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Risk Sensitive Portfolio Optimization in aSemi-Markov Modulated Market 1
Anindya Goswami
30th December 2014
1Coauthor: M. K. Ghosh and Suresh K. Kumar
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
ProblemsNotationsValue of a Portfolio
“all the eggs should not be placed in the same basket”
A financial market consists of numerous securities.
A typical investor invests his/her initial capital in differentsecurities and carry out continuous trading of them toincrease the final wealth of the portfolio.
Q1. Which choice of portfolio would result in the best return?
Q2. Assets are risky. How to manage the risk?
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
ProblemsNotationsValue of a Portfolio
Notations
S0t := locally risk free asset price at time t
S it := risky asset prices at time t, i = 1, . . . , n
N i (t):= number of units invested in the i th asset at t
Vt :=∑n
i=0 Ni (t)S i
t , value of the portfolio at t
ui (t) := N i (t)S it
Vt, the fraction invested in the i th asset.
Thenn∑
i=0
ui (t) = 1 .
Hence
u0(t) = 1 −n∑
i=1
ui (t).
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
ProblemsNotationsValue of a Portfolio
SDE Satisfied by the Value of Portfolio
Let u(t) = [u1(t), . . . , un(t)]′ (∈ A ⊆ Rn) be the portfolio at timet. Hence u(t) is an A-valued process.Self-financing condition implies that
dVt =n∑
i=0
N i (t)dS it .
Then the value of the portfolio or the wealth process denoted byV u
t takes the form
dV ut
V ut
=n∑
i=0
ui (t)dS i
t
S it
.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Semi-Markov Process
{Xt}t≥0 be a semi-Markov process taking values inX = {1, 2, . . . , k}.Xt is modeled to present hypothetical states of the market attime t.
P(XTn+1 = x ′,Tn+1 − Tn ≤ y | XTn = x) = pxx ′F (y | x).
The transition matrix (pxx ′) is irreducible.
F (y | x) < 1 for each x and for all y ∈ [0,∞).
F (· | x) has continuously differentiable density f (· | x).
Define Yt := t − Tn(t) [Holding Time]
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Semi-Markov Process
{Xt}t≥0 be a semi-Markov process taking values inX = {1, 2, . . . , k}.Xt is modeled to present hypothetical states of the market attime t.
P(XTn+1 = x ′,Tn+1 − Tn ≤ y | XTn = x) = pxx ′F (y | x).
The transition matrix (pxx ′) is irreducible.
F (y | x) < 1 for each x and for all y ∈ [0,∞).
F (· | x) has continuously differentiable density f (· | x).
Define Yt := t − Tn(t) [Holding Time]
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Semi-Markov Modulated Market Model
dS0t = r(t,Xt)S0
t dt, S00 = s0 > 0 ,
dS it = S i
t
µi (t,Xt)dt +n∑
j=1
σij (t,Xt) dW jt
, S i0 = si > 0,
σ(t, x) = [σij (t, x)]n×n, x = 1, · · · , k ,b(t, x) = [µ1(t, x)− r(t, x), · · · , µn(t, x)− r(t, x)].
The wealth V u corresponding to the portfolio u satisfies
dV ut = V u
t [(r(t,Xt) + b(t,Xt)u(t)) dt + u(t)′σ(t,Xt)dWt ]. (1)
(i) r(t, x), µi (t, x), σij (t, x) are continuous on [0,T ] ∀x , i , j .(ii) a(t, x) := σ(t, x)σ(t, x)′ continuous on [0,T ]and ∃ a δ > 0 such that a(t, x) ≥ δI ∀x(iii){Xt} and {Wt} are independent.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Derivation - Infinitesimal Generator
For a fixed u ∈ A the process {(V ut ,Xt ,Yt)}t≥0 is Markov. The
infinitesimal generator of the process is the family of operatorsAu
t : C 2,1(R×X × (0, t))⋂C (R×X × [0, t])→ C (R×X × [0, t]),
given by
Autϕ(v , x , y) :=
∂
∂yϕ(v , x , y) + (r(t, x) + b(t, x)u)v
∂
∂vϕ(v , x , y)
+1
2(u′a(t, x)u)v2 ∂
2
∂v2ϕ(v , x , y)
+f (y | x)
1− F (y | x)
∑j 6=x
pxj [ϕ(v , j , 0)− ϕ(v , x , y)].
ϕ ∈ Dom(Aut ), v ∈ R, x ∈ X , y ∈ (0, t).
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Expected Terminal Utility - Optimization - HJB
Let U be a utility function satisfying the usual condition.
