Optimal Investment with Partial Information Tomas Bj ork - PDF document

Optimal Investment with Partial Information Tomas Bj¨ ork Stockholm School of Economics Mark Davis Imperial College Camilla Land´ en Royal Institute of Technology Tomas Bj¨ ork, 2007

Standard Problem Maximize utility of final wealth. max E P [ U ( X T )] Model: dS t = αS t dt + S t σdW t , dB t = rB t dt X t = portfolio value at t u t = relative portfolio weight in stock at t Wealth dynamics dX t = X t { u t ( α − r ) + r } dt + u t X t σdW t Standard approaches: • Dynamic programming. (Merton etc) • Martingale methods. (Huang etc) Tomas Bj¨ ork, 2007 1

Standard assumption: • The volatility σ and the mean rate of return α are known . Standard results: • Very explicit results. • Nice mathematics. Sad facts from real life: • The volatility σ can be estimated with some precision. • The mean rate of return α can not be estimated at all . Example: If σ = 20% and we want a 95% confidence interval for α , we have to observe S for 1600 years. Tomas Bj¨ ork, 2007 2

Reformulated Problem • Model α as random variable or random process. • Take the estimation procedure explicitly into account in the optimization problem. Tomas Bj¨ ork, 2007 3

Extended Standard Problem Model: dS t = α ( t, Y t ) S t dt + S t σ ( t, Y t ) dW t , • Y is a “hidden Markov process” which cannot be observed directly. • We can only observe S . Tomas Bj¨ ork, 2007 4

Previous Studies • Power or exponential utility. • Y is a diffusion: (Genotte, Brennan, Brendle) • Y is a finite state Markov chain: (B¨ auerle–Rieder, Nagai–Runggaldier, Haussmann– Sass). Technique: • Filtering theory. • Use conditional density as extended state. • Dynamic programming. Results: • Very nice explicit results. • Sometimes a bit messy. • Separate study for each model. Tomas Bj¨ ork, 2007 5

Object of Present Study • Study a more general problem • Avoid DynP (regularity, viscosity solutions etc). • Investigate the general structure. Tomas Bj¨ ork, 2007 6

Related Zariphopoulou Problem � 1 � γX γ max E P T dS t = α ( t, Y t ) S t dt + S t σ t ( t, Y t ) dW t , dY t = µ ( t, Y t ) dt + b ( t, Y t ) dW t . Note: Both S and Y are observable . Same W driving S and Y . (Zariphopoulou allows for general correlation) Wealth dynamics dX t = X t { u t ( α t − r ) + r } dt + u t X t σdW t For simplicity we put r = 0 Tomas Bj¨ ork, 2007 7

 � � uαxF x + 1 2 u 2 σ 2 x 2 F xx + µF y + 1   2 b 2 F yy + uxσbF xy F t + sup = 0 ,  u F ( T, s, y ) = x γ    γ . Ansatz: F ( t, x, y ) = x γ γ G ( t, y ) , PDE: � � 2(1 − γ ) · G 2 γα 2 γb 2 G t + 1 γαb y 2 b 2 G yy + µ + G y + 2 σ 2 (1 − γ ) G + G = 0 σ (1 − γ ) Non linear! We have a problem! Tomas Bj¨ ork, 2007 8

PDE: � � G t + 1 γαb 2 b 2 G yy + µ + G y σ (1 − γ ) 2(1 − γ ) · G 2 γα 2 γb 2 y + 2 σ 2 (1 − γ ) G + G = 0 Clever idea by Zariphopoulou: G ( t, y ) = H ( t, y ) 1 − γ � � βα 2 µ + αβ H y + 1 2 b 2 H yy + H t + σ b 2 σ 2 (1 − γ ) H = 0 , H ( T, y ) = 1 . Linear! Feynman-Kac representation. Tomas Bj¨ ork, 2007 9

Zariphopoulou Result • Optimal value function V ( t, x, y ) = x γ γ H ( t, y ) 1 − γ , • H is given by PDE or by � � �� � T βα 2 1 H ( t, y ) = E 0 exp (1 − γ ) σ 2 dt , t,y 2 t where the measure Q 0 has likelihood dynamics of the form � αβ � dL 0 t = L 0 dW t . t σ • The optimal control is given by σ 2 (1 − γ ) + b α σ · H y u ∗ ( t, x, y ) = H . Tomas Bj¨ ork, 2007 10

What on earth is going on? Tomas Bj¨ ork, 2007 11

Present Paper Model: (Ω , F , P, F ) dS t = α t S t dt + S t σ t dW t , • α and σ are general F -adapted • F S t ⊆ F t • The short rate is assumed to be zero. Wealth dynamics: dX t = u t α t X t dt + u t X t σ t dW t , Problem: E P [ U ( X T )] max u over F S -adapted portfolios. Tomas Bj¨ ork, 2007 12

Strategy • Start by analyzing the completely observable case. • Go on to partially observable model. • Use filtering results to reduce the problem to the completely observable case. Tomas Bj¨ ork, 2007 13

Completely observable case Model: (Ω , F , P, F ) dS t = α t S t dt + S t σ t dW t , • F t = F W t • α and σ are general F W -adapted Wealth dynamics: dX t = u t α t X t dt + u t X t σ t dW t , Problem: E P [ U ( X T )] max u over F W -adapted portfolios. Tomas Bj¨ ork, 2007 14

