Optimal Portfolio Selection when Stock Price …...2 Stochastic Control vs Martingale Approach 3...

Optimal Portfolio Selection when Stock PriceReturns follow Jump-Diffusions

Daniel Michelbrink

School of Mathematical SciencesThe University of Nottingham, UK

[email protected]

Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 1 / 27

Outline

1 Introduction and Model Setup

2 Stochastic Control vs Martingale Approach

3 Admissible Strategies and Budget Constraints

4 The Optimisation Problem

5 Example: Power Utility


Outline







How should I invest my money?

An investor wants to invest money and needs some decision criteria.During the investment he/she also might want to withdraw some moneyfor consumption (e.g. buying a new car)The investor’s wealth is the collection of all the investor’s assets.Key question: How to choose an appropriate portfolio?

Expected Utility Maximisation

max E(∫ T

0U1(t , ct )dt + U2(VT )

)


Utility Functions

DefinitionA utility function U : R+ → R is a C1 function such that

1 U(·) is strictly increasing and strictly concave,2 U ′(∞) = 0 and U ′(0+) =∞.

Measures the amount of ‘utility’ an individual gains from some good.Utility of money: Money becomes less important the more we have.


Some Example Utility Functions

Logarithmic Utility

U(x) = log(x)

Used to solve the St. Petersburg paradox and important for maximising longterm growth rate.

Exponential Utility

U(x) = −e−αx

Power Utitlity

U(x) =xp

p

Used in Merton’s portfolio problem. Markowitz mean-variance problem isequavalent to a quadratic utility problem (p=2)!


The Market Model

Stock Price ModeldSt

St−= αtdt + σtdBt +

∫Eγ(t , z)N(dt ,dz)

where N(dt ,dz) is a Poisson random measure with intensity ν(dz). E is theset of possible jump sizes and γ(t , z) ≥ −1.

Bond Price ModeldS0

t

S0t

= rtdt

Wealth Process

dVt = rtVt−dt − ctdt + πtVt−

((αt − rt

)dt + σtdBt +

∫Eγ(t , z)N(dt ,dz)

)V0 = x (initial endowment)


Outline







How to solve the problem?

1 Stochastic ControlMerton (1971): Geometric Brownian motion modelKaratzas, Lehoczky, Sethi, Shreve (1990): Geometric Brownian motionmodelFramstad, Øksendal, Sulem (1999): Jump-Diffusion model

2 Martingale ApproachKaratzas, Lehoczky, Shreve (1987): Geometric Brownian motion modelKaratzas, Lehoczky, Shreve, Xu (1991):Geometric Brownian motion model in incomplete marketBardhan, Chao (1995): Jump-Diffusion model in complete marketBarbachan (2003): Levy process model


How does stochastic control work?

Hamilton-Jacobi-Bellman EquationSolve

Aπ,cφ(t , v) + U1(t , c) ≤ 0 t ≤ T ,Aπ

∗,c∗φ(t , v) + U1(t , c∗) = 0 t ≤ T ,φ(T , v) = U2(v).

where Aπ,c is the generator of the wealth process, e.g. in case of a Ito-Levymodel

Aπ,cφ(s, x) = ∂tφ+(π(α− r)x − c

)∂xφ

+12π2x2σ2∂xxφ+

∫E

(φ(s, x + πxz)− φ(s, x)

)ν(dz)

Recall:

Af (s, x) := limt↓0

Es,x(f (t ,Xt )

)− f (s, x)

t


How does the martingale approach work?Using tools from stochastic calculus to find a solution.

Ito Formula for Jump-DiffusionsLet f ∈ C1,2([0,∞)× R) and X (t) as above. Then

df (t ,X ) =∂

∂tf (t ,X )dt + α(t ,X )

∂

∂xf (t ,X )dB +

12σ(t ,X )2 ∂

2

∂x2 f (t ,X )dt

+f (t ,X + ∆X )− f (t ,X ).

Martingale Representation TheoremAny (P,Ft )-martingale M(t) has the representation

M(t) = M(0) +

∫ t

0aD(s)dB(s) +

∫ t

0

∫E

aJ(s, y)N(ds,dy)

where aD is predictable and square integrable and aJ is a Ft -predictable,marked process, that is integrable with respect to ν(ds,dy).


Change of Probablity Measure (Girsanov’s Theorem)

We can change the probability measure from P to Q by using a martingle Z asRadon-Nikodym density:

dQdP

= ZT .

In our case Z will be

dZt = ZtθDt dB − Zt

∫E

(1− θJ(t , y)

)N(dt ,dy); Z0 = 1.

New Brownian motion and Poisson MeasureUnder the new measure Q we have

dBQ = dB − θDdt is a Q-Brownian motionNQ(dt ,dz) = N(dt ,dz)− θJ(z)ν(dz)dt is a Q-Poisson randommeasure.


Measure Change and the Wealth Process

Wealth Process under P

dVt = rtVtdt − ctdt + πtVt

((αt − rt

)dt + σtdBt +

∫Eγ(t , y)N(dt ,dy)

)

Discounted Wealth Process under Q

dVt

S0t

= − ct

S0t

dt + πtVt

S0t

(σtdBQ

t +

∫Eγ(t , y)NQ(dt ,dy)

)

Conditions on θD and θJ

αt − rt + σtθDt +

∫Eγ(t , y)θJ(t , y)ν(dy) = 0


Outline







Admissible Portfolio-Consumption Strategies

Admissible Investment-Consumption StrategiesA pair (π, c) is called admissible, write (π, c) ∈ A, if

Vt ≥ 0, t ∈ [0,T ].

Define the state price density as H(t) := Z (t)/S0(t), where Z (t) is theprocess from the measure change dQ

dP |Ft = Zt .

