Optimal portfolio strategies under partial information with expert opinions

Transcription

1 1 / 35 Optimal portfolio strategies under partial information with expert opinions Ralf Wunderlich Brandenburg University of Technology Cottbus, Germany Joint work with Rüdiger Frey Research Seminar WU Wien, December 7, 2012

2 Agenda 2 / 35 1 Dynamic Portfolio Optimization 2 Partial Information and Expert Opinions 3 Optimization for Power Utility 4 Approximation of the Optimal Strategy

3 Dynamic Portfolio Optimization 3 / 35 Initial capital x 0 > 0 Horizon [0, T ] Aim Problem maximize expected utility of terminal wealth find an optimal investment strategy How many shares of which asset have to be held at which time by the portfolio manager? Market model continuously tradable assets drift depends on unobservable finite-state Markov chain investor only observes stock prices and expert opinions

4 Classical Black-Scholes Model of Financial Market 4 / 35 (Ω, G = (G t ) t [0,T ], P) filtered probability space Bond S 0 t = e rt, r risk-free interest rate Stocks prices S t = (S 1 t,..., Sn t ), returns dr i t = dsi t S i t dr t = µ dt + σ dw t µ R n average stock return, drift σ R n n volatility W t n-dimensional Brownian motion parameters µ and σ are constant and known Generalization time-dependent (non-random) parameters µ, σ, r

5 5 / 35 Portfolio Initial capital X 0 = x 0 > 0 Wealth at time t invested in X t = X t ( ht 0 + }{{} ht 1 }{{} ht n }{{} ) bond stock 1 stock n h k t fractions of wealth invested in asset k Strategy h t = (h 1 t,..., hn t ) Self financing condition (assume r = 0 for simplicity) Wealth equation X t satisfies linear SDE with initial value x 0 dx (h) t = X (h) t h t (µ dt + σdw t ) X (h) 0 = x 0

6 Utility Function U : [0, ) R { } strictly increasing and concave Inada conditions lim U (x) = and lim U (x) = 0 x 0 x U(x) = { x θ θ for θ (, 1) \ {0} power utility log x for θ = 0 log-utility x θ 1 θ 6 / 35

7 Optimization Problem Wealth dx (h) t = X (h) t h t (µ dt + σdw t ), X (h) 0 = x 0 Admissible Strategies H = {(h t ) t [0,T ] h t R n, E [ exp { T 0 h t 2 dt }] < } Reward function Value function v(t, x, h) = E t,x [ U(X (h) T ) ] for h H V (t, x) = sup h H v(t, x, h) Find optimal strategy h H such that V (0, x 0 ) = v(0, x 0, h ) Solution optimal fractions of wealth h t = 1 1 θ (σσ ) 1 µ = const Merton (1969,1973) using methods from dynamic programming 7 / 35

8 Drawbacks of the Merton Strategy 8 / 35 Sensitive dependence of investment strategies on the drift µ of assets Drift is hard to estimate empirically need data over long time horizons (other than volatility estimation) is not constant depends on the state of the economy Non-intuitive strategies for constant fraction of wealth (0, 1) = sell stocks when prices increase buy stocks when prices decrease = Model drift as stochastic process, not directly observable

9 9 / 35 Models With Partial Information on the Drift Drift depends on an additional source of randomness µ = µ t = µ(y t ) with factor process Y t Investor is not informed about factor process Y t, he only observes Stock prices S t or equivalently stock returns R t Expert opinions news, company reports recommendations of analysts or rating agencies own view about future performance = Model with partial information Problem Investor needs to learn the drift from observable quantities Find an estimate or filter for µ(y t )

10 10 / 35 Models With Partial Information on the Drift (cont.) Linear Gaussian Model Lakner (1998), Nagai, Peng (2002), Brendle (2006) Drift µ(y t ) = Y t is a mean-reversion process where W 1 t dy t = α(µ Y t )dt + βdw 1 t is a Brownian motion (in)dependent of W t

11 Models With Partial Information on the Drift (cont.) 11 / 35 Hidden Markov Model (HMM) Sass, Haussmann (2004), Rieder, Bäuerle (2005), Nagai, Rungaldier (2008) Factor process Y t finite-state Markov chain, independent of W t state space {e 1,..., e d }, unit vectors in R d states of drift µ(y t ) = MY t where M = (µ 1,..., µ d ) generator or rate matrix Q R d d diagonal: Q kk = λ k exponential rate of leaving state k conditional transition prob. P(Y t = e l Y t = k, Y t Y t ) = Q kl /λ k initial distribution (π 1,..., π d )

12 HMM Filtering 12 / 35 Returns dr t = ds t S t = µ(y t ) dt + σ dw t observations Drift µ(y t ) = M Y t non-observable (hidden) state Investor Filtration F = (F t ) t [0,T ] with F t = σ(s u : u t) G t Filter p k t := P(Y t = e k F t ) Innovations process µ(y t ) := E[µ(Y t ) F t ] = µ(p t ) = d p j t µ j j=1 B t := σ 1 ( R t t 0 µ(y s )ds ) is an F-BM HMM filter Liptser, Shiryaev (1974), Wonham (1965), Elliot (1993) p0 k = π k d dpt k = Q jk p j t dt + β k(p t ) db t j=1 where β k (p) = p k σ 1( µ k d ) p j µ j j=1

