Introduction Fractional Stochastic Volatility under Power Utility General Utility Optimal Portfolio under Fractional Stochastic Environment Ruimeng Hu Joint work with Jean-Pierre Fouque Department of Statistics and Applied Probability University of California, Santa Barbara 8th Western Conference in Mathematical Finance, March 24–25, 2017 Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 1 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Portfolio Optimization: Merton’s Problem An investor manages her portfolio by investing on a riskless asset B t and one 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 π d X π t = ( rX π X π t + π t ( µ − r )) d t + π t σ d W t , 0 = x Objective: E [ U ( X π T ) | X π M ( t, x ; λ ) := sup t = x ] π ∈A ( x,t ) where A ( x ) contains all admissible π and U ( x ) is a utility function on R + Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 2 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Portfolio Optimization: Merton’s Problem An investor manages her portfolio by investing on a riskless asset B t and one 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 π d X π t = ( rX π X π t + π t ( µ − r )) d t + π t σ d W t , 0 = x Objective: E [ U ( X π T ) | X π M ( t, x ; λ ) := sup t = x ] π ∈A ( x,t ) where A ( x ) contains all admissible π and U ( x ) is a utility function on R + Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 2 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility 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 Jaisson, Rosenbaum ’16 Omar, Masaaki, Rosenbaum ’16 We work with the following slowly varying fractional stochastic factor 1 � t e − δa ( t − s ) d W ( H ) Z δ,H := δ H , H ∈ (0 , 1) t s −∞ 1 Garnier Solna ’15: for linear problem of option pricing Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 3 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility 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 Jaisson, Rosenbaum ’16 Omar, Masaaki, Rosenbaum ’16 We work with the following slowly varying fractional stochastic factor 1 � t e − δa ( t − s ) d W ( H ) Z δ,H := δ H , H ∈ (0 , 1) t s −∞ 1 Garnier Solna ’15: for linear problem of option pricing Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 3 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Fractional BM and Fractional OU A fractional Brownian motion W ( H ) , H ∈ (0 , 1) t a continuous Gaussian process zero mean � � � | t | 2 H + | s | 2 H − | t − s | 2 H � = σ 2 W ( H ) W ( H ) E H s t 2 H < 1 / 2 : short-range correlation; H > 1 / 2 : long-range correlation Consider the Langevin equation driven by fractional Brownian motion t d t + d W ( H ) d Z H = − aZ H t t � t � t −∞ e − a ( t − s ) d W ( H ) stationary solution Z H −∞ K ( t − s ) d W Z = = s t s � W, W Z � correlated with risky asset d t = ρ d t Gaussian process with zero mean and constant variance Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 4 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Fractional BM and Fractional OU A fractional Brownian motion W ( H ) , H ∈ (0 , 1) t a continuous Gaussian process zero mean � � � | t | 2 H + | s | 2 H − | t − s | 2 H � = σ 2 W ( H ) W ( H ) E H s t 2 H < 1 / 2 : short-range correlation; H > 1 / 2 : long-range correlation Consider the Langevin equation driven by fractional Brownian motion t d t + d W ( H ) d Z H = − aZ H t t � t � t −∞ e − a ( t − s ) d W ( H ) stationary solution Z H −∞ K ( t − s ) d W Z = = s t s � W, W Z � correlated with risky asset d t = ρ d t Gaussian process with zero mean and constant variance Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 4 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Fractional BM and Fractional OU A fractional Brownian motion W ( H ) , H ∈ (0 , 1) t a continuous Gaussian process zero mean � � � | t | 2 H + | s | 2 H − | t − s | 2 H � = σ 2 W ( H ) W ( H ) E H s t 2 H < 1 / 2 : short-range correlation; H > 1 / 2 : long-range correlation Consider the Langevin equation driven by fractional Brownian motion t d t + d W ( H ) d Z H = − aZ H t t � t � t −∞ e − a ( t − s ) d W ( H ) stationary solution Z H −∞ K ( t − s ) d W Z = = s t s � W, W Z � correlated with risky asset d t = ρ d t Gaussian process with zero mean and constant variance Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 4 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Our Study Gives.... Under the slowly varying fSV model and power utility � � � µ ( Z δ,H ) d t + σ ( Z δ,H � W, W Z � d S t = S t ) d W t , t t d t = ρ d t. � t Z δ,H −∞ K δ ( t − s ) d W Z = s , t The value process V δ t E [ U ( X π t := sup π ∈A δ T ) | F t ] The corresponding optimal strategy π ∗ First order approximations to V δ t and π ∗ A practical strategy to generate this approximated value process Z δ,H is not Markovian nor a semi-martingale ⇒ HJB PDE is not available t Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 5 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility 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 . Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 6 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Proposition: Martingale Distortion Transformation 2 The value process V t is given by t λ 2 ( Y s ) d s � � � �� q V t = X 1 − γ � T 1 − γ � � t e � G t E 2 qγ 1 − γ � t 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 e � G t 2 qγ 2 Tehranchi ’04: different utility function, proof and assumptions Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 7 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility Remarks only works for one factor model 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 Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 8 / 19
Introduction Fractional Stochastic Volatility under Power Utility General Utility 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 . ⇒ 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 Ruimeng Hu (UCSB) Optimal Portfolio under Fractional Environment March 24, 2017 9 / 19
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