Worst-Case Value-at-Risk of Derivative Portfolios

Transcription

1 Worst-Case Value-at-Risk of Derivative Portfolios Steve Zymler Berç Rustem Daniel Kuhn Department of Computing Imperial College London Thalesians Seminar Series, November 2009

2 Risk Management is a Hot Topic Since the meltdown of some of Wall Street s biggest names, there has been a great deal of talk, even in quant circles, that this widespread institutional reliance on VaR was a terrible mistake. At the very least, the risks that VaR measured did not include the biggest risk of all: the possibility of a financial meltdown. Wall Street s Era of Risk Mismanagement, New York Times, DealBook, January 2009

3 Risk Management is a Hot Topic To me, VaR is charlatanism because it tries to estimate something that is not scientifically possible to estimate, namely the risks of rare events. It gives people misleading precision that could lead to the buildup of positions by hedgers. It lulls people to sleep. All that because there are financial stakes involved. Nassim Taleb The World According to Nassim Taleb, DerivativesStrategy.com, January 1997

4 Outline 1 Introduction to Convex Optimization 2 Portfolio Optimization and Value-at-Risk 3 Worst-Case Value-at-Risk and Robust Optimization 4 5 Numerical Results: Beating the S&P Conclusions

5 Convex Optimization Consider the general optimization problem: minimize f 0 (x) subject to f i (x) 0, i = 1,..., m x R n (P) where: f0 (x) is the objective function fi (x) 0 are the constraints x R n are the decision variables If f 0, f 1,..., f m are convex in x then: (P) is polynomial time solvable (tractable) If not then: (P) is a global optimization problem NP-hard (intractable)

6 Convex Conic Optimization The class of Conic Programs is comprised of: Linear Programs (LP): minimize subject to c T x Ax b Second-Order Cone Programs (SOCP): minimize c T x subject to A i x + b i 2 f T i x + d i, i = 1,..., m Semidefinite Programs (SDP): minimize c T x n subject to F 0 + F i x i 0 Can all be solved using very efficient software packages i=1

7 Second-Order Cone or Ice Cream Cone { (x, y, z) : } x 2 + y 2 z z y x

8 Positive Semidefinite Cone {[ ] x y S 2 : y z [ ] x y y z } z y x 0.5 1

9 Linear Programming Duality For every Primal Problem a Dual Problem can be constructed: Primal Problem: minimize c T x subject to Ax b, x 0 (P) Dual Problem: maximize b T y subject to A T y c, y 0 (D) If (P) is feasible, i.e.: ˆx R n such that Aˆx b, ˆx 0 then Opt(P) = Opt(D) (Strong Duality)

10 Portfolio Optimization Consider a market consisting of m assets. Optimal Asset Allocation Problem Choose the weights vector w R m to make the portfolio return high, whilst keeping the associated risk ρ(w) low. Portfolio optimization problem: Popular risk measures ρ: minimize ρ(w) w R m subject to w W. Variance Markowitz model (1952) Value-at-Risk Focus of this talk

11 Value-at-Risk: Definition Let r denote the random returns of the m assets. The portfolio return is therefore w T r. Value-at-Risk (VaR) The minimal level γ R such that the probability of w T r exceeding γ is smaller than ɛ. { { VaR ɛ (w) = min γ : P γ w T r } } ɛ

12 Theoretical and Practical Problems of VaR VaR lacks some desirable theoretical properties: Not a coherent risk measure. Needs precise knowledge of the distribution of r. Non-convex function of w VaR minimization intractable. To optimize VaR: resort to VaR approximations. Example: assume r N (µ r, Σ r ), then VaR ɛ (w) = µ T r w Φ 1 (ɛ) w T Σ r w, Normality assumption unrealistic (CLT not applicable) may underestimate the actual VaR! Very difficult to know true distribution: model risk

13 Worst-Case Value-at-Risk Only know means µ r and covariance matrix Σ r 0 of r. Let P r be the set of all distributions of r with mean µ r and covariance matrix Σ r. Worst-Case Value-at-Risk (WCVaR) { { WCVaR ɛ (w) = min γ : P γ w T r } } ɛ P P r WCVaR is immunized against uncertainty in P: distributionally robust. Unless the most pessimistic distribution in P r is the true distribution, actual VaR will be lower than WCVaR.

