On Historical Value at Risk under Distribution Uncertainty

http://dx.doi.org/10.4236/jmf.2015.52010

On Historical Value at Risk under

Distribution Uncertainty

Atsushi Iizuka1_{, Yumiharu Nakano}2 1_{Informatix Inc., Kanagawa, Japan}

2_{Tokyo Institute of Technology, Tokyo, Japan}

Email: [email protected], [email protected]

Received 11February 2015; accepted 7April 2015; published 10April 2015

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Abstract

We investigate the asymptotics of the historical value-at-risk under capacities defined by sublin-ear expectations. By generalizing Glivenko-Cantelli lemma, we show that the historical value-at- risk eventually lies between the upper and lower value-at-risks quasi surely.

Keywords

Value-at-Risk, Sublinear Expectation, Capacities, Glivenko-Cantelli Lemma

1. Introduction

In financial industry, the value-at-risk has been one of main tools for risk management (see, e.g., McNeil et al.

[1], and Föllmer and Schied [2]). In this framework, the random variables for assets or asset returns are assumed to have distributions without uncertainty. In other words, it is implicitly assumed that there are true asset distri-butions and the estimation difficulty comes from our limited capability. However, it should be remarked that there is a possibility that the assets have the distribution uncertainty, i.e., the assets may have Knightian uncer-tainty (see Knight [3]).

To capture the distribution uncertainty, the theory of sublinear expectation is introduced and developed (see

Peng [4] [5] and the references therein). In this theory, the term probability is replaced by the ones of the upper

and lower capacities induced by the upper and lower expectations, respectively, and the distribution uncertainty is described by the gap between the upper and lower expectations.

uncertainty, and then show that the historical value-at-risk eventually lies in between the upper and lower value- at-risks quasi surely.

This paper is organized as follows: In Section 2, we recall the theory of sublinear expectation. Section 3 is devoted to the statement of the main results and its proofs.

2. Sublinear Expectation and Capacities

In this section, we recall the basis of the sublinear expectation, introduced by Peng [4]. Let Ω be a given set and

 a linear space of -valued functions on Ω. We assume that ϕ

(

X1,,Xn

)

∈ whenever X1,,Xn∈ and ϕ is a bounded function on n or ,

( )

n l Lip C

ϕ∈  where ,

( )

n l Lip

C  denotes the linear space of func-tions ϕ on n satisfying

( ) ( )

x y C

(

1 xm ym

)

x y, ,x y n,

ϕ −ϕ ≤ + + − ∈_

for some C>0 and m∈ depending on ϕ. We call an element in  a random variable.

We consider a sublinear expectation :→, in the sense of [4]. Namely, E is assumed to be satisfy the following conditions: for any X Y, ∈,

1) Monotonicity: if X ≤Y then 

[ ]

X ≤

[ ]

Y .

2) Constant preserving: 

[ ]

x =x for x∈.

3) Subadditivity: 

[ ] [ ]

X − Y ≤

[

X −Y

]

.

4) Positive homogeneity: 

[ ]

λX =λ

[ ]

X for λ ≥0.

Moreover, we assume that 

[ ]

Xi →0 for

{ }

Xi ⊂ with Xi

( )

ω 0 for each ω ∈Ω. Then, by

Theo-rem 2.1 and Remark 2.2 in [5], there exists a set  of probability measures on

(

Ω,σ

( )



)

such that

[ ]

sup P

[ ]

, ,

P

X E X X

∈

= ∈



 

where EP denotes the linear expectation with respect to P∈. Then, the

[ ]

0,1 -valued set functions

( )

[ ]

1 , A

( )

[ ]

1 , A

( )

, C∗ A = C∗ A = − − A∈σ 

define capacities, where 1A denotes the indicator function of a set A. That is, each C∈

{

C C∗, ∗

}

satisfies the following:

1) C

( )

∅ =0, C

( )

Ω =1.

2) If A B, ∈σ

( )

 satisfy A⊂B then C A

( )

≤C B

( )

.

We refer to Denenberg [6] for the theory of capacities. Throughout this paper, we assume that each

{

,

}

C∈ C C∗ _∗ satisfies the following:

3) If

{ }

( )

1

n _n

A ∞₌ ⊂σ  satisfies A1⊂A2⊂, then limn C A

( )

n C

(

k 1Ak

)

∞

→∞ =



= .

4) If

{ }

An n₁ σ

( )

∞

= ⊂  satisfies A1⊃A2⊃, then limn C A

( )

n C

(

k 1Ak

)

∞

→∞ =



₌ .

