Qualitative properties of optimal portfolios in log-normal markets - PowerPoint PPT Presentation

Qualitative properties of optimal portfolios in log-normal markets ProCoFin Conference New York, June 2012 Thaleia Zariphopoulou U.T. Austin and OMI 1

References Work in progress • Temporal and spatial properties of optimal portfolios in log-normal markets (with S. Kallblad) • Complete monotonicity and marginal utilities (with S. Kallblad) • The optimal wealth process in log-normal markets (with P. Monin) 2

The classical Merton problem 3

The classical Merton problem • (Ω , F , P ) ; W standard Brownian motion • Traded securities � dS t = µS t dt + σS t dW t , S 0 > 0 dB t = 0 , B 0 = 1 π 0 • Self-financing strategies t (bond allocation), π t (stock allocation) X t = π 0 • Value of allocation t + π t λ = µ dX t = σπ t ( λ dt + dW t ) ; σ 4

Value function • Trading horizon [0 , T ] , T < ∞ • Utility function at T : U ( x ) , x ≥ 0 • Value function � U ( X T ) /X t = x � V ( x, t ) = sup E P A • Set of admissible strategies A � T � � s ds < + ∞ , X π ≥ 0 , a.e. π 2 A = π : π s ∈ F s , E P t 5

Optimality and HJB equation • The value function V : [0 , ∞ ) × [0 , T ] → [0 , ∞ )  � 1 � 2 σ 2 π 2 V xx + µπV x  V t + max = 0  π (HJB)   V ( x, T ) = U ( x ) • Optimal feedback controls π ∗ ( x, t ) = − λ V x ( x, t ) σ V xx ( x, t ) • Optimal wealth process dX ∗ s = µπ ∗ ( X s , s ) ds + σπ ∗ ( X s , s ) dW s ; X t = x π 0 , ∗ = X ∗ s − π ∗ π ∗ s = π ∗ ( X ∗ • Optimal allocations : s (bond), s , s ) (stock) s 6

Questions The optimal feedback portfolio and investment weight are given by π ∗ ( x, t ; T ) = λ w ∗ ( x, t ; T ) = λ r ( x, t ; T ) σ r ( x, t ; T ) and , σ x where r is the local risk tolerance function, r ( x, t ; T ) = − V x ( x, t ; T ) V xx ( x, t ; T ) We want to investigate for π ∗ ( x, t ; T ) , w ∗ ( x, t ; T ) and r ( x, t ; T ) • Spatial monotonicity • Spatial concavity/convexity • Temporal monotonicity • Sensitivities w.r.t. market parameters and horizon (portfolio greeks) 7

Fundamental Question Which properties, qualitative and structural, of quantities prescribed at T (e.g. risk aversion, risk tolerance, utility, marginal utility, inverse marginal utility, prudence,...) are propagated to the analogous quantities at previous trading times? Previous work • Spatial monotonicity (Borell; same model) • Time monotonicity (Gollier; discrete time) • Rich body of work in one-period models (Arrow, Ross, Kimball, Mossin, Roll, Pratt,...) 8

Optimal quantities and related partial differential equations 9

Related PDE • Value function V ( x, t ) — HJB equation 2 λ 2 V 2 V t − 1 x = 0 ; V ( x, T ) = U ( x ) V xx • Wealth function H ( x, t ) — heat equation � = H x ( x, t ) � H ( x, t ) , t r H t + 1 2 λ 2 H xx = 0 H ( x, T ) = I ( e − x ) I = ( U ′ ) ( − 1) ; , • Risk tolerance r ( x, t ) — fast diffusion equation r ( x, T ) = − U ′ ( x ) r t + 1 2 λ 2 r 2 r xx = 0 ; U ′′ ( x ) • Risk aversion γ ( x, t ) — porous medium equation � 1 � γ ( x, T ) = − U ′′ ( x ) γ t − 1 2 λ 2 = 0 ; U ′ ( x ) γ xx 10

Related PDE and optimal processes • Wealth function H ( x, t ) — heat equation � = H x ( x, t ) � H ( x, t ) , t r H t + 1 2 λ 2 H xx = 0 H ( x, T ) = I ( e − x ) I = ( U ′ ) ( − 1) ; , • Optimal wealth process (for convenience, initial time is set at zero) � � X ∗ ,x H ( − 1) ( x, 0) + λ 2 t + λW t , t = H t • Optimal stock allocation process = λ � � = λ � � π ∗ ,x H ( − 1) ( X ∗ ,x H ( − 1) ( x, 0) + λ 2 t + λW t , t σ H x , t ) , t σ H x t t The above optimal processes, X ∗ ,x and π ∗ ,x , are readily constructed via dual- t t ity arguments but the above alternative representations are quite convenient for addressing the questions herein. 11

Temporal and spatial properties of optimal portfolios 12

Spatial monotonicity of local risk tolerance Result: If the investor’s risk tolerance RT ( x ) = − U ′ ( x ) U ′′ ( x ) is increasing, then, for all t ∈ [0 , T ) , the local risk tolerance r ( x, t ) is also increasing in x . Proof: Recall that r ( H ( x, t ) , t ) = H x ( x, t ) with  H t + 1 H ( x, T ) = I ( e − x ) 2 λ 2 H xx = 0  ;   H xt + 1 2 λ 2 H xxx = 0 H x ( x, T ) = − e − x I ′ ( e − x ) > 0  ;   r x ( x, t ) = H xx ( H ( − 1) ( x, t ) , t ) Therefore, H x ( H ( − 1) ( x, t ) , t ) RT ′ ( x ) = H xx ( H ( − 1) ( x, T ) , T ) and RT ′ ( x ) > 0 Similarly, H x ( H ( − 1) ( x, T ) , T ) A direct application of the comparison principle for the heat equations satisfied by H x and H xx yields the result. The above provides a short proof of Borell’s result. 13

