SPDE and portfolio choice (joint work with M. Musiela) Princeton University. Thaleia Zariphopoulou The University of Texas at Austin

Transcription

1 SPDE and portfolio choice (joint work with M. Musiela) Princeton University November 2007 Thaleia Zariphopoulou The University of Texas at Austin 1

2 Performance measurement of investment strategies 2

3 Market environment Riskless and risky securities (Ω, F, P) ; W =(W 1,...,W d ) standard Brownian Motion Traded securities 1 i k dst i = St (μ i tdt + σt i ) dw t, S0 i > 0 db t = r t B t dt, B 0 =1 μ t,r t R, σ i t Rd bounded and F t -measurable stochastic processes Postulate existence of an F t -measurable stochastic process λ t R d satisfying μ t r t 1=σ T t λ t No assumptions on market completeness 3

4 Market environment Self-financing investment strategies π 0 t, π t =(π 1 t,...,πi t,...,πk t ) Present value of this allocation X t = k πt i i=0 dx t = k i=1 π i tσ i t (λ t dt + dw t ) = σ t π t (λ t dt + dw t ) 4

5 Traditional framework A (deterministic) utility datum u T (x) is assigned at the end of a fixed investment horizon U T (x) =u T (x) No market input to the choice of terminal utility Backwards in time generation of the indirect utility V s (x) =sup π E P (u T (X π T ) F s; X π s = x) V s (x) =sup π E P (V t (X π t ) F s ; X π s = x) (DPP) V s (x) =E P (V t (Xt π ) F s ; Xs π = x) The value function process becomes the intermediate utility for all t [0,T) 5

6 Investment performance process V s (x) F s V t (x) F t 0 s t u T (x) T For each self-financing strategy, represented by π, the associated wealth X π t satisfies E P (V t (X π t ) F s ) V s (X π s ), 0 s t T There exists a self-financing strategy, represented by π,forwhichthe associated wealth Xt π satisfies E P (V t (Xt π ) F s )=V s (Xs π ), 0 s t T 6

7 Investment performance process V t,t (x) F t, 0 t T V t,t (X π t ) is a supermartingale V t,t (Xt π ) is a martingale V t,t (x) is the terminal utility in trading subintervals [s, t], 0 s t Observations V T,T (x) is chosen exogeneously to the market Choice of horizon possibly restrictive More realistic to have random terminal data, V T,T (x, ω) =U(x, ω) 7

8 Investment performance process U t (x) is an F t -adapted process, t 0 The mapping x U t (x) is increasing and concave For each self-financing strategy, represented by π, the associated (discounted) wealth X π t satisfies E P (U t (X π t ) F s ) U s (X π s ), 0 s t There exists a self-financing strategy, represented by π,forwhich the associated (discounted) wealth Xt π satisfies E P (U t (Xt π ) F s )=U s (Xs π ), 0 s t 8

9 Optimality across times 0 U s (x) F s 0 U t (x) F t U t (x) F t U s (x) =sup A E(U t (X π t ) F s,x s = x) Does such a process aways exist? Is it unique? 9

10 Forward performance process Adatumu 0 (x) is assigned at the beginning of the trading horizon, t =0 U 0 (x) =u 0 (x) Forward in time generation of optimal performance E P (U t (X π t ) F s) U s (X π s ), E P (U t (Xt π ) F s )=U s (Xs π ), 0 s t 0 s t Many difficulties due to inverse in time nature of the problem 10

11 The stochastic PDE of the forward performance process 11

12 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 du = 1 2 σσ + A (Uλ+ a) 2 A 2 U dt + a dw 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. 12

13 At the optimum The optimal portfolio vector π is given in the feedback form πt = π (Xt,t)= σ +A (Uλ+ a) A 2 U (X t,t) The optimal wealth process X solves dxt = σσ +A (Uλ+ a) A 2 U (X t,t)(λdt + dw t ) 13

14 Intuition for the structure of the forward performance process Assume that U = U (x, t) solves du (x, t) =b (x, t) dt + a (x, t) dw t where b, a are F t measurable processes. Recall that for an arbitrary admissible portfolio π, the associated wealth process, X π, solves dx π t = σ t π t (λ t dt + dw t ) Apply the Ito-Ventzell formula to U (X π t,t) we obtain du (X π t,t) = b (X π t,t) dt + a (X π t,t) dw t +U x (X π t,t) dx π t U xx (X π t,t) d X π t + a x (X π t,t) d W, X π t = ( b (X π t,t)+u x (X π t,t) σ t π t λ t + σ t π t a x (X π t,t)+ 1 2 U xx (X π t,t) σ t π t 2) dt +(a (X π t,t)+u x (X π t,t) σ t π t ) dw t 14

15 Intuition (continued) By the monotonicity and concavity assumptions, the quantity sup (U x (Xt π,t) σ t π t λ t + σ t π t a x (Xt π,t)+ 1 π 2 U xx (Xt π,t) σ t π t 2) is well defined. Calculating the optimum π yields π t = σ + t M ( Xt π,t ) = ( U x X π t,t ) ( λ t + a x X π t,t ) ( U xx X π t,t ) Deduce that the above supremum is given by σ t σ t + Choose the drift coefficient ( ( Ux X π t,t ) ( λ t + a x X π t,t )) 2 ( 2U xx X π t,t ) b (x, t) = M (x, t) 15