Martingale approach Complete market, so we can separate choice of optimal wealth profile X T from optimal portfolio choice. E P [ U ( X )] max X ∈F T s.t. budget constraint E Q [ X ] = x, Rewrite budget as E P [ L T X ] = x, where L t = dQ dP , on F t Lagrangian relaxation � � L = E P [ U ( X )] − λ E P [ L T X ] − x , Tomas Bj¨ ork, 2007 15

Relaxed problem � max { U ( X ) − λ ( L T X − x ) } dP. X Ω Separable problem with solution U ′ ( X ) = λL T Optimal wealth: X = F ( λL T ) , where F = ( U ′ ) − 1 The Lagrange multiplier is determined by the budget constraint E P [ L T X ] = x . Tomas Bj¨ ork, 2007 16

Power utility 1 F ( y ) = y − X = F ( λL T ) , 1 − γ , Easy calculation gives us. Result: • Optimal wealth is given by X = x 1 − 1 − γ · L , T H 0 • H 0 is given by H 0 = E P � � γ L − β , β = T 1 − γ • Optimal expected utility V 0 is given by V 0 = x γ γ H 1 − γ . 0 • This is where the fun starts. Tomas Bj¨ ork, 2007 17

H 0 = E P � � γ L − β , β = T 1 − γ Recall � � � T � T α 2 σdW t − 1 α L T = exp − σ 2 dt . 2 0 0 Thus �� T � � T βα 2 βα σ dW t + 1 L − β = exp σ 2 dt . T 2 0 0 Define the P - martingale L 0 by �� t � � βα � � βα � 2 � t dW s − 1 L 0 t = exp ds σ 2 σ 0 0 We can then write � � � T βα 2 1 L − β = L 0 T exp (1 − γ ) σ 2 dt . T 2 0 Tomas Bj¨ ork, 2007 18

� � �� � T βα 2 1 H 0 = E P L 0 T exp (1 − γ ) σ 2 dt , 2 0 Since L 0 is a martingale, it defines a change of measure t = dQ 0 L 0 dP , on F t , Thus � � �� � T βα 2 1 H 0 = E 0 exp (1 − γ ) σ 2 dt , 2 0 where L 0 has P -dynamics � βα � dL 0 t = L 0 dW t , t σ Tomas Bj¨ ork, 2007 19

Results • Optimal wealth is given by X = x 1 − 1 − γ · L , T H 0 • H 0 is given by � � �� � T βα 2 1 H 0 = E 0 t exp dt , (1 − γ ) σ 2 2 0 t • L 0 = dQ 0 /dP has dynamics � βα t � dL 0 t = L 0 dW t , t σ t • Optimal expected utility V 0 is given by V 0 = x γ γ H 1 − γ . 0 This can in fact be extended Tomas Bj¨ ork, 2007 20

Results in the observable case • The optimal wealth process is given by t = xH t 1 − X ⋆ 1 − γ · L , t H 0 • H t is given by �� � � � � T � βα 2 1 � H t = E 0 s exp ds � F t , � 2 (1 − γ ) σ 2 t s • The optimal expected utility process V t is given by t ) γ V t = ( X ⋆ H 1 − γ . t γ • L 0 = dQ 0 /dP has dynamics � βα t � dL 0 t = L 0 dW t , t σ t Tomas Bj¨ ork, 2007 21

Furthermore • The optimal portfolio process is given by t (1 − γ ) + 1 α t σ H u ∗ t = σ 2 σ t H where dH t = µ H dt + σ H dW t Tomas Bj¨ ork, 2007 22

Partially observable case Model: (Ω , F , P, F ) dS t = α t S t dt + S t σ t dW t , • F S t ⊆ F t • α is only F -adapted and thus not directly observable. • σ is F S t -adapted (WLOG). Wealth dynamics: dX t = u t α t X t dt + u t X t σ t dW t , Problem: E P [ U ( X T )] max u over F S -adapted portfolios. Tomas Bj¨ ork, 2007 23

Recap on FKK filtering theory Given some filtration F : dY t = a t dt + dM t dZ T = b t dt + dW t Here all processes are F adapted and Y = signal process, Z = observation process, M = martingale w.r.t. F W = Wiener w.r.t. F We assume (for the moment) that M and W are independent . Problem: Compute (recursively) the filter estimate � � ˆ Y t | F Z Y t = E t Tomas Bj¨ ork, 2007 24

The innovations process Recall F -dynamics of Z dZ t = b t dt + dW t Our best guess of b t is ˆ b t , so the genuinely new information should be dZ t − ˆ b t dt The innovations process ¯ W is defined by W t = dZ t − ˆ ¯ b t dt Theorem: The process ¯ W is F Z -Wiener. Thus the F Z -dynamics of Z are dZ t = � b t dt + d ¯ W t Tomas Bj¨ ork, 2007 25

Back to the model dS t = α t S t dt + S t σ t dW t , Define Z by 1 dZ t = dS t S t σ t i.e. dZ t = α t dt + dW t σ t We then have dZ t = � α t dt + d ¯ W t σ t where ¯ W is F S -Wiener. Thus we have price dynamics α t S t dt + S t σ t d ¯ dS t = � W t , We are back in the completely observable case! Tomas Bj¨ ork, 2007 26

The mathbfF S martingale measure ¯ Q is defined by d ¯ Q dP = ¯ on F S L t , t , (1) with L given by � � − ˆ α d ¯ L t = ¯ d ¯ L t W t . (2) σ Q 0 is defined by The measure ¯ d ¯ Q 0 dP = ¯ L 0 on F S t , t , L 0 given by with ¯ � ˆ � αβ d ¯ t = ¯ d ¯ L 0 L 0 W t . t σ Tomas Bj¨ ork, 2007 27