Robust Optimisation & its Guarantees

Transcription

1 Imperial College London APMOD 9-11 April, 2014 Warwick Business School ness with D. Kuhn, P. Parpas, W. Wiesemann, R. Fonseca, M. Kapsos, S.Žaković, S. Zymler

2 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

3 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

4 Introduction to Generic problem: x X : decision variables y: problem specific data Uncertainty in y due to: Inaccurate forecasts min f (x, y) x X Inaccurate assumptions (e.g. distributions) etc. ness Disregarding uncertainty = bad decisions...

5 Deterministic ness alive (E [position]) = true, but E [alive (position)] = false!

6 Example: Macroeconomic Policy with Rival Models minimise x {λ J HMT (Y HMT (x), x)+(1 λ) J NIESR (Y NIESR (x), x)} 1, 2 ness J(HMT) J(NIESR) λ J(HMT) + (1 λ) J(NIESR) λ 1 Becker et al. [1986] 2 R et al. [2000]

7 Uncertainty Set Set U for y, y U with high confidence. Typical U: Discrete: U = {ŷ 0,..., ŷ i,..., ŷ k }, i I. Interval: U = {y : y y y}. Ellipsoid: U = {y : Ay 2 δ} worst-case : best decision x X in view of worst possible scenario y U Minimax problem: ness min max x X y U f (x, y).

8 Discrete Minimax { min x X max λ i λi J i (x) } i λi = 1, λ i 0, i min x X max i U {J i (x)} 3 min x X,v R 1 {v v J i (x), i U} ness 3 Obasanjo et al. [2010]

9 Saddlepoint Solution f (x, y) convex in x concave in y minimax saddlepoint elegant models powerful algorithms ness f (x, y) f (x, y ) f (x, y ) x X, y U.

10 LP Duality For every Primal a Dual can be constructed: Primal: minimise c T x subject to Ax b, x 0 Dual: maximise b T y subject to A T y c, y 0 (P) feasible: ˆx R n such that Aˆx b, ˆx 0 (P) (D) ness = Opt(P) = Opt(D) (Strong Duality)

11 Dualising minimax Original: Inner: Inner dual: Original equivalent: min x 0:Ax b max y 0:W y h ct x + d T y + x T Qy }{{} max y 0:W y h ct x + (Qx + d) T y }{{} min s.t. h T λ + c T x λ 0, W T λ Qx + d { } min c T x + h T λ λ 0, W T λ Qx + d x 0:Ax b ness

12 Multiple Maxima Global optimisation: Generally, multiple (global) maxima for y: ness f (x, y) = x 2 + y 2 f (x, y ) = 4 x = 0, y = 2 2

13 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

14 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

15 Mean-Variance Portfolio Optimal Asset Allocation Compute w R n for high return low risk ρ(w) Mean-Variance Portfolio : { } max w T µ λw T Σw w R w W n Expected return: w T µ Risk: ρ(w) w T Σw ness Risk aversion: λ λ : max w R n{w T µ w T Σw υ; w W}

16 Mean-Variance Portfolio Risk and Return 4 w: optimal risk-return Γ, r: worst-case Worst-case optimal w, r, Γ: min w R n max w T Σ(Γ, r)w = w T (Γ rr T )w Γ U Γ, r U r s.t Γ: Second moment; Γ rr T 0 min w T r R, r U r w W. Consistent mean covariance ness 4 Ye et al. [2012]

17 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

18 Minimum risk rival covariances [Kapsos R: 2014] Covariances unknown! Estimated with error. statistics: shrinkage to reduce estimation error. Example: Shrinkage: Averaging sample structured estimator Step 1 Step 2 Σ S = δ ˆΣ 1 + (1 δ )ˆΣ 2 δ = arg min δ E( δ ˆΣ 1 + (1 δ)ˆσ 2 Σ 2 ) min w W {w T Σ S w} ness

