Optimal Investment under Dynamic Risk Constraints and Partial Information

Transcription

1 Opimal Invesmen under Dynamic Risk Consrains and Parial Informaion Wolfgang Puschögl Johann Radon Insiue for Compuaional and Applied Mahemaics (RICAM) Ausrian Academy of Sciences 2 h Sepember 27 Join work wih J. Saß (RICAM), Suppored by FWF, Projec P17947-N12 Workshop and Mid-Term Conference on Advanced Mahemaical Mehods for Finance Vienna

2 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 2 / 34

3 Model Seup Filered probabiliy space: (Ω, F = (F ) [,T ], P) Finie ime horizon: T > Money marke: bond wih sochasic ineres raes r Z ds () = S () r d, S () = 1, i.e., S () = exp r s ds, r uniformly bounded and progressively measurable w.r.. F Sock marke: n socks wih price process S = (S (1),..., S (n) ), reurn R, and excess reurn R, where ds = Diag(S )(µ d + σ dw ), dr = µ d + σ dw, d R = dr r d. W n-dimensional sandard Brownian moion w.r.. F and P drif µ R n F -adaped and independen of W volailiy σ R n n progressively measurable w.r.. F S, σ non-singular, and σ 1 uniformly bounded. 3 / 34

4 Risk Neural Probabiliy Measure We inroduce he risk neural probabiliy measure ( for filering and opimizaion). Definiion Maringale densiy process Z = exp Z θ s dw s 1 2 Z «θ s 2 ds wih θ = σ 1 (µ r 1 n) he marke price of risk Risk neural probabiliy measure P defined by d P dp := Z T Ẽ expecaion operaor under P Girsanov s heorem: defines a P-Brownian moion Z W := W + θ s ds 4 / 34

5 Parial Informaion Remark We consider he case of parial informaion: we can only observe ineres raes and sock prices (F r,s ) bu no he drif The porfolio has o be adaped o F r,s we need he condiional densiy ζ = EˆZ F S we need he filer for he drif ˆµ = Eˆµ F S Assumpion The ineres raes r are F S -adaped F r,s = F S Z is a maringale w.r.. F and P Lemma We have F S = F W = F R he marke is complee w.r.. F S 5 / 34

6 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 6 / 34

7 Consumpion and Trading Sraegy Definiion Trading sraegy π : n-dimensional, F S -adaped, measurable Iniial capial x > Wealh process X π saisfies dx π = π (µ d + σ dw ) + (X π 1 n π )r d X π = x A sraegy is admissible if X π a.s. for all [, T ] π represens he wealh invesed in he socks a ime η π = π /X π denoes he corresponding fracion of wealh 7 / 34

8 Uiliy Funcions Definiion U : [, ) R { } is a uiliy funcion, if U is sricly increasing, sricly concave, wice coninuously differeniable on (, ), and saisfies he Inada condiions: U ( ) = lim x U (x) =, U (+) = lim x U (x) =. I denoes he inverse funcion of U. Assumpion I(y) Ky a, I (y) Ky b for all y (, ) and a, b, K > Example Logarihmic uiliy U(x) = log(x) Power uiliy U(x) = x α /α for α < 1, α. 8 / 34

9 Opimizaion Problem Opimizaion Problem We opimize under parial informaion! Objecive: Maximize he expeced uiliy from erminal wealh, i.e., maximize EˆU(X T ) under (risk) consrains we sill have o specify. The opimizaion problem consiss of wo seps: 1. Find he opimal erminal wealh 2. Find he corresponding rading sraegy 9 / 34

10 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 1 / 34

11 Time-Dependen Convex Consrains We can wrie our model under full informaion wih respec o F R as dr = ˆµ d + σ dv, [, T ]. where he innovaion process V = (V ) [,T ] is a P-Brownian moion defined by Z Z V = W + σs 1 (µ s ˆµ s) ds = Z σs 1 dr s σs 1 ˆµ s ds. K represens he consrains on porfolio proporions a ime η π K K is a F -progressively measurable closed convex se = K R n ha conains For each we define he suppor funcion δ : R n R {+ } of K by δ (y) is F -progressively measurable δ (y) = sup x K ( x y), y R n. y δ (y) is a lower semiconinuous, proper, convex funcion on is effecive domain K = {y R n : δ (y) < } 11 / 34