Budget ConstraintIf (π, c) ∈ A. Then

E(∫ T

0Htctdt + HT VT

)≤ x


Reversing the Budget Constraint

For the geometric Brownian motion model or in the case that the stock pricealso has a Poisson jump with a single jump size it is possible to reverse thebudget constraint in the following way:

LemmaLet c be a consumption process and Y a FT measurable random variablesuch that

E(∫ T

0Htctdt + HT Y

)= x

Then there exists a portfolio process π such that (π, c) ∈ A and Y = V (T ).

However, in more sophisticated models like for jump-diffusions it is moredifficult to prove the Lemma.


Conditions on the trading strategy

Let φ(t) denote the amount of money invested into the stock. Then we needthat

φ(t)σ(t) =1

H(t)

((J(t)−M(t)

)θD(t) + aD(t)

),

φ(t)γ(t , y) =1

H(t)

((M(t)− J(t)

)1− θJ(t , y)

θJ(t , y)+

aJ(t , y)

θJ(t , y)

), ∀y ∈ E ,

where

M(t) := E(∫ T

0H(s)c(s)ds + H(T )Y

∣∣Ft

)J(t) :=

∫ t

0H(s)c(s)ds

and aD and aJ are the martinale representation coefficent of M(t).


Conditions on the trading strategy (2)

We have 2 equations for 3 unknown: φ, θD, and θJ

φ(t)σ(t) =1

H(t)

((J(t)−M(t)

)θD(t) + aD(t)

),

φ(t)γ(t , y) =1

H(t)

((M(t)− J(t)

)1− θJ(t , y)

θJ(t , y)+

aJ(t , y)

θJ(t , y)

), ∀y ∈ E ,

However, we also have the following condition on θD and θJ :

α(t)− r(t) + σ(t)θD(t) +

∫Eγ(t , y)θJ(t , y)ν(dy) = 0


Outline







The Problem

Optimisation Problem and admissible StrategiesMaximise

J(x ;π, c) = E(∫ T

0U1(s, cs)ds + U2(VT )

)with repect to π and c that satisfy

A(x) :={

(π, c) ∈ A(x) : E(∫ T

0U1(s, cs)−ds + U2(VT )−

)> −∞

}

Optimal Value FunctionFor the above problem, we define the optimal value function as

Φ(x) := sup(π,c)∈A(x)

J(x ;π, c) = J(x , ;π∗, c∗)


Solving the Optimisation Problem

Transfer to Constraint Optmisation

max(π,c)∈A E(∫ T


)s.t.

E(∫ T

0H(u)cudu + H(T )V (T )

)= x

Use constraint optimisation techniques: Lagrange multiplier

Lagrange Multiplier

L(c,Y , λ) = E(∫ T


)−λ[E(∫ T

0H(t)ctdt + H(T )V (T )

)− x

]Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 21 / 27

The Solution

Optimal Values

Optimal consumption: c∗(t) = I1(

t ,H(t)X−1(x))

Optimal terminal wealth: V ∗(T ) = I2(

H(T )X−1(x))

Optimal value: Φ(x) = Y(X−1(x)

)where X and Y are some known functions. I1 and I2 are the inverse of U ′1 andU ′2 respectively.


Outline







Example: Maximising with discounted Power Utility

Portfolio Wealth Process:

dV (t) =(

rtV (t)− ct

)dt

+π(t)V (t)((αt − rt

)dt + σtdBt +

∫Eγ(t , y)N(dt ,dy)

)

The maximisation Problem

max1β

E(∫ T

0e−δ1tcβt dt + e−δ2T Vβ

T

).

with δi ≥ 0, i = 1,2, and β < 1. Note β = 0 leads to log utility.


The trading strategy and the market prices of risk

Conditions on φ, θD, θJ

φ(t)σ(t) =1

H(t)

((J(t)−M(t)

)θD(t) + aD(t)

),

φ(t)γ(t , y) =1

H(t)

((M(t)− J(t)

)1− θJ(t , y)

θJ(t , y)+

aJ(t , y)

θJ(t , y)

), ∀y ∈ E ,

α(t)− r(t) + σ(t)θD(t) +

∫Eγ(t , y)θJ(t , y)ν(dy) = 0

Conditions on π, θD, θJ for Power Utility Problem

π(t)ξ(t) =1

β − 1θD(t), (1)

π(t)γ(t , y) = θJ(t , y)1/β−1 − 1, y ∈ E , (2)

α(t)− r(t) + σ(t)θD(t) +

∫Eγ(t , y)θJ(t , y)ν(dy) = 0 (3)


Optimal trading strategy

Optimality condition on π

The optimal strategy π(t) satisfies

α(t)− r(t) + σ(t)2(β − 1)π(t) +

∫Eγ(t , y)

(1 + π(t)γ(t , y)

)1−βν(dy) = 0.

Note that a similar result can be found in Framstad, Øksendal, Sulem (1999).


References

Barbachan, Jose F. (2003): Optimal Consumption and Investment withLevy Processes. Revista Brasileira de Economia, Vol. 57, 4.

Bardhan, Indrajit & Chao, Xiuli (1995): Martingale Analysis for Assets withDiscontinuous Returns. Mathematics of Operations Research, Vol.20, 1,243-256.

Karatzas, Ioannis, Lehoczky, John P., Sethi, Suresh P., and Shreve, SteveE. (1990): Explicit solution of a general consumption/investment problem.Journal of Mathematical Operations Research, Vol.11, Issue 2, 261-294.

Øksendal, Bernt & Sulem, Agnes (2007): Applied Stochastic Control ofJump Diffusions. Berlin Heidelberg New York: Springer.


Optimal Portfolio Selection when Stock Price …...2 Stochastic Control vs Martingale Approach 3...

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