19 Simultaneously: best w worst case Σ Estimator Σ S = m i=1 δ i ˆΣ i ; δ {[0, 1], i δ i = 1}. model 5 min θ θ R,w W min max w W δ min max w W δ w Σ S (δ)w δ i w ˆΣ i w min max w ˆΣ i w. w W i i s.t. w ˆΣ i w θ, i = 1,..., m. ness 5 Kapsos and R [2014]

20 Equally-weighted Risk Contribution [Kapsos, Christofides, Parpas R: 2012] Assets contribute equal risk Risk measure: variance minimum var + diversification In real life... Σ unknown Estimation error. Time-varying ρ(w) w i = ρ(w) w j, i, j ρ(w) = w T Σw min w { w T Σw i ln w i c } Model { } min max w T Σw ln w w W Σ S i c i ness Discrete or continuous uncertainty sets for Σ: 6 6 Kapsos et al. [2012]

21 Equally-weighted Risk Contribution Discrete uncertainty Convex min θ n s.t. ln w i c i=1 w T Σ j w θ, j = 1,..., m w 0 S = { Σ j }, j = 1,..., m. Continuous uncertainty SIP min max w T Σw w Σ n s.t. ln w i c i=1 Σ l Σ Σ u w 0 Σ 0. ness

22 Z Ω ratio [Kapsos, Zymler, Christofides R: 2011] Performance measure for non-normal return distribution Ω ratio = Light Grey Area Dark Grey Area ness Maximising Ω ratio max w { + τ [1 F (w y)]dy τ F (w y)dy } w W Quasi-convex problem Solution: family of convex problems or fractional LP 0 1 Y X 1 0 1

23 Ω ratio maximisation Family of convex problems - continuous distributions 7 { } max δ(w E p (r) τ) (1 δ)e p ([τ w r] + ) w W w solve for varying δ - keep solution with max Ω ratio Fractional LP - discrete distributions { } w T r τ max w j [τ w T r j ] + w p j i = 1; w w w i ness 7 Kapsos et al. [2011]

24 Worst-case Ω ratio [Kapsos, Christofides R: 2014] Definition Worst-case Ω (WCΩR) - fixed w W - wrt set of probability distributions P or Π 8 : Discrete analogue w E p (r) τ WCΩR(w) inf p P E p ([τ w r] + ), WCΩR(w) inf π Π w (R π) τ ] +. π [τ1 (Rw) ness Density functions only known to belong to set P or Π. 8 Kapsos et al. [2014]

25 Mixture distribution uncertainty { l } p(r) P = λ i p i (r) : λ i Λ, i=1 λ i : unknown mixture weight for probability distribution p i (r). max θ w W,θ R ness s.t. δ (w T E p i (r) τ) (1 δ) E p i ([τ w T r] + ) θ i = 1,..., l.

26 Two distributions with same mean variance Figure: Two distributions with same mean and variance. dotted distribution is a symmetric normal distribution. dark line shows a negatively skewed distribution with fat tails. Sharpe ratio is indifferent between the two. A rational investor will always prefer the red distribution. ness

27 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

28 Portfolio [Zymler, Kuhn R: 2010] r: Asset returns. Portfolio return: w T r. Max return 9,10 : max w W w T r r U r {r (r µ) T Σ 1 (r µ) δ 2 } optimisation worst-case: ness max min w W r U r w T r max w W w T µ δ Σ 1/2 w 2. 9 Ben-Tal and Nemirovski [1999] 10 R and Howe [2002]

29 Known means µ Σ 0, r, but not entire distribution. P set of all distributions with mean µ cov Σ. For any w W, p distributions P δ = p/(1 p) = inf P{w T r min P P r U r with probability p better return than worst-case 11. Non-inferiority insurance: w T r} = p θ = max w W min r U r w T r = w T r θ r U r. ness 11 El Ghaoui et al. [2003]

30 Support Information Support on r: r: realization of r. Support with U r : B = {r : l r u} (or: B = {r : r 0}) U r = {r B (r µ) T Σ 1 (r µ) δ 2 } ness

31 Duality Strong convex duality: max min w T r max w W r U r w W,s 0 µt (w s) δ s: dual variable. Consider ρ: ρ(w) = min s 0 ρ coherent risk-measure Σ 1/2 (w s). 2 µ T (w s) + δ Σ 1/2 (w s). 2 ness max worst-case return min coherent risk!