14 Worst-Case Value-at-Risk Only know means µ r and covariance matrix Σ r 0 of r. Let P r be the set of all distributions of r with mean µ r and covariance matrix Σ r. Worst-Case Value-at-Risk (WCVaR) { { } } WCVaR ɛ (w) = min γ : max P P P r γ w T r ɛ WCVaR is immunized against uncertainty in P: distributionally robust. Unless the most pessimistic distribution in P r is the true distribution, actual VaR will be lower than WCVaR.

15 Worst-Case Value-at-Risk El Ghaoui et al. have shown that WCVaR ɛ (w) = µ T w + κ(ɛ) w T Σw, where κ(ɛ) = (1 ɛ)/ɛ. Key idea behind proof: For any fixed γ R and w W, define S as S = { r R m : γ w T r } Then we have We also define I S as max P { γ w T r } = max P { r S} P P r P P r I S (r) = { 1 if r S 0 otherwise

16 Resolving Integration via Dualization We can write π = max P P r π = max µ M + s. t. Key idea: construct the dual! P { r S} as I S (r)µ(dr) R m µ(dr) = 1 Rm Rm rµ(dr) = µ r R m rr T µ(dr) = Σ r + µ r µ T r, (P) min y 0 + y T µ r + Y, Σ r + µ r µ T r s. t. y 0 R, y R m, Y S m y 0 + y T r + Y, rr T I S (r) r R m (D) It can be shown that: Opt(P) = Opt(D)

17 Robust Optimization Perspective on WCVaR It can be shown that: WCVaR ɛ (w) = max r U ɛ w T r, where the ellipsoidal uncertainty set U ɛ is defined as { U ɛ = r R m : (r µ r ) T Σ 1 r (r µ r ) κ(ɛ) 2} Therefore, min WCVaR ɛ(w) min max w T r w W w W r U ɛ Can be interpretted as a game versus nature which acts as a purely adverserial player.

18 Robust Optimization Perspective on WCVaR Stock C Stock A Stock B

19 Extensions Incorporate Moment Uncertainty: Mean Uncertainty: U µ = { µ : µ µ µ } Covariance Matrix Uncertainty: U Σ = { Σ 0 : Σ Σ Σ } Inject moment uncertainty into ellipsoid: U ɛ = { r R m : Σ r U Σ, (r µ r ) T Σ 1 r (r µ r ) κ(ɛ) 2} Incorporate Support Information: Let B be the smallest set such that P { r B} = 1 Can be used to make WCVaR less conservative: U ɛ = { r B : (r µ r ) T Σ 1 r (r µ r ) κ(ɛ) 2}.

20 Worst-Case VaR for Derivative Portfolios Assume that the market consists of: n m basic assets with returns ξ, and m n derivatives with returns η. ξ are only risk factors. We partition asset returns as r = ( ξ, η). Derivative returns η are uniquely determined by basic asset returns ξ. There exists f : R n R m with r = f ( ξ). f is highly non-linear and can be inferred from: Contractual specifications (option payoffs) Derivative pricing models

21 Worst-Case VaR for Derivative Portfolios WCVaR is applicable but not suitable for portfolios containing derivatives: Moments of η are difficult to estimate accurately. Disregards perfect dependencies between η and ξ. WCVaR severly overestimates the actual VaR, because: Σr only accounts for linear dependencies Uɛ is symmetric but derivative returns are skewed

22 Generalized Worst-Case VaR Framework We develop two new Worst-Case VaR models that: Use first- and second-order moments of ξ but not η. Incorporate the non-linear dependencies f Generalized Worst-Case VaR Let P denote set of all distributions of ξ with mean µ and covariance matrix Σ. { { } } min γ : max P γ w T f ( ξ) ɛ P P When f ( ξ) is: convex polyhedral Worst-Case Polyhedral VaR (SOCP) nonconvex quadratic Worst-Case Quadratic VaR (SDP)

