Portfolio choice under space - time monotone criteria Kick - off - PowerPoint PPT Presentation

Portfolio choice under space - time monotone criteria Kick - off Workshop RICAM September 2008 Thaleia Zariphopoulou The University of Texas at Austin 1

Performance measurement of investment strategies 2

Market environment Riskless and risky securities W = ( W 1 , . . . , W d ) • (Ω , F , P ) ; standard Brownian Motion • Traded securities � � ⎧ dS i t = S i µ i t dt + σ i S i t · dW t 0 > 0 , ⎪ t ⎨ 1 ≤ i ≤ k ⎪ dB t = r t B t dt , B 0 = 1 ⎩ µ t , r t ∈ R , σ i t ∈ R d bounded and F t -measurable stochastic processes • Postulate existence of an F t -measurable stochastic process λ t ∈ R d satisfying 1 = σ T µ t − r t 1 t λ t • No assumptions on market completeness 3

Market environment π 0 t , π t = ( π 1 t , . . . , π i t , . . . , π k • Self-financing investment strategies t ) • Present value of this allocation k π i � X t = t i =0 k π i t σ i � dX t = t · ( λ t dt + dW t ) i =1 = σ t π t · ( λ t dt + dW t ) 4

Investment performance process U ( x, t ) is an F t -adapted process, t ≥ 0 • The mapping x → U ( x, t ) is increasing and concave • For each self-financing strategy, represented by π , the associated (discounted) wealth X π t satisfies E P ( U ( X π t , t ) | F s ) ≤ U ( X π s , s ) , 0 ≤ s ≤ t • There exists a self-financing strategy, represented by π ∗ , for which the associated (discounted) wealth X π ∗ satisfies t E P ( U ( X π ∗ t , t ) | F s ) = U ( X π ∗ s , s ) , 0 ≤ s ≤ t 5

Optimality across times U ( x, t ) ∈ F t | | 0 U ( x, s ) ∈ F s U ( x, t ) ∈ F t | | | 0 E ( U ( X π U ( x, s ) = sup t , t ) |F s , X s = x ) A The traditional value function is a special case of the above, defined only for t ∈ [0 , T ] and with (terminal) condition U ( x, T ) = u ( x ) , U ( x, T ) ∈ F 0 6

The forward performance SPDE 7

The forward performance SPDE Let U ( x, t ) be an F t − measurable process such that the mapping x → U ( x, t ) is increasing and concave. Let also U = U ( x, t ) be the solution of the stochastic partial differential equation 2 � � � σσ + A ( Uλ + a ) dU = 1 � � � dt + a · dW A 2 U 2 where a = a ( x, t ) is an F t − adapted process, while A = ∂ ∂x . Then U ( x, t ) is a forward performance process. • The process a may depend on t, x, U, its spatial derivatives etc. • Note that in the traditional (backward) case, the volatility process is fixed 8

Optimal portfolios and wealth At the optimum • The optimal portfolio vector π ∗ is given in the feedback form t , t ) = − σ + A ( Uλ + a ) t = π ∗ ( X ∗ π ∗ ( X ∗ t , t ) A 2 U • The optimal wealth process X ∗ solves t = − σσ + A ( Uλ + a ) dX ∗ ( X ∗ t , t ) ( λdt + dW t ) A 2 U 9

The zero volatility case: a ( x, t ) ≡ 0 10

Space-time monotone performance process The forward performance SPDE simplifies to 2 � � � σσ + A ( Uλ ) dU = 1 � � � dt A 2 U 2 The process � t 2 ds � � σ s σ + � U ( x, t ) = u ( x, A t ) with A t = s λ s � � � 0 with u : R × [0 , + ∞ ) → R , increasing and concave with respect to x , and solving u t u xx = 1 2 u 2 x is a solution. MZ (2006) Berrier, Rogers and Tehranchi (2007) 11

Performance measurement time t 1 , information F t 1 110 risk premium 109.8 � t 1 109.6 | λ | 2 ds A t 1 = u(x,t 1 ) 109.4 0 109.2 109 1 0.8 1 0.6 0.8 0.6 0.4 0.4 Time 0.2 Wealth 0.2 0 0 ��� ��� A t 1 + u ( x, t 1 ) � � � � U ( x, t 1 ) = u ( x, A t 1 ) ∈ F t 1 12

Performance measurement time t 2 , information F t 2 110 risk premium 105 � t 2 | λ | 2 ds 100 A t 2 = u(x,t 2 ) 0 95 90 1 0.8 1 0.6 0.8 0.6 0.4 0.4 Time 0.2 Wealth 0.2 0 0 ��� ��� A t 2 + u ( x, t 2 ) � � � � U ( x, t 2 ) = u ( x, A t 2 ) ∈ F t 2 13

Performance measurement time t 3 , information F t 3 110 105 risk premium 100 � t 3 95 | λ | 2 ds A t 3 = 90 u(x,t 3 ) 0 85 80 75 1 0.8 1 0.6 0.8 0.6 0.4 0.4 Wealth 0.2 0.2 Time 0 0 ��� ��� A t 3 + u ( x, t 3 ) � � � � U ( x, t 3 ) = u ( x, A t 3 ) ∈ F t 3 14

Forward performance measurement time t , information F t u(x,t) market Time Wealth ��� ��� MI ( t ) u ( x, t ) + � � � � 15 U ( x, t ) = u ( x, A t ) ∈ F t

Properties of the performance process U ( x, t ) = u ( x, A t ) • the deterministic risk preferences u ( x, t ) are compiled with � t 0 | λ | 2 ds the stochastic market input A t = • the evolution of preferences is “deterministic” • the dynamic risk preferences u ( x, t ) reflect the risk tolerance and the impatience of the investor 16