32 Modelling Option Returns Option weights: w d Returns r d f ( r) Call j strike K j call price C j on underlying i, price S i 0 : r d j = f j ( r) = max { 0, S i 0 r i K j } C j = max {0, a j + b j r i } ; a j = K j C j, b j = S i 0 C j. Put j with premium P j : r d j General form: = f j ( r) = max {0, a j + b j r i } ; a j = K j P j, b j = S i 0 P j. ness r d = f ( r) = max{0, a + B r}

33 Incorporating Options in Framework Portfolio return r p = w T r + (w d ) T r d. w d 0; 1 T w + 1 T w d = 1 - else, too risky nonconvex max-min: Equivalent SIP: maximise w,w d,φ max (w,w d ) W φ min w T r + (w d ) T r d r U r, r d =f (r) s.t. w T r + (w d ) T r d φ r U r, r d = f (r) (w, w d ) W ness At optimality φ worst-case portfolio return, r U r.

34 Incorporating Options in Framework Portfolio return r p = w T r + (w d ) T r d. w d 0, 1 T w + 1 T w d = 1. max-min: max (w,w d ) W Equivalent SOCP: min w T r + (w d ) T r d r U r, r d =f (r) ness max w,w d,φ, y, s s.t. φ µ T (w + B T y s) δ Σ 1/2 (w + B T y s) + a T y φ 2 (w, w d ) W, 0 y w d, s 0 At optimality φ worst-case portfolio return, r U r.

35 Insured Portfolio Non-inferiority guarantee at optimality : w T r + (w d ) T r d φ r U r, r d = f (r) Extreme events: r outside U r no guarantees! Control deterioration below φ for any realisation r: w T r + (w d ) T r d θφ r B, r d = f (r), θ [0, 1]. Insurance guarantee expressed as fraction of φ: Non-inferiority guarantee is no hedge against extremes Prevents overly expensive insurance. ness

36 Guarantee Tradeoff Insured robust portfolio optimisation: max w,w d,φ φ subject to w T r + (w d ) T r d φ r U r, r d = f (r) w T r + (w d ) T r d θφ (w, w d ) W. Has SOCP reformulation tractable. r B, r d = f (r) Exposes tradeoff: non-inferiority vs insurance guarantees: U r increases φ decreases. { insurance level θφ φ decreases associated insurance cost/premium ness

37 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

38 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

39 FX - Triangulation [Fonseca, Wiesemann, Zymler, Kuhn R, 2011] n currencies: E i domestic/unit ith foreign E 0 i E i : today future spot rate e i = E i /E 0 i : currency i return - uncertain EUR/USD GBP/USD cross-rate EUR/GBP No-arbitrage: non-convex constraint ness e i 1 n(n 1) ce ij = 1 i, j = 1,..., n; (ij) = 1,..., e j 2 Uncertainty interval for cross-rates ce ij = e j e i ; Convex: n(n 1) inequalities for cross rates ce ce ij ce ce e i e j ce e i, e i 0

40 Currency Return Uncertainty Θ e Θ e = {e 0 (e ē) Σ 1 (e ē) δ 2 Ae 0} A: triangular relationship among rates ness max w R n { min w T e e Θ e } w T 1 = 1, w, 0

41 Solution To solve problem ness max w R n min e Θ e w T e s. t. w T 1 = 1 w 0

42 Solution To solve problem Start inner min wrt FX return max w R n min e Θ e w T e s. t. w T 1 = 1 w 0 min e R n w T e s. t. Σ 1/2 (e ē) δ Ae 0 e 0 ness

43 Dual Problem SOCP: primal dual have same objective value. Dual: max v,k,y ē T (w s) δv s. t. Σ 1/2 (w s) = v s w s, v 0 A T k + y = s ness

