Backward and Forward Preferences Undergraduate scholars: Gongyi - PowerPoint PPT Presentation

Backward and Forward Preferences Undergraduate scholars: Gongyi Chen, Zihe Wang Graduate mentor: Bhanu Sehgal Faculty mentor: Alfred Chong University of Illinois at Urbana-Champaign Illinois Risk Lab Illinois Geometry Lab Final Presentation December 13th, 2018

Introduction Consider an investor who has a portfolio consisting of two assets and he wants to optimize his wealth. Select a time period [ 0 , T ] , adopt the binomial model, and pre-specify the time T utility function of the investor at time 0. After the exogenous triplet is pre-committed at time 0, we then solve for optimal investment strategies and derive the value functions/preferences before time T . We will then use hypothetical value for the model and S&P 500 data to simulate the value functions.

Two-period Binomial Model Let (Ω , F , P ) be the probability space. � � Let F = F t 2 , T be the filtration generated by t = 0 , T ξ ∗ = ξ ∗ � � 2 , T . t t = 0 , T The two random variable ξ ∗ 2 and ξ ∗ T are given by T  ξ u ∗ 2 = P ( ξ ∗ 2 = ξ u ∗ p u 2 |F 0 ) 2 , T 0 , T T T  ξ ∗ 2 = T ξ d ∗ 2 = P ( ξ ∗ 2 = ξ d ∗ p d 2 |F 0 ) 2 ,  T 0 , T T T  ξ u ∗ 2 , T = P ( ξ ∗ T = ξ u ∗ p u T |F T 2 ) T , T  ξ ∗ T = ξ d ∗ 2 , T = P ( ξ ∗ T = ξ d ∗ p d T |F T 2 ) T ,  T

Two-period Binomial Model Suppose there are two types of assets, a risk-free asset that offers a return of r and a risky asset. Denote the price of the risky asset to be S ∗ = S ∗ � � 2 , T . t t = 0 , T 2 < e r T The arbitrage-free conditions are 0 < ξ d ∗ 2 < ξ u ∗ 2 and T T T < e r T 0 < ξ d ∗ 2 < ξ u ∗ T .

Two-period Binomial Model The following relationships for the price hold in the two period binomial model.   S ∗ 0 ξ u ∗ S ∗ 2 ξ u ∗ 2 = p u 2 , T = p u p 0 , T p T 2 , T , T 0 , T T T   2 , T S ∗ S ∗ 2 = T = 2 T S ∗ 0 ξ d ∗ S ∗ 2 ξ d ∗ 2 = p d 2 , T = p d p 0 , T p T 2 , T ,  T 0 , T  T T 2 , T 2 where S ∗ 0 ∈ F 0 , S ∗ 2 and S ∗ 2 ∈ F T T ∈ F T . T

Two-period Binomial Model t = T t = 0 t = T 2 0 ξ u ∗ ξ u ∗ S ∗ T T 2 p u 0 ξ u ∗ S ∗ T T 2 , T 2 0 ξ u ∗ ξ d ∗ S ∗ T T p d 2 T p u 2 , T S ∗ 0 , T 0 2 0 ξ d ∗ ξ ∗ u S ∗ T T 2 p d 0 , T 0 ξ d ∗ p u S ∗ 2 T T 2 , T 2 p d S 0 ξ d ∗ ξ d ∗ T 2 , T T T 2

Two-period Binomial Model Consider an investor who holds α shares of risky asset S , and risk-free asset B , with initial wealth x . X ∗ α ; x � � Let 2 , T be a wealth process of the investor t = 0 , T t employing control α with initial wealth x at t = 0. Therefore, X ∗ α ; x = x , 0 e − r T 2 e − r T 2 X ∗ α ; x = x + α 0 ( S ∗ 2 − S ∗ 0 ) , T T 2 = e − r T T e − rT − S ∗ 2 e − r T e − rT X ∗ α ; x 2 X ∗ α ; x 2 ( S ∗ 2 ) . + α T T T T 2

Two-period Binomial Model For simplicity, we’ll use the discount price of the risky e − rt S ∗ � � � � asset. Denote it by S = S t 2 , T = 2 , T . t = 0 , T t t = 0 , T Let the up and down factor for discounted price to be t e − r T ξ ∗ � � � 2 � ξ t 2 , T = 2 , T . t = T t = T The arbitrage free conditions become 0 < ξ d 2 < 1 < ξ u 2 and T T 0 < ξ d T < 1 < ξ u T .

Two-period Binomial Model X α ; x � � Define a discounted wealth process to be 2 , T , t = 0 , T t changing S ∗ to S . X α ; x = x , 0 X α ; x = x + α 0 ( S T 2 − S 0 ) , T 2 X α ; x = X α ; x + α T 2 ( S T − S T 2 ) . T T 2

Two-period Binomial Model X α ; x , t � � Denote another wealth process s = t ... T be a wealth s process of the investor employing control α with the initial wealth x at time t . Therefore, α ; x , T X 2 = x + α T 2 ( S T − S T 2 ) , T α ; X α ; x , T T 2 X α ; x = X α ; x + α T 2 ( S T − S T 2 ) = X 2 , T T T 2 α 0 ; x , T ; X α T α 0 ,α T ; x T 2 2 X 2 = X 2 . T T

Two-period Binomial Model The value function at time T is given by V ( x , T ) = U T ( x ) . The objective function is E [ U T ( X α ; x )] . T Therefore, his value function at time t = 0 is α E [ U T ( X α ; x V ( x , 0 ) = sup )] . T And the value function at t = T 2 is given by V ( x , T α ; x , T 2 ) = sup α E [ U T ( X 2 ) |F T 2 ] . T

Two-period Binomial Model By the definition and Dynamic Programming Principle, V ( x , T ) = U T ( x ) , ; x ; T α T V ( x , T 2 2 ) = sup E [ V ( X 2 , T ) |F T 2 ] , T α T 2 , T E [ V ( X α 0 ; x V ( x , 0 ) = sup 2 )] . T α 0 2 We can solve the value functions recursively, and this is why � � V = V ( ω, x , t ) 2 , T are coined as backward ω ∈ Ω , x ∈ R , t = 0 , T preferences. α ∗ α ∗ T ⇐ V ( x , T 0 2 V ( x , 0 ) 2 ) ⇐ V ( x , T ) .

