Worst-Case Value-at-Risk of Non-Linear Portfolios

8/2/2019 Worst-Case Value-at-Risk of Non-Linear Portfolios

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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



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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.



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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.



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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.


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


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


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)


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

.


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

.


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() .


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


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.


Robust Optimization Perspective on WCPVaR


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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!




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Example: WCPVaR vs WCVaR


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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.




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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.


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


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.


Robust Optimization Perspect on WCQVaR


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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.




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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!


Robust Optimization Perspective on WCQVaR


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Robust Optimization Perspective on WCQVaR


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.


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.


Questions?


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Paper available on optimization-online.

c 2007 Salvador Dal, Gala-Salvador Dal Foundation/Artists Rights Society (ARS), New York


Worst-Case Value-at-Risk of Non-Linear Portfolios

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