# Crash Hedging Strategies and Optimal Portfolios by dfgh4bnmu

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```									Crash Hedging Strategies
and
Optimal Portfolios

by

Olaf Menkens
School of Mathematical Sciences
Dublin City University (DCU)

St. Andrews, March 17th, 2009
Crash Hedging Strategies and Optimal Portfolios                                1

1 Introduction

1.1 Optimal Investment in the Black–Scholes
Setting

Most basic setting:

dP0,0(t) = P0,0(t) r0 dt ,                        P0,0(0) = 1 ,        “bond”
dP0,1(t) = P0,1(t) [µ0 dt + σ0 dW0(t)] ,          P0,1(0) = p1 ,      “stock”
with constant market coeﬃcients µ0, r0, σ0 = 0 and where W0 is a
Brownian motion on a complete probability space (Ω, F , P ).

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                2

The optimal portfolio problem in this setting is to ﬁnd a solution
of

sup            π
E U X0 (T )     ,
π(·)∈A0(x)

π
where U is the utility function of the investor, and X0 denotes the
wealth process of the investor given the portfolio strategy π. More
speciﬁc, the wealth process satisﬁes
π        π
dX0 (t) = X0 (t) [(r0 + π(t) [µ0 − r0]) dt + π(t)σ0 dW0(t)] ,
π
X0 (0) = x.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                    3

Classical solution methods are
• the Martingale method (Pliska (1986), Karatzas et al. (1987),
and Cox and Huang (1989)) and
• the stochastic control method (Merton (1969 and 1971)).

A solution of the optimal portfolio problem is called an optimal
portfolio strategy and will be denoted by π ∗. The most used
utility functions and the corresponding optimal portfolio strategies
are
i) Logarithmic utility: U (x) = ln(x) with π0 = µ0−r0 ,
∗
2
σ0

ii) HARA–utility: U (x) = 1 xγ with γ < 1, γ = 0 and with
γ
π0,γ = 1−γ µ0−r0 ,
∗      1
2
σ0

iii) Exponential utility: U (x) = − exp (−λx) with λ > 0 and with
∗
π0,λ(t) =     π∗
1        · µ0−r0 · exp (−r0 (T − t)).
2
σ0
λX 0,λ (t)
c Olaf Menkens                                   School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                    4

Well–known empirical fact: Black–Scholes–price–model cannot
explain large movements of real stock prices (the so–called “jumps”
or “crashes”).

Examples: Merton (1976) or Aase (1984).

dP0,1(t) = P0,1(t) [µ0 dt + σ0 dW0(t) − k dN (t)] ,             P0,1(0) = p1 .
where N is a Poisson process with intensity λ and k > 0 is the
jump size. In the logarithmic–utility case U (x) = ln(x) the optimal
∗
portfolio strategy πp calculates to

2                                        2                        2
∗ = σ 0 + k (µ 0 − r 0 ) −
πp
kλ − (µ0 − r0)
+
σ0 + k (µ0 − r0)
.
2kσ0 2                    2
kσ0              2kσ02

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                                                       5

∗
Level Lines of the Optimal Portfolio Strategie πP
1                                                                                               1                         1
1
1

0
0
0.9

1
0
Probability of no crash occurring within the next year

The market coefficients are r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125.
−1

0.8                                                                                         −1
0

−1
1

∗
0.7

−1                                                                                −2

−2
0

0.6                                                                   −2
−1

−2
1

0.5                                                                                                              −3

−3
0

−3
0.4

0.1          0.2       0.3       0.4             0.5           0.6       0.7           0.8       0.9             1
k −− Crash height
The variables are the crash height k and the probability that no crash occurs within the
next year. The market coeﬃcients are assumed to be r0 = 0.05, µ0 = 0.1 and σ0 = 0.2.

c Olaf Menkens                                                                                                                                       School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                                6

∗
The Dependence of the Optimal Portfolio Strategie πP
on the Crash Height and the Crash Intensity

The market coefficients are r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125.
2
π −− Wealth fraction invested in the risky asset

1

∗
0

−1

−2

−3

−4
0

0.2

0.4
1
0.9
0.6                                                                       0.8
0.7
0.6
0.5
k −− Crash height   0.8                             0.4
0.3
0.2               λ −− Crash intensity
1       0.1
0

∗
This graphic shows the dependence of πP from the crash height k and the crash intensity
λ. The market coeﬃcients are assumed to be r0 = 0.05, µ0 = 0.1 and σ0 = 0.2.
c Olaf Menkens                                                                                                             School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                7

Remarks:

• Although these processes deliver a better ﬁt, they do not help
to determine optimal portfolio strategies under the threat of a
crash!

• The time to maturity is very important in crash modelling.
However, this variable is neglected in traditional crash modelling.

• Asymmetry of risk is not taken into consideration:

– For π ∈ [0, 1], you can loose at most your investment in the
stock.

– For π < 0 or π > 1, however, you can loose much more than
just your investment in the stock!
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                        8

1.3 Alternative Crash Modelling
1. Hua and Wilmott (1997) → Number and size of crashes in a given time
interval are bounded. ⇒ No probabilistic assumptions on height, number
and times of occurence of crashes.

