Optimal Investment under Dynamic Risk Constraints and Partial - PowerPoint PPT Presentation

Optimal Investment under Dynamic Risk Constraints and Partial Information Wolfgang Putschögl Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences www.ricam.oeaw.ac.at 20 th September 2007 Joint work with J. Saß (RICAM), Supported by FWF, Project P17947-N12 Workshop and Mid-Term Conference on Advanced Mathematical Methods for Finance Vienna

Outline Model Setup Problem formulation Time-Dependent Convex Constraints Dynamic Risk Constraints Gaussian Dynamics for the Drift A hidden Markov Model (HMM) for the Drift Example 2 / 34

Model Setup ◮ Filtered probability space : (Ω , F = ( F t ) t ∈ [ 0 , T ] , P ) ◮ Finite time horizon: T > 0 ◮ Money market : bond with stochastic interest rates r “Z t ” d S ( 0 ) = S ( 0 ) S ( 0 ) S ( 0 ) r t d t , = 1 , = exp , i.e., r s d s t t 0 t 0 r uniformly bounded and progressively measurable w.r.t. F ◮ Stock market : n stocks with price process S t = ( S ( 1 ) , . . . , S ( n ) ) ⊤ , return R t , and t t excess return ˜ R t , where d ˜ d S t = Diag ( S t )( µ t d t + σ t d W t ) , d R t = µ t d t + σ t d W t , R t = d R t − r t d t . W n -dimensional standard Brownian motion w.r.t. F and P drift µ t ∈ R n F t -adapted and independent of W volatility σ t ∈ R n × n progressively measurable w.r.t. F S t , σ t non-singular, and σ − 1 uniformly bounded. t 3 / 34

Risk Neutral Probability Measure We introduce the risk neutral probability measure ( → for filtering and optimization). Definition ◮ Martingale density process Z t Z t „ « s d W s − 1 � θ s � 2 d s θ ⊤ Z t = exp − 2 0 0 with θ t = σ − 1 ( µ t − r t 1 n ) the market price of risk t ◮ Risk neutral probability measure ˜ P defined by d ˜ P dP := Z T E expectation operator under ˜ ˜ P ◮ Girsanov’s theorem: Z t ˜ W t := W t + θ s d s 0 defines a ˜ P-Brownian motion 4 / 34

Partial Information Remark ◮ We consider the case of partial information: → we can only observe interest rates and stock prices ( F r , S ) but not the drift ◮ The portfolio has to be adapted to F r , S ˆ Z t |F S ˜ → we need the conditional density ζ t = E t ˆ µ t |F S ˜ → we need the filter for the drift ˆ µ t = E t Assumption ◮ The interest rates r are F S -adapted → F r , S = F S ◮ Z is a martingale w.r.t. F and P Lemma ◮ We have F S = F W = F R → the market is complete w.r.t. F S ˜ ˜ 5 / 34

Outline Model Setup Problem formulation Time-Dependent Convex Constraints Dynamic Risk Constraints Gaussian Dynamics for the Drift A hidden Markov Model (HMM) for the Drift Example 6 / 34

Consumption and Trading Strategy Definition ◮ Trading strategy π t : n -dimensional, F S -adapted, measurable ◮ Initial capital x 0 > 0 ◮ Wealth process X π satisfies d X π t = π ⊤ t ( µ t d t + σ t d W t ) + ( X π t − 1 ⊤ n π t ) r t d t X π 0 = x 0 ◮ A strategy is admissible if X π t ≥ 0 a.s. for all t ∈ [ 0 , T ] π t represents the wealth invested in the stocks at time t η π t = π t / X π t denotes the corresponding fraction of wealth 7 / 34

Utility Functions Definition U : [ 0 , ∞ ) → R ∪ {−∞} is a utility function, if U is strictly increasing, strictly concave, twice continuously differentiable on ( 0 , ∞ ) , and satisfies the Inada conditions: U ′ ( ∞ ) = lim x →∞ U ′ ( x ) = 0 , U ′ ( 0 +) = lim x ↓ 0 U ′ ( x ) = ∞ . I denotes the inverse function of U ′ . Assumption | I ′ ( y ) | ≤ Ky − b for all y ∈ ( 0 , ∞ ) and a , b , K > 0 I ( y ) ≤ Ky a , Example Logarithmic utility U ( x ) = log ( x ) Power utility U ( x ) = x α /α for α < 1 , α � = 0. 8 / 34

Optimization Problem Optimization Problem We optimize under partial information! Objective: Maximize the expected utility from terminal wealth, i.e., ˆ ˜ maximize E U ( X T ) under (risk) constraints we still have to specify. The optimization problem consists of two steps: 1. Find the optimal terminal wealth 2. Find the corresponding trading strategy 9 / 34

Outline Model Setup Problem formulation Time-Dependent Convex Constraints Dynamic Risk Constraints Gaussian Dynamics for the Drift A hidden Markov Model (HMM) for the Drift Example 10 / 34