Let us recall several concepts in the sublinear expectation theory. The random variable Yn is said to be independent of X:=

(

Y Y1, 2,,Yn−1

)

, where Yi∈, i=1,,n, if

(

)

(

)

( )

1

,

, n , n , l Lip n .

x X

X Y x Y C

ϕ ϕ ϕ +

=

 

 =   ∈

  _   _ 

  

We say that X∈ and Y∈ have the same distribution if

( )

X

( )

Y , Cl Lip,

( )

.

ϕ = ϕ ϕ∈

   

    

 

A sequence

{ }

Xi i ₁ ∞

= is called the one of independent, identically distributed random variables if Xi and

1

X have the same distribution, and if Xi+1 is independent of Y:=

(

X1,,Xi

)

for any i≥1. As in the linear

case, we call a sequence of independent, identically distributed random variables an IID sequence. We say that the distribution of X has an uncertainty if _ϕ

( )

X _ is nonlinear in ϕ. In particular, set

[ ]

[ ]

2 2 2 2

, , , .

X X X X

Then if µ µ≠ , then we say that X has the mean uncertainty. Similarly, X is said to have the volatility or variance uncertainty if σ σ≠ .

3. Main Resutls

For any X∈, we define the functions F∗,F∗:→

[ ]

0,1 by

( )

(

)

sup

[

]

,

( )

(

)

inf

[

]

, .

P P

F∗ x C∗ X x P X x F x∗ C∗ X x _∈ P X x x

∈

= ≤ = ≤ = ≤ = ≤ ∈_



 (1)

Proposition 3.1 Let X∈ and let F∗ and F_∗ be as in (1). Then each F∈

{

F∗,F_∗

}

satisfies the fol-lowing:

1) F x

( )

1 ≤F x

( )

2 for x x1, 2∈ with x1≤x2.

2) limx→−∞F x

( )

=0, limx→∞F x

( )

=1.

3) lim_yxF y

( )

=F x

( )

for x∈.

Proof. The assertion (1) follows from the monotonicity of C∗ and C_∗.

To prove (2), take C∈

{

C C∗, _∗

}

and set F x

( )

=C X

(

≤x

)

. We will see limn→∞F x

( )

n =0 for any se-quence

{ }

xn with x1>x2>→ −∞. By setting An= −∞

(

,xn

]

we have F x

( )

n =C A

( )

n and

1 2 , , n1 n

A ⊃A ⊃



∞₌ A = ∅. It follows from the definition of the capacity that

( )

(

)

lim n lim n 0.

n→∞F x =C n→∞A =

Similarly, limx→∞F x

( )

=1 follows.

Finally, by an argument similar to the proof of (2) with An= −∞ +

(

,x εn

]

, we can show

(

)

( )

limn→∞F x+εn =F x for any sequence

{ }

εn with ε1>ε2>→0, implying (3). □ The proposition above justifies the following definition:

Definition 3.2 For a random variable X, we call the function F∗ and F_∗ as in (1) the upper and lower cumulative distribution functions of X, respectively.

For an IID sequence

{ }

Xi i ₁

∞

= , the empirical distribution function Fˆ :n × Ω →

[ ]

0,1 of

{ }

Xi is defined by

(

)

_{( )}, ₎

( )

1 1

ˆ _{, , , , .}

i n

n X

i

F x I x x n

n ω

ω _{ ∞} ω

=

=

∑

∈ ∈Ω ∈

If

{ }

X_{i i}∞₌₁ satisfies 1

p

X  _{ < ∞}  

 for some p>1, then by Strong law of large number under sublinear ex-pectation (see Theorem 1 in Chen [7]),

( )

_liminf ˆ

( )

_limsup ˆ

( )

( )

_{1, .}

n n

n _n

C∗ F x∗ _→∞ F x F x F∗ x x

→∞

 _{≤ ≤ ≤} _{= ∈}

 

   (2)

Indeed,

[ , )

( )

(

[ , )

( )

)

(

)

( )

1 1 1 ,

i i i

X x C X x C X x F x

∗ ∗ ∗ ∞ ∞  _{ = = = ≤ =}    [ , )

( )

(

[ , )

( )

)

(

)

( )

1 1 1 .

i i i

X ∞ x C∗ X ∞ x C∗ X x F x∗

 

−_− _= = = ≤ =

We show a stronger result, which is a generalization of Glivenko-Cantelli lemma.

