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 opportunities 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 Specify the investment universe, i.e. the dynamics of assets would be 2 traded, and investment constraints. 3 Solve for a self-financing strategy selection which maximizes the expected utility of the terminal wealth Analyze properties of the optimal solution 4 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 Can the same utility function be used for all time horizons ? 2 No- in fact the investor gets more value (in terms of the value function) 3 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 In the classical formulation the utility refers to the utility for the last 1 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 To be indifferent to investment horizon, it needs to maintain a 3 intertemporal consistency Only at the optimum the investor achieves on the average his 4 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 Need to be stable with respect of classical operation in the market as 5 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 d � σ i , j t dW j d ξ i t = ξ i t [ b i d ξ 0 t = ξ 0 t dt + t ] , t r t dt i = 1 Risk premium vector, η ( t ) with b ( t ) − r ( t ) 1 = σ t η ( t ) Self-financing strategy starting from x at time r t dt + π ∗ dX π t = r t X π , X π t σ t ( dW t + η t dt ) 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 ) = sup π ∈A E ( 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 π in A E P ( u ( t , X π t ) / F s ) ≤ u ( s , X π s ) , ∀ s ≤ t or 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 : E P ( u ( t , X π ∗ ) / F s ) = u ( s , X π ∗ s ) , ∀ s ≤ t 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 ) , E ( u ( T , X r , x ,π u ( r , x ) = sup ) / F r ) , ∀ T ≥ r T π ∈A ( r , x ) 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 C 2 - utility function and Z a positive semimartingale, with drift λ t and volatility γ t . Change of probability def Let u be the adapted process defined by u ( t , x ) = Z t v ( 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 � t 0 ( r s ds + η ∗ s dW s + 1 2 || η s || 2 ds ) . or the discounted measure H t = exp ( − 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 v x ( t , x ) 2 xv xx ( t , x ) || Proj A ( η t + γ t ) || 2 = 0 xv x ( t , x ) µ t + r t − The property holds true If v ( x ) = x 1 − α / 1 − α (Power utility) and 1 2 α ||� η t + γ t � A || 2 , then u is a forward utility. µ t / ( α − 1 ) + r t − If v ( x ) = exp − c x is a forward utility if r = 0 ,and µ t = 1 2 ||� η t + γ t � A || 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 α t and volatility δ t . Change of numéraire def Let u be u ( t , x ) = v ( x / Y t ) . 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 optimal portfolio. , η − δ ∈ ( K σ t ) ⊥ , δ ∈ ( K σ t ) We have r t = α t − < δ t , η t >, By Itô’s formula, the volatility of the forward utility is Γ( t , x ) = − x u x ( 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 L ξ w.r. ξ . Admissible portfolios are stable w. r. to the initial condition t , X r , x ,π ,π X r , x ,π π ∈ A ( t , X r , x ,π = X t , ) t + h t + h 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
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