Progressive Dynamic Utilities with given optimal portfolio El - - PowerPoint PPT Presentation

Progressive Dynamic Utilities with given optimal portfolio

El Karoui Nicole & M’RAD Mohamed

UnivParis VI / École Polytechnique,CMAP elkaroui@cmapx.polytechnique.fr with the financial support of the "Fondation du Risque" and the Fédération des banques Françaises

18 May 2009

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 1 / 32

Plan

1

Utility forward Framework and definition

2

Forward Stochastic Utilities Definition

3

Non linear Stochastic PDE Utility Volatility

4

Forward utilities with given optimal portfolio New approach by stochastic flows

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 2 / 32

Investment Banking and Utility Theory

Some remarks on utility functions and their dynamic properties from M.Musiela, T.Zariphopoulo, C.Rogers +alii (2005-2009) No clear idea how to specify the utility function Classical or recursive utility are defined in isolation to the investment

• pportunities given to an agent

Explicit solutions to optimal investment problems can only be derived under very restrictive model and utility assumptions - dependence on the Markovian assumption and HJB equations In non-Markovian framework, theory is concentrated on the problem of existence and uniqueness of an optimal solution, often via the dual representation of utility. Main Drawbacks Not easy to develop pratical intuition on asset allocation Creates potential intertemporal inconsistency

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 3 / 32

The classical formulation

Different steps

1

Choose a utility function,U(x) (concave et strictly increasing) for a fixed investment horizon T

2

Specify the investment universe, i.e. the dynamics of assets would be traded, and investment constraints.

3

Solve for a self-financing strategy selection which maximizes the expected utility of the terminal wealth

4

Analyze properties of the optimal solution

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 4 / 32

Shortcomings

Intertemporality

1

The investor may want to use intertemporal diversification, i.e., implement short, medium and long term strategies

2

Can the same utility function be used for all time horizons ?

3

No- in fact the investor gets more value (in terms of the value function) from a longer term investment.

4

At the optimum the investor should become indifferent to the investment

• horizon. .

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 5 / 32

Dynamic programming and Intertemporality I

1

In the classical formulation the utility refers to the utility for the last rebalancing period

2

The mathematical version is the Dynamic programming principle (in Markovian setup for simplicity) : Let V(t,x,U,T) be the maximal expected utility for a initial wealth x at time t, and a terminal utility function U(x, T), then V(t, x, U, T) = V(t, x, V(t + h, ., U, T), t + h) The value function V(t + h, ., U, T) is the implied utility for the maturity t + h

3

To be indifferent to investment horizon, it needs to maintain a intertemporal consistency

4

Only at the optimum the investor achieves on the average his performance objectives. Sub optimally he experiences decreasing future expected performance.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 6 / 32

Dynamic programming and Intertemporality II

5

Need to be stable with respect of classical operation in the market as change of numéraire.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 7 / 32

Forward Dynamic Utility

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 8 / 32

Utility forward Framework and definition

Investment Universe

Asset dynamics dξi

t = ξi t[bi tdt + d

• i=1

σi,j

t dW j t ],

dξ0

t = ξ0 t rtdt

Risk premium vector, η(t) with b(t) − r(t)1 = σtη(t) Self-financing strategy starting from x at time r dX π

t = rtX π t dt + π∗ t σt(dWt + ηtdt)

, X π

r = x

The set of admissible strategies is a vector space (cone) denoted by A.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 9 / 32

Utility forward Framework and definition

Classical optimization problem

Classical problem

Given a utility function U(T, x), maximize : V(r, x) = supπ∈AE(U(X π

T ))

(1) The choice of numéraire is not really discussed Backward problem since the solution is obtained by recursive procedure from the horizon. In the forward point of view, a given utility function is randomly diffused, but with the constrained to be at any time a utility function.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 10 / 32

Forward Stochastic Utilities Definition

Forward Utility

Definition (Forward Utility)

A forward dynamic utility process starting from the given utility U(r, x), is an adapted process u(t, x) s.t. i) Concavity assumption u(r, .) = U(r.), and for t ≥ r, x → u(t, x) is increasing concave function, ii) Consistency with the investment universe For any admissible strategy πinA EP(u(t, X π

t )/Fs) ≤ u(s, X π s ), ∀s ≤ t

• r equivalently (u(t, X π

t ); t ≥ r) is a supermartingale.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 11 / 32