2. Korn and Wilmott (2002) → Determine worst case bounds for the perfor-
mance of optimal investment.

For simplicity: One bond, one stock, at most one crash in [0, T ] with a crash
height of k with 0 < k∗ ≤ k ≤ k∗ < 1. Security prices in “normal times”:
dP0,0 (t) = P0,0 (t) r0 dt ,                  P0,0 (0) = 1 ,     “bond”
dP0,1 (t) = P0,1 (t) [µ0 dt + σ0 dW0 (t)] ,   P0,1 (0) = p1 ,    “stock”

At crash time: stock price falls by a factor of k ∈ [k∗ , k∗ ].
π
Consequence: The wealth process X0 (t) at crash time t satisﬁes:
π                    π             π
X0 (t−) = (1 − π(t)) X0 (t−) + π(t)X0 (t−)
π             π
=⇒          (1 − π(t)) X0 (t−) + π(t)X0 (t−) (1 − k)

π         π
=        (1 − π(t)k) · X0 (t−) = X0 (t).


c Olaf Menkens                                       School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                9

Thus: Following the portfolio process π(·) if a crash of size k
happens at time t leads to a ﬁnal wealth of
π
X π (T ) = (1 − π(t)k) · X0 (T ) ,
π
if X0 (·) denotes the wealth process in the model without any crash.

Hence:
• “High” values of π(·) lead to a high ﬁnal wealth if no crash
occurs at all, but to a high loss at the crash time.
• “Low” values of π(·) lead to a low ﬁnal wealth if no crash occurs
at all, but to a small loss (or even no loss at all!) at the crash
time.

Moral: We have two competing aspects (“Hedging vs. Return”)
for two diﬀerent scenarios (“Crash or not”) and are therefore faced
with a balanced problem between crash impact minimization
and return maximization.
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               10

Aim: Find the best uniform worst case bound, e.g. solve
'                                        \$

sup       inf E [U (X π (T ))] ,
π(·)∈A0(x) 0≤τ ≤T
k∈K
&                                        %

where the ﬁnal wealth satisﬁes X π (T ) = (1 − π(τ )k) X0 (T ) in the
π

case of a crash of size k at stopping time τ . Moreover, K =
{0} ∪ [k∗, k∗].

1
Note: To avoid bankruptcy we require π(t) < k∗ for all t ∈ [0, T ].

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               11

Important Remarks: (!)
• We do not (!) compare two diﬀerent strategies scenario-wise.
Typically, two diﬀerent strategies have two diﬀerent worst case
scenarios!

• The worst case bounds do not depend on the probability of the
worst case!

• Assuming µ0 > r0, we do not have to consider portfolio pro-
cesses π(t) that can attain negative values since the utility func-
tion is increasing in x.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                 12

Two extreme strategies (in the logarithmic utility case):

1. “Playing safe”:
π(t) ≡ 0 =⇒ worst case scenario: no crash (!), leading to the
following worst case bound of

W CB0 = E ln X 0 (T )      = ln(x) + r0T.

2. “Optimal investment in the crash–free world”:
π(t) ≡ π0 = µ0−r0 =⇒ worst case scenario: a crash of maximum
∗
2    σ0
size k∗ (at any arbitrary time (!)), leading to the following worst
case bound of
2
∗                          1 µ0 − r 0
W CBπ∗ = E ln        X π0   (T )   = ln(x) + r0T +               T
0                                       2     σ0
+ ln 1 − π0k∗ .
∗

c Olaf Menkens                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               13

Insights:

• it depends on the time to maturity which one of the above
strategies is better.

• strategy 1 takes too few risk to be good if no crash occurs while
strategy 2 is too risky to perform well if a crash occurs =⇒ the
optimal strategy should balance this out!

• a constant portfolio process cannot be the optimal one.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               14

2 Optimal Investment under the Threat
of a Crash
This chapter is based on Korn and Wilmott (2002), Korn and M.
(2005), and M. (2006).

2.1 The Set up

U (x) = ln(x), the price of the bond and the risky asset are assumed
to be given by

dP1,0(t) = P1,0(t) r1 dt ,                        P1,0(τ ) = P0,0(τ ) ,
dP1,1(t) = P1,1(t) [µ1 dt + σ1 dW1(t)] ,           P1,1(τ ) = (1 − k) P0,1(τ ) ,
respectively, with constant market coeﬃcients r1, µ1 and σ1 = 0
after a possible crash of size k at time τ .

For simplicity, the initial market will also be called market 0, while
the market after a crash will be called market 1.
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                   15

Definition 2.1
1. The wealth process X π (t) in the crash model is deﬁned as
π
X0 (t)                             for 0 ≤ t < τ
X π (t) =                          π
π,τ,X0 (τ )                              (1)
[1 − π(τ )k]   X 1             (t) for t ≥ τ ≥ 0 ,
given the occurrence of a jump of height k at time τ , is strictly
positive.
2. The problem to solve

sup        inf E [ln (X π (T ))] ,                   (2)
π(·)∈A(x) 0≤τ ≤T,
k∈K
where the ﬁnal wealth X π (T ) in the case of a crash of size k at
time τ is given by
π
π (T ) = 1 − π(τ )k X π,τ,X0 (τ ) T ,                    (3)
X         [          ] 1           ( )
π
π,τ,X0 (τ )
with X1           (t) as above, is called the worst case scenario
portfolio problem.
c Olaf Menkens                                   School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                      16

Definition 2.2
1. The value function to the above problem is deﬁned via

νc(t, x) =      sup        inf E ln X π,t,x(T )         .         (4)
π(·)∈A(t,x) t≤τ ≤T,
k∈K