23 Piecewise Linear Portfolio Model Assume that the m n derivatives are European put/call options maturing at the end of the investment horizon T. Basic asset returns: r j = f j ( ξ) = ξ j for j = 1,..., n. Assume option j is a call with strike k j and premium c j on basic asset i with initial price s i, then r j is f j ( ξ) = 1 max {0, s i (1 + c ξ } i ) k j 1 j { } = max 1, a j + b j ξi 1, where a j = s i k j, b j = s i. c j c j Likewise, if option j is a put with premium p j, then r j is { } f j ( ξ) = max 1, a j + b j ξi 1, where a j = k j s i, b j = s i. p j p j

24 Piecewise Linear Portfolio Model In compact notation, we can write r as ( ) ξ r = f ( ξ) = { }. max e, a + B ξ e Partition weights vector as w = (w ξ, w η ). No derivative short-sales: w W = w η 0. Portfolio return of w W can be expressed as w T r = w T f ( ξ) = (w ξ ) T ξ + (w η ) T max { } e, a + B ξ e.

25 Worst-Case Polyhedral VaR Use the piecewise linear portfolio model: { } w T f ( ξ) = (w ξ ) T ξ + (w η ) T max e, a + B ξ e. Worst-Case Polyhedral VaR (WCPVaR) For any w W, we define WCPVaR ɛ (w) as { { } } WCPVaR ɛ (w) = min γ : max P γ w T f ( ξ) ɛ. P P

26 Worst-Case Polyhedral VaR: Convex Reformulations Theorem: SDP Reformulation of WCPVaR WCPVaR of w can be computed as an SDP: WCPVaR ɛ(w) = min γ s. t. M S n+1, y R m n, τ R, γ R Ω, M τɛ, M 0, τ 0, 0 y w η» 0 w ξ + B T y M + (w ξ + B T y) T τ + 2(γ + y T a e T w η 0 ) Where we use the second-order moment matrix Ω:» Σ + µµ T µ Ω = µ T 1

27 Worst-Case Polyhedral VaR: Convex Reformulations Theorem: SOCP Reformulation of WCPVaR WCPVaR of w can be computed as an SOCP: WCPVaR ɛ(w) = min 0 g w µt (w ξ + B T g) + κ(ɛ) Σ 1/2 (w ξ + B T g)... η 2... a T g + e T w η SOCP has better scalability properties than SDP.

28 Robust Optimization Perspective on WCPVaR WCPVaR minimization is equivalent to: min max w W r Uɛ p w T r. where the uncertainty set Uɛ p R m is defined as ξ R n such that Uɛ p = r Rm : (ξ µ) T Σ 1 (ξ µ) κ(ɛ) 2 and r = f (ξ) Unlike U ɛ, the set U p ɛ is not symmetric!

29 Robust Optimization Perspective on WCPVaR

30 Example: WCPVaR vs WCVaR Consider Black-Scholes Economy containing: Stocks A and B, a call on stock A, and a put on stock B. Stocks have drifts of 12% and 8%, and volatilities of 30% and 20%, with instantaneous correlation of 20%. Stocks are both $100. Options mature in 21 days and have strike prices $100. Assume we hold equally weighted portfolio. Goal: calculate VaR of portfolio in 21 days. Generate 5,000,000 end-of-period stock and option prices. Calculate first- and second-order moments from returns. Estimate VaR using: Monte-Carlo VaR, WCVaR, and WCPVaR.

