BACHELIER FINANCE SOCIETY 4 th World Congress Tokyo, 26 Investments and forward utilities Thaleia Zariphopoulou The University of Texas at Austin 1
Topics Utility-based measurement of performance Utilities and market dynamics Optimal allocations and their stochastic evolution Efficient frontier 2
References Joint work with M. Musiela (BNP Paribas, London) Investments and forward utilities Preprint 26 Backward and forward dynamic utilities and their associated pricing systems: Case study of the binomial model Indifference Pricing, PUP (23, 25) Investment and valuation under backward and forward dynamic utilities in a stochastic factor model to appear in Dilip Madan s Festschrift (26) 3
Utility-based measurement of performance 4
Dynamic utility measurement time t 1,informationF t1 asset returns 11 constraints 19.8 market view away from equilibrium benchmark calendar time numeraire subordination 19.6 19.4 19.2 19 1 u(x,t 1 ).8.6.4 Time.2.2.4.6 Wealth.8 1 MI(t 1 ) + u(x, t 1 ) U(x, t 1 ; MI) F t1 π(x, t 1 ; MI) F t1
Dynamic utility measurement time t 2,informationF t2 asset returns constraints market view away from equilibrium benchmark numeraire 11 15 1 95 u(x,t 2 ) calendar time subordination 9 1.8.6.4 Time.2.2.4.6 Wealth.8 1 MI(t 2 ) + u(x, t 2 ) U(x, t 2 ; MI) F t2 π(x, t 2 ; MI) F t2
Dynamic utility measurement time t 3,informationF t3 asset returns 11 constraints market view away from equilibrium benchmark calendar time numeraire subordination 15 1 95 9 85 8 75 u(x,t 3 ) 1.8.6.4.2 Time.2.4.6 Wealth.8 1 MI(t 3 ) + u(x, t 3 ) U(x, t 3 ; MI) F t3 π(x, t 3 ; MI) F t3
Forward 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
Forward 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
Forward 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
Forward 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
Utility traits u(x, t) : x wealth and t time Monotonicty u x (x, t) > Risk aversion u xx (x, t) < Impatience u t (x, t) < Fisher (1913, 1918), Koopmans (1951), Koopmans-Diamond-Williamson (1964)... 12
Modelling and management of uncertainty 13
Investment universe Riskless and risky assets (Ω, F, P) ; W =(W 1,...,W d ) standard Brownian Motion Prices 1 i k ds i t = S i t(µ i tdt + σ i t dw t ), S i > db t = r t B t dt, B =1 µ t,r t R, σ i t Rd bounded and F t -measurable stochastic processes Postulate existence of a F t -measurable stochastic process λ t R d satisfying µ t r t 1=σ T t λ t 14
Investment universe Self-financing investment strategies π t,πi t, i =1,...,k Present value of this allocation X t = k πt i i= dx t = k i= π i t(µ i t r t ) dt + k i= π i tσ i t dw t = σ t π t (λ t dt + dw t ) π t =(π 1 t,...,πk t ), µ t r t 1=σ T t λ t 15
Forward dynamic utilities 16
Fundamental characteristics of a forward dynamic utility structure 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 17
Forward dynamic utility U(x, t) is an F t -adapted process As a function of x, U is increasing and concave Utility datum U(x, ) = u (x) assigned at forward normalization point t = 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 18
Construction of forward dynamic utilities 19
Forward dynamic utilities Creating the martingale that yields the optimal utility volume No Markovian assumptions Stochastic optimization problem inverse in time Stochastic input Market Key idea Variational input Individual Maximal utility Optimal allocation 2
Forward dynamic utilities Stochastic input : (Y t,z t,a t ) Variational input : u(x, t) Time change A t Benchmark Y t utility surface u(x, t) Market view Z t U(x, t) =u( x Y t,a t )Z t 21
Market input processes 22