44 Dual Problem SOCP: primal dual have same objective value. Dual: max v,k,y ē T (w s) δv s. t. Σ 1/2 (w s) = v s w s, v 0 A T k + y = s Replace original problem: max w,k,y s. t. ē T (w s) δ Σ 1/2 (w s) φ φ s w w T 1 = 1 w, s 0 A T k + y = s ness

45 Hedging Integrating Options Option returns: e d f (e) = max{0, a p + b p e}, a p = K p { e d f (e) = max 0, K E 0 } e p, b p = E 0 To guard FX returns outside Θ e, investing in currency O s, with minimum return guarantee: ρ. p ness

46 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

47 International Portfolio n assets m currencies - both returns uncertain Allocation matrix O: { 1, if ith asset in jth currency o ij = 0, otherwise 2 returns for each asset i: Local asset: ra i = P i /Pi 0 Currency: re j = E j /Ej 0 Hedging: Quanto options - linking foreign equity with forward FX Basic max w min r a,r e Ξ { [diag(r a)or e] T w) } w T 1 = 1, w 0 ness { ([ Ξ = r a, r e 0 : Ar e 0 r a r e ] ]) T ([ [ rā Σ 1 r e r a r e ] ]) } [ rā δ 2 r e

48 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

49 One-Stage to Two-Stages x ξ y(ξ) t first stage second stage decision decision Approximating nonlinear decision rule by affine rules. ness (a) Nonlinear decision rule a T x + c T y(ξ) b(ξ), ξ Ξ (b) Affine decision rule a T x + c T (y 0 + Y ξ) b(ξ), ξ Ξ; (y(ξ) y 0 + Y ξ) optimisation to reformulate the constraints.

50 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

51 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

52 Example: Minimax Hedging Strategy Problem Option = contract entitling holder to buy/sell specific # shares at a certain time - for agreed price Hedging option risk mainly confined to option seller due to liability contingent on asset underlying option Seller of option needs to position to minimise potential negative impact of such liability Selling option risky with potentially unlimited loss - buying option mainly nonrisky - insurance at a price and minor risk is potential loss if option not exercised ness Strategy minimises (by choosing # shares to hold - instead of all contracted shares) worst-case potential hedging error (wrt future stock price).

53 Example: Minimax Hedging Strategy Problem formulation 12 : min x t s.t. max f (x t, yt+1), S yt+1 S y S,lower t+1 yt+1 S y S,upper t+1 ness 12 R and Howe [2002]

54 Example: Hedging Error Hedging error : HE = N(B t B t+1 (y S t+1 )) + x t(y S t+1 y S t ) N : contracted # shares B t : call price x t : # shares to hold y S t+1 Rk : stock price U d R k+1 : desired potential HE transaction c Minimax hedging strategy Minimise max potential HE between t t + 1: f (x t, y S t+1) = 1 2 < U(x t, y S t+1) U d, Q(U(x t, y S t+1) U d ) > ness

55 Example: Hedging Error r: risk free interest rate, t: hedging interval ˆK: transaction cost (% of transaction volume) U(x t, y S t+1) = U 1 (x t, y S t+1) = k i=1 U 1 (x t, y S t+1 ). U 2 (x t ) x i,t (y S i,t+1 y S i,t) + N i (B S i,t B i,t (y S + k i=1 i,t+1)) [ (x i,t x i,t 1 )y S i,t + C i,t 1 (1 + r t)].r t ness

56 Example: Hedging Error C i,t 1 = C i,t 2 (1 + r t) U 2 (x t ) = (x i,t 1 x i,t 2 )yi,t 1 S ˆK (x i,t 1 x i,t 2 )yi,t 1. S U 1,2 (x 1,t ). U k,2 (x k,t ) ness with U i,2 (x i,t ) = ˆK(x i,t x i,t 1 )y S i,t.