Forward Stochastic Utilities Definition

Definition

iii) Existence of optimal There exists an optimal admissible self-financing strategy π∗, for which the utility of the optimal wealth is a martingale : EP(u(t, X π∗

t

)/Fs) = u(s, X π∗

s ), ∀s ≤ t

iv) In short for any admissible strategy, u(t, X π

t ) is a supermartingale, and a

martingale for the optimal strategy π∗ and then : u(r, x) is the value function of the optimization program with terminal random utility function u(T, x), u(r, x) = sup

π∈A(r,x)

E(u(T, X r,x,π

T

)/Fr), ∀T ≥ r where A(r, x) is the set of admissible strategies

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 12 / 32

Forward Stochastic Utilities Definition

Change of probability in standard utility function

Let v be C2- utility function and Z a positive semimartingale, with drift λt and volatility γt.

Change of probability

Let u be the adapted process defined by u(t, x)

def

= Ztv(x). u(t, x) is an adapted concave and increasing random field Consistency with Investment Universe The supermartingale property for u(t, X π

t ) holds true when Z is the discounted density of martingale

measure Ht = exp(− t

0(rsds + η∗ sdWs + 1 2||ηs||2ds). or the discounted

density of any equivalent martingale measure.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 13 / 32

Forward Stochastic Utilities Definition

The condition is not necessary, since by standard calculation, if v xvx(t, x)µt + rt − vx(t, x) 2xvxx(t, x)||ProjA(ηt + γt)||2 = 0 The property holds true If v(x) = x1−α/1 − α (Power utility) and µt/(α − 1) + rt −

1 2α||ηt + γtA||2, then u is a forward utility.

If v(x) = exp −c x is a forward utility if r = 0 ,and µt = 1

2||ηt + γtA||2

In the other cases, the martingale is the only solution....

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 14 / 32

Forward Stochastic Utilities Definition

Change of numéraire

Let Y a positive process with return αt and volatility δt.

Change of numéraire

Let u be u(t, x)

def

= v(x/Yt). u(t, x) is an adapted concave and increasing random field The supermartingale property holds true if Y is the inverse of discounted density of martingale measure, known as Market numéraire, or Growth

• ptimal portfolio.

We have rt = αt− < δt, ηt >, , η − δ ∈ (Kσt)⊥, δ ∈ (Kσt) By Itô’s formula, the volatility of the forward utility is Γ(t, x) = −x ux(t, x)δ

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 15 / 32

Non linear Stochastic PDE

Markovian case

We first consider the Markovian case where all parameters are functions of the time and of the state variables. The diffusion generator is the elliptic operator Lξ w.r. ξ. Admissible portfolios are stable w. r. to the initial condition X r,x,π

t+h

= X

t,X r,x,π

t

,π t+h

, π ∈ A(t, X r,x,π

t

) What is HJB equation for Markovian forward utility ?

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 16 / 32

Non linear Stochastic PDE

Example (HJB PDE)

Let u(t, ., ξ) be a Markov forward utility with initial condition u(r, x), concave w.r. to x. Then ut(t, x, ξ) + Lξu(t, x, ξ) + H(t, x, ξ, u′, u”, ∆σ

x,ξu)(t, x)) = 0

The Hamiltonian is defined for w < 0 by H(t, x, ξ, p, p′, w) = sup

π∈At

• < σ∗π, pη + p′ > +1/2 π∗σσ∗π

H(t, x, ξ, p, p′, w) = − 1

2w ||ProjKt(ηp + p′)||2 is the Hamiltonian taken at the

• ptimal

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 17 / 32

Non linear Stochastic PDE Utility Volatility

Optimal Portfolio and Volatility

Optimal portfolio ˜ π˜ σ(t, x, ξ) = − 1 uxx(t, x, ξ)ProjKt(ηu′ + ∆σ

x,ξu).

Volatility Parameters The utility volatility is Γ(t, x, ξt) is What is HJB equation for Markovian forward utility ? Γ(t, x, ξ) = (∇ξu(t, x, ξ))∗σ(t, ξ), Γx = ∂ ∂x Γu.

Theorem (Non Linear Dynamics, u(r, x) = U(r, x))

du(t, x, ξt) = ||ProjKt(ux(t, x)ηt + Γx(t, x, ξt))||2 2uxx(t, x, ξt) dt + Γ(t, x, ξt)dWt

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 18 / 32

Non linear Stochastic PDE Utility Volatility

Stochastic PDE

In the general case of forward utility, same ideas than for the stochastic PDE’s

• f interest rate or implied volatility.

