Optimal investments under dynamic performance critria. Lecture IV

Transcription

1 Optimal investments under dynamic performance critria Lecture IV 1

2 Utility-based measurement of performance 2

3 Deterministic environment Utility traits u(x, t) : x wealth and t time Monotonicity u x (x, t) > Risk aversion u xx (x, t) < Impatience u t (x, t) < Fisher (1913, 1918), Koopmans (1951), Koopmans-Diamond-Williamson (1964)... 3

4 Stochastic environment Important ingredients Time evolution concurrent with the one of the investment universe Consistency with up to date information Incorporation of available opportunities and constraints Meaningful optimal utility volume 4

5 Dynamic utility U(x, t) is an F t -adapted process As a function of x, U is increasing and concave For each self-financing strategy, represented by π, the associated (discounted) wealth X t satisfies E P (U(X π t,t) F s ) U(X π s,s) s t There exists a self-financing strategy, represented by π,forwhich the associated (discounted) wealth Xt π satisfies E P (U(Xt π,t) F s ) = U(Xs π,s) s t 5

6 Traditional framework A deterministic utility datum u T (x) is assigned at the end of a fixed investment horizon U(x, T )=u T (x) Backwards in time generation of optimal utility volume V (x, t) =sup π E P (u(x π T,T) F t; X π t = x) V (x, t) =sup π E P (V (X π s,s) F t ; X π t = x) (DPP) V (x, t) =E P (V (Xs π,s) F t ; Xt π = x) U(x, t) V (x, t) t<t The dynamic utility coincides with the traditional value function 6

7 A deterministic utility datum u (x) is assigned at the beginning of the trading horizon, t = U(x, ) = u (x) Forward in time generation of optimal utility volume U(X π s,s)=e P (U(X π t,t) F s ) s t Dynamic utility can be defined for all trading horizons Utility and allocations take a very intuitive form Difficulties due to the inverse in time nature of the problem Utility is not exogeneously given but is implied/calibrated w.r.t. investment opportunities 7

8 Motivational examples 8

9 An incomplete multiperiod binomial example Exponential utility datum Traded security: S t,t=, 1,... ξ t+1 = S t+1 S t,ξ t+1 = ξ d t+1,ξu t+1 with <ξ d t+1 < 1 <ξu t+1 Second traded asset is riskless yielding zero interest rate Stochastic factor: Y t,t=, 1,... η t+1 = Y t+1 Y t,η t+1 = η d t+1,ηu t+1 with η d t <η u t Probability space (Ω, (F t ), P) {S t,y t : t =, 1,...} : a two-dimensional stochastic process 9

10 State wealth process: X t, t = s +1,s+2,...,... α i : the number of shares of the traded security held in this portfolio over thetimeperiod[i 1,i] X t = X s + t i=s+1 α i S i Forward dynamic exponential utility U(X α s,s)=e P (U(X α t,t) F s ) U(x, ) = e γx, γ > 1

11 A forward dynamic utility U(x, t) = e γx if t = e γx+ t i=1 h i if t 1 Auxiliary quantities : local entropies h i with h i = q i log q i P (A i F i 1 ) +(1 q i)log 1 q i 1 P (A i F i 1 ) A i = {ξ i = ξ u i } and q i = Q (A i F i 1 ) for i =, 1,.. and Q being the minimal relative entropy measure 11

12 Important insights The forward utility process U(x, t) = e γx+ t i=1 h i is of the form U(x, t) =u(x, A t ) where u(x, t) is the deterministic utility function u(x, t) = e γx+1 2 t and A t corresponds to a time change depending on the market input A t =2 t h i i=1 12

13 Important insights (continued) The variational utility input u(x, t) = e γx+1 2 t solves the partial differential equation u t u xx = 1 2 u2 x u(x, ) = e γx The stochastic market input A t =2 t h i i=1 plays now the role of time. It depends exclusively on the market parameters. 13

14 A continuous-time example Investment opportunities Riskless bond : r = Risky security : ds t = σ t S t (λ t dt + dw t ) Utility datum at t =: u (x) Wealth process dx t = σ t π t (λ t dt + dw t ) X = x Market input : λ t, A t da t = λ 2 t dt A = 14