13 HMM Filtering: Example 13 / 35

14 HMM Filtering: Example 14 / 35

15 Expert Opinions Academic literature: drift is driven by unobservable factors Models with partial information, apply filtering techniques Linear Gaussian models Hidden Markov models Practitioners use static Black-Litterman model Apply Bayesian updating to combine subjective views (such as asset 1 will grow by 5% ) with empirical or implied drift estimates Present paper combines the two approaches consider dynamic models with partial observation including expert opinions 15 / 35

16 Expert Opinions Modelled by marked point process I = (T n, Z n ) I(dt, dz) At random points in time T n Poi(λ) investor observes r.v. Z n Z Z n depends on current state Y Tn, density f (Y Tn, z) (Z n ) cond. independent given F Y T = σ(y s : s [0, T ]) Examples Absolute view: Z n = µ(y Tn ) + σ ε ε n, (ε n ) i.i.d. N(0, 1) The view S will grow by 5% is modelled by Z n = 0.05 σ ε models confidence of investor Relative view (2 assets): Z n = µ 1 (Y Tn ) µ 2 (Y Tn ) + σ ε ε n Investor filtration F = (F t ) with F t = σ(s u : u t; (T n, Z n ): T n t) 16 / 35

17 HMM Filtering - Including Expert Opinions 17 / 35 Extra information has no impact on filter p t between information dates T n Bayesian updating at t = T n : p k T n p k T n f (e k, Z n ) recall: f (Y Tn, z) is density of Z n given Y Tn with normalizer d p j T f (e n j, Z n ) =: f (p Tn, Z n ) j=1 HMM filter p0 k = π k d dpt k = Q jk p j t dt + β k(p t ) db t + pt k j=1 Compensated measure Z ( ) f (e k,z) 1 Ĩ(dt dz) f (p t,z) d Ĩ(dt dz) := I(dt dz) λdt pt f k (e k, z) dz k=1 } {{ } compensator

18 Filter: Example 18 / 35

19 Filter: Example 19 / 35

20 Optimization Problem Under Partial Information Wealth dx (h) t = X (h) t h t (µ(y t ) dt + σdw t ), X (h) 0 = x 0 Admissible Strategies H = {(h t ) t [0,T ] h t K R n with K compact h is F-adapted } Reward function Value function v(t, x, h) = E t,x [ U(X (h) T ) ] for h H V (t, x) = sup h H v(t, x, h) Find optimal strategy h H such that V (0, x 0 ) = v(0, x 0, h ) 20 / 35

21 Reduction to an OP Under Full Information 21 / 35 Consider augmented state process (X t, p t ) Wealth dt + σdb t ), X =M p t Filter dpt k = d Q jk p j t dt + β k(p t ) db t Reward function Value function dx (h) t = X (h) t h t ( µ(y t ) }{{} j=1 +p k t Z ( ) f (e k,z) 1 Ĩ(dt dz), f (p t,z) v(t, x, p, h) = E t,x,p [ U(X (h) T ) ] for h H V (t, x, p) = sup h H v(t, x, p, h) Find h H(0) such that V (0, x 0, π) = v(0, x 0, π, h ) (h) 0 = x 0 p k 0 = πk

22 Logarithmic Utility U(X (h) T (h) ) = log(x T ) = log x 0 + T 0 (h µ(y s s ) 1 ) T 2 h s σσ h s ds + h s σdb s 0 [ E[U(X (h) T T )] = log x 0 + E (h µ(y s s ) 1 ) ] 0 2 h s σσ h s ds + 0 Optimal Strategy h t = (σσ ) 1 µ(yt ). Certainty equivalence principle h is obtained by replacing in the optimal strategy under full information h full t = (σσ ) 1 µ(y t ) the unknown drift µ(y t ) by its filter µ(y t ) 22 / 35

23 Solution for Power Utility 23 / 35 Risk-sensitive control problem Nagai & Runggaldier (2008), Davis & Lleo (2011) { T T Let Z h := exp θ h s σdb s θ2 } h s σσ h s ds 2 0 Change of measure: P (h) (A) = E[Z (h) 1 A ] for A F T Reward function E t,x,p [U(X (h) T )] = x θ θ E (h) t,p 0 T [ { }] exp b(p s, h s )ds t }{{} =: v(t, p, h) independent of x where b(p, h) := ( θ h Mp 1 θ ) 2 h σσ h Value function V (t, p) = sup h H v(t, p, h) for 0 < θ < 1 Find h H such that V (0, π) = v(0, π, h )