Market input processes (Y t,z t,a t ) These F t -mble processes do not depend on the investor s variational utility They reflect and represent, respectively Y t : benchmark numeraire Z t : market view away from market equilibrium feasibility and trading constraints A t : subordination 23
Market input processes Market environment 1 i k dst i = St i ( µ i t dt + σt i ) dw t db t = r t B t dt µ t r t 1=σt T λ t, µ t,σ t,r t F t σ + t : Moore-Penrose matrix inverse σ + t = lim ε (σ t σ t + εi) 1 σ t, σ t : conjugate transpose of σ t Properties : σ t σ + t σ t = σ t, σ + t σ tσ + t = σ + t (σ t σ + t )T = σ t σ + t, (σ + t σ t) T = σ + t σ t 24
Market input processes Asset price coefficients λ t, σ t F t Benchmark and/or numeraire A replicable process Y t satisfying dy t = Y t δ t (λ t dt + dw t ) Y =1 δ t F t, σ t σ + t δ t = δ t 25
Market input processes Market views, feasibility and trading constraints An exponential martingale Z t satisfying dz t = Z t φ t dw t Z =1, φ t F t Subordination A non-decreasing process A t solving da t = δ t σ t σ + t (λ t + φ t ) 2 dt A = 26
Variational input utility surfaces 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) 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 + 1 2 r(x, t)u x = u(x, ) = u (x) Characteristic curves: dx(t) dt = 1 2 r(x(t),t) 29
Construction of utility surface u(x, t) using characteristics dx(t) dt = 1 2 r(x(t),t) Utility datum u (x) 3
Construction of characteristics dx(t) dt = 1 2 r(x(t),t) Utility datum u(x, ) Characteristic curves 31
Propagation of utility datum along characteristics 32
Propagation of utility datum along characteristics 33
Utility surface u(x, t) 34
Two related pdes Fast diffusion equation for risk tolerance r t + 1 2 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 35
Difficulties Utility equation: u t u xx = 1 2 u2 x Inverse problem and fully nonlinear Utility transport equation: u t + 1 2 r(x, t)u x = Shocks, solutions past singularities Fast diffusion equation: r t + 1 2 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< 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 r(x, t;1, ) = x u(x, t) =log x t r(x, t; α, ) = α x u(x, t) = γ 1xγ e 2(1 γ) t, γ = α 1 γ α 37
Risk tolerance r(x, t) =.5x 2 +15.5e.5t 4.2 4.1 4 3.9 3.8 1.8.6 Time.4.2 1 5 Wealth 5 1 38
Utility surface u(x, t) generated by risk tolerance r(x, t) =.5x 2 +15.5e.5t 2 4 6 8 1 6 4 Time 2.5 1 1.5 Wealth 2 2.5 Characteristics: dx(t) dt = 1 2.5x(t) 2 +15.5e.5t 39
Risk tolerance r(x, t) = 1x 2 + e 1t 2.2 2 1.8 1.6 1.4 1.2 1.8 1.8.6 Time.4.2 1 5 Wealth 5 1 4
Utility surface u(x, t) generated by risk tolerance r(x, t) = 1x 2 + e 1t.26.24.22.2.18.16.14.12 1.5 1.5 5 1 15 Time Wealth Characteristics: dx(t) dt = 1 2 1x(t) 2 + e 1t 41
Risk tolerance r(x, t) = x 2 +1=1 2 1.5 1.5 1.8.6 Time.4.2 1 5 Wealth 5 1 42
Utility surface u(x, t) = e x+t generated by risk tolerance r(x, t) =1 2 4 6 8 1 6 4 Time 2.5 1 1.5 Wealth 2 2.5 Characteristics: dx(t) dt = 1 2 43
Risk tolerance r(x, t) = x 2 +e t = x 6 5 4 3 2 1 1.8.6 Time.4.2 1 5 Wealth 5 1 44
Utility surface u(x, t) =logx t, x> generated by risk tolerance r(x) =x 4 2 2 4 6 6 4 Time 2 5 1 Wealth 15 2 Characteristics: dx(t) dt = 1 2 x(t) 45
Risk tolerance r(x, t) = 4x 2 +e 4t =2 x 6 5 4 3 2 1 1.8.6 Time.4.2 1 5 Wealth 5 1 46
Utility surface u(x, t) =2 xe t 2, x> generated by risk tolerance r(x, t) =2x.12.11.1.9.8 1.5 1 Time.5 5 1 Wealth 15 2 Characteristics: dx(t) dt = x(t) 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) = 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 +t m(x; α) =x, n(t; β) =1 u(x, t) =log x 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 81