Two-period Binomial Model Assume V ( x , T ) = U T ( x ) = − e − γ x , γ > 0. − γ x − E Q [ h T V ( x , T |F T ] 2 ) = − e , 2 2 where  q u q d � � � � h u = q u + q d p u 2 , T = p u 2 = ξ u T ln T T ln T , 2 , T = P ( p T 2 , T | ξ T );  p u T p d T T T   T 2 T 2  2 , T 2 , T h T 2 = q u q d � � � � h u = q u + q d p u 2 , T = p u 2 = ξ d T ln T T ln T , 2 , T = P ( p T 2 , T | ξ T ) ,  p u  T p d T T T  T  2 T 2 2 , T 2 , T and q u T , q d T are the conditional probabilities of going up and down under measure Q in [ T 2 , T ] . � pu � pd T T � � 2 , T 2 , T ln − ln qu qd The optimal strategy is α ∗ T = T . T ( ξ u T − ξ d γ S T T ) 2 2

Two-period Binomial Model − γ x − E Q [ h 0 + h T ] , V ( x , 0 ) = − e 2 � q u � q d T T where h 0 = q u + q d and q u 2 , q d � � 2 ln 2 2 ln 2 2 are the p u p u T T T T 0 , T 0 , T 2 2 conditional probabilities of going up and down under measure Q in [ 0 , T 2 ] . � pu � pu 0 , T 0 , T � � 2 2 − h u + h d ln − ln qu qd T T T T 2 2 The optimal strategy α ∗ 2 0 = 2 . γ S 0 ( ξ u − ξ d ) T T 2 2

Two-period Binomial Model First, if the stock price goes up initially at t = T 2 = ξ u 2 ( ξ T 2 ) , T

Two-period Binomial Model And if the price of the risky asset goes up at t = T ( ξ T = ξ u T ) ; Or if the price of risky asset goes down at t = T , ( ξ T = ξ d T ) ,

Two-period Binomial Model Second, if the stock price goes down initially at t = T 2 = ξ d 2 ( ξ T 2 ) , T

Two-period Binomial Model And if the price of the risky asset goes up at t = T ( ξ T = ξ u T ) ; Or if the price of risky asset goes down at t = T ( ξ T = ξ d T ) ,

Multi-period Model Consider there are N periods, where t = 0 , T N , 2 T N ... T . For simplicity, denote h = T N so that t = 0 , h , 2 h ..., T . Let (Ω , F , P , F ) be the filtered probability space. Suppose there are only one risk-free asset and one risky asset.

Multi-period Model ∀ t = 0 , h , ..., ( N − 1 ) h , in the interval [ t , t + h ] , the price could go up or down with respective probabilities, � ξ u p u t , t + h = P ( ξ t + h = ξ u t + h |F t ) t + h , ( ξ t + h |F t ) = ξ d p d t , t + h = P ( ξ t + h = ξ d t + h , t + h |F t ) where ξ u t + h , ξ d t + h and p u t , t + h , p d t , t + h ∈ F t .

Multi-period Model ∀ t = 0 , h , ..., ( N − 1 ) h , we used the money account as the numeraire. The arbitrage-free condition is 0 < ξ d t + h < 1 < ξ u t + h . The risk-neutral probability of realizing up in [ t , t + h ] is 1 − ξ d q u t , t + h = Q ( ξ t + h = ξ u t + h |F t ) = t + h t + h . ξ u t + h − ξ d We assume ξ u t + h , ξ d t + h , p u t , t + h , p d t , t + h , q u t , t + h and q d t , t + h ∈ F 0 to simplify the simulation process.

Multi-period Model � � Denote α t t = 0 , h ,..., ( N − 1 ) h be a stochastic control process. Let X α ; x = X α ; x � � t = 0 , h ,..., ( N − 1 ) h , T be the wealth process of the t investor using the control α with initial wealth x at time 0. Therefore, X α ; x = x , 0 X α ; x t + h = X α ; x + α t ( S t + h − S t ) , ∀ t = 0 , h , ..., ( N − 1 ) h . t

Multi-period Model ∀ s = 0 , h , ... T , let X α ; x , s = X α ; x , s � � t = s , s + h ,..., T , be the wealth t process of the investor using the control α with initial wealth x at time s . Therefore, X α ; x , 0 = X α ; x , X α ; x , s = x , s X α ; x , s = X α ; x , s + α t ( S t + h − S t ) , ∀ t = s , s + h , ..., ( N − 1 ) h . t + h t And more importantly, α ; X α ; x , t t + h , t + h X α ; x , t = X ∀ t = s , s + h , ..., ( N − 1 ) h . , T T

Multi-period Model The value function at time T is given by V ( x , T ) = U T ( x ) . The objective function is E [ U T ( X α ; x )] . T Therefore, his value function at time t = 0 is α E [ U T ( X α ; x V ( x , 0 ) = sup )] . T ∀ t = 0 , h , ..., T , value function at time t is given by α E [ U T ( X α ; x , t V ( x , t ) = sup ) |F t ] . T