12 Time-Dependen Convex Consrains Definiion A rading sraegy η π is called K -admissible for iniial capial x > if X π η π K for all [, T ]. a.s. and We denoe he class of admissible rading sraegies for iniial capial x by A K (x ). We inroduce he se H of dual processes ν : [, T ] Ω K which are F R -progressively measurable processes, saisfying EˆR T ` ν 2 + δ (ν ) d <. For each dual process ν H we inroduce a new ineres rae process r ν = r + δ (ν ). a new drif process ˆµ ν = ˆµ + ν + δ (ν )1 n. a new marke price of risk θ ν = σ 1 (ˆµ r + ν ) a new densiy process ζ ν given by dζ ν = θ ν ζ ν dv Then: Soluion under consrains = soluion under no consrains wih new marke coefficiens! Problem: Find opimal ν! 12 / 34

13 Time-Dependen Convex Consrains Proposiion Suppose x > and E[U (X η T )] < for all ηπ A K (x ). A rading sraegy η π A K (x ) is opimal, if for some y >, ν H where ζ T = condiion X π T = I(y ζ T ), X ν (y ) = x, ν ζ T. Furher, η π and ν have o saisfy he complemenary slackness δ (ν ) + (η π ) ν =, [, T ]. y, ν solve he dual problem Ṽ (y) = inf ν H EˆŨ(y ζ ν T ), where Ũ(y) = sup x> U(x) xy, y > is he convex dual funcion of U. If F R = F V holds, hen an opimal rading sraegy exiss. 13 / 34

14 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 14 / 34

15 Limied Expeced Loss & Limied Expeced Shorfall Suppose we canno rade in [, + ]. Then X π = X π + X π Z = X π + exp exp 1 2 Z r s ds X π + + exp Z + Nex, we impose he relaive LEL consrain wih ε = LX π. Ẽˆ( X π Definiion K LEL := η π R n Ẽˆ( X π ) F S < ε diag(σ sσ s ) ds + ) F S < ε, Z + r s ds (η π ) X π σ s d W s / 34

16 Limied Expeced Loss & Limied Expeced Shorfall We inroduce he relaive LES consrain as an exension o he LEL consrain Ẽˆ( X π + q ) F S < ε, wih ε = L 1 X π and q = L 2 X π. LES wih L 2 = corresponds o LEL wih L = L 1. LEL: any loss in [, + ] can be hedged wih L% of he porfolio value. LES: any loss greaer L 2 % of he porfolio value in [, + ] can be hedged wih L 1 % of he porfolio value. LEL & LES: For hedging we can use sandard European call and pu opions. Definiion Lemma K LEL K LES := η π R n Ẽˆ( X π and K LES are convex. + q ) F S < ε For n = 1 we obain he inerval K LES = [η l, η u ]. 16 / 34

17 bounds on η π for LEL and LES 1 1 Bounds on η π L (in % of Wealh) 5 (in days) Bounds on η π L 1 L / 34

18 bounds on η π for LEL Upper bound on η π L = 2% L = 1.2% L =.4% (in days) Lower bound on η π L = 2% L = 1.2% L =.4% (in days) 18 / 34

19 Oher consrains Value-a-Risk consrain: Under he original measure X π X π is given by Z = X π + exp r s ds X π + (η π ) X π Z + `µs 1 Z + 2 diag(σsσ s ) ds + exp Z + σ s dw s exp r s ds. We impose for n = 1 he relaive VaR consrain on he loss ( X π ), P`( X π ) > LX π F S, µ = ˆµ < γ. VaR is compued under he original measure P. Under parial informaion we need he (unknown) value of he drif use e.g. µ = ˆµ. For n = 1 we obain he inerval K VaR = [η l, η u ]. If n > 2 hen K VaR may no be convex! Possible o apply a large class of oher risk consrains e.g. CVaR. 19 / 34

20 Sraegy Corollary (Logarihmic uiliy) U(x) = log(x), n = 1, no consrains: η o := η π = 1 (ˆµ σ 2 r ). Wih consrains: 8 η >< u if η o > η u, η c := η π = η o if η o ˆη, l η u, >: η l if η o < η l. Hence, we cu off he sraegy obained under no consrains if i exceeds or falls below a cerain hreshold. 2 / 34

21 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 21 / 34

22 Gaussian Dynamics (GD) for he Drif Drif: modeled as he soluion of he sochasic differenial equaion (cf. Lakner 98) dµ = κ( µ µ ) d + υ d W, µ N (ˆµ, ρ ), n-dimensional, W is a n-dimensional Brownian moion wih respec o (F, P), We are in he siuaion of Kalman-filering wih signal µ, observaion R, and filer ˆµ = Eˆµ F S. Filer: ˆµ is he unique F S -measurable soluion of dˆµ = ˆ` κ ρ (σ σ ) 1 ˆµ + κ µ d + ρ (σ σ ) 1 dr, ρ = ρ (σ σ ) 1 ρ κρ ρ κ + υυ, wih iniial condiion (ˆµ, ρ ). ζ 1 saisfies dζ 1 = ζ 1 (ˆµ r 1 n) (σ ) 1 d W. Proposiion F S = F R = F W = F V an opimal rading sraegy exiss. 22 / 34

