Post on 06-Apr-2018
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Worst-Case Value-at-Risk of Non-Linear
Portfolios
Steve Zymler Daniel Kuhn Ber Rustem
Department of ComputingImperial College London
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
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Portfolio Optimization
Consider a market consisting of massets.
Optimal Asset Allocation Problem
Choose the weights vector w Rm to make the portfolio returnhigh, whilst keeping the associated risk (w) low.
Portfolio optimization problem:
minimizewRm
(w)
subject to w
W.
Popular risk measures :
Variance Markowitz model Value-at-Risk Focus of this talk
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Value-at-Risk: Definition
Let r denote the random returns of the massets.
The portfolio return is therefore wTr.
Value-at-Risk (VaR)
The minimal level R such that the probability of wTrexceeding is smaller than .
VaR(w) = min
: P
wTr
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
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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(r,r), then
VaR(w) =Tr
w
1()wTrw,
Normality assumption unrealistic may underestimate the actual VaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
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Worst-Case Value-at-Risk
Only know means r and covariance matrix r 0 of r. Let Pr be the set of all distributions of r with mean r and
covariance matrix r.
Worst-Case Value-at-Risk (WCVaR)
WCVaR(w) = min
: sup
PPr
P
wTr
WCVaR is immunized against uncertainty in P:
distributionally robust.
Unless the most pessimistic distribution in Pr is the truedistribution, actual VaR will be lower than WCVaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
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Robust Optimization Perspective on WCVaR
El Ghaoui et al. have shown that
WCVaR(w) = Tw+ ()wTw,
where () =
(1 )/. Connection to robust optimization:
WCVaR(w) = maxrU
wTr,
where the ellipsoidal uncertainty setU is defined as
U = r : (r r)
T1
r
(rr)
()2 .
Therefore,
minwW
WCVaR(w) minwW
maxrU
wTr.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
C
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Worst-Case VaR for Derivative Portfolios
Assume that the market consists of:
n mbasic assets with returns , and m nderivatives with returns . are only risk factors.
We partition asset returns as r = (, ).
Derivative returns are uniquely determined by basic
asset returns . There exists f : Rn Rm with r = f(). f is highly non-linear and can be inferred from:
Contractual specifications (option payoffs) Derivative pricing models
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
W C V R f D i i P f li
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Worst-Case VaR for Derivative Portfolios
WCVaR is applicable but not suitable for portfolioscontaining derivatives:
Moments of are difficult to estimate accurately.
Disregards perfect dependencies between and .
WCVaR severly overestimates the actual VaR, because:
r only accounts for linear dependencies
U is symmetric but derivative returns are skewed
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
G li d W t C V R F k
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Generalized Worst-Case VaR Framework
We develop two new Worst-Case VaR models that:
Use first- and second-order moments of but not . Incorporate the non-linear dependencies f
Generalized Worst-Case VaR
Let Pdenote set of all distributions of
with mean andcovariance matrix .
min
: sup
PP
P
wTf()
When f() is: convex polyhedral Worst-Case Polyhedral VaR (SOCP) nonconvex quadratic Worst-Case Quadratic VaR (SDP)
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Pi i Li P tf li M d l
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Piecewise Linear Portfolio Model
Assume that the m nderivatives are European put/call optionsmaturing at the end of the investment horizon T.
Basic asset returns: rj = fj() = j for j = 1, . . . , n.
Assume option j is a call with strike kj and premium cj on basicasset i with initial price si, then rj is
fj() = 1cj
max
0, si(1 + i) kj 1
= max1, aj + bji 1
, where aj =
si kjcj
, bj =si
cj.
Likewise, if option j is a put with premium pj, then rj is
fj() = max1, aj + bji 1
, where aj =
kj sipj
, bj = sipj
.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Pi i Li P tf li M d l
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Piecewise Linear Portfolio Model
In compact notation, we can write r as
r = f() =
maxe,a+ B e
.
Partition weights vector as w = (w,w).
No derivative short-sales: w W = w 0. Portfolio return of w Wcan be expressed as
wTr = wTf()
= (w)T + (w)T maxe,a+ B e
.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Worst Case Polyhedral VaR
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Worst-Case Polyhedral VaR
Use the piecewise linear portfolio model:
wTf() = (w)T + (w)T max
e,a+ B e
.
Worst-Case Polyhedral VaR (WCPVaR)
For any w W, we define WCPVaR(w) as
WCPVaR(w) = min : supPP
P wTf() .
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Worst Case Polyhedral VaR: Convex Reformulations
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Worst-Case Polyhedral VaR: Convex Reformulations
Theorem: SDP Reformulation of WCPVaRWCPVaR of w can be computed as an SDP:
WCPVaR(w) = min
s. t. M Sn+1, y Rmn, R, R
, M , M
0, 0, 0 y w
M +
0 w + BTy
(w + BTy)T + 2( + yTa eTw)
0
Where we use the second-order moment matrix :
=+ T
T 1
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Worst Case Polyhedral VaR: Convex Reformulations
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Worst-Case Polyhedral VaR: Convex Reformulations
Theorem: SOCP Reformulation of WCPVaR
WCPVaR of w can be computed as an SOCP:
WCPVaR(w) = min0gw
T(w + BTg) + () 1/2(w + BTg)2 . . .