31 Example: WCPVaR vs WCVaR probability (%) portfolio loss At confidence level ɛ = 1%: VaR WCVaR unrealistically high: 497%. WCVaR is 7 times larger than WCPVaR. WCPVaR is much closer to actual VaR. Monte-Carlo VaR Worst-Case VaR Worst-Case Polyhedral VaR Confidence level (1-ε)%

32 Delta-Gamma Portfolio Model m n derivatives can be exotic with arbitrary maturity time. Value of asset i = 1... m is representable as v i ( ξ, t). For short horizon time T, second-order Taylor expansion is accurate approximation of r i : r i = f i ( ξ) θ i + T i ξ ξ T Γ i ξ i = 1,..., m. Portfolio return approximated by (possibly non-convex): w T r = w T f (ξ) θ(w) + (w) T ξ ξ T Γ(w) ξ, where we use the auxiliary functions m m θ(w) = w i θ i, (w) = w i i, Γ(w) = i=1 i=1 We now allow short-sales of options in w m w i Γ i. i=1

33 Worst-Case Quadratic VaR Worst-Case Quadratic VaR (WCQVaR) For any w W, we define WCQVaR as { { min γ : max P γ θ(w) (w) T ξ 1 } P P 2 ξ T Γ(w) ξ Theorem: SDP Reformulation of WCQVaR WCQVaR can be found by solving an SDP: WCQVaR ɛ (w) = min γ s. t. M S n+1, τ R, γ R } ɛ Ω, M τɛ, M 0, τ 0, [ ] Γ(w) (w) M + (w) T 0 τ + 2(γ + θ(w)) There seems to be no SOCP reformulation of WCQVaR.

34 Robust Optimization Perspect on WCQVaR WCQVaR minimization is equivalent to: where min max w W Z Uɛ q Q(w), Z [ 1 Q(w) = 2 Γ(w) 1 2 (w) ] 1, 2 (w)t θ(w) and the uncertainty set Uɛ q S n+1 is defined as { [ ] } Uɛ q X ξ = Z = ξ T S n+1 : Ω ɛz 0, Z 0 1 U q ɛ is lifted into S n+1 to compensate for non-convexity.

35 Robust Optimization Perspect on WCQVaR There is a connection between U ɛ R m and U q ɛ S n+1. If we impose: w W = Γ(w) 0 then robust optimization problem reduces to: min w W max w T r r Uɛ q where the uncertainty set U q ɛ Uɛ q = r Rm : R m is defined as ξ R n such that (ξ µ) T Σ 1 (ξ µ) κ(ɛ) 2 and r i = θ i + ξ T i ξt Γ i ξ i = 1,..., m Unlike U ɛ, the set U q ɛ is not symmetric!

36 Robust Optimization Perspective on WCQVaR

37 Example: WCQVaR vs WCVaR Now we want to estimate VaR after 2 days (not 21 days). VaR not evaluated at option maturity times use WCQVaR (not WCPVaR). Use Black-Scholes to calculate prices and greeks. probability (%) VaR Monte-Carlo VaR Worst-Case VaR Worst-Case Quadratic VaR portfolio loss Confidence level (1-ε)% At ɛ = 1%: WCVaR still 3 times larger than WCQVaR.

38 Index Tracking using Worst-Case Quadratic VaR Total test period: Jan. 2nd, 2004 Oct. 10th, Estimation Window: 600 days. Out-of-sample returns: robust strategy with options robust strategy without options S&P 500 (benchmark) Relative Wealth Period Outperformance: option strat 56%, stock-only strat 12%. Sharpe Ratio: option strat 0.97, stock-only strat Allocation option strategy: 89% stocks, 11% options.

39 Extensions Introduce additional risk-factors such as stochastic volatility. No new theory needed: delta-gamma-vega-vomma approximation is still quadratic. Copulas also model non-linear dependencies. They could be integrated into Worst-Case VaR framework.

40 Conclusions Worst-Case VaR makes weak assumptions about probability distributions. Can be used to compute as well as optimize the risk of large scale derivative portfolios. Options are risky, but their use in a robust optimization framework can offer substantial benefits. No quantitative risk method can completely replace sound investment analysis!

41 Questions? Paper available on optimization-online.org c 2007 Salvador Dalí, Gala-Salvador Dalí Foundation/Artists Rights Society (ARS), New York