23 Bayesian case The Bayesian case is a special case of he Gaussian dynamics for he drif. Drif: µ µ = (µ (1),..., µ(n) ) is an (unobservable) F -measurable Gaussian random variable wih known mean vecor ˆµ and covariance marix ρ. Filer: Explici soluion: Z 1 Z ˆµ = 1 n n + ρ (σ sσs ) 1 ds ˆµ + ρ (σ sσs ) 1 dr s, Z 1ρ ρ = 1 n n + ρ (σ sσs ) ds / 34

24 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 24 / 34

25 HMM: The Drif The drif process µ of he reurn, is a coninuous ime Markov chain given by µ = BY, B R n d, where Y is a coninuous ime Markov chain wih sae space he sandard uni vecors {e 1,..., e d } in R d, and rae marix Q R d d, where Q kl is he jump rae or ransiion rae from e k o e l, λ k = Q kk = P d l=1,l k Q kl is he rae of leaving e k, he waiing ime for he nex jump is exponenially disribued wih parameer λ k and Q kl /λ k is he probabiliy ha he chain jumps o e l when leaving e k for l k. The differen saes of he drif are he columns of B. We can wrie he marke price of risk as θ = σ 1 (µ r 1 n) = Θ Y, where Θ := σ 1 (B r 1 n d ). 25 / 34

26 HMM: Filering We are in he siuaion of HMM filering since R = R BYs ds + R σs dws. We need he condiional densiy ζ = (ζ ) [,T ] = EˆZ F S = 1, 1 d E 1 he unnormalized filer E = (E ) [,T ] = ẼˆZT Y F S, he normalized filer Ŷ = (Ŷ) [,T ] = EˆY F S = = ζ E. E 1 d E Theorem (Wonham/Ellio) Proposiion E = E[Y ] + Z Z Q E s ds + Diag(E s)θ s F S = F R = F W = F V an opimal rading sraegy exiss. d W s 26 / 34

27 Ouline Model Seup Problem formulaion Time-Dependen Convex Consrains Dynamic Risk Consrains Gaussian Dynamics for he Drif A hidden Markov Model (HMM) for he Drif Example 27 / 34

28 Example (1/3) S Time µ BŶ Time We consider he HMM for he drif. 28 / 34

29 Example c d (2/3).35.3 σ Time 1 2 η η l η u Time For he volailiy we consider he Hobson-Rogers model. 29 / 34

30 Example c d (3/3) X π Time 3 / 34

31 Numerical Resuls (1/2) We consider 2 socks of he Dow Jones Indusrial Index We use daily prices (adjused for dividends and splis) for 3 years, Parameer esimaes are based on five years wih saring year 1972, 1973,..., 1996 using a Markov Chain Mone Carlo algorihm. We apply he sraegy in he subsequen year we perform 5 experimens whose oucomes we average. We consider LEL- and LES-consrain. 31 / 34

32 Numerical Resuls c d (2/2) U( ˆX T ) mean median s.dev. abored unconsrained b&h Meron GD Bayes HMM LEL risk consrain (L=.5%) GD Bayes HMM LES risk consrain (L1=.1%,L2=5%) GD Bayes HMM LEL and LES improve he performance of all models. Wih LEL and LES we don go bankrup anymore. The HMM sraegy wih risk consrains ouperforms all oher sraegies. 32 / 34

33 Conclusion & Oulook Conclusion We show how o apply dynamic risk consrains using ime-dependen convex consrains. We derive explici rading sraegies wih dynamic risk consrains under parial informaion. The numerical resuls indicae ha dynamic risk consrains can reduce he risk and improve he performance. Oulook Allow for consumpion. More deailed analysis of he mulidimensional case. Explici sraegies for general uiliy. 33 / 34

34 Furher Reading D. Cuoco, H. He, and S. Issaenko, Opimal Dynamic Trading Sraegies wih Risk Limis, FAME, Inernaional Cener for Financial Asse Managemen and Engineering, 22. K. F. C. Yiu, Opimal porfolios under a value-a-risk consrain, J. Econom. Dynam. Conrol 28 (24), no. 7, , Mahemaical programming. W. Puschögl and J. Sass, Opimal Invesmen under Dynamic Risk Consrains and Parial Informaion, (27), working paper. 34 / 34