. . . aTg+ eTw
SOCP has better scalability properties than SDP.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Robust Optimization Perspective on WCPVaR
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Robust Optimization Perspective on WCPVaR
WCPVaR minimization is equivalent to:
minwW
maxrU
p
wTr.
where the uncertainty setUp R
m
is defined as
Up
=
r Rm :
Rn such that( )T1( ) ()2 andr = f()
UnlikeU, the setUp is not symmetric!
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Robust Optimization Perspective on WCPVaR
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Robust Optimization Perspective on WCPVaR
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Example: WCPVaR vs WCVaR
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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, andWCPVaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Example: WCPVaR vs WCVaR
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Example: WCPVaR vs WCVaR
0
2
4
6
8
10
12
14
16
-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1
probability(%)
portfolio loss
0
0.51
1.5
2
2.5
3
3.5
4
4.5
5
80 82 84 86 88 90 92 94 96 98 100
VaR
Confidence level (15)%
MonteCarlo VaRWorstCase VaR
WorstCase Polyhedral VaR
At confidence level = 1%:
WCVaR unrealistically high: 497%. WCVaR is 7 times larger than WCPVaR.
WCPVaR is much closer to actual VaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Delta-Gamma Portfolio Model
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Delta Gamma Portfolio Model
m nderivatives can be exotic with arbitrary maturity time.Value of asset i = 1 . . . m is representable as vi(, t).
For short horizon time T, second-order Taylor expansion is
accurate approximation of ri:
ri = fi() i +Ti +1
2Ti i = 1, . . . , m.
Portfolio return approximated by (possibly non-convex):
wTr = wTf() (w) +(w)T + 1
2T(w),
where we use the auxiliary functions
(w) =
mi=1
wii, (w) =
mi=1
wii, (w) =
mi=1
wii.
We now allow short-sales of options in w
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Worst-Case Quadratic VaR
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Worst Case Quadratic VaR
Worst-Case Quadratic VaR (WCQVaR)
For any w
W, we define WCQVaR as
min
: sup
PP
P
(w)(w)T 1
2T(w)
Theorem: SDP Reformulation of WCQVaR
WCQVaR can be found by solving an SDP:
WCQVaR(w) = min
s. t. M Sn+1, R, R, M , M 0, 0,M +
(w) (w)
(w)T + 2(+ (w)) 0
There seems to be no SOCP reformulation of WCQVaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Robust Optimization Perspect on WCQVaR
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Robust Optimization Perspect on WCQVaR
WCQVaR minimization is equivalent to:
minwW
maxZU
q
Q(w), Z
where
Q(w) = 1
2
(w) 1
2
(w)12(w)
T (w)
,
and the uncertainty setUq Sn+1 is defined as
Uq
=Z
= X T 1
Sn+1 : Z 0, Z 0 Uq is lifted into Sn+1 to compensate for non-convexity.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Robust Optimization Perspect on WCQVaR
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Robust Optimization Perspect on WCQVaR
There is a connection betweenU Rm andUq Sn+1. If we impose: w W = (w) 0 then robust
optimization problem reduces to:
minwW
maxrU
q
wTr
where the uncertainty setUq Rm is defined as
Uq
= r R
m :
Rn such that()T1(
)
()2 and
ri = i + Ti + 12Ti i = 1, . . . , m
UnlikeU, the setUq
is not symmetric!
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Robust Optimization Perspective on WCQVaR
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Robust Optimization Perspective on WCQVaR
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Example: WCQVaR vs WCVaR
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p Q
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.
0
1
2
3
4
56
7
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6
probability(%)
portfolio loss
0
0.2
0.4
0.6
0.8
11.2
1.4
80 82 84 86 88 90 92 94 96 98 100
VaR
Confidence level (1-)%
Monte-Carlo VaR
Worst-Case VaRWorst-Case Quadratic VaR
At = 1%: WCVaR still 3 times larger than WCQVaR.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Index Tracking using Worst-Case Quadratic VaR
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g g
Total test period: Jan. 2nd, 2004 Oct. 10th, 2008. Estimation Window: 600 days. Out-of-sample returns: 581.
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 100 200 300 400 500 600
RelativeWealth
Period
robust strategy with optionsrobust strategy without options
S&P 500 (benchmark)
Outperformance: option strat 56%, stock-only strat 12%. Sharpe Ratio: option strat 0.97, stock-only strat 0.13. Allocation option strategy: 89% stocks, 11% options.
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios
Questions?
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Paper available on optimization-online.
c 2007 Salvador Dal, Gala-Salvador Dal Foundation/Artists Rights Society (ARS), New York
Zymler, Kuhn and Rustem Worst-Case Value-at-Risk of Non-Linear Portfolios