2. The value function in the crash–free setting of market i will
be denoted by
π,t,x
νi(t, x) =        sup       E ln Xi     (T )
π(·)∈Ai(t,x)
for i = 0, 1.

c Olaf Menkens                                      School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                 17

Definition 2.3
For i = 0, 1 let us name

1. the optimal portfolio strategy in market i, assuming that no
crash will happen, by
∗ := µi − ri .
πi       2
σi

2. Moreover,
2      2
1 µi − r i        σi
Ψi := ri +             = ri +    πi 2
∗
2   σi            2
will be called the utility growth potential or earning potential
of market i.

c Olaf Menkens                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                         18

Deﬁne for an arbitrary admissible portfolio strategy π(t)
π,t,x
νπ (t, x) := E ln X0 (T )
                                                            
T
1
= ln (x) + E                                           2
π(s) (µ0 − r0) + r0 − π 2(s)σ0 ds
                                            
2
t
                                   
2              T
σ0                               2
= ln (x) −    E               π(s) − π0 2 − 2 Ψ0 ds
∗            
2                                σ0
t
                                       
T            2
σ0
=   ln (x) + E            Ψ0 −    π(s) − π0 2 ds .
∗
                                 
2
t

In particular, the value function of market i given that no crash
occurs is
νi(t, x) = ln (x) + Ψi (T − t) .

c Olaf Menkens                                        School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               19

2.2 A Main Result

Definition 2.4
ˆ
1. A portfolio strategy π determined via the equation

ν1 (t, x (1 − π (t)k∗))
ˆ                ˆ
for π (t) ≥ 0
ˆ
ν (t, x) =                                                for all t ∈ [0, T ]
ˆ
ν1 (t, x (1 − π (t)k∗))        ˆ
for π (t) < 0
will be called a crash hedging strategy.

˜
2. A portfolio strategy π is a partial crash hedging strategy, if
˜
there exists an S ∈ (0, T ) such that π is a crash hedging strategy
on [0, S] and is a solution to the worst case scenario portfolio
problem on [S, T ].

ˆ
Hereby, the convention ν (t, x) := νπ (t, x) is used.
ˆ

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               20

Rewriting the determining equation for the non–negative crash hedg-
ˆ
ing strategy π gives

ν (t, x) = ν1 t, x 1 − π (t)k∗
ˆ                      ˆ
T         2
σ0
⇐⇒        ln (x) +       Ψ0 −    π (s) − π0 2 ds
ˆ        ∗
2
t
= ln (x) + ln 1 − π (t)k∗ + Ψ1 (T − t)
ˆ
T               2
σ0
⇐⇒      ln 1 − π (t)k∗
ˆ          =       Ψ0 − Ψ 1 −    π (s) − π0 2 ds . (5)
ˆ        ∗
2
t

ˆ
Assuming that π is diﬀerentiable, diﬀerentiating with respect to t
yields
−ˆ′(t)k∗
π             2
σ0
=     π (t) − π0 2 + Ψ1 − Ψ0
ˆ        ∗
1 − π (t)k∗
ˆ          2
1     2
σ0
⇐⇒               ˆ
π ′(t) =  π (t) − ∗
ˆ                π (t) − π0 2 + Ψ1 − Ψ0 .
ˆ        ∗
k     2
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                      21

Theorem 2.5
ˆ
1. If Ψ1 ≥ r0, then there exists a unique crash hedging strategy π , which is
given by the solution of the diﬀerential equation
1     2
σ0
′                                  ∗ 2
ˆ
π (t) =    ˆ
π (t) − ∗       (ˆ(t) − π0 ) + Ψ1 − Ψ0 ,
π                               (6)
k     2
and   ˆ
π (T ) = 0 .                                                (7)
1
Moreover, this crash hedging strategy is bounded by 0 ≤ π ≤ k∗ , if Ψ1 > Ψ0 .
ˆ
In the case of Ψ1 ≤ Ψ0 , the crash hedging strategy is bounded by 0 ≤ π ≤ˆ
∗    2
π0 − σ2 (Ψ0 − Ψ1 ).
0

ˆ
2. If Ψ1 < r0, then there exists a unique crash hedging strategy π , which is
given by the solution of the diﬀerential equation
1     2
σ0
′                                  ∗ 2
ˆ
π (t) =    ˆ
π (t) −         (ˆ(t) − π0 ) + Ψ1 − Ψ0 ,
π                               (8)
k∗   2
and   ˆ
π (T ) = 0 .                                                (9)
∗         2
Furthermore, this crash hedging strategy is bounded by π0 −        2
σ0
(Ψ0 − Ψ1 ) ≤
ˆ
π (t) < 0 for t ∈ [0, T ).

c Olaf Menkens                                    School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                          22

∗
3. If Ψ1 < Ψ0 and π0 < 0, there exists a partial crash hedging strategy π (which
˜
ˆ
is diﬀerent from π ), if
∗
ln 1 − π0 k∗
S := T −                    > 0.                      (10)
Ψ0 − Ψ1
˜
With this, π is on [0, S] given by the unique solution of the diﬀerential
equation
1              2
σ0
′                                        ∗ 2
˜
π (t) =  ˜
π (t) −                  (˜(t) − π0 ) + Ψ1 − Ψ0 ,
π                             (11)
k∗            2
∗
and   ˜
π (S) = π0 .                                                     (12)
˜         ∗
On [S, T ] set π (t) := π0 . This partial crash hedging strategy is bounded by
∗     2                      ∗
π0 −    2
σ0
˜
(Ψ0 − Ψ1 ) ≤ π ≤ π0 < 0.