15 Building the martingale U(Xt π,t) Assume that we can construct U(x, t) via U(Xt π,t)=u(xt π,a t ) U(x, ) = u(x, ) = u (x) where u(x, t) is the variational utility input and A t the stochastic market input du(xt π,t)=u x(x t,a t )σ t π t dw t +(u t (Xt π,a t)λ 2 t + u x(xt π,a t)σ t π t λ t u xx(xt π,a t)σt 2π2 t )dt }{{} 15

16 Variational utility input condition u t u xx = 1 2 u2 x u(x, ) = u (x) The optimal allocations in stock, π t π t = σ 1 t π, t = X π t, and in bond, π, t, u x (X λ t π,a t ) t u xx (X π,a t ) = σ 1 t λ t R t t σt 1 λ t R t R t = r(x π t,a t ) ; r(x, t) = u x(x, t) u xx (x, t) The local risk tolerance r(x, t) and the subordinated risk tolerance process R t emerge as important quantities 18

17 Dynamic utility measurement time t 1,informationF t1 asset returns 11 constraints 19.8 market view away from equilibrium benchmark calendar time numeraire subordination u(x,t 1 ) Time Wealth.8 1 MI(t 1 ) + u(x, t 1 ) U(x, t 1 ; MI) F t1 π(x, t 1 ; MI) F t1 19

18 Dynamic utility measurement time t 2,informationF t2 asset returns constraints market view away from equilibrium benchmark numeraire u(x,t 2 ) calendar time subordination Time Wealth.8 1 MI(t 2 ) + u(x, t 2 ) U(x, t 2 ; MI) F t2 π(x, t 2 ; MI) F t2 2

19 Dynamic utility measurement time t 3,informationF t3 asset returns 11 constraints market view away from equilibrium benchmark calendar time numeraire subordination u(x,t 3 ) Time Wealth.8 1 MI(t 3 ) + u(x, t 3 ) U(x, t 3 ; MI) F t3 π(x, t 3 ; MI) F t3 21

20 Dynamic utility measurement time t, informationf t u(x,t) asset returns additional market input Time Wealth MI(t) + u(x, t) U(X t,t) F t π (X t,t) F t 22

21 Dynamic utility measurement time t 1,informationF t1 u(x,t) asset returns additional market input Time Wealth MI(t 1 ) + u(x, t 1 ) U(X t 1,t 1 ) F t1 π (X t 1,t 1 ) F t1 23

22 Dynamic utility measurement time t 2,informationF t2 u(x,t) asset returns additional market input Time Wealth MI(t 2 ) + u(x, t 2 ) U(X t 2,t 2 ) F t2 π (X t 2,t 2 ) F t2 24

23 Dynamic utility measurement time t 3,informationF t3 u(x,t) asset returns additional market input Time Wealth MI(t 3 ) + u(x, t 3 ) U(X t 3,t 3 ) F t3 π (X t 3,t 3 ) F t3 25

24 Construction of a class of forward dynamic utilities 24

25 Creating the martingale that yields the optimal utility volume Minimal model assumptions Stochastic optimization problem inverse in time Key idea Stochastic input Market Variational input Individual Maximal utility Optimal allocation 25

26 Variational input utility surfaces 26

27 Utility surface A model independent variational constraint on impatience, risk aversion and monotonicity Initial utility datum u (x) =u(x, ) Fully non-linear pde u t u xx = 1 2 u2 x u(x, ) = u (x) 27

28 Utility transport equation The utility equation can be alternatively viewed as a transport equation with slope of its characteristics equal to (half of) the risk tolerance r(x, t) = u x(x, t) u xx (x, t) u t r(x, t)u x = u(x, ) = u (x) Characteristic curves: dx(t) dt = 1 2 r(x(t),t) 28

29 Construction of utility surface u(x, t) using characteristics dx(t) dt = 1 2 r(x(t),t) Utility datum u (x) 29

30 Construction of characteristics dx(t) dt = 1 2 r(x(t),t) Utility datum u(x, ) Characteristic curves 3

31 Propagation of utility datum along characteristics 31

32 Propagation of utility datum along characteristics 32

33 Utility surface u(x, t) 33

34 Two related pdes Fast diffusion equation for risk tolerance r t r2 r xx = r(x, ) = r (x) (FDE) Conductivity : r 2 Porous medium equation for risk aversion γ t = γ(x, t) = 1 r(x, t) ( ) 1 γ xx γ(x, ) = 1 r (x) (PME) Pressure : r 2 and (PME) exponent: m = 1 34

