Stochastic Partial Differential Equations and Portfolio Choice. Crete, May Thaleia Zariphopoulou

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1 Stochastic Partial Differential Equations and Portfolio Choice Crete, May 2011 Thaleia Zariphopoulou Oxford-Man Institute and Mathematical Institute University of Oxford and Mathematics and IROM, The University of Texas at Austin 1

2 Joint work with M. Musiela References Stochastic PDE and portfolio choice (2007 and 2009) To appear in Volume in honor of Eckhard Platen s 60 th birthday Portfolio choice under dynamic investment performance criteria (2008), Quantitative Finance Portfolio choice under space-time monotone performance criteria (2009), SIAM Journal on Financial Mathematics, inprint 2

3 Performance measurement of investment strategies 3

4 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 -predictable stochastic processes Postulate existence of an F t -predictable stochastic process λ t R d satisfying μ t r t 1=σ T t λ t No assumptions on market completeness 4

5 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 ) 5

6 Investment performance measurement 6

7 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) 7

8 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 8

9 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) 9

10 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 10

11 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? 11

12 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 12

13 The stochastic PDE of the forward performance process 13

14 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. 14

15 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 ) 15

16 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 16

17 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) 17

18 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 18

19 The volatility of the investment performance process This is the novel element in the new approach of forward investment performance measurement The volatility models how the current shape of the performance process is going to diffuse in the next trading period The volatility is up to the investor to choose, in contrast to the classical approach in which it is uniquely determined via the backward construction of the value function process When the volatility is not state-dependent, we are essentially in the zero volatility case The volatility process results in non-myopic portfolios The volatility s dependence on the risk premium is intriguing The process a may depend on t, x, U, its spatial derivatives etc. Specifying the appropriate class of volatility processes is very challenging but extremely didactic! 19

20 The zero volatility case: a(x, t) 0 20

21 Space-time monotone performance process The forward performance SPDE simplifies to The process du = 1 2 σσ + A (Uλ) A 2 U U (x, t) =u (x, A t ) with A t = 2 dt 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 (2009) u t u xx = 1 2 u2 x 21

22 Performance measurement time t 1,informationF t1 risk premium A t1 = t1 0 λ 2 ds u(x,t 1 ) Time Wealth A t1 + u(x, t 1 ) U(x, t 1 )=u(x, A t1 ) F t1 22

23 Performance measurement time t 2,informationF t2 risk premium A t2 = t2 0 λ 2 ds u(x,t 2 ) Time Wealth A t2 + u(x, t 2 ) U(x, t 2 )=u(x, A t2 ) F t2 23

24 Performance measurement time t 3,informationF t3 A t3 = risk premium t3 0 λ 2 ds u(x,t 3 ) Time Wealth A t3 + u(x, t 3 ) U(x, t 3 )=u(x, A t3 ) F t3 24

25 Forward performance measurement time t, informationf t u(x,t) market Time Wealth MI(t) + u(x, t) U(x, t) =u(x, A t ) F t 25

26 Properties of the performance process U (x, t) =u (x, A t ) U(x, t) is decreasing in time if λ 0 the deterministic risk preferences u (x, t) are compiled with the stochastic market input A t = t 0 λ 2 ds the evolution of preferences is deterministic if λ =0, U(x, t) =U(x, 0) if λ large, the investor is heavily penalized if he does not invest The investment performance criterion evolves together with the market 26

27 Optimal allocations 27

28 Optimal allocations Let X t be the optimal wealth, and A t the time-rescaling processes dx t = σ tπ t (λ tdt + dw t ) da t = λ t 2 dt Define R t r(x t,a t ) r(x, t) = u x(x, t) u xx (x, t) Optimal portfolios π t = σ+ t λ tr t The optimal portfolio is always myopic 28

29 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 tr t The optimal wealth and portfolios are explicitly constructed if the function r(x, t) is known. Should we model r(x, t) instead of u(x, t)? 29

30 Concave utility inputs and increasing harmonic functions 30

31 Concave utility inputs and increasing harmonic functions There is a one-to-one correspondence between strictly concave solutions u(x, t) to u t = 1 2 u 2 x u xx and strictly increasing solutions to h t h xx =0 via the transformation u(h(x, t),t)=e x+ 2 t 31

32 Concave utility inputs and increasing harmonic functions x h(x, t) u(h(x, t),t) The harmonic function h(x, t) is defined on R [0 + ) and represents the investor s wealth The range ofh(x, t) reflects the wealth state constraints (e.g., wealth bounded from below) The harmonic function h(x, t) is increasing in x 32

33 Concave utility inputs and increasing harmonic functions If h(x, t) harmonic then h x (x, t) is also harmonic (h x ) t (h xx) x =0 Because h x (x, t) is positive harmonic it can be represented via Widder s theorem as h x (x, t) = R exy 1 2 y2t ν(dy) The wealth function h(x, t) is then constructed from h x (x, t) Boundary and asymptotic behavior of h x (x, t) not obvious 33