The trader’s objective: maximize expected terminal utility
Ju(t, v , x , y) := Eu[U(V uT ) | Vt = v ,Xt = x ,Yt = y ]
ϕ(t, v , x , y) := supu
Ju(t, v , x , y)
Supremum is taken over all admissible portfolio strategies.
Using DPP and the verification theorem of controlled Markovprocesses, the HJB equation for ϕ is given by
∂
∂tϕ(t, v , x , y) + sup
u∈AAu
tϕ(t, v , x , y) = 0, ϕ(T , v , x , y) = U(v).
Can we solve it?
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Expected Terminal Utility - Optimization - HJB
Let U be a utility function satisfying the usual condition.
The trader’s objective: maximize expected terminal utility
Ju(t, v , x , y) := Eu[U(V uT ) | Vt = v ,Xt = x ,Yt = y ]
ϕ(t, v , x , y) := supu
Ju(t, v , x , y)
Supremum is taken over all admissible portfolio strategies.
Using DPP and the verification theorem of controlled Markovprocesses, the HJB equation for ϕ is given by
∂
∂tϕ(t, v , x , y) + sup
u∈AAu
tϕ(t, v , x , y) = 0, ϕ(T , v , x , y) = U(v).
Can we solve it?
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Examples: Power Utility and Logarithmic Utility
If U(v) = log v , v > 0 or U(v) = 1γ v
γ , 0 < γ < 1 then thecorresponding HJB equation has the classical solution.Sketch of the Proof
1 Using an appropriate trial solution we separate the variable v .Obtain a non-local differential equation involving variablest, x , y .
2 Using Feynman-Kac formula, we obtain a mild solution of thenew equation.
3 Using stochastic analysis the mild solution is shown to satisfya Volterra equation of second kind.
4 The desired smoothness of the solution is obtained bystudying the Volterra equation.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Risk Sensitive Criterion: Definitions & Motivation
For a particular θ( 6= 0) (to be set by the investor according to hisrisk aversion) the risk sensitive criterion on finite and infinite timehorizon are defined as
Ju,Tθ (t, v , x , y) := −2
θlog Eu
[V u
T
−θ2 | Vt = v ,Xt = x ,Yt = y
]and
Juθ (t, v , x , y) := −2
θlim infT→∞
1
Tlog Eu
[V u
T
−θ2 | Vt = v ,Xt = x ,Yt = y
].
Taylor series expansion of JTθ about θ = 0 gives
−2
θlog Eu
[e(− θ
2) log V u
T
]= Eu logV u
T −θ
4Varu(logV u
T ) + o(θ).
A risk neutral (i.e., θ = 0) investor would effectively optimize theexpected logarithmic utility of terminal wealth.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Market ModelOptimization of Terminal UtilityRisk Sensitive Criterion
Risk Sensitive Criterion: Definitions & Motivation
For a particular θ( 6= 0) (to be set by the investor according to hisrisk aversion) the risk sensitive criterion on finite and infinite timehorizon are defined as
Ju,Tθ (t, v , x , y) := −2
θlog Eu
[V u
T
−θ2 | Vt = v ,Xt = x ,Yt = y
]and
Juθ (t, v , x , y) := −2
θlim infT→∞
1
Tlog Eu
[V u
T
−θ2 | Vt = v ,Xt = x ,Yt = y
].
Taylor series expansion of JTθ about θ = 0 gives
−2
θlog Eu
[e(− θ
2) log V u
T
]= Eu logV u
T −θ
4Varu(logV u
T ) + o(θ).
A risk neutral (i.e., θ = 0) investor would effectively optimize theexpected logarithmic utility of terminal wealth.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Finite Horizon Case
Theorem
(i) Let ψθ(t, x , y) = E [e∫ T
t hθ(s,Xs )ds | Xt = x ,Yt = y ] where x ∈ X ; 0 < y < t ≤ T ;
hθ(t, x) = θ2
infu∈A
[− r(t, x)− b(t, x) u + 1
2( θ
2+ 1)[u′a(t, x)u]
].
Then, ψθ(t, x , y) is a mild solution of(∂
∂t+
∂
∂y
)ψθ(t, x , y) +
f (y | x)
1− F (y | x)
∑j 6=x
pxj (ψθ(t, j , 0)− ψθ(t, x , y))
+θ
2inf
u∈A
[− r(t, x)− b(t, x)u +
1
2(θ
2+ 1)u′a(t, x)u
]ψθ(t, x , y) = 0
ψθ(T , x , y) = 1.