24 HJB-Equation State dp t = α(p t, h t )dt + β (p t )db t + Z γ(p t, z)ĩ(dt dz) Generator L h g(p) = 1 2 tr[ β (p)β(p)d 2 g ] + α (p, h) g +λ Z {g(p + γ(p, z)) g(p)}f (p, z)dz { } V t (t, p) + sup h R n L h V (t, p) b(p, h)v (t, p) = 0 terminal condition V (T, p) = 1 Candidate for the Optimal Strategy h = h (t, p) = 1 (1 θ) (σσ ) 1{ Mp + 1 } V (t, p) σβ(p) pv (t, p) }{{} myopic strategy + correction Certainty equivalence principle does not hold 24 / 35

25 Justification of HJB-Equation 25 / 35 Standard verification arguments fail, since we cannot guarantee uniform ellipticity of the diffusion part: tr [ β (p)β(p)d 2 V ] ξ β (p)β(p)ξ c ξ 2 for some c > 0 and all ξ R d satisfiable only if number of assets n number of drift states d Applying results and techniques from Pham (1998) = V is a unique continuous viscosity solution of the HJB-equation

26 Regularization of HJB-Equation 26 / 35 Add a small Gaussian perturbation 1 m d B t first d 1 components of the filter to the SDE for the Consider control problem for the modified dynamics of the filter Modified HJB-equation has an additional diffusion term 1 2m V m (t, p) = uniform ellipticity Applying results from Davis & Lleo (2011) = classical solution V m (t, p) to the modified HJB-equation Standard verification results can be applied Convergence results for m : optimal strategy ε-optimal strategy to the modified control problem is an to the original control problem

27 Approximation of the optimal strategy 27 / 35 Policy Improvement Numerical solution of HJB equation Feynman-Kac formula for linearized HJB equation

28 Policy Improvement Starting approximation is the myopic strategy h (0) t = 1 1 θ (σσ ) 1 Mp t The corresponding reward function is [ ( T )] V (0) (t, p) := v(t, p, h (0) ) = E t,p exp b(p (h(0) ) s, h (0) s )ds t Consider the following optimization problem { max L h V (0) (t, p) b(p, h)v (0) (t, p) } h with the maximizer h (1) (t, p) = h (0) (t, p) + 1 (1 θ)v (0) (t, p) (σ ) 1 β(p) p V (0) (t, p) For the corresponding reward function V (1) (t, p) := v(t, p, h (1) ) it holds Lemma ( h (1) is an improvement of h (0) ) V (1) (t, p) V (0) (t, p) 28 / 35

29 Policy Improvement (cont.) Policy improvement requires Monte-Carlo approximation of reward function [ ( T )] V (0) (t, p) = E t,p exp b(p (h(0) ) s, h (0) s )ds ]. t Generate N paths of ps h(0) starting at time t with p = p t Estimate expectation E t,p [ ] Approximate partial derivatives V (0) p k (t, p) by finite differences Compute first iterate h (1) 29 / 35

30 Numerical Results 30 / 35

31 Numerical Results For t = T n : nearly full information = correction 0 31 / 35

32 Numerical solution of HJB equation HJB Equation { } V t (t, p) + sup h R n L h V (t, p) b(p, h)v (t, p) = 0 terminal condition V (T, p) = 1 Generator L h g(p) = 1 2 tr[ β (p)β(p)d 2 g ] + α (p, h) g Plugging in the optimal strategy +λ Z {g(p + γ(p, z)) g(p)}f (p, z)dz h = h 1 (t, p) = (1 θ) (σσ ) 1{ Mp + 1 } V (t, p) σβ(p) pv (t, p) }{{} myopic strategy + correction yields a nonlinear partial integro-differential equation Normalization of p = reduction to d 1 spatial variables d = 2: only one spatial variable, ellipticity condition is satisfied 32 / 35

33 Numerical solution of HJB equation (cont.) 33 / 35

34 Conclusion 34 / 35 Portfolio optimization under partial information on the drift Investor observes stock prices and expert opinions Non-linear HJB-equation with a jump part Computation of the optimal strategy

35 References 35 / 35 Davis, M. and Lleo, S. (2011). Jump-Diffusion Risk-Sensitive Asset Management II: Jump-Diffusion Factor Model, arxiv: v1. Frey, R., Gabih, A. and Wunderlich, R. (2012): Portfolio optimization under partial information with expert opinions. International Journal of Theoretical and Applied Finance,Vol. 15, No. 1. Nagai, H. and Runggaldier, W.J. (2008): PDE approach to utility maximization for market models with hidden Markov factors. In: Seminar on Stochastic Analysis, Random Fields and Applications V (R.C.Dalang, M.Dozzi, F.Russo, eds.). Progress in Probability, Vol.59, Birkhäuser Verlag, Pham, H. (1998): Optimal Stopping of Controlled Jump Diffusion Processes: A viscosity Solution Approach. Journal of Mathematical Systems, estimations, and Control Vol.8, No.1, Rieder, U. and Bäuerle, N. (2005): Portfolio optimization with unobservable Markov-modulated drift process. Journal of Applied Probability 43, Sass, J. and Haussmann, U.G (2004): Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain. Finance and Stochastics 8,