34 Concave utility inputs and increasing harmonic functions The measure ν becomes the defining element Its support plays a key role in the form of the range of h(x, t) and, as a result, in the form of the domain and range of u(x, t) as well as in its asymptotic behavior (Inada conditions) It defines the class of initial conditions Can it be inferred from the investor s desired investment targets? 34

35 Examples: h has infinite range 35

36 Range of h(x, t) =(, + ) Assumption on ν: + eyx ν(dy) < +, x R ν({0}) > 0 ν({0}) =0,ν ( (, 0) ) ν ( (0, + ) ) > 0 ν({0}) =0,ν ( (, 0) ) =0,ν ( (0, + ) ) > 0 and ν(dy) y =+ ν({0}) =0,ν ( (0, + ) ) =0,ν ( (, 0) ) > 0 and 0 ν(dy) y = 36

37 Concave utility inputs and increasing harmonic functions Increasing harmonic function h : R [0, + ) R is represented as h (x, t) = R e yx 1 2 y2t 1 ν (dy) y The associated utility input u : R [0, + ) R is then given by the concave function u (x, t) = 1 2 t e h( 1) (x,s)+ s ( 2h x h ( 1) (x, s),s ) x ds e h( 1) (z,0) dz 37

38 Measure ν has compact support ν(dy) =δ 0,whereδ 0 is a Dirac measure at 0 Then, and h (x, t) = R e yx 1 2 y2t 1 δ y 0 = x u (x, t) = 1 2 t 0 e x+s 2ds + x 0 e z dz =1 e x+ t 2 38

39 Measure ν has compact support ν (dy) = b 2 (δ a + δ a ), a,b > 0 Then, If, a =1,then u (x, t) = 1 2 If a 1,then ( ln u(x, t) = ( a) 1+ 1 a a 1 ( x + e 1 a 2 t h (x, t) = b a e 1 2 a2t sinh (ax) x 2 + b 2 e t) et b 2x ( x x 2 + b 2 e t) t ) 2 β a e at +(1+ ( ax+ ) a )x ax 2 + βe at ( ax+ ax 2 + βe at ) 1+ 1 a Generalized power and logarithmic cases 39

40 Measure ν has infinite support ν(dy) = 1 2π e 1 2 y2 dy Then h(x, t) =F ( ) x t +1 F (x) = x 0 ez2 2 dz and u(x, t) =F ( F ( 1) (x) t +1 ) 40

41 Examples: h has semi-infinite range 41

42 Range of h(x, t) =(0, + ) ν ( (, 0) ) =0, ν({0}) =0, ν ( (0, + ) ) > 0 + ν(dy) 0 + y ν(dy) < + Then h(x, t) = + e yx y 2 y2 t ν(dy) 42

43 Range of h(x, t) =(0, + ) ν ( (0, 1] ) =0 and ν(dy) y 1 < + h(x, t) = + e yx y 2 y2 t ν(dy) u(x, t) = 1 2 t 0 e h( 1) (x,s)+ s 2 h x (h ( 1) (x, s),s) ds + x 0 e h( 1) (z,0) dz lim u(x, t) =0, lim x 0 u x(x, t) =+, x 0 lim x + u x(x, t) =0 43

44 Example (no mass in (0, 1]) ν(dy) =δ γ, γ > 1 h(x, t) = + e yx y 2 y2 t ν(dy) = 1 γ eγx 1 2 γ2 t u(x, t) = γ γ 1 γ xγ 1 γ γ 1 e γ 1 2 t lim u(x, t) =0, t 0 x 0 44

45 Range of h(x, t) =(0, + ) ν ( (0, 1] ) > 0 or ν ( (0, 1] ) =0 and ν(dy) y 1 =+ h(x, t) = + e yx y 2 y2 t ν(dy) u(x, t) = 1 2 t 0 e h( 1) (x,s)+ s 2 h x (h ( 1) (x, s),s) ds + x x 0 e h( 1) dz, x0 > 0 lim u(x, t) =, lim x 0 u x(x, t) =+, x 0 lim x + u x(x, t) =0 45

46 Examples with ν ( (0, 1] ) > 0 ν(dy) =δ γ γ =1 h(x, t) = + e yx y 2 y2 t ν(dy) =e x 1 2 t u(x, t) =ln x x 0 t 2 ν(dy) =δ γ γ (0, 1) u(x, t) = γ γ 1 γ xγ 1 γ 1 γ e 1 γ 2 t + γ γ 1 γ γ 1 1 γ x γ 0 46

47 Optimal processes and increasing harmonic functions 47

48 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 r2 r xx =0 r(x, t) = u x(x, t) u xx (x, t) 48

49 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 (x, t) =h x ( h ( 1) (x, t),t ) = R eyh( 1) (x,t) 1 2 y2t ν (dy) is a risk tolerance function solving the FDE 49

