Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Optimal Portfolio under Fractional Stochastic Environment Jean-Pierre Fouque Joint work with Ruimeng Hu Department of Statistics and Applied Probability University of California, Santa Barbara Jim Gatheral’s 60th Birthday Conference NYU October 13-15, 2017 Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 1 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Portfolio Optimization: Merton’s Problem An investor manages her portfolio by investing in a riskless asset B t and in a risky asset S t (single asset for simplicity) � d B t = rB t d t d S t = µS t d t + σS t d W t π t – amount of wealth invested in the risky asset at time t X π t – the wealth process associated to the strategy π d S t + X π t = π t t − π t d X π d B t (self-financing) S t B t =( rX π t + π t ( µ − r )) d t + π t σ d W t Objective: Sharpe ratio: λ = µ E [ U ( X π T ) | X π M ( t, x ; λ ) := sup t = x ] , σ π ∈A ( x,t ) where A ( x ) contains all admissible π and U ( x ) is a utility function on R + Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 2 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Portfolio Optimization: Merton’s Problem An investor manages her portfolio by investing in a riskless asset B t and in a risky asset S t (single asset for simplicity) � d B t = rB t d t d S t = µS t d t + σS t d W t π t – amount of wealth invested in the risky asset at time t X π t – the wealth process associated to the strategy π d S t + X π t = π t t − π t d X π d B t (self-financing) S t B t =( rX π t + π t ( µ − r )) d t + π t σ d W t Objective: Sharpe ratio: λ = µ E [ U ( X π T ) | X π M ( t, x ; λ ) := sup t = x ] , σ π ∈A ( x,t ) where A ( x ) contains all admissible π and U ( x ) is a utility function on R + Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 2 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Stochastic Volatility In Merton’s work, µ and σ are constant, complete market Empirical studies reveals that σ exhibits “random” variation Implied volatility skew or smile Stochastic volatility model: µ ( Y t ) , σ ( Y t ) → incomplete market Rough Fractional Stochastic volatility: Gatheral, Jaisson and Rosenbaum ’14 ( “volatility is rough” ) Jaisson, Rosenbaum ’16 ( “from Hawkes processes to fractional diffusions” ) Omar, Masaaki and Rosenbaum ’16 ( “leveraged rough volatility” ) We study the Merton problem under slowly varying fractional stochastic environment: Nonlinear + Non-Markovian → HJB PDE not avaialbe Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 3 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Stochastic Volatility In Merton’s work, µ and σ are constant, complete market Empirical studies reveals that σ exhibits “random” variation Implied volatility skew or smile Stochastic volatility model: µ ( Y t ) , σ ( Y t ) → incomplete market Rough Fractional Stochastic volatility: Gatheral, Jaisson and Rosenbaum ’14 ( “volatility is rough” ) Jaisson, Rosenbaum ’16 ( “from Hawkes processes to fractional diffusions” ) Omar, Masaaki and Rosenbaum ’16 ( “leveraged rough volatility” ) We study the Merton problem under slowly varying fractional stochastic environment: Nonlinear + Non-Markovian → HJB PDE not avaialbe Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 3 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Related Literature Option Pricing + Markovian modeling: Fouque, Papanicolaou, Sircar and Solna ’11 (CUP) Portfolio Optimization + Markovian modeling: Fouque, Sircar and Zariphopoulou ’13 (MF) Fouque and Hu ’16 (SICON) Option Pricing + Non-Markovian modeling: Garnier and Solna ’15 (SIFIN), ’16 (MF) Portfolio Optimization + Non-Markovian modeling: Fouque and Hu arXiv:1703.06969 (slow factor) Fouque and Hu arXiv:1706.03139 (fast factor) Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 4 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV A General Non-Markovian Model Dynamics of the risky asset S t � d S t = S t [ µ ( Y t ) d t + σ ( Y t ) d W t ] , �� � W Y � Y t : a general stochastic process, G t := σ -adapted, 0 ≤ u ≤ t � W, W Y � with d t = ρ d t . Dynamics of the wealth process X t (assume r = 0 for simplicity): d X π t = π t µ ( Y t ) d t + π t σ ( Y t ) d W t Define the value process V t by E [ U ( X π V t := sup T ) | F t ] π ∈A t where U ( x ) is of power type U ( x ) = x 1 − γ 1 − γ , γ > 0 . Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 5 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Proposition: Martingale Distortion Transformation 1 The value process V t is given by t λ 2 ( Y s ) d s � � � �� q V t = X 1 − γ � T λ ( y ) = µ ( y ) 1 − γ � t � E e � G t , 2 qγ 1 − γ σ ( y ) � t where under � P , � W Y := W Y t + 0 a s d s is a BM. t The optimal strategy π ∗ is � λ ( Y t ) � ρqξ t π ∗ t = γσ ( Y t ) + X t γσ ( Y t ) where ξ t is given by the martingale representation d M t = M t ξ t d � W Y t and M t is � 0 λ 2 ( Y s ) d s � � � T � 1 − γ M t = � e � G t E 2 qγ 1 Tehranchi ’04: different utility function, proof and assumptions Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 6 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Remarks only works for one factor models assumptions: integrability conditions of ξ t , X π t and π t γ = 1 → case of log utility, can be treated separately degenerate case λ ( y ) = λ 0 , M t is a constant martingale, ξ t = 0 V t = X 1 − γ λ 0 1 − γ 2 γ λ 2 0 ( T − t ) , t π ∗ 1 − γ e t = γσ ( Y t ) X t . uncorrelated case ρ = 0 , the problem is “linear” since q = 1 t λ 2 ( Y s ) d s � � � V t = X 1 − γ � T t = λ ( Y t ) 1 − γ � t π ∗ 1 − γ E e � G t , γσ ( Y t ) X t . 2 γ Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 7 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Sketch of Proof (Verification) V t is a supermartingale for any admissible control π V t is a true martingale following π ∗ π ∗ is admissible Define α t = π t /X t , then d V t = V t D t ( α t ) d t + d Martingale with the drift factor D t ( α t ) t σ 2 − λ 2 D t ( α t ) := α t µ − γ 1 − γ a t ξ t + q ( q − 1) q 2 α 2 2(1 − γ ) ξ 2 2 γ + t + ρqα t σξ t maximize D t ⇒ α ∗ t and D t ( α ∗ t ) = 0 with the right choice of a t and q : � 1 − γ � γ a t = − ρ λ ( Y t ) , q = γ + (1 − γ ) ρ 2 . γ Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 8 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Sketch of Proof (Verification) V t is a supermartingale for any admissible control π V t is a true martingale following π ∗ π ∗ is admissible Define α t = π t /X t , then d V t = V t D t ( α t ) d t + d Martingale with the drift factor D t ( α t ) t σ 2 − λ 2 D t ( α t ) := α t µ − γ 1 − γ a t ξ t + q ( q − 1) q 2 α 2 2(1 − γ ) ξ 2 2 γ + t + ρqα t σξ t maximize D t ⇒ α ∗ t and D t ( α ∗ t ) = 0 with the right choice of a t and q : � 1 − γ � γ a t = − ρ λ ( Y t ) , q = γ + (1 − γ ) ρ 2 . γ Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 8 / 25
Introduction Martingale Distortion Transformation Merton Problem with Slowly Varying fSV General Utility Fast fSV Sketch of Proof (Verification) V t is a supermartingale for any admissible control π V t is a true martingale following π ∗ π ∗ is admissible Define α t = π t /X t , then d V t = V t D t ( α t ) d t + d Martingale with the drift factor D t ( α t ) t σ 2 − λ 2 D t ( α t ) := α t µ − γ 1 − γ a t ξ t + q ( q − 1) q 2 α 2 2(1 − γ ) ξ 2 2 γ + t + ρqα t σξ t maximize D t ⇒ α ∗ t and D t ( α ∗ t ) = 0 with the right choice of a t and q : � 1 − γ � γ a t = − ρ λ ( Y t ) , q = γ + (1 − γ ) ρ 2 . γ Jean-Pierre Fouque (UCSB) Optimal Portfolio under Fractional Environment October 14, 2017 8 / 25
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