The optimal portfolio strategy for an investor, who wants to maximize her worst
case scenario portfolio problem, is given by
∗
¯                   ˜
π (t) := min {ˆ(t), π (t), π0 }
π                       for all t ∈ [0, T ],          (13)
˜                                              ¯
where π (t) is only taken into account if it exists. π will be named the optimal
crash hedging strategy.

c Olaf Menkens                                          School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                                                      23

Geometric Interpretation of the Crash Hedging Strategy
2.5

The initial market coefficients are given by r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125.
π hat

The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2,
1
0
2
iii)                    vi)

ν < ν hat − B           ν <ν

1
π             π        π        0
π −− Wealth fraction invested in the risky asset

0
∗
i)

ν > ν hat − B

1
π              π

0
1.5

π∗ = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k∗ = 0.2.
π∗
0

0
ii) crash is negative

0
1                                                                                              ν > ν hat − B
π             π

∗
v) crash is positive                                     iv) crash is positive
0.5       νπ < ν0                                                  νπ < ν hat − Bπ

1
1
0
0             5            10   15                20              25                 30        35                 40
Time in years

c Olaf Menkens                                                                                                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                24

∗
Example for Ψ1 = Ψ0 and π0 ≥ 0
1.4

The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.2, π0 = 1.25, Ψ0 = 0.08125.
The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.15, σ1 = 0.4,
1.2
π −− Wealth fraction invested in the risky asset

1

∗
0.8

π1 = 0.625, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2.
∗
0.6

0.4

0.2

∗
0
0   10   20   30   40        50         60     70     80    90    100
Time in years

ˆ    ¯    ˆ
This graphic shows π = π = φ0 (blue dash–dotted line with black background), ϕ = π0 ˆ    ∗

¯               ˆ                               ∗
(blue dotted line), ϕ (green line), φ1 (red dash–dotted line), and π1 (red dotted line).

c Olaf Menkens                                                                                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                              25

∗
Example for Ψ1 > Ψ0 and π0 ≥ 0
6

The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.2, π0 = 1.25, Ψ0 = 0.08125.
The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2,
1
5

π1 = 1.75, Ψ1 = 0.09125. Moreover, k∗ = 0.05 and k = 0.2. Observe that t0 = 89.5149.
1
π −− Wealth fraction invested in the risky asset

∗
4

1
3

∗
2

1

∗
0
0   10   20   30   40        50         60     70     80    90    100
Time in years

ˆ                      ¯               ¯               ˆ
This graphic shows π (black dashed line), π (black line), ϕ (green line), φ0 (blue dash–
ˆ                           ∗                          ∗
dotted line), φ1 (red dash–dotted line), π0 (blue dotted line), and π1 (red dotted line).

c Olaf Menkens                                                                                             School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                         26

∗
Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 ≥ 0
2

1.8

The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2,
The initial market coefficients are given by r0 = 0.02, µ0 = 0.1, σ0 = 0.2, π0 = 2, Ψ0 = 0.1.
1.6
π −− Wealth fraction invested in the risky asset

∗
1.4

1.2

π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2.
1

∗
0.8

0.6

0.4

0.2

∗
0
0   2      4   6   8        10         12     14     16    18    20
Time in years

ˆ    ¯             ˆ                     ¯               ˆ
This graphic shows π = π (black line), ϕ (cyan dotted line), ϕ (green line), φ0 (blue
ˆ                           ∗                               ∗
dash–dotted line), φ1 (red dash–dotted line), π0 = 2 (blue dotted line), and π1 (red
dotted line).
c Olaf Menkens                                                                                              School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                           27

∗
Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 ≥ 0, the long term behaviour
2

1.8

The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2,
The initial market coefficients are given by r0 = 0.02, µ0 = 0.1, σ0 = 0.2, π0 = 2, Ψ0 = 0.1.
1.6
π −− Wealth fraction invested in the risky asset

∗
1.4

1.2

π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2.
1

∗
0.8

0.6

0.4

0.2

∗
0
0   10   20   30   40        50         60     70     80    90    100
Time in years

ˆ   ¯              ˆ
This graphic shows the long term behaviour of π = π (black line), ϕ (cyan dotted line),
¯                                           ˆ
ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 = 2 (blue dotted
ϕ (green line), φ                                                       ∗
∗
line), and π1 (red dotted line).
c Olaf Menkens                                                                                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                              28

∗
Example for Ψ1 < r0 and π0 ≥ 0
1.5

The initial market coefficients are given by r0 = 0.1, µ0 = 0.15, σ0 = 0.2, π∗ = 1.25, Ψ0 = 0.13125.
The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.05, σ1 = 0.2,
1
π −− Wealth fraction invested in the risky asset

0
0.5

π1 = 0.5, Ψ1 = 0.035. Moreover, k∗ = 0.05 and k = 0.2.
∗
0

−0.5

∗
−1
0   2    4   6   8        10         12      14    16    18     20
Time in years

This graphic shows π = π (black line), ϕ = ϕ (green line with cyan dotted points), φ0
ˆ    ¯             ¯   ˆ                                         ˆ
ˆ                           ∗ (blue dotted line), and π ∗ (red
(blue dash–dotted line), φ1 (red dash–dotted line), π0                          1
dotted line).
c Olaf Menkens                                                                                             School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                         29