50 Optimal portfolio and optimal wealth Let h be an increasing solution of the backward heat equation h t h xx =0 and h ( 1) stands for its spatial inverse Let the market input processes A and M by defined by A t = t 0 λ s 2 ds and M t = t 0 λ s dw s Then the optimal wealth and optimal portfolio processes are given by and X,x t = h ( h ( 1) (x, 0) + A t + M t,a t ) πt ( = h x h ( 1) ( Xt,x ) ),A t,at σ + t λ t 50

51 Complete construction Utility inputs and harmonic functions u t = 1 u 2 x h t u xx 2 h xx =0 Harmonic functions and positive Borel measures h(x, t) ν(dy) Optimal wealth process Xt,x = h ( h ( 1) ) (x, 0) + A t + M t,a t M = π,x t Optimal portfolio process t ( = h x h ( 1) ( Xt,x ) ),A t,at σ + t λ t 0 λ s dw s, M t = A t 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? 51

52 An extended class of forward investment performance processes 52

53 Co-linear volatility a(x, t) =φ t U(x, t)+δ t xu x (x, t) φ t,δ t F t ; σ t σ + t δ t = δ t ( ) x U(x, t) =u Yt,A t Z t Benchmark/numeraire Y t dy t = Y t δ t (λ t dt + dw t ), Y 0 =1 Market view dz t = Z t φ t dw t, Z 0 =1 Time-rescaling A t = t 0 λ s + φ s δ s 2 ds 53

54 Inferring investor s preferences 54

55 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? Desired future expected wealth Investor s investment targets Desired distribution References Sharpe (2006) Sharpe-Golstein (2005) 55

56 Distributional properties of the optimal wealth process The case of deterministic market price of risk Using the explicit representation of X,x we can compute the distribution, density, quantile and moments of the optimal wealth process. P ( X,x t y ) = N h( 1) (y, A t ) h ( 1) (x, 0) A t At f X,x t (y) =n h( 1) (y, A t ) h ( 1) (x, 0) A t At 1 r (y, A t ) y p = h ( h ( 1) (x, 0) + A t + A t N ( 1) (p),a t ) 56

57 Properties of the expected optimal wealth process EX,x t = h(h ( 1) (x, 0) + A t, 0) x E(X,x t )= r(e(x,x t ), 0) r(x, 0) = r(h(h( 1) (x, 0) + A t, 0), 0) r(x, 0) E(r(Xt,x,A t )) = r(e(xt,x ), 0) 57

58 Target: The mapping x E ( Xt,x ) is linear, for all x>0. Then, there exists a positive constant γ>0 such that the investor s forward investment performance process is given by and by Moreover, U (x, t) = γ γ 1 xγ 1 γ e 1 2 (γ 1)A t, if γ 1 U t (x) =lnx 1 2 A t, if γ =1 E ( Xt,x ) = xe γa t Calibrating the investor s preferences consists of choosing a time horizon, T, and the level of the mean, mx (m >1).Then, the corresponding γ must satisfy xe γa T = mx and, thus, is given by γ = ln m A T The investor can calibrate his expected wealth only for a single time horizon. 58

59 Relaxing the linearity assumption The linearity of the mapping x E ( Xt,x ) is a very strong assumption. It only allows for calibration of a single parameter, namely, the slope, and only at a single time horizon. Therefore, if one intends to calibrate the investor s preferences to more refined information, then one needs to accept a more complicated dependence of E ( Xt,x ) on x. Target: Fix x 0 and consider calibration to E ( X,x ) 0 t, for t 0 The investor then chooses an increasing function m (t) (with m (t) > 1) to represent E ( X,x ) 0 t, E ( X,x ) 0 t = m (t), for t 0. What does it say about his preferences? Moreover, can he choose an arbitrary increasing function m (t)? 59

60 Can the agent s investment wish be granted? For simplicity, assume x 0 = 1 and that ν is a probability measure. h ( 1) (1, 0) = 0 and we deduce that Then, E ( Xt,1 ) = m(t) =h (At, 0) = 0 e ya t ν (dy) Clearly, the investor may only wish for a function m (t),t>0, which can be represented, for some probability measure ν in the form m (t) = 0 e ya t ν (dy), t 0 60

61 Summary Space-time monotone investment performance criteria (a(x, t) =0) Explicit construction of forward performance process Connection with space-time harmonic functions The measure in the harmonic function becomes the defining element of the entire construction Explicit construction of the optimal wealth and optimal portfolio processes Calibration of the measure to the market opportunities and the individual investment targets 61

62 Open problems Characterization of the appropriate class of volatility processes (El Karoui) Axiomatic construction of the forward investment process (Z.-Zitkovic, Zitkovic) Valuation with forward investment performance criteria (Musiela-Z., Musiela-Sokolova-Z., Leung-Sircar-Z.) Behavioral finance (Z.-Zhou) 62