optimal portfolio selection when stock price …...2 stochastic control vs martingale approach 3...
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Optimal Portfolio Selection when Stock PriceReturns follow Jump-Diffusions
Daniel Michelbrink
School of Mathematical SciencesThe University of Nottingham, UK
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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 2 / 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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 3 / 27
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 )
)
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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.
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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)!
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 6 / 27
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)
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 7 / 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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 8 / 27
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
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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
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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).
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 11 / 27
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.
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 12 / 27
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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 13 / 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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 14 / 27
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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 15 / 27
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.
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 16 / 27
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).
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 17 / 27
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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 18 / 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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 19 / 27
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∗)
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 20 / 27
Solving the Optimisation Problem
Transfer to Constraint Optmisation
max(π,c)∈A E(∫ T
0U1(s, cs)ds + U2(VT )
)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
0U1(s, cs)ds + U2(VT )
)−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.
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 22 / 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
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 23 / 27
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.
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 24 / 27
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)
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 25 / 27
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).
Daniel Michelbrink (Uni Nottingham) Portfolio Selection with Jump-Diffusions 17/03/2009 26 / 27
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.
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