Time-Dependent Convex Constraints ◮ We can write our model under full information with respect to F R as d R t = ˆ µ t d t + σ t d V t , t ∈ [ 0 , T ] . where the innovation process V = ( V t ) t ∈ [ 0 , T ] is a P-Brownian motion defined by Z t Z t Z t σ − 1 σ − 1 σ − 1 V t = W t + s ( µ s − ˆ µ s ) d s = d R s − µ s d s . ˆ s s 0 0 0 ◮ K t represents the constraints on portfolio proportions at time t → η π t ∈ K t K t is a F t -progressively measurable closed convex set ∅ � = K t ⊆ R n that contains 0 ◮ For each t we define the support function δ t : R n �→ R ∪ { + ∞} of − K t by y ∈ R n . ( − x ⊤ y ) , δ t ( y ) = sup x ∈ K t → δ t ( y ) is F t -progressively measurable → y �→ δ t ( y ) is a lower semicontinuous, proper, convex function on its effective K t = { y ∈ R n : δ t ( y ) < ∞} domain ˜ 11 / 34

Time-Dependent Convex Constraints Definition A trading strategy η π is called K t -admissible for initial capital x 0 > 0 if X π t ≥ 0 a.s. and η π t ∈ K t for all t ∈ [ 0 , T ] . We denote the class of admissible trading strategies for initial capital x 0 by A K t ( x 0 ) . We introduce the set H of dual processes ν t : [ 0 , T ] × Ω �→ ˜ K t which are ˆR T � ν t � 2 + δ t ( ν t ) F R ` ´ ˜ t -progressively measurable processes, satisfying E d t < ∞ . 0 For each dual process ν ∈ H we introduce ◮ a new interest rate process r ν t = r t + δ t ( ν t ) . ◮ a new drift process ˆ µ ν t = ˆ µ t + ν t + δ t ( ν t ) 1 n . t = σ − 1 ◮ a new market price of risk θ ν (ˆ µ t − r t + ν t ) t ◮ a new density process ζ ν given by d ζ ν t = − θ ν t ζ ν t d V t Then: Solution under constraints = solution under no constraints with new market coefficients! Problem: Find optimal ν ! 12 / 34

Time-Dependent Convex Constraints Proposition T )] < ∞ for all η π ∈ A K ( x 0 ) . Suppose x 0 > 0 and E [ U − ( X η ◮ A trading strategy η π ∈ A K ( x 0 ) is optimal, if for some y ∗ > 0 , ν ∗ ∈ H X ν ∗ ( y ∗ ) = x 0 , T = I ( y ∗ ˜ ζ ∗ X π T ) , T . Further, η π and ν ∗ have to satisfy the complementary slackness ζ ν ∗ where ˜ T = ˜ ζ ∗ condition δ t ( ν ∗ t ) ⊤ ν ∗ t ) + ( η π t = 0 , t ∈ [ 0 , T ] . ◮ y ∗ , ν ∗ solve the dual problem ˜ ˆ ˜ U ( y ˜ ζ ν ˜ V ( y ) = inf ν ∈H E T ) , where ˜ ˘ ¯ U ( y ) = sup x > 0 U ( x ) − xy , y > 0 is the convex dual function of U. ◮ If F R = F V holds, then an optimal trading strategy exists. 13 / 34

Outline Model Setup Problem formulation Time-Dependent Convex Constraints Dynamic Risk Constraints Gaussian Dynamics for the Drift A hidden Markov Model (HMM) for the Drift Example 14 / 34

Limited Expected Loss & Limited Expected Shortfall Suppose we cannot trade in [ t , t + ∆ t ] . Then “Z t +∆ t “Z t +∆ t ” ” t ) ⊤ X π ∆ X π t = X π t +∆ t − X π t = X π − X π ( η π t exp r s d s t + exp r s d s t t t Z t +∆ t Z t +∆ t − 1 “ “ ” ” diag ( σ s σ ⊤ σ s d ˜ × exp s ) d s + W s − 1 . 2 t t Next, we impose the relative LEL constraint ˜ t ) − |F S ˆ (∆ X π ˜ E < ε t , t with ε t = LX π t . Definition ˛ ˜ K LEL ˘ η π t ∈ R n ˛ ˆ (∆ X π t ) − |F S ˜ ¯ := E < ε t t t 15 / 34

Limited Expected Loss & Limited Expected Shortfall We introduce the relative LES constraint as an extension to the LEL constraint ˜ t + q t ) − |F S ˆ (∆ X π ˜ E < ε t , t with ε t = L 1 X π t and q t = L 2 X π t . ◮ LES with L 2 = 0 corresponds to LEL with L = L 1 . ◮ LEL: any loss in [ t , t + ∆ t ] can be hedged with L % of the portfolio value. ◮ LES: any loss greater L 2 % of the portfolio value in [ t , t + ∆ t ] can be hedged with L 1 % of the portfolio value. ◮ LEL & LES: For hedging we can use standard European call and put options. Definition ˛ ˜ K LES ˘ t ∈ R n ˛ ˆ t + q t ) − |F S ˜ ¯ := η π E (∆ X π < ε t t t Lemma K LEL and K LES are convex. t t For n = 1 we obtain the interval K LES = [ η l t , η u t ] . t 16 / 34

bounds on η π for LEL and LES 10 10 Bounds on η π Bounds on η π 0 0 −10 5 10 −10 2 2 5 1 1 0 0 0 0 ∆ t (in days) L (in % of Wealth) L 2 L 1 17 / 34

bounds on η π for LEL 8 0 L = 2% L = 1 . 2% L = 0 . 4% 6 −2 Upper bound on η π Lower bound on η π 4 −4 2 −6 L = 2% L = 1 . 2% L = 0 . 4% 0 −8 0 1 2 3 4 5 0 1 2 3 4 5 ∆ t (in days) ∆ t (in days) 18 / 34