16 Solutions to the forward performance SPDE du = 1 2 σσ + A (Uλ+ a) 2 A 2 U dt + a dw Local differential coefficients a (x, t) =F (x, t, U (x, t),u x (x, t)) Difficulties The equation is fully nonlinear The diffusion coefficient depends, in general, on U x and U xx The equation is not (degenerate) elliptic 16

17 Choices of volatility coefficient The deterministic case: a (x, t) =0 The forward performance SPDE simplifies to The process du = 1 σσ + A (Uλ) 2 2 A 2 dt U U (x, t) =u (x, A t ) with A t = t 0 σ s σ + s λ s 2 ds with u : R [0, + ) R, increasing and concave with respect to x, and solving is a solution. MZ (2006) Berrier, Rogers and Tehranchi (2007) u t u xx = 1 2 u2 x 17

18 a (x, t) =0 σ, λ constants and u separable (in space and time) The forward performance process reduces to a deterministic function. U (x, t) =u (x, t) u (x, t) = e x+ t 2 or u (x, t) = 1 γ xγ e γ 2(1 γ) λ2 t Horizon-unbiased utilities Henderson-Hobson (2006) a (x, t) =k, k R U(x, t) =u(x, A t )+kw t 18

19 The market-view case a = Uφ, φ is a d dim F t adapted process The forward performance SPDE becomes du = 1 σσ + AU (λ + φ) 2 2 A 2 dt + Uφ dw U Define the processes Z and A by and The process U = U (x, t) dz = Zφ dw and Z 0 =1 A t = t 0 σ s σ + s (λ s + φ s ) 2 ds with u solving U (x, t) =u (x, A t ) Z t u t u xx = 1 2 u2 x is a solution 19

20 The benchmark case a (x, t) = xu (x, t) δ, The forward performance SPDE becomes δ is a d dim F t adapted process du (x, t) = 1 σ t σ t + (U x (x, t)(λ t δ t ) xu xx (x, t)) 2 dt xu x (x, t) δ t dw t 2 U xx (x, t) Define the processes Y and A by and Assume σσ + δ = δ dy t = Y t δ t (λ t dt + dw t ) with Y 0 =1 A t = t 0 σ s σ + s λ s δ s 2 ds. The process ( ) x U = U (x, t) =u Yt,A t with u as before is a forward performance. 20

21 A general case a (x, t) = xu x (x, t) δ + U (x, t) φ The forward performance SPDE becomes du (x, t) = 1 2 σ t σ + t (U x (x, t)((λ t + φ t ) δ t ) xu xx (x, t) δ t ) 2 U xx (x) +( xu x (x, t) δ t + U (x, t) φ t ) dw t Recall the benchmark and market view processes and dy t = Y t δ t (λ t dt + dw t ) with Y =1 dz t = Z t φ t dw t with Z =1 dt 21

22 Define the process A t = t 0 σ s σ + s (λ s + φ s ) δ s 2 ds The process ( ) x U = U (x, t) =u Yt,A t is a forward performance Z t MZ (2006, 2007) 22

23 The u-pde An important differential object is the fully non-linear pde u t u xx = 1 2 u2 x t>0, with u 0 (x) =U (x, 0). The local risk tolerance A quantity that enters in the explicit representation of the optimal portfolios r = u x u xx Modelling considerations 23

24 Three related pdes Fast diffusion equation for risk tolerance r t r2 r xx =0 r(x, 0) = r 0 (x) (FDE) Conductivity : r 2 The transport equation u t ru x =0 with u 0 such that r 0 = r (x, 0) = u 0 (x) u 0 (x) Porous medium equation for risk aversion γ = r 1 γ t = 1 2 F (γ) xx with F (γ) =γ 1 24

25 Difficulties Differential input equation: u t u xx = 1 2 u2 x Inverse problem and fully nonlinear Transport equation: u t ru x =0 Shocks, solutions past singularities Fast diffusion equation: r t r2 r xx =0 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 = 1 2 (1 γ ) xx Majority of results for (PME), γ t =(γ m ) xx, are for m>1, partialresultsfor 1 <m<0 25

26 An example of local risk tolerance (MZ (2006) and Z-Zhou (2007)) r(x, t; α, β) = αx 2 + βe αt α, β > 0 (Very) special cases r(x, t;0,β)= β u(x, t) = e x β + t2, x R r(x, t;1, 0) = x u(x, t) =log x t 2, x>0 r(x, t; α, 0) = α x u(x, t) = γ 1xγ e 2(1 γ) t,x 0, γ= α 1 γ α 26