∗
Example for Ψ1 > Ψ0 and π0 < 0
5

The initial market coefficients are given by r0 = 0.05, µ0 = 0, σ0 = 0.2, π0 = −1.25, Ψ0 = 0.08125.
The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2,
4

1        1
π −− Wealth fraction invested in the risky asset

3

∗

1
2

π∗ = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k∗ = 0.2.
1

∗
0

−1

1
1
−2
0   1    2   3   4         5         6      7     8     9     10
Time in years

ˆ                       ¯    ¯     ∗
This graphic shows π (black dashed line), π = ϕ = π0 (blue dotted line with black
ˆ                           ˆ                               ∗
background), φ0 (blue dash–dotted line), φ1 (red dash–dotted line), and π1 (red dotted
line).
c Olaf Menkens                                                                                          School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                       30

∗
Example for r0 ≤ Ψ1 ≤ Ψ0 and π0 < 0
5

The initial market coefficients are given by r = 0.05, µ = −0.05, σ = 0.2, π = −2.5, Ψ = 0.175.
4

The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2,
0
1
π = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k = 0.2. Observe that S = 8.5936.
3

1
0
∗
π −− Wealth fraction invested in the risky asset

2

1
0
1

0
0

∗
0
−1

∗
−2

−3

1
−4

1
∗
−5
0   1      2   3   4         5         6      7     8     9     10
Time in years

ˆ                       ¯   ˜               ˆ
This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ     ¯
ˆ
ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line),
(green line), φ                                                       ∗
∗
and π1 (red dotted line).
c Olaf Menkens                                                                                            School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                      31

∗
Example for Ψ1 < r0 and π0 < 0, the long term behaviour
5

The initial market coefficients are given by r = 0.1, µ = 0.05, σ = 0.2, π = −1.25, Ψ = 0.13125.
The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.05, σ = 0.2,
4

1
0

π = 0.5, Ψ = 0.035. Moreover, k = 0.05 and k = 0.2. Observe that S = 19.3701.
3

1
π −− Wealth fraction invested in the risky asset

0
∗
2

1
0
1

0

∗
0

0
−1

∗
−2

−3

1
1
∗
−4
0   2   4   6   8        10         12     14     16    18    20
Time in years

ˆ                       ¯   ˜               ˆ
This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ     ¯
ˆ
ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line),
(green line), φ                                                       ∗
∗
and π1 (red dotted line).
c Olaf Menkens                                                                                          School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                      32

∗
Example for Ψ1 < r0 and π0 < 0
0.5

The initial market coefficients are given by r0 = 0.1, µ0 = 0.05, σ0 = 0.2, π∗ = −1.25, Ψ0 = 0.13125.
The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.05, σ1 = 0.2,
0

π1 = 0.5, Ψ1 = 0.035. Moreover, k∗ = 0.05 and k = 0.2. Observe that S = 1.3701.
−0.5
π −− Wealth fraction invested in the risky asset

0
−1

−1.5

∗
−2

−2.5

−3

∗
−3.5
0   0.2   0.4   0.6   0.8         1         1.2      1.4   1.6   1.8    2
Time in years

ˆ                       ¯   ˜               ˆ
This graphic shows π (black dashed line), π = π (black line), ϕ (cyan dotted line), ϕ     ¯
ˆ
ˆ0 (blue dash–dotted line), φ1 (red dash–dotted line), π0 (blue dotted line),
(green line), φ                                                       ∗
∗
and π1 (red dotted line).
c Olaf Menkens                                                                                                     School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                      33

3.2 Optimal Portfolios Given the Probability of a
Crash
In this section, let us suppose that the investor knows the probability
of a crash occurring. Let p, with p ∈ [0, 1], be the probability of a
crash happening. In this situation, the optimization problem writes
to
sup        inf Ep ln X π,t,x (T )
t≤τ ≤T,
π(·)∈A(t,x)
k∈K

π,t,x
:=           sup          p·     inf E ln X π,t,x (T )   + (1 − p) E ln X0 (T )
t≤τ ≤T,
π(·)∈A(t,x)
k∈K

π,t,x
=    sup            p·    inf E ν1 τ, X0 (τ ) (1 − π(τ )k)        + (1 − p) E [νπ (t, x)]   .
t≤τ ≤T,
π(·)∈A(t,x)
k∈K

c Olaf Menkens                                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                               34

Observe that the two extremes, p ∈ {0, 1} are straightforward to
solve:

• p = 1:           sup        inf E1 ln X π,t,x (T )       =      sup        inf E ln X π,t,x (T )       .
t≤τ ≤T,                                        t≤τ ≤T,
π(·)∈A(t,x)                                    π(·)∈A(t,x)
k∈K                                            k∈K

Thus, this is the original worst case scenario portfolio problem.
π,t,x
• p = 0:             sup          inf E0 ln X π,t,x (T )   =        sup         E ln X0 (T )         ,
t≤τ ≤T,
π(·)∈A(t,x)                                    π(·)∈A(t,x)
k∈K

which is the classical optimal portfolio problem. The solution
is well–known and is given in our notation (see Deﬁnition 1) by
∗
π0 .

c Olaf Menkens                                                  School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                35

Let us now consider the case p ∈ (0, 1). Denoting the crash hedging
ˆ
strategy in this situation by πp and the corresponding utility function
ˆ
by νp (t, x) := νπp (t, x), the deﬁning equilibrium equation for the
ˆ
crash hedging strategy can be written as