1

Assume du(t, x) = β(t, x)dt + Γ(t, x)dWt, u(r, x) = u(x),

2

Use Itô-Ventzell-Kunita formula to the random field u(t, x) and optimal portfolio to express martingale property. We obtain very similar stochastic PDE

Theorem

The general case : Drift Constraint Assume that u is a forward utility function with decomposition du(t, x) = β(t, x)dt + Γ(t, x)dWt, u(r, x) = u(x), then β(t, x) = ux(t, x). ux(t, x) 2uxx(t, x)||ProjKt(ηt + Γx(t, x) ux(t, x))||2

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 19 / 32

Non linear Stochastic PDE Utility Volatility

Open Questions ?

What about the volatility of the utility ? Under which assumptions, how can be sure that solutions are concave and increasing, with Inada condition and asymptotic elasticity constraint.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 20 / 32

Non linear Stochastic PDE Utility Volatility

Decreasing forward Utility I

Zariphopoulo, C.Rogers and alii

Theorem

Assume the volatility ∀t, ∀xΓ(t, x) = 0. Then u is decreasing in time, du(t, x) = u2

x

2uxx(t, x)||ηt||2dt u is a forward utility iff there exist C and ν , a finite measure with support in [0, +∞) (ν(0) = 0), such that the Fenchel transform of u, v(t, x) verifies u(t, y) =

• 1

1 − r (1 − y1−re

r(1−r) 2

R t

0 |ηs|2dsν(dr) + C

This result is based on the result of Widder (1963) characterizing positive space-time harmonic function.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 21 / 32

Non linear Stochastic PDE Utility Volatility

Change of numéraire

Theorem (Stability by change of numéraire)

Let u(t,x) be a forward utility and Yt a numéraire. Then ˆ u(t, ˆ x) = u(t, x/Yt) is a progressive utility for the investment universe associated with the change of numéraire (Xt/Yt), with initial condition ˆ u(0, ˆ x) = u(0, yˆ x) With the market numéraire as numéraire, the market has no risk premium, and the ration

Γx ux has the same impact than a risk premium,

but depending of the level of the wealth x at time t.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 22 / 32

Forward utilities with given optimal portfolio

Forward Utility with given optimal portfolio

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 23 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Conditions of optimality : martingale market

Definition (Conditions (O∗) )

Let (X x

t ; t ≥ 0) and

• Y(t, x); t ≥ 0
• be two given stochastic fields.

Conditions (O∗) are : (O1) For all x > 0, (X x

t ; t ≥ 0) is a local martingale and an admissible wealth

process. (O2) For all x > 0, The process (Y(t, x); t ≥ 0) is a local martingale. (O3) The process

• X x

t Y(t, x); t ≥ 0

• is a martingale.

(O4) For any admissible portfolio π and any initial wealth x′, the process (X x′,π

t

; t ≥ 0) is a positive local martingale, (x, x′) > 0. (X x′,π

t

Y(t, x)); t ≥ 0) is a positive submartingale. (local martingale if K an s-e-v).

Proposition (Optimality Conditions)

Let u a forward dynamic utility, (X x

t ; t ≥ 0) the optimal portfolio and Y(t, x) the

process defined by Y(t, x) = u′(t, X x

t )

then, the random fields (X x, Y(., x) satisfy Conditions (O∗) .

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 24 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Existence of forward utilities with given optimal portfolio

Hypothesis

Assume the optimal portfolio X ∗

t (x)

to be continuously increasing with respect to the initial wealth (true in all examples, may be a consequence of no arbitrage opportunity) to be martingale Denote by X(t, z) its inverse flow defined as X(t, z) = (X ∗

t (.))−1(z).

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 25 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Theorem

Then for all given utility function U such that U′(X(t, z)) is locally integrable near z = 0, the stochastic process u defined by u(t, x) = x U′(X(t, z))dz, u(t, 0) = 0 (2) is a progressive utility in the martingale market constrained by K. The associated optimal wealth process is X ∗ and the optimal martingale measure is constant equal to 1.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 26 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Proof of theorem 8 : Step 1/2 I

Lemma

For all admissible wealth process (X π

t (s, x); s ≤ t), we have

E

• u(t, X π

t (s, x)/Fs

• ≤ E
• u(t, X ∗

t (s, x))/Fs

• Proof : By concavity of the process u(t, x), we have

u

• t, X π

t (s, x)

• − u
• t, X ∗

t (s, x)

• ≤
• X π

t (s, x) − X ∗ t (s, x)

• u′

t, X ∗

t (s, x)

• .

