Exploring non linearities in Hedge Funds

Transcription

1 Exploring non linearities in Hedge Funds An application of Particle Filters to Hedge Fund Replication Guillaume Weisang 1 Thierry Roncalli 2 1 Bentley University, Waltham, MA 2 Lyxor AM January 28-29, 2010 G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

2 Literature Review Overview of the Factor Approach Overview of the Factor Approach A hedge fund portfolio: r HF t = ω it r it i I Assumption The structure of all asset returns can be summarized by a set of risk factors {F j } j=1,...,m : with m t r it = α i + β i j F jt + ξ it j=1 E(ξ it F 1t...F mt ) = 0 A typical factor model One assumes such that rt HF = αt HF + risk exposures m j=1 α HF w HF jt F jt + ε t t = α i ω it i I w HF jt = ω it β i j i I ε t = ω it ξ it i I G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

3 Literature Review Overview of the Factor Approach Literature Review of the Factor Approach Summary Static Linear factor models [Amenc et al., 2007] Lack reactivity Fail the test of robustness, giving poor out-of-sample results Factor selection [Fung and Hsieh, 1997] [Lo, 2008] In static models, economic selection of factors significant improvement over other methodologies for out-of-sample robustness test. In dynamic models, [Darolles and Mero, 2007] uses a PCA-based factor evaluation methodology [Bai and Ng, 2006] on rolling OLS regressions. Improvement over naive inclusion of all relevant economic factors Poor Interpretability of the evaluated factors Dynamic linear models [Roncalli and Teiletche, 2008] [Lo, 2008] [Jaeger, 2009]: Capturing the unobservable dynamic allocation using traditional (OLS) methods is Very difficult Estimates can vary greatly at balancing dates Nonlinear models methodological challenge [Amenc et al., 2008] [Diez de los Rios and Garcia, 2008] G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

4 Research Program and Methodology Hedge Fund Replication The Nonlinear Non-Gaussian Case Why It is Interesting HF Returns are not Gaussian negative skewness and positive excess kurtosis. Nonlinearities in HF Returns Nonlinearities documented from the very start of hedge-fund replication see, e.g., [Fung and Hsieh, 1997]. Nonlinearities are important for some strategies but not for the entire industry [Diez de los Rios and Garcia, 2008]. may be due to positions in derivative instruments or un-captured dynamic strategies see, e.g., [Merton, 1981]. No successful hedge fund replication using non-linear models has ever been done G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

5 Research Program and Methodology Methodology Tracking Problems Definition (Tracking Problem) The following two equations define a tracking problem (TP) [Arulampalam et al., 2002]: { xk = f (t k, x k 1, ν k ) (Transition Equation) z k = h(t k, x k, η k ) (Measurement Equation) where shadow object x k R nx is the state vector, and z k R nz the measurement vector at step k. ν k et η k are mutually independent i.i.d noise processes. The functions f and h can be non-linear functions. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

6 Research Program and Methodology Methodology Tracking Problems and Tactical Allocation Tracking Systems Discrete case, at time step k Outputs Inputs Tracking Error e k = z k ẑ k k 1 e k = rk HF rk Clone Censored measurement z k z k = rk HF HFR: x k = (w HF 1k,...,wHF mk ) Inputs Exogenous signals ψ k = (x 0,η 1:k,ν 1:k ) HF changes in allocation, strategies or reporting Tracking System Controlled input u k Assumption: u k = K k z k Adjustments to the replication portfolio s risk exposures ψ k u k Γ k Controller K k risk exposures Outputs z k e k G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

7 Research Program and Methodology Methodology Bayesian Filters Optimal Control Theory Under some general assumptions, one can prove tracking error e k = T eψk ψ k exogenous signals with transfer function T eψk = Γ eψk + Γ euk K k ( I Γzuk ) 1 Γzψk The role of the controller K k is to stabilize the system make T eψ small in an appropriate sense. Definition (Stability) A system is said to be marginally stable if the state x is bounded for all time t and for all bounded initial states x 0. Bayesian Filters are algorithms which provide the optimal estimators of the state x k Advantages of Bayesian Filters: no assumption of stationarity G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

8 Research Program and Methodology Methodology Bayesian Filters Solving Tracking Problems At time step k Prediction equation for the prior density p(x k z 1:k 1 ) = p(x k x k 1 ) p(x k 1 z 1:k 1 ) dx k 1 Update equation for the posterior density p(x k z 1:k ) p(z k x k ) p(x k z 1:k 1 ) Best estimates ˆx k k 1 = E[x k z 1:k 1 ] ˆx k k = E[x k z 1:k ] Implementation Kalman Filter (KF): linear Gaussian case H Filters or Particle Filters (PF): nonlinear or non Gaussian case G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