νp (t, x) = p · ν1 t, x 1 − πp(t)k∗
ˆ                           ˆ         + (1 − p) νπp (t, x)
ˆ
⇐⇒            νp (t, x) = p · ν1 t, x 1 − πp(t)k∗
ˆ                           ˆ                   ˆ
+ (1 − p) νp(t, x)
⇐⇒            νp (t, x) = ν1 t, x 1 − πp(t)k∗ ,
ˆ                       ˆ
ˆ      ˆ
hence πp ≡ π . This result shows that the crash hedging strategy
remains the same even if the probability of a crash is known. Thus,
this result justiﬁes the wording worst case scenario of the above
developed concept. This is due to the fact that the worst case
scenario should be independent of the probability of the worst case
and which has been shown above. Let us summarize this result in
a proposition.

c Olaf Menkens                                School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               36

Proposition 3.1
Given that the probability of a crash is positive, the worst case
scenario portfolio problem as it has been deﬁned in Deﬁnition 2.1
is independent of the probability of the worst case.

If the probability of a crash is zero, the worst case scenario portfolio
problem reduces to the classical crash–free portfolio problem.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               37

3.3 The q –quantile crash hedging strategy

Obviously, the concept of the worst case scenario has the disadvan-
tage that additional information (namely the given probability of a
crash) is not used. However, if the probability of a crash and the
probability of the crash size is known, it is possible to construct the
(lower) q –quantile crash hedging strategy.

Assume that pc(t) ∈ [0, 1] is the probability of a crash at time t ∈
[0, T ] and let p(k, t) ∈ [0, 1] be the density of the distribution function
for a crash of size k ∈ [k∗, k∗] at time t. Moreover, suppose that a
function q : [0, T ] −→ [0, 1] is given. With this deﬁne

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                 38

                                                                                
 0
                                                           if 1 − pc (t) ≥ q(t) 



                                  kq                                            


 inf
       kq : 1 − pc (t) + pc (t)        p(k, t) dk ≥ q(t)   if 1 − pc (t) < q(t) 

kq (t; π) :=                                      k∗                       and π ≥ 0
k∗

                                                                                


 sup k : 1 − p (t) + p (t) p(k, t) dk ≥ q(t)                                    
else



      q       c       c                                                         


kq

for any given portfolio strategy π. This has the following interpre-
tation. The probability that at most a crash of size kq (t) at time
t happens is q(t). Equivalently, the probability that a crash higher
than kq (t) will happen at time t is less than 1 − q(t). Obviously, this
is a Value at Risk approach.

c Olaf Menkens                                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               39

Notice that the worst case of a nonnegative portfolio strategy is
either a crash of size k∗ or no crash. On the other hand, the worst
case of a negative portfolio strategy is either a crash of size k∗
or no crash. Correspondingly, the q–quantile calculates diﬀerently
for negative portfolio strategies (see the third row) than for the
nonnegative portfolio strategies (see the second row). Furthermore,
denote by
                                                
 {0}
                  if kq (t) = 0                 

Kq (t) := {0} ∪ [k∗, kq (t)] if kq (t) = 0 and π ≥ 0   .
 {0} ∪ k (t), k ∗ else
[ q        ]
                                           


c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                 40

Definition 3.2
1. The problem to solve

sup       inf     E [ln (X π (T ))] ,               (14)
π(·)∈A(x) 0≤τ ≤T,
k∈Kq (t)

where the ﬁnal wealth X π (T ) in the case of a crash of size k at
time s is given by
π
π (T ) = 1 − π(τ )k X π,τ,X0 (τ ) T ,                  (15)
X         [          ] 1           ( )
π
π,τ,X0 (τ )
with X1           (t) as above, is called the (lower)           q –quantile
scenario portfolio problem.

c Olaf Menkens                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                       41

2. The value function to the above problem is deﬁned via

νq (t, x) =      sup       inf      E ln X π,t,x(T )      .       (16)
π(·)∈A(t,x) t≤τ ≤T,
k∈Kq (t)

ˆ
3. A portfolio strategy πq determined via the equation

ˆ
νπq (t, x) = ν1 (t, x (1 − πq (t)kq (t)))
ˆ                                               for all t ∈ [0, T ] with kq (t) > 0
will be called a (lower) q –quantile crash hedging strategy.

c Olaf Menkens                                       School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                42

Remark 3.3
1. It is straightforward to see that the 1–quantile scenario portfolio
problem is equivalent to the worst case scenario portfolio prob-
lem in Deﬁnition 2.1. Moreover, the 1–quantile crash hedging
strategy is equivalent to the crash hedging strategy in Deﬁnition
3.1 in M. (2006), p. 602.
2. Notice that the q–quantile scenario portfolio problem is only a
q–quantile concerning the crash. The randomness of the market
movement represented in the model by a geometric Brownian
motion has been averaged out, namely by taking the expectation
– and not the q–quantile.