57 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

58 Example: Global Strucural Model Distinguishability Modelling Modelling process systems e.g chemical reactors, crystallisation un, fermentation Possible to propose more than one mathematical model to describe underlying system We wish to determine whether the mathematical structure of these models can be distinguished from one another Goal: determine best model Two models fermentation of baker s yeast in a batch reactor ness Approximate solution of these models by their Fleiss functional expansions

59 Example: Global Strucural Model Distinguishability Structural distinguishability problem 13 : min max θ,θ x Φ D = 4 [L [1] k k=1 (x, θ) L[2] k (x, θ )] 2 s.t θ i 1.0, i = 1,..., θ i 1.0, i = 1,..., x , 10 2 x ness θ θ : parameter vectors from two models x: vector of state/response variables. 13 Žaković and R [2003]

60 Example: Global Strucural Model Distinguishability L k i, i = 1, 2, 3, 4, k = 1, 2: coefficients of functional expansions for solution trajectories of two models: k = 1 L [1] 1 = ( θ 1x 2 θ 2 + x 2 θ 4 )x 1 L [1] 2 = ( θ 1x 2 θ 4 )x 1 x 1 x 2 ( θ 1x 2 θ 2 + x 2 θ 2 + x 2 (θ 2 + x 2 ) 2 ) L [1] 3 = θ 1x 1 x 2 θ 3 (θ 2 + x 2 ) L [1] 4 = θ 1 x 1 x 2 θ 3 (θ 2 + x 2 ) x θ 1 1x 2 ( θ 3 (θ 2 + x 2 ) + θ 1 x 2 θ 3 (θ 2 + x 2 ) 2 ) θ 1 ness

61 Example: Global Strucural Model Distinguishability k = 2 L [2] 1 = (θ1x 2 θ3)x 1 L [2] 2 = (θ1x 2 θ3)x 1 x 2 θ1x 1 L [2] 3 = θ 1 x 2x 1 θ 2 L [2] 4 = 2 θ 1 x 2x 1 θ 2 ness

62 Outline ness Stock-Only Risk Parity Ω Ratio Additional via Options Currency-Only Stocks Currencies Hedging Structural Model Distinguishability ness

63 : Under Uncertainty Intuitive approach to data uncertainty Immunises against effects of uncertainty Out-of-sample improvements with RO Non-inferiority property further guarantees. No substitute to wisdom! Algorithms ness Good algorithms for convex-concave problems. Multiple global optima issues for nonconvex problems.

64 R. G. Becker, B. Dwolatzky, E. Karakos, and B. R. simultaneous use of rival models in policy optimisation. Economic Journal, 96(382):pp , A. Ben-Tal and A. Nemirovski. solutions of uncertain linear programs. Oper. Res. Lett., 25:1 13, Laurent El Ghaoui, Maksim Oks, and Francois Oustry. Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res., 51(4): , M. Kapsos and B. R. minimum variance and shrinkage methods. Working paper, M. Kapsos, S. Zymler, N. Christofides, and B. R. Optimizing the omega ratio using linear programming. Journal of Computational Finance, M. Kapsos, N. Christofides, P. Parpas, and B. R. equally-weighted risk contribution. Under revision, M. Kapsos, N. Christofides, and B. R. Worst-case robust omega ratio. European Journal of Operational Research, ness

65 Title E. Obasanjo, G. Tzallas-Regas, and B. R. An interior-point algorithm for nonlinear minimax problems. J. Optim. ory Appl., 144(2): , R and Melendres Howe. Algorithms for worst-case design and applications to risk management. Princeton University Press, Princeton, NJ, R, Robin G. Becker, and Wolfgang Marty. min-max portfolio strategies for rival forecast and risk scenarios. J. Econom. Dynam. Control, 24(11-12): , K. Ye, P. Parpas, and B. R. portfolio optimization: a conic programming approach. Computational Optimization and Applications, 52(2): , Stanislav Žaković and Berc R. Semi-infinite programming and applications to minimax problems. Ann. Oper. Res., 124:81 110, ness