(ii) ψθ satisfies the following Volterra equation of second kind
ψθ(t, x , y) =1− F (T − t + y | x)
1− F (y | x)e∫ T
t hθ(s,x)ds +
∫ T−t
0
f (y + α | x)
1− F (y | x)×e
∫ t+αt hθ(s,x)ds
∑j
pxjψθ(t + α, j , 0)
dα. (2)
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Finite Horizon Case
Theorem
(iii)ψθ(t, x , y) is the classical solution of the Cauchy problem.(iv) The risk sensitive optimal expected utility is given by
ϕθ(t, v , x , y) := supu
Ju,Tθ (t, v , x , y) = log v −
2
θlog(ψθ(t, x , y)).
(v) The optimal strategy for the risk sensitive criterion on finite horizonCase 1: A = Rn, the minimizing u∗θ (t, x) in the PDE is given by
u∗θ (t, x) = 1
1+ θ2
a(t, x)−1 b(t, x)′,
Case 2: For A an n-dimensional rectangle, i.e., A =∏i≤n
[ci , di ], then
u∗θ,i (t, x) =
1
1+ θ2
(a(t, x)−1 b(t, x)′)i if 1
1+ θ2
(a(t, x)−1 b(t, x)′)i ∈ [ci , di ]
ci if 1
1+ θ2
(a(t, x)−1 b(t, x)′)i < ci
di if 1
1+ θ2
(a(t, x)−1 b(t, x)′)i > di
.
(3)
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30 35 40 45 50
0 < θ < 500 < θ < 500 < θ < 500 < θ < 50
Ris
k S
en
se
tiv
e C
rite
ria
at
x
=1
, t=
0, y
=0
Regime1 Regime2 Regime3
Figure : Risk Sensitive Optimal Expected Utility
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Infinite Horizon Case: Optimal Strategy
We restrict ourselves to the autonomous model.
1
TJ
u∗θ ,Tθ (v , x , y) ≥ 1
TJu,Tθ (v , x , y)
for any admissible strategy u. Hence
lim infT→∞
1
TJ
u∗θ ,Tθ (v , x , y) ≥ lim inf
T→∞
1
TJu,Tθ (v , x , y) (4)
for all admissible strategy u. Thus u∗θ is optimal for the respectiveaction spaces for the infinite horizon case as well.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Optimal Value: Large Deviation Principle
If the limit exists
limT→∞
1
TJ
u∗θ ,Tθ (v , x , y) = −2
θlim
T→∞
1
Tlog(E [e
∫ T0
hθ(Xs )ds | X0 = x ,Y0 = y ])
where
hθ(x) =θ
2infu∈A
[− r(x)− b(x) u +
1
2(θ
2+ 1)[u′a(x)u]
].
Right side is the large deviation limit for semi-Markov process. Forsemi-Markov case the existence of the limit and the representationthereof in terms of some relative entropy is not available in theliterature.
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Subcase: Irreducible Markov State
E [e∫ T
thθ(Xs )ds | Xt = x ] =
k∑j=1
exp(
(Λ + diag(hθ(·)))(T − t))
(x , j).
Limit exists if the matrix Λ + diag(hθ(·)) has a principal eigenvalue η(say). Then due to the multiplicative nature of the exponential function
limT→∞
1
Tlog E [e
∫ T0
hθ(Xs )ds | X0 = x ]
= limT→∞
1
Tlog
k∑j=1
exp(
(Λ + diag(hθ(·)))(T ))
(x , j) = η.
We prove the existence of η. Let c := minx (λxx + hθ(x)) and
Λθ := Λ + diag(hθ(·))− cIk×k .
By Perron-Frobenius theorem the spectral radius of Λθ is an eigenvalue ρ.
Therefore, η := ρ+ c is the principal eigenvalue of Λ + diag(hθ(·)).
Anindya Goswami Semi-Markov Process in Portfolio Optimization
MotivationSemi-Markov Modulated Market
Optimization of Risk Sensitive Criterion
Optimal Portfolio StrategiesNumerical Example for Finite Horizon CaseInfinite Horizon Case
Subcase: Irreducible Markov State: Numerical Results
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 5 10 15 20 25 30 35 40 45 50
0 < θθθθ < 50
Infi
nit
e H
oriz
on
Ris
k S
en
seti
ve C
rit
eria
Figure : Infinite Horizon Risk Sensitive Optimal Portfolio Growth RateAnindya Goswami Semi-Markov Process in Portfolio Optimization
M. K. Ghosh, A. Goswami and Suresh K. Kumar, Portfoliooptimization in a semi-Markov modulated market, AppliedMathematics and Optimization 60 (2009) 275-296.
Thank You