Risk tolerance r(x, t) = ϕ(φ 1 (x;.5) t +5.2.15.1.5 1.5 1 Time.5 5 1 Wealth 15 2 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 +5 2.5 2 1.5 1.5 1.5 1 Time.5 5 1 Wealth 15 2 Characteristics: dx(t) dt = ϕ(φ 1 (x(t);.5)) t +5 5
Utility function u(x, t ) (fixed time) t =2 5 4.8 4.6 4.4 4.2 Utility 4 3.8 3.6 3.4 3.2 2 4 6 8 1 12 14 16 18 2 Wealth 51
Utility function u(x,t) (fixed wealth level) x =3.5 3.8 3.6 3.4 3.2 Utility 3 2.8 2.6 2.4 2.2.5 1 1.5 2 2.5 3 Time 52
Optimal utility volume Optimal asset allocation 53
Optimal utility volume Stochastic market input λ t,σ t benchmark, views subordination Variational input x, r (x) = u (x) u (x) fast diffusion eqn transport eqn (Y t,z t,a t ) u(x, t) 7 U(x, t) =u( x A t,y t )Z t Model independent construction! 54
What is the optimal allocation? Optimal portfolio processes π t =(π t,π 1 t,...,π k t ) can be directly and explicitly characterized along with the construction of the forward utility! 55
Stochastic input Market The structure of optimal portfolios dx t = σ t π t (λ t dt + dw t ) Variational input Individual (Y t,z t,a t ) wealth x λ t,σ t,δ t,φ t risk tolerance r(x, t) 7 1 πt is a linear combination Y t of (benchmarked) optimal wealth and subordinated (benchmarked) risk tolerance 56
Optimal asset allocation Let X t be the optimal wealth, Y t the benchmark and A t the subordination processes 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 Define r t the subordinated (benchmarked) risk tolerance r t = r ( X t Y t,a t ) Optimal (benchmarked) portfolios 1 Y t π t = σ + t ( (λ t + φ t )r t + δ t ( X t Y t r t )) 57
Wealth-risk tolerance stochastic evolution 58
A system of SDEs at optimum utility volume X t = X t Y t and r t = r( X t,a t ) d X t = r 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 Separability of wealth dynamics in terms of risk tolerance and market input Sensitivity of risk tolerance in terms of its spatial gradient and changes in optimal wealth Utility functional has essentially vanished Universal representation, no Markovian assumptions 59
An efficient frontier Optimal wealth-risk tolerance ( X t, r t ) system of SDEs in original market configuration d X t = r t (σ tσ + 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 change of measure historical benchmarked change of time Levy s theorem 6
An efficient frontier Optimal wealth-risk tolerance (x 1 t,x2 t ) system of SDEs in canonical market configuration x 1 t = ( X t Y t ) A ( 1) t ( ) x 2 X t = r t,a t Y t A ( 1) t M t = A t w t = M A ( 1) dx 1 t = x2 t dw t dx 2 t = r x(x 1 t,t)x2 t dw t x 1 = x y, x2 = r x( x y, ) 61
Analytic solution of the efficient frontier SDE system dx 1 t = x2 t dw t dx 2 t = r x(x 1 t,t)x2 t dw t Define the budget capacity function h(x, t) via x = h(x,t) x du h(x,t) r(u, t) = γ(u, t)du x x : related to symmetry properties of risk tolerance, reflection point of its spatial derivative and risk aversion front 62
Analytic solutions The budget capacity function h solves the (inverse) heat equation h t + 1 2 h xx 1 2 r x(x,t)h x = h(x, ) = h (x), x = h (x) x du r(u, ) Solution of the efficient frontier SDE system x 1 t = h(z t,t) x 2 t = h z(z t,t) z t = h 1 (x) t 1 2 r x(x,s)ds + w t Using equivalent measure transformations and time change we recover the original pair of optimal (benchmarked) wealth and (benchmark) risk tolerance 63
Utility-based performance measurement Market Benchmark, views, constraints Investor Wealth, risk tolerance Market input processes Subordination Fast diffusion eqn Transport eqn Forward evolution > Y t,z t,a t < x, r(x, t), u(x, t) Optimal utility volume and optimal portfolios measure change time change Efficient frontier SDE system Heat eqn Fast diffusion eqn Universal analytic solutions
T h a n k y o u!