Theorem 3.3 Let

{ }

Xi i ₁

∞

= be an IID sequence with 1

p

X  _{ < ∞}  

 for some p>1. Denote by F∗ and F_∗ the upper and lower cumulative distribution functions of X respectively, and denote by Fˆn

( )

x the empirical

disribution function of

{ }

Xi i ₁

∞

= . Then,

( )

_liminf ˆ

( )

_limsup ˆ

( )

( )

_{, 1.}

n n

n _n

C∗ F x∗ _→∞ F x F x F∗ x x

→∞

 _{≤ ≤ ≤ ∈} ₌

 

 

We need the following lemma for the proof of the theorem.

{ }

k ₀

{ }

j _j

t ₌ ⊂ ∪ ±∞ such that −∞ = < <t0 t1 < = ∞tk and

(

j1

)

( )

j ,

(

j1

)

( )

j , 0 1, F∗ t ₊ − −F∗ t ≤ε F t_{∗ +} − −F t_∗ ≤ε ≤ ≤ −j k

where F s

( )

− =lim_usF u

( )

, s∈, for each F∈

{

F∗,F_∗

}

.

Proof. Let ε∈

( )

0,1 . Then there exists t1> −∞ such that F

( ) ( )

t1 ,F t1 ε

∗

∗ ≤ . Starting with t0= −∞, we recursively define

{ }

t_j by

( )

( )

( )

( )

{

}

1 sup : , , 1, 2,

j j j j

t + = z≥t F∗ z ≤F∗ t +ε F z∗ ≤F t∗ +ε j= 

By this recursion, we can find k≥2 such that F

( )

tk 1 1 ε

∗

− ≥ − or F t∗

( )

k−1 ≥ −1 ε hold, and set tk = +∞. With this sequence, the lemma follows. □

With the help of Lemma 3.4, we can show Theorem 3.3.

Proof of Theorem 3.3. Let ε >0 be fixed. By Lemma 3.4, there exists a partion of  such that

0 1 k

t t t

−∞ = < <_< = ∞ and

(

j1

)

( )

j ,

(

j1

)

( )

j , 0 1.

F∗ t ₊ − −F∗ t ≤ε F t_{∗ +} − −F t_∗ ≤ε ≤ ≤ −j k (3)

By (2), we have, C_*

( )

A_q =1 for any q∈, where

( )

( )

( )

( )

{

_liminf ˆ _limsup ˆ

}

_{, .}

s n n

n _n

A F s∗ _→∞ F s F s F∗ s s

→∞

= ≤ ≤ ≤ ∈

Thus, P A

( )

_q =1 for any P∈ and so P

(



q∈Aq

)

=1 for any P∈. Hence

( )

t_j 1,

( )

t_j 1, , 1, , 1

P A = P A − = P∈ j=  k− ,

where for s∈

( )

( )

( )

( )

{

_liminf ˆ _limsup ˆ

}

_.

s n n

n _n

A− F s∗ _→∞ F s F s F∗ s

→∞

= − ≤ − ≤ − ≤ −

So we have

(

)

(

0 j j

)

1.

k

t t

j

C∗



= A ∩A − =

Now, for any x∈ there exists j such that x∈ t_j−1,t_j

)

. Thus, by (3),

( )

( )

( )

( )

1

( )

( )

ˆ ˆ ˆ _,

n n j j n j j

F x −F∗ x ≤F t − −F∗ t₋ ≤F t − −F∗ t − +ε

( )

( )

( ) ( )

1 *

( )

1

( )

1

ˆ ˆ _,

n n j j n j j

F x −F x_∗ ≥F t₋ −F t − ≥F t₋ −F t_{∗ −} −ε

so

( )

ˆn

( )

j1

( )

j1 ˆn

( )

( )

ˆn

( )

j X

( )

j .

F x_∗ +F t₋ −F∗ t₋ − ≤ε F x ≤F∗ x +F t− −F t− +ε

Therefore, on



kj₌₀

(

At_j ∩At_j−

)

, letting n→ ∞ we get F x∗

( )

− ≤ε liminfn→∞F xˆn

( )

and

( )

_limsup ˆ

( )

n n

F∗ x + ≥ε →∞F x for all x∈. Thus

( )

_liminf ˆ

( )

_limsup ˆ

( )

( )

_{, 1, 0.}

n n

n _n

C∗ F x∗ ε _→∞ F x F x F∗ x ε x ε

→∞

 _{− ≤ ≤ ≤ + ∈} _{= >}

 