27 Optimal allocations 27

28 Optimal portfolio vector The SPDE for the forward performance process du = 1 2 σσ + A (Uλ+ a) 2 A 2 U dt + a dw The optimal portfolio vector πt = π (t, Xt )= σ +A (Uλ+ a) A 2 U (X t,t) The optimal wealth process dxt = σσ +A (Uλ+ a) A 2 U (X t,t)(λdt + dw t ) 28

29 Optimal portfolios in the MZ example 29

30 The structure of optimal portfolios Stochastic input Market dx t = σ t π t (λ t dt + dw t ) Differential input Individual (Y t,z t ) wealth x λ t,σ t,δ t,φ t risk tolerance r(x, t) A t r t r2 r xx =0 7 U(x, t) =u ( x Yt,A t ) Zt 1 πt is a linear combination Y t of (benchmarked) optimal wealth and subordinated (benchmarked) risk tolerance 30

31 Optimal asset allocation Let X t be the optimal wealth, Y t the benchmark and A t the time-rescaling processes Define dx t = σ tπ t (λ tdt + dw t ) dy t = Y t δ t (λ t dt + dw t ) da t = σ t σ + t (λ t + φ t ) δ t 2 dt X t X t Y t and R t r( X t,a t ) Optimal (benchmarked) portfolios ˆπ t 1 Y t π t = m t X t + n t R t m t = σ + t δ t n t = σ + t (λ t + φ t δ t ) 31

32 Stochastic evolution of wealth-risk tolerance Explicit construction of optimal processes 32

33 A system of SDEs at the optimum X t = X t Y t and R t = r( X t,a t ) d X t = r( X t,a t )(σ t σ + t (λ t + φ t ) δ t ) ((λ t δ t ) dt + dw t ) d R t = r x ( X t,a t )d X t The optimal wealth and portfolios are explicitly constructed if the function r(x, t) is known 33

34 Solutions of the fast diffusion risk tolerance pde r t r2 r xx =0 Positive and increasing space-time harmonic functions Assume that h(x, t) is positive, increasing in x, and satisfies h t h xx =0 Then, it follows from Widder s theorem, that there exists a finite positive Borel measure such that h(x, t) = 0 e yx 1 2 y2t ν(dy) 34

35 Risk tolerance function Take a positive and increasing space time harmonic function h(x, t) Define the risk tolerance function r(x, t) by r(x, t) =h x (h 1 (x, t),t) Then r(x, t) solves the FDE r t r2 r xx =0, r(0,t)=0 35

36 The differential input function u Define the function Then u solves u(x, t) = x 0 exp ( h 1 (y, t)+ 1 2 t ) u t u xx = 1 2 u2 x dy Alternatively, use r(x, t) =h x (h 1 (x, t),t) and the transport equation u t ru x =0 36

37 Example Consider the case when the positive Borel measure is a Dirac delta, i.e., ν = δ γ, γ > 0 Then h(x, t) =e γx 1 2 γ2t, h 1 (x, t) = 1 γ ( log x + 1 ) 2 γ2 t, r(x, t) =λx, u(x, t) = γ xγ 1 γ γ 1 e 1 2 (γ 1)t 37

38 Globally defined solutions to the u-pde and the FDE Assume that for a finite positive Borel measure on R R e yx ( 1+ y + 1 y ) ν(dy) < Assume that the equation below has a solution b (t) = 1 2 R e yb(t) 1 2 y2 t ν(dy) R e yb(t) 1 2 y2t yν(dy), b(0) = b 0 38

39 Increasing space-time harmonic functions Define the function h(x, t) = R ( e yx 1 2 y2t + e yb(t) 1 2 y2 t ) 1 y v(dy) The above function satisfies h t h xx =0, h(b(t),t)=0 b(t) =h 1 (0,t) Risk tolerance function The solution to the fast diffusion risk tolerance pde is given by r(x, t) =h x (h 1 (x, t),t) 39

40 Example For positive constants a and b define h(x, t) = b a exp ( 1 2 a2 t ) sinh(ax) Observe that r(x, t) = a 2 x 2 + b 2 exp( a 2 t) The corresponding u(x, t) function can be calculated explicitly The above class covers the classical exponential, logarithmic and power cases Notice that r(x, t) is globally defined 40

41 Optimal wealth and risk tolerance processes Define the process Note that M t = t 0 σσ+ λ s dw s Optimal wealth process Risk tolerance process R x, t A t = M t X x, t = h(h 1 (x, 0) + A t + M t,a t ) = r(x x, t,a t )=h x (h 1 (x, 0) + A t + M t,a t ) 41

42 Construction Initial data u 0 (x), orr 0 (x), yields h(x, 0) Backward heat equation for h Solution h(x, t) Risk tolerance function r(x, t) =h x (h 1 (x, t),t) Market input M t = t 0 σ sσ + s λ s dw s 42

43 Construction Optimal wealth X x, t = h ( h 1 (x, 0) + M t + M t, M t ) Optimal risk tolerance r(xt x, (,t)=h x h 1 (Xt x,,t),t ) Optimal portfolio π t = k 1 t X x, t Distributional properties of optimal wealth + kt 2 r(xt x,,t) Specification of initial data h(x, 0)? Inference of initial data from the investor s wish list. 43