From the definition 2 of u, we get that u′ t, X ∗

t (s, x)

• = u′(s, x). The inequality

bellow becomes u

• t, X π

t (s, x)

• − u
• t, X ∗

t (s, x)

• ≤
• X π

t (s, x) − X ∗ t (s, x)

• u′(s, x).

(3) We have also that X ∗

t (s, x) is martingale by assumption and X π . is a

supermartingale because X π

t (s, x) is an admissible wealth in the martingale

• market. Those properties, together wilth (4), imply

E

• u
• t, X π

t (s, x)

• − u
• t, X ∗

t (s, x)

• /Fs
• ≤ E
• (X π

t (s, x) − X ∗ t (s, x)

• /Fs
• u′

t, X ∗

t (s, x

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 27 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Proof of theorem 8 : Step 1/2 I

Lemma

For all admissible wealth process (X π

t (s, x); s ≤ t), we have

E

• u(t, X π

t (s, x)/Fs

• ≤ E
• u(t, X ∗

t (s, x))/Fs

• Proof : By concavity of the process u(t, x), we have

u

• t, X π

t (s, x)

• − u
• t, X ∗

t (s, x)

• ≤
• X π

t (s, x) − X ∗ t (s, x)

• u′

t, X ∗

t (s, x)

• .

From the definition of u, we get that u′ t, X ∗

t (s, x)

• = u′(s, x). The inequality

below becomes u

• t, X π

t (s, x)

• − u
• t, X ∗

t (s, x)

• ≤
• X π

t (s, x) − X ∗ t (s, x)

• u′(s, x).

(4) Since X ∗

t (s, x) is a martingale by assumption and X π . a supermartingale as

positive local martingale, together wilth (4)then E(u(t, X π

t (s, x)) − u(t, X ∗ t (s, x))/Fs) ≤ E

• X π

t (s, x) − X ∗ t (s, x)/Fs

• u′

s, x)

• ≤ 0.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 28 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Construction of all progressive utilities with given

• ptimal portfolio

Let (X ∗, Y) a pair of process satisfying

Hypothesis (C1)

(A1) The process (X ∗

t (x); x ≥ 0, t ≥ 0) strictly increasing from 0 to +∞

(A2) while (Y(t, x); x ≥ 0, t ≥ 0) is strictly decreasing from +∞ to 0 such that Y(t, x) is locally integrable near x = 0. (A3) Y(0, x) is a decreasing function, denoted by U′(x).

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 29 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Construction (suite)

Hypothesis (C2, Technical)

H1 local) For all x, there exist an integrable positif adapted process, Ut(x) > 0 such that, if we denote by B(x, α) the ball of radius α > 0 centered at x, ∀y, y′ ∈ B(x, α), |X ∗

t (y) − X ∗ t (y′)| < |y − y′| Dt(x), t ≥ 0

(5) H2 global) Dt(x) is increasing with respect to x and DI

t(x) =

x Y(t, z)Dt(z)dz is integrable for all t ≥ 0.

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 30 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Proposition

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 31 / 32

Forward utilities with given optimal portfolio New approach by stochastic flows

Characterisation of all forward utilities with given

• ptimal portfolio

Theorem

Let (X ∗, Y) a pair of stricly positif process satisfying assymptions C2 and C3, such that Y(t, x) is locally integrable near x = 0 and Y(t, 0) = +∞, Y(t, +∞) = 0. Let X the inverse flow of X ∗ and define the concave increasing process u by u(t, x) = x Y(t, X(t, z))dz (7) If (X ∗, Y) satisfy the optimality conditions (O∗) then u(t, x) is a progressive dynamic utility on the martingale market with u0 as the initial function, X ∗ as the optimal wealth process. The optimal martingale measure is Y(t, x)/Y(0, x) and the convex dual is given by ˜ u(t, y) = +∞

y

X ∗

t ((Y)−1(t, z))dz.

(8)

El Karoui Nicole & M’RAD Mohamed (CMAP) Conf Risk Crisis, Septembre 2009 18 May 2009 32 / 32