 

If we write Bε for the event inside the brace above and denote B m1B1m

∞ =

=

_

, then by the continuity of the capacity

( )

(

lim 1m

)

lim

( )

1m 1,

m m

C B∗ =C∗ _→∞B = _→∞C B∗ =

Recall that for a function F:→

[ ]

0,1 satisfying (1)-(3) in Proposition 3.1 and for α∈

( )

0,1 , the α- quantile qα

( )

F of F is defined by

( )

inf

{

:

( )

}

.

q_α F = x∈ F x >α

Then we have the following: Theorem 3.5 Let

{ }

Xi i ₁

∞

= be an IID sequence with 1

p

X  _{ < ∞}  

 for some p>1. Denote by F∗ and F_∗ the upper and lower cumulative distribution functions of X respectively, and denote by Fˆn

( )

x the

empirical disribution function of

{ }

Xi i ₁ ∞

= . Consider the upper, lower, and historical value-at-risk defined respectively by

( )

( )

( )

( )



_{( )}

( )

ˆ

VaR∗ α := −q_α F∗ , VaR_∗ α := −q_α F_∗ , VaRn α := −q_α F_n .

Suppose that for α∈

( )

0,1

( )

, VaR

( )

.

F∗ t <α t< − ∗ α (4)

Then, the historical value-at-risk eventually lies in between the upper and lower value-at-risk, i.e.,

( )



_{( )}



_{( )}

_{( )}

_{( )}

VaR liminf VaRn limsupVaRn VaR 1, 0,1 .

n _n

C∗ ∗ α _→∞ α α ∗ α α

→∞

 _{≤ ≤ ≤} _{= ∈}

 

 

Proof. Consider the event A defined by

( )

( )

( )

( )

{

_liminf ˆ _limsupˆ _,

}

_.

n n

n _n

A F x∗ _→∞ F x F x F∗ x x

→∞

= ≤ ≤ ≤ ∈

In view of Theorem 3.3, it suffices to show that for a given ε >0 and ω ∈A there exists N=N

( )

ω such that

( )

(

ˆ

( )

_,

)

( )

_{, .} n

q_α F∗ − ≤ε q_α F ⋅ω ≤q_α F∗ +ε n≥N (5)

To this end, fix ω ∈A and set q∗ =qα

( )

F∗ , q∗=qα

( )

F∗ . By the definition of the infimum and the

condition (4),

(

2

)

(

2 .

)

F∗ q∗−ε < <α F q_{∗ ∗}+ε

Thus we can take δ >0 such that

(

2

)

(

2 .

)

F∗ q∗−ε < − < + <α δ α δ F q_{∗ ∗}+ε

Next, take N=N

( )

ω such that

( )

ˆ

(

_,

)

( )

_{, , ,} n

F x∗ − ≤δ F xω ≤F∗ x +δ x∈ n≥N

and set qn

( )

ω =qα

(

Fˆn

( )

ω

)

. Then,

( )

(

₂

)

ˆ

(

( )

_{2 ,}

)

(

_{2 .}

)

n n n

F q_∗ ω −ε ≤F q ω −ε ω + ≤ − <δ α δ F q_{∗ ∗}+ε

So we have qn

( )

ω <q∗−ε. Similarly, we see

( )

(

₂

)

ˆ

(

( )

_{2 ,}

)

(

_{2 ,}

)

n n n

F∗ q ω +ε ≥F q ω +ε ω δ α δ− > − >F∗ q∗−ε

leading to qn

( )

ω q ε

∗

> − . Thus (5) follows. □

References

[1] McNeil, A. J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton University Press, Princeton.

[2] Föllmer, H. and Schied, A. (2004) Stochastic Finance: An Introduction in Discrete Time. 2nd Edition, Walter de Gruyter, Berlin.http://dx.doi.org/10.1515/9783110212075

[4] Peng, S. (2006) G-Expectation, G-Brownian Motion and Related Stochastic Calculus of Itô’s type, In: Benth, F.E., et al.,Eds., Stochastic Analysis and Applications: The Abel Symposium 2005, Springer-Verlag, Berlin, 541-567.

[5] Peng, S. (2010) Nonlinear Expectations and Stochastic Calculus under Uncertainty. arXiv:1002.4546[math.PR]

[6] Denneberg, D. (1994) Non-Additive Measure and Integral. Kluwer Academic Publishers, Dordrecht.

http://dx.doi.org/10.1007/978-94-017-2434-0