Deﬁne the support of kq to be

supp (kq ) := {t ∈ [0, T ] : kq (t) > 0} .

c Olaf Menkens                                School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                   43

Theorem 3.4
Let us suppose that kq is continuously diﬀerentiable on supp (kq )
with respect to t.
1. Then there exists a unique (lower) q-quantile crash hedging
ˆ
strategy πq , which is on supp (kq ) given by the solution of the
diﬀerential equation
1        2
σ0
ˆ′
πq (t) =   πq (t) −
ˆ                     πq (t) − π0 2 + Ψ1 − Ψ0 − πq (t)kq (t),
ˆ         ∗               ˆ      ′
kq (t)    2
ˆ
πq (T ) = 0 .
∗
For t ∈ [0, T ] \ supp (kq ) set πq (t) := π0.
ˆ

Moreover, the q–quantile crash hedging strategy is for t ∈ supp (kq )
bounded by
1      1
ˆ
0 ≤ πq (t) <          ≤         if Ψ1 ≥ r0.
kq (t)   k∗

c Olaf Menkens                                   School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                44

∗
Additionally, if Ψ1 ≤ Ψ0 and π0 ≥ 0, the q–quantile crash hedg-
ing strategy has another upper bound with πq < π0− 2 (Ψ0 − Ψ1).
ˆ     ∗
σ2          0

On the other side, if Ψ1 < r0 the q–quantile crash hedging
strategy is bounded by

∗     2
π0 −                ˆ
Ψ − Ψ1) < πq (t) < 0
2( 0
for t ∈ [0, T ).
σ0

∗
2. If Ψ1 < Ψ0 and π0 < 0, there exists a partial q–quantile crash
˜                                    ˆ
hedging strategy πq at time t (which is diﬀerent from πq ), if
∗
ln 1 − π0kq (t)
Sq (t) := T −                     >0       for t ∈ supp (kq ) .      (17)
Ψ 0 − Ψ1
˜
With this, πq (t) is given by the unique solution of the diﬀerential
equation

c Olaf Menkens                                School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                 45

1         2
σ0
˜′
πq (t) =   πq (t) −
˜                      πq (t) − π0 2 + Ψ1 − Ψ0 − πq (t)kq (t),
˜         ∗               ˜      ′
kq (t)     2
∗
πq (Sq (t)) = π0 .
˜
∗
For Sq (t) ≤ 0 set πq (t) := π0. This partial crash hedging strategy
˜
is bounded by

∗      2                    ∗
π0 −     2 (Ψ0 − Ψ1) < πq ≤ π0 < 0.
˜
σ0

If kq is independent of the time t, the optimal portfolio strategy for
an investor, who wants to maximize her q–quantile scenario portfolio
problem, is given by

¯                      ˜        ∗
πq (t) := min {ˆq (t), πq (t), π0}
π                        for all t ∈ [0, T ],       (18)
˜                                          ¯
where πq will be taken into account, if it exists. πq will also be
called the optimal q –quantile crash hedging strategy.

c Olaf Menkens                                 School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               46

Remark 3.5
1. It is also possible to solve the above problem if kq is not contin-
ˆ
uously diﬀerentiable. In order to verify this deﬁne πk to be the
unique solution of
1       2
σ0
ˆ
πk′ (t) = ˆ
πk (t) −           πk (t) − π0 2 + Ψ1 − Ψ0
ˆ         ∗                        and (19)
k      2
ˆ
πk (T ) = 0,                                                           (20)
ˆ          ˆ
for k > 0. Set then πq (t) := πkq (t)(t) where the convention
∗
π0(t) := π0 is used in order to include the case kq (t) = 0. Note
ˆ
that this procedure is also possible for continuously diﬀeren-
tiable kq . However, only if kq is continuously diﬀerentiable, it is
ˆ
possible that πq is also continuously diﬀerentiable.
ˆ′    ˆ′
2. Notice that πk < πk for k1 < k2. Hence, πk1 ≥ πk2 with strict
ˆ     ˆ
1     2
ˆ        ˆ
inequality applying on [0, T ). Thus, in particular, πq (t) > π (t)
for t ∈ [0, T ) for any q which satisﬁes q(t) < 1 for t ∈ [0, T ).
Moreover, πq1 (t) ≤ πq2 (t), if q1 > q2.
ˆ        ˆ
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               47

3. For this remark, let us suppose that the market conditions do
not change, hence Ψ1 = Ψ0. Moreover, keep in mind that any
∗
πk is bounded by π0 from above. Thus, it is clear that
ˆ

0 for t = T
ψ(t) :=     ∗
π0 else
ˆ
is an upper bound for any πk with k > 0. Unfortunately, it is
not possible to show that

ˆ
πk∗ −→ ψ
for k∗ ↓ 0 with k∗ = 0, since πk∗ is only known implicitly and
ˆ
not explicitly. However, this is exactly what can be observed in
practice.
∗
Moreover, keep in mind that the case k = 0 yields π0 as the
∗
optimal portfolio with π0 ≡ ψ.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                         48

∗
Example of k −→ 0 for Ψ1 = Ψ0 and π0 ≥ 0
0.7

The initial market coefficients are given by r0 = 0.05, µ0 = 0.15, σ0 = 0.4, π0 = 0.625, Ψ0 = 0.08125.
The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.2,
0.6
π −− Wealth fraction invested in the risky asset

∗
0.5

0.4

π1 = 1.25, Ψ1 = 0.08125. Moreover, k∗ = 0.05 and k = 0.2.
∗
0.3

0.2

0.1

∗
0
0     1     2   3   4           5         6          7     8     9        10
Time in years

This graphic shows π = πk∗ (black dashed line), π k∗ (red dashed line), π k∗ (blue dash–
ˆ   ˆ                        ˆ                       ˆ
2                      10
dotted line), π k∗ (cyan dotted line), π
ˆ                        ˆ                                                                             ∗
(green solid line), and π0 (black dotted line).
k∗
100               1000

c Olaf Menkens                                                                                                    School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                        49