Spatial concavity/convexity of local risk tolerance Result: If the investor’s risk tolerance RT ( x ) is concave/convex, then, for all t ∈ [0 , T ) , the local risk tolerance r ( x, t ) is also concave/convex. Proof: Using again that r ( H ( x, t ) , t ) = H x ( x, t ) , we deduce � � H x ( H ( − 1) , t ) H xx ( H ( − 1) , t ) � � 1 � � r xx ( x, t ) = r 2 ( x, t ) det � � � H xx ( H ( − 1) , t ) H xxx ( H ( − 1) , t ) � � � Similarly � � H x ( H ( − 1) , T ) H xx ( H ( − 1) , T ) � � 1 � � RT ′′ ( x ) = RT 2 ( x ) det � � � � H xx ( H ( − 1) , T ) H xxx ( H ( − 1) , T ) � � The sign of the above Hankel determinants depends on the log concavity/log convexity of the function H x ( x, t ) , 0 ≤ t ≤ T . 14

Proof (con’d) On the other hand, H x solves the heat equation H xt + 1 H x ( x, T ) = − e − x I ′ ( e − x ) 2 λ 2 H xxx = 0 ; Moreover, RT ( x ) is concave/convex iff H x ( x, T ) is log concave/log convex. Classical results for the heat equation (e.g., Keady (1990)) yield the preservation of log concavity/log convexity of the solution H x ( x, t ) . 15

Temporal monotonicity of risk tolerance Result: If the investor’s risk tolerance RT ( x ) is concave/convex, then, the local risk tolerance r ( x, t ) is increasing/decreasing with respect to time. Proof: The fast diffusion equation yields r t + 1 2 λ 2 r 2 r xx = 0 ; r ( x, T ) = RT ( x ) If RT ( x ) is concave/convex, the previous result yields that r ( x, t ) is also concave/convex. Then, the above equation gives that r t > 0 ( < 0 ). Therefore, if the investor’s risk tolerance RT ( x ) is concave/convex, then, the optimal feedback stock allocation, π ∗ ( x, t ) = λ σ r ( x, t ) , increases/decreases as the time to maturity decreases. 16

Robustness of risk tolerance and dependence on market parameters 17

Comparison result Result: Assume that RT 1 ( x ) ≤ RT 2 ( x ) , all x ≥ 0 . Then, for all x ≥ 0 , r 1 ( x, t ) ≤ r 2 ( x, t ) , t ∈ [0 , T ) . Proof: Recall that r solves r t + 1 2 λ 2 r 2 r xx = 0 . Comparison for such equations might not hold. Let F ( x, t ) = r 2 ( x, t ) . Then F solves the quasilinear equation F t + 1 2 FF xx − 1 4 F 2 F ( x, t ) = RT 2 ( x ) x = 0 ; Establish comparison for the above equation (use results of Fukuda et al. (1993)). Use positivity of risk tolerance to conclude. Previous comparison results were produced for RT i ( x ) being linear ((Huang-Z.), (Back et al.)). The above result was proved by a combination of duality and penalization arguments by Xia. 18

Consequences of the comparison result Recall that π ∗ ( x, t ) and r ( x, t ) solve t + 1 π ∗ ( x, T ) = λ π ∗ 2 σ 2 π ∗ π ∗ xx = 0 ; σ RT ( x ) r t + 1 2 λ 2 r 2 r xx = 0 ; r ( x, t ) = RT ( x ) Result: If RT ( x ) is concave/convex, then r ( x, t ) is increasing/decreasing with respect to the stock’s Sharpe ratio λ . Proof: RT ( x ) concave − → r ( x, t ) concave. If λ 1 ≤ λ 2 , then r 1 ( x, t ) satisfies r 1 ,t + 1 1 r 1 ,xx = r 1 ,t + 1 1 r 1 ,xx + 1 ≥ r 1 ,t + 1 2 λ 2 1 r 2 2 λ 2 2 r 2 2 ( λ 2 1 − λ 2 2 ) r 2 2 λ 2 2 r 2 1 r 1 ,xx 1 r 1 ,xx . � �� � > 0 Therefore, r 1 is a subsolution to the equation satisfied by r 2 , and, thus r 1 ( x, t ) ≤ r 2 ( x, t ) 19

Consequences of the comparison result (con’d) • If RT ( x ) is concave/convex, then r ( x, t ) is increasing/decreasing with respect to the mean rate of return, µ , and decreasing/increasing with respect to the volatility σ . • The optimal portfolio π ∗ ( x, t ; σ, λ ) = λ σ r ( x, t ; σ, λ ) is always increasing in λ and decreasing in σ . • If RT ( x ) is concave/convex, then for all ( x, t ) , π ∗ ( x, t ) ≤ λ RT ′ (0) x σ RT ′ (0) x r ( x, t ) ≤ and ( ≥ ) 20

The optimal wealth process and space-time harmonic functions 21

The optimal wealth process � � X ∗ ,x H ( − 1) ( x, 0) + λ 2 t + λW t , t = H t where H t + 1 2 λ 2 H xx = 0 H ( x, T ) = I ( e − x ) ; • Therefore, the process H ( − 1) ( X ∗ ,x ) − H ( − 1) ( x, 0) = λ 2 t + λW t t is independent of risk preferences, across all investors! • The function H ( − 1) plays a very important role in several key calculations. (See, also, a recent preprint of Shkolnikov (2012)) 22