35 Difficulties Utility equation: u t u xx = 1 2 u2 x Inverse problem and fully nonlinear Utility transport equation: u t r(x, t)u x = Shocks, solutions past singularities Fast diffusion equation: r t r2 r xx = Inverse problem and backward parabolic, solutions might not exist, locally integrable data might not produce locally bounded slns in finite time Porous medium equation: γ t =( γ 1) xx Majority of results for (PME), γ t =(γ m ) xx, are for m>1, partialresultsfor 1 <m< 35

36 A rich class of risk tolerance inputs Addititively separable risk tolerance r 2 (x, t; α, β) =m(x; α, β)+n(t; α, β) m(x; α, β) =αx 2 r(x, t; α, β) = Example n(x; α, β) =βe αt αx 2 + βe αt α, β > (Very) special cases r(x, t;,β)= β u(x, t) = e x β + t 2 r(x, t;1, ) = x u(x, t) =log x t 2 r(x, t; α, ) = α x u(x, t) = γ 1xγ e 2(1 γ) t, γ = α 1 γ α 36

37 Risk tolerance r(x, t) =.5x e.5t Time Wealth

38 Utility surface u(x, t) generated by risk tolerance r(x, t) =.5x e.5t Time Wealth Characteristics: dx(t) dt = 1 2.5x(t) e.5t 38

39 Risk tolerance r(x, t) = 1x 2 + e 1t Time Wealth

40 Utility surface u(x, t) generated by risk tolerance r(x, t) = 1x 2 + e 1t Time Wealth Characteristics: dx(t) dt = 1 2 1x(t) 2 + e 1t 4

41 Risk tolerance r(x, t;, 1) = x 2 +1= Time Wealth

42 Utility surface u(x, t) = e x+ t 2 generated by risk tolerance r(x, t) = Time Wealth Characteristics: dx(t) dt =

43 Risk tolerance r(x, t;1, ) = x 2 +e t = x Time Wealth

44 Utility surface u(x, t) =logx 2 t, x> generated by risk tolerance r(x) =x Time Wealth 15 2 Characteristics: dx(t) dt = 1 2 x(t) 44

45 Risk tolerance r(x, t;4, ) = 4x 2 +e 4t =2 x Time Wealth

46 Utility surface u(x, t) =2 xe t 2, x> generated by risk tolerance r(x, t) =2x Time Wealth 15 2 Characteristics: dx(t) dt = x(t) 46

47 Multiplicatively separable risk tolerance r(x, t; α, β) =m(x; α)n(t; β) Example m(x; α) =ϕ(φ 1 (x; α)) n(t; β) = 1 t + β, β > Φ(x; α) = x α ez2 /2 dz ϕ =Φ r(x, t; α, β) =ϕ(φ 1 (x; α)) (Very) special cases m(x; α) =α, n(t; β) =1 u(x, t) = e x α + 2 t m(x; α) =x, n(t; β) =1 u(x, t) =log x 2 t m(x; α) =αx, n(t; β) =1 u(x, t) = 1 γ xγ e γ 2(1 γ) t, γ = α 1 α

48 Risk tolerance r(x, t) = ϕ(φ 1 (x;.5) t Time Wealth

49 Utility surface u(x, t) =Φ(Φ 1 (x;.5) t +5) generated by risk tolerance r(x, t) = ϕ(φ 1 (x;.5)) t Time Wealth 15 2 Characteristics: dx(t) dt = ϕ(φ 1 (x(t);.5)) t +5 49

50 Utility function u(x, t ) (fixed time) t = Utility Wealth 5

51 Utility function u(x,t) (fixed wealth level) x = Utility Time 51

52 Summary on variational utility input Key state variables: wealth and risk tolerance Risk tolerance solves a fast diffusion equation posed inversely in time r t r2 r xx = r(x, ) = u (x) u (x) Utility surface generated by a transport equation u t r(x, t)u x = u(x, ) = u (x) Forward dynamic utility process constructed by compiling variational utility input and stochastic market input 52