The Range of (Optimal) q –Quantile Crash Hedging
∗
Strategies for Ψ1 = Ψ0 and π0 ≥ 0
0.8

The market coefficients after a possible crash are assumed to be r1 = 0.05, µ1 = 0.1, σ1 = 0.25,
The initial market coefficients are given by r0 = 0.05, µ0 = 0.1, σ0 = 0.25, π∗ = 0.8, Ψ0 = 0.07.
0.7

0.6
π −− Wealth fraction invested in the risky asset

0
0.5

π∗ = 0.8, Ψ1 = 0.07. Moreover, k∗ = 0.05 and k∗ = 0.2.
0.4

0.3

0.2

0.1

1
0
0   1   2   3   4         5         6     7    8     9     10
Time in years

This graphic shows πk∗ (black solid line), πk∗ (black dashed line), the range of possible
ˆ                       ˆ
q–quantile crash hedging strategies (light grey and dark grey area), the range of possible
∗
optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line).
c Olaf Menkens                                                                                             School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                    50

The Range of (Optimal) q –Quantile Crash Hedging
∗
Strategies for Ψ1 > Ψ0 and π0 ≥ 0
25

The initial market coefficients are given by r = 0.05, µ = 0.1, σ = 0.2, π = 1.25, Ψ = 0.08125.
The market coefficients after a possible crash are assumed to be r = 0.03, µ = 0.1, σ = 0.2,
1
0
20

1
π −− Wealth fraction invested in the risky asset

0
∗

1
0
15

π = 1.75, Ψ = 0.09125. Moreover, k = 0.05 and k = 0.2.
0

∗
0
10

∗
5

1
1
∗
0
0   5   10   15        20         25        30     35     40
Time in years

ˆ
This graphic shows πk∗ (black solid line), πk∗ (black dashed line), the range of possible
ˆ
q–quantile crash hedging strategies (light grey and dark grey area), the range of possible
∗
optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line).
c Olaf Menkens                                                                                           School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                                                                                                                                                                                                                                    51

The Range of (Optimal) q –Quantile Crash Hedging
∗
Strategies for r0 < Ψ1 < Ψ0 and π0 < 0
25

The initial market coefficients are given by r = 0.05, µ = −0.05, σ = 0.2, π∗ = −2.5, Ψ = 0.175.
The market coefficients after a possible crash are assumed to be r1 = 0.03, µ1 = 0.1, σ1 = 0.2,
0
20

0
π −− Wealth fraction invested in the risky asset

15

0

π1 = 1.75, Ψ1 = 0.09125. Moreover, k∗ = 0.05 and k = 0.2.
0
10

∗

Observe that S = 8.5936 and that S = 5.1586.
0

∗
5

0

∗
−5
0   1   2   3   4         5         6    7     8    9     10
Time in years

ˆ
This graphic shows πk∗ and πk∗ (black solid line), πk∗ and πk∗ (black dashed line), the range
¯                       ˆ       ¯
of possible q–quantile crash hedging strategies (light grey area), the range of possible
∗
optimal q–quantile crash hedging strategies (dark grey area), and π0 (black dotted line).
c Olaf Menkens                                                                                            School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               52

4 Extensions

Possible extensions are

• More crashes (see Korn and Wilmott (2002), Korn and M.
(2005), M. (2004)).
=⇒ System of diﬀerential equations.

• More stocks (see e. g. Hua and Wilmott (1997)).
=⇒ Numerical methods and crash coeﬃcients.

• General utility functions (see Korn and M. (2005), M. (2004)).
=⇒ Stochastic control approach.

c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               53

• Connection to problems in actuarial mathematics (see Korn
(2005)).
=⇒ Investing in the presence of additional risk processes.

• Worst case scenario optimization for reinsurance (see Korn, M.
, Steﬀensen, work in progress)

• Costs and beneﬁts of crash hedging (see M. (2004)).
=⇒ Calculating the costs and the potential beneﬁts of crash
hedging.

• Diﬀerential games (see Korn and Steﬀensen (2005))

• Market coeﬃcients after a crash depend on the crash size k (see
M. (2004)).
=⇒ Diﬀerential equations for π and ˆ, the worst case crash size.
ˆ     k
c Olaf Menkens                               School of Mathematical Sciences, DCU
Crash Hedging Strategies and Optimal Portfolios                               54

4 References
1. Ralf Korn and Paul Wilmott, Optimal Portfolios under the Threat
of a Crash, International Journal of Theoretical and Applied Fi-
nance, 5(2):171 – 187, 2002.
2. Ralf Korn, Worst–Case Scenario Investment for Insurers, In-
surance: Mathematics and Economics, 36(1):1 – 11, February
2005.
3. Ralf Korn and Olaf Menkens, Worst–Case Scenario Portfolio
Optimization: A New Stochastic Control Approach, Journal of
Mathematical Methods of Operations Research, 62(1):123 –
140, 2005.
4. Ralf Korn and Mogens Steﬀensen, On Worst–Case Portfolio
Optimization, SIAM J. Control Optim., 46(6):2013 - 2030,
2007.
5. Olaf Menkens, Crash Hedging Strategies and Worst–Case Sce-
nario Portfolio Optimization, International Journal of Theorec-
tical and Applied Finance, June, 9(4):597 – 618, 2006.
c Olaf Menkens                               School of Mathematical Sciences, DCU

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