9 Research Program and Methodology Methodology Particle Filters If the posterior density p(x k z 1:k ) π (x k ) such that π (x) is easy to evaluate but difficult to draw sample from. Let { x i Ns k} be samples from an importance density q( ) i=1 The posterior density at time k can then be approximated as where, using Bayes rule, N s p(x k z 1:k ) w i k δ ( x k x i ) k i=1 w i k wi k 1 p ( z k xi k) ( p x i xi ) k 1:k 1 q ( x i k xi 1:k 1, z ) k (1) The set of support points { x i Ns k} i=1 and their associated weights { w i k,i = 1,...,N } s characterizes the posterior density at time step k Equation (1) is at the core of Particle Filters (PF). Considering different assumptions leads to different numerical algorithms (SIS, GPF, SIR, RPF, etc.). G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

10 Research Program and Methodology Objectives Hedge Fund Replication Non-Gaussian Nonlinear Case Objectives Non-Gaussian or Nonlinear Case Non Gaussian w HF k = w HF k 1 + ν k r k HF = r k whf k + η k η k H with H non Gaussian Nonlinear w HF k = w HF k 1 + ν k rk HF = r k whf k + w HF k,(m+1) r k,(m+1) (s k ) + η k with r k,(m+1) (s k ) nonlinear The objectives of this paper are to e.g., option on S&P 500 explore the nature of HF nonlinearities 1. non Gaussian errors 2. nonlinear factor explore possible remedy for Hedge Fund replication: PF G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

11 Findings: Key Points and Future Developments The Gaussian Distribution Assumption The Gaussian Distribution Assumption Framework Consider rk HF = r k β k + η k β k = β k 1 + ν k η k H with H non Gaussian May be solved using Particle Filters. Assume H is a Skew t distribution S ( µ η,σ η,α η,ν η ) 3 estimation methods (PF #1) ML on parameters of H using the Kalman Filter (KF) tracking errors. (PF #2) GMM to estimate m + 3 parameters (classical MM + two moments for skewness and kurtosis). (PF #3) Same as (PF #2) except ˆα η is forced to 10. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

12 Findings: Key Points and Future Developments The Gaussian Distribution Assumption The Gaussian Distribution Assumption Results with SIR algorithm and particles ˆµ 1Y ˆσ 1Y s γ 1 γ 2 HF LKF PF # PF # PF # π AB σ TE ρ τ ρ S LKF PF # PF # PF # Conclusion With linear assets, higher kurtosis and negative skewness come at the cost of a higher tracking error σ TE. It is not the right way to get at nonlinearities. 1 LKF = linear 6F model estimated using Kalman filter. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

13 Findings: Key Points and Future Developments Nonlinear Assets Taking into account Nonlinear Assets Idea Build Option Factors where w HF k = w HF k 1 + ν k rk HF = r k whf k + w HF k 1,(m+1) r k,(m+1) (s k ) + η k r k,(m+1) (s k ) nonlinear, (e.g., the return of a systematic one-month option selling strategy on S&P 500) s k is the strike of the option at time index k. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

14 Findings: Key Points and Future Developments Nonlinear Assets Exogenous Option Factors Problem Results are data dependent: liquidity, bid/ask spread, size amount (e.g., backtest with VIX). Example Systematic selling at end of month 1M put (resp. call) options with s k = 95% (respectively 100%). Conclusion Dependent on Rebalancing dates (e.g., end of month certainly a most favorable time for selling put options). Implied volatility data and on skew s and bid/ask spread s assumptions. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

15 Findings: Key Points and Future Developments Nonlinear Assets Endogenous Option Factors Estimation procedure Two ways to estimate these strikes (a) Estimate the option strikes separately from the tracking problem: endogenous to the estimation but exogenous to the filter (which can then use KF). (b) The option strike belongs to the state vector of a nonlinear TP system ( ) ( ) ( ) wk wk 1 νk = + s k s k 1 r (HF) m k = w (i) k r(i) k i=1 ε k + w (m+1) k r (m+1) k (s k ) + η k (2) PF η k N ( 0,ση 2 ) and ( ) (( νk 0 N ε k 0 ) ( Q 0, 0 σs 2 )) with Q = diag ( σ 2 1,...,σ m,σ 2 m+1). The vector of unkown parameters to estimate is then θ = { σ 1,...,σ m,σ m+1,σ s,σ η }. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

