r As Σs dWs −
kAs Σs k ds ,
2 t
t
where (At )t∈[0,T ] is a deterministic process with values in Rd×d and bounded by A∗ ∈ Rd×d . Then
(Zt )t∈[0,T ] is a martingale.
Proof. By Lemma 4.2. in ? we get that (Zt ) is a martingale, if there is constant C 0 ∈ R+ s.t.
q
p
T r (θ (Σ) θT (Σ) θ (Σ) θT (Σ)) ≤ C 0 T r (ΣΣ)
(8.3)
with θ (Σ) = AΣ1/2 . Consider the left-hand side of this inequality:
T r θ (Σ) θT (Σ) θ (Σ) θT (Σ)
=T r(AΣAT AΣAT ) ≤ λmax T r AΣΣAT = λmax T r ΣAT AΣ ≤ λ2max T r (ΣΣ) ,
where λmax is the largest eigenvalue of AT A. The second last inequality follows from the fact
that the trace of a matrix is the sum of its eigenvalues and
AΣAT AΣAT = AΣOΛOT ΣAT λmax AΣΣAT ,
where OΛOT is the spectral decomposition of AT A. The last inequality follows in the same way,
i.e.
ΣAT AΣ = ΣOΛOT Σ λmax ΣΣ.
Since As ∈ Rd×d is bounded on [0, T ], λmax is also bounded on [0, T ] and we denote its upper
bound by λ∗max . Then one concludes that (8.3) is satisfied with C 0 = λ∗max , which implies that
(Zt ) is a martingale.
Proposition 8.2. Let us denote
Z T
Z
1 T
T 1/2
S
T 1/2 2
Zt := exp
As Σs dWs −
kAs Σs k ds ,
2 t
t
t ∈ [0, T ]
where (At )t∈[0,T ] is a deterministic process with values in Rd which is bounded by A∗ ∈ Rd . Then
(Zt )t∈[0,T ] is a martingale.
14
N. BÄUERLE AND Z. LI
S , W σ >=
Proof. First note that it is sufficient to show that E Zt = 1 for t ∈ [0, T ]. Since < Wt,k
t,kj
p
d
S
σ
2
ρj we obtain (Wt,k ) = (ρ1 Wt,k1 + 1 − ρ1 Ŵt,k1 ) where (Ŵt ) is a d × d Brownian motion
matrix,
independent of (Wtσ ). Thus, when we denote vρ := (ρ1 , 0, . . . , 0)T ∈ Rd and v̄ρ :=
p
( 1 − ρ21 , 0, . . . , 0)T ∈ Rd we obtain
d
1/2
1/2
1/2
ATt Σt dWtS = T r vρ ATt Σt dWtσ + T r v̄ρ ATt Σt dŴt
and
1/2
1/2
1/2
1/2
kATt Σt k2 = ρ21 + (1 − ρ21 ) kATt Σt k2 = kvρ ATt Σt k2 + kv̄ρ ATt Σt k2 .
Hence we can write
1Z T
Z T
d
T 1/2
σ
T r vρ As Σs dWs −
Zt = exp
kvρ ATs Σs1/2 k2 ds
2 t
t
1Z T
Z T
T r v̄ρ ATs Σs1/2 dŴs −
kv̄ρ ATs Σs1/2 k2 ds
· exp
2 t
t
Z T
Z T
T r v̄ρ ATs Σs1/2 dŴs
E
T r vρ ATs Σs1/2 dWsσ
= E
t
·
·
t
where E denotes the stochastic exponential. Now we obtain
σ E Zt = E E[Zt |FTW ]
h Z T
h Z T
ii
σ
T 1/2
σ
= E E
T r vρ As Σs dWs
E E
T r v̄ρ ATs Σs1/2 dŴs
|FTW
.
·
t
·
t
(Wtσ )
Since (Ŵt ) and
are independent, the inner conditional expectation is equal to 1 due to
Example 4 in ?. From Proposition 8.1 we conclude that the remaining expression is also 1. (N. Bäuerle) Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe,
Germany
E-mail address: [email protected]
(Z. Li) Institute for Stochastics, Karlsruhe Institute of Technology, D-76128 Karlsruhe, Germany
E-mail address: [email protected]