Optimal allocations 17

Optimal allocations • Let X ∗ t be the optimal wealth, and A t the time-rescaling processes dX ∗ t = σ t π ∗ t · ( λ t dt + dW t ) dA t = | λ t | 2 dt • Define r ( x, t ) = − u x ( x, t ) R ∗ t � r ( X ∗ t , A t ) u xx ( x, t ) Optimal portfolios π ∗ t = σ + t λ t R ∗ t 18

A system of SDEs at the optimum dX ∗ t = r ( X ∗ ⎧ t , A t ) λ t · ( λ t dt + dW t ) ⎨ dR ∗ t = r x ( X ∗ t , A t ) dX ∗ ⎩ t π ∗ t = σ + t λ t R ∗ t The optimal wealth and portfolios are explicitly constructed if the function r ( x, t ) is known 19

Concave utility inputs and increasing harmonic functions 20

Concave utility inputs and increasing harmonic functions There is a one-to-one correspondence between strictly concave solutions u ( x, t ) to u 2 u t = 1 x 2 u xx and strictly increasing solutions to h t + 1 2 h xx = 0 21

Concave utility inputs and increasing harmonic functions • Increasing harmonic function h : R × [0 , + ∞ ) → R is represented as e yx − 1 2 y 2 t − 1 � h ( x, t ) = ν ( dy ) y R • The associated utility input u : R × [0 , + ∞ ) → R is then given by the concave function � t � x u ( x, t ) = − 1 0 e − h ( − 1) ( x,s )+ s 0 e − h ( − 1) ( z, 0) dz h ( − 1) ( x, s ) , s � � ds + 2 h x 2 The support of the measure ν plays a key role in the form of the range of h and, as a result, in the form of the domain and range of u as well as in its asymptotic behavior (Inada conditions) 22

Examples 23

Measure ν has compact support ν ( dy ) = δ 0 , where δ 0 is a Dirac measure at 0 Then, e yx − 1 2 y 2 t − 1 � h ( x, t ) = δ 0 = x y R and � t � x u ( x, t ) = − 1 0 e − x + s 0 e − z dz = 1 − e − x + t 2 ds + 2 2 24

Measure ν has compact support ν ( dy ) = b 2 ( δ a + δ − a ) , a, b > 0 δ ± a is a Dirac measure at ± a Then, h ( x, t ) = b ae − 1 2 a 2 t sinh ( ax ) If, a = 1 , then − e t � � u ( x, t ) = 1 − t � � � � � � x 2 + b 2 e − t x 2 + b 2 e − t ln x + x − b 2 x 2 2 If a � = 1 , then √ a e − at + (1 + √ a ) x � √ a x + � � 1+ 1 ax 2 + βe − at u ( x, t ) = ( √ a ) β √ a 1 −√ a t e 2 � 1+ 1 a − 1 √ a � √ a x + � ax 2 + βe − at 25

Measure ν has infinite support 2 y 2 dy 1 2 π e − 1 √ ν ( dy ) = Then � x � � z 2 x 2 dz √ t + 1 h ( x, t ) = F F ( x ) = 0 e and √ � � F ( − 1) ( x ) − u ( x, t ) = F t + 1 26

Optimal processes and increasing harmonic functions 27

Optimal processes and risk tolerance ⎧ dX ∗ t = r ( X ∗ t , A t ) λ t · ( λ t dt + dW t ) ⎨ dR ∗ t = r x ( X t , A t ) dX ∗ ⎩ t Local risk tolerance function and fast diffusion equation r t + 1 2 r 2 r xx = 0 r ( x, t ) = − u x ( x, t ) u xx ( x, t ) 28

Local risk tolerance and increasing harmonic functions If h : R × [0 , + ∞ ) → R is an increasing harmonic function then r : R × [0 , + ∞ ) → R + given by R e yh ( − 1) ( x,t ) − 1 2 y 2 t ν ( dy ) � h ( − 1) ( x, t ) , t � � r ( x, t ) = h x = is a risk tolerance function solving the FDE 29

Optimal portfolio and optimal wealth • Let h be an increasing solution of the backward heat equation h t + 1 2 h xx = 0 and h ( − 1) stands for its spatial inverse • Let the market input processes A and M by defined by � t � t 0 | λ | 2 ds A t = and M t = 0 λ · dW • Then the optimal wealth and optimal portfolio processes are given by h ( − 1) ( x, 0) + A t + M t , A t X ∗ ,x � � = h t and X ∗ ,x π ∗ � h ( − 1) � � � σ + t = h x , A t , A t t λ t t 30

Complete construction Utility inputs and harmonic functions u 2 u t = 1 h t + 1 x ⇐ ⇒ 2 h xx = 0 2 u xx Harmonic functions and positive Borel measures h ( x, t ) ⇐ ⇒ ν ( dy ) Optimal wealth process � t X ∗ ,x = h h ( − 1) ( x, 0) + A + M, A � � M = 0 λ · dW s , � M � = A Optimal portfolio process π ∗ ,x = h x h ( − 1) ( X ∗ ,x , A ) , A σ + λ � � The measure ν emerges as the defining element ⇒ ⇒ ν h u How do we choose ν and what does it represent for the investor’s risk attitude ? 31

Inferring investor’s preferences 32

Calibration of risk preferences to the market Given the desired distributional properties of his/her optimal wealth in a specific market environment, what can we say about the investor’s risk preferences? Investor’s investment targets • Desired future expected wealth • Desired distribution References Sharpe (2006) Sharpe-Golstein (2005) 33