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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




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).





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

.




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]





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.





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).





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?





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.





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.




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)





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)





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





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.





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.





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(·)).





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.

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