16 Findings: Key Points and Future Developments Nonlinear Assets Endogenous Option Factors Estimation procedure (Cont d) Direct estimation of system (2) s parameters by PF and (relative) scarcity of data on HF returns is very difficult = use of yet another estimation method Step 1 σ η and σ i for i = 1,...,m: ML considering the linear factor model Step 2 σ m+1 and σ s : grid-based method conditionally on the previous estimates. If f denotes the statistic of interest in the maximization (or minimization) of and if Ω denotes the set of grid points: { ˆσ m+1, ˆσ s } = argmax f (σ m+1,σ s ˆσ 1,..., ˆσ m, ˆσ η ) u.c. (σ m+1,σ s ) Ω. Results are biased yet consistent w.r.t. linear model G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

17 Findings: Key Points and Future Developments Nonlinear Assets Endogenous Option Factors Example: Grid approach applied to the HFRI RV index G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

18 Findings: Key Points and Future Developments Nonlinear Assets Endogenous Option Factors (a) Exposures of the linear assets for the HFRI RV index (b) Option exposures and strikes for the HFRI RV index G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

19 Findings: Key Points and Future Developments Nonlinear Assets HFR Non-Gaussian Nonlinear Case Key points and Future Developments Gaussian assumption KF s tracking errors have skew and excess kurtosis. A remedy: Skew t distribution 1. very difficult direct estimation of parameters in PF 2. no luck with two-step procedure (KF + GMM) = Skew, TE Nonlinear Factor Endogenous and exogenous Robust Methodology 1. Exogenous factors are extremely data dependent 2. Endogenous factors: some success using a grid-based approach and KF; PF code has to be parallelized 3. For now, purely academic exercise 1. TO BE DEVELOPED 2. H Filters minimize worst cases robust to violations of Gaussian and linearity assumptions G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

20 Bibliography Selected References I Thierry Roncalli and Guillaume Weisang Tracking Problems, Hedge Fund Replication and Alternative Beta. Working Paper, available on SSRN, Andrew Lo Hedge Funds: An Analytic Perspective. Princeton University Press, N. Amenc, W. Géhin, L. Martellini, and J-C. Meyfredi. The Myths and Limits of Passive Hedge Fund Replication. Working Paper, June N. Amenc, L. Martellini, J-C. Meyfredi and V. Ziemann. Passive Hedge Fund Replication Beyong the Linear Case. Working Paper, January S. Darolles and G. Mero. Hedge Fund Replication and Factor Models. Working Paper, G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

21 Bibliography Selected References II J. Bai and S. Ng. Evaluating Latent and Observed Factors in Macroeconomics and Finance. Journal of Econometrics, 131(1-2): , A. Diez de los Rios and R. Garcia. Assessing and Valuing the Non-Linear Structure of Hedge Fund Returns. Working Paper, W. Fung and D. A. Hsieh. Empirical Characteristics of Dynamic Trading Strategies: the Case of Hedge Funds. Review of Financial Studies, 10: , T. Roncalli and J. Teiletche. An Alternative Approach to Alternative Beta. Journal of Financial Transformation, 24:43-52, Available at SSRN: G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

22 Bibliography Selected References III R. C. Merton. On Market Timing and Investment Performance. I. An Equilibrium Theory of Value for Market Forecasts. Journal of Business, 54(3): , S. Arulampalam, S. Maskell, N.J. Gordon and T. Clapp. A Tutorial on Particle Filters for Online nonlinear/non-gaussian Bayesian Tracking. IEEE Transaction on Signal Processing, 50(2): , Februrary Lars Jaeger. Alternative Beta Strategies and Hedge Fund Replication. John Wiley & Sons, NY, 1 st edition, G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22

23 Statistics Description Statistics Description ˆµ 1Y is the annualized performance; π AB the proportion of the HFRI index performance explained by the clone; σ TE is the yearly tracking error; ρ, τ and ρ S are respectively the linear correlation, the Kendall tau and the Spearman rho between the monthly returns of the clone and the HFRI index; s is the sharpe ratio; γ 1 is the skewness; γ 2 is the excess kurtosis. G. Weisang, T. Roncalli (Bentley Univ., Lyxor AM) Exploring non linearities in HF: Particle Filters January 28-29, / 22