- What is V aR? - V alue at Risk is the loss, whih is - - PowerPoint PPT Presentation

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- What is V aR? - V alue at Risk is the loss, whih is - - PowerPoint PPT Presentation

- Qualit y of V alue-at-Risk App ro ximations b y Numerial Solutions of SDEs - Denis T ala y and Ziyu Zheng This w o rk is a joint ollab o ration b et w een the RiskLab and INRIA Sophia Antip olis.


slide-1
SLIDE 1
  • Qualit
y
  • f
V alue-at-Risk App ro ximations b y Numeri al Solutions
  • f
SDEs
  • Denis
T ala y and Ziyu Zheng This w
  • rk
is a joint
  • llab
  • ration
b et w een the RiskLab and INRIA Sophia Antip
  • lis.
(Institut National de Re her he en Info rmatique et en Automatique) RiskLab Risk Da y 2000 :0

slide-2
SLIDE 2
  • What
is V aR?
  • V
alue at Risk is the loss, whi h is ex eeded with some given p roba- bilit y α ,
  • ver
a given ho rizon. Mathemati ally , V alue at Risk is the quantile
  • f
a random va riable, whi h des rib es the p
  • ssible
loss in the future. RiskLab Risk Da y 2000 :1

slide-3
SLIDE 3
  • A
t ypi al example
  • A
nan ial ma rk et
  • nsists
in N + M + 1 se urities: N sto ks, M b
  • nds
and an instantaneous riskless saving a ount. An investment a tivit y is mo delled b y a N + M + 2 dimensional sto hasti dierential equation, whose solution rep resents the p ri es
  • f N
sto ks, the p ri es
  • f M
b
  • nds,
the saving a ount and the w ealth p ro ess X(·)
  • f
the trader's p
  • rtfolio.
The V aR(α) fo r this investment is the la rgest value su h that

P (X(T) − X(0) ≤ V aR(α)) ≤ α.

RiskLab Risk Da y 2000 :2

slide-4
SLIDE 4
  • Cal ulation
  • f
V aR
  • The
b est w a y to al ulate V aR,
  • f
  • urse,
w
  • uld
b e to use an analyti fo rmula. F
  • r
a mo del whose
  • e ients
a re
  • mplex
fun tions
  • r
a high di- mensional mo del, it is imp
  • ssible
to al ulate the V aR expli itly . Solution: App ro ximate the V aR b y numeri al metho ds. RiskLab Risk Da y 2000 :3

slide-5
SLIDE 5
  • Evaluating
existing metho ds
  • When
  • ne
ho
  • ses
a numeri al metho d fo r V aR al ulation,
  • ne
has to mak e a tradeo b et w een a ura y , numeri al
  • st
and generalit y . W e give some
  • mments
  • n
some standa rd metho ds. Delta metho d, Delta-gamma metho d and related metho ds a re fast but not very a urate, and they require a simple mo del (whi h de- reases the global a ura y). F ull Monte Ca rlo metho d gives a b etter a ura y , but it requires exa t p ri ing fo rmula (whi h restri ts the hoi e
  • f
the mo dels), and is time- onsuming. RiskLab Risk Da y 2000 :4

slide-6
SLIDE 6
  • Evaluating
existing metho ds ( ont.)
  • The
Grid Monte Ca rlo metho d gives go
  • d
a ura y and an b e applied to general nonlinea r mo dels, but w e re ommend it fo r lo w dimensional p roblems
  • nly
, fo r a p
  • rtfolio
with a la rge numb er
  • f
se urities, the numeri al
  • st
is very exp ensive. RiskLab Risk Da y 2000 :5

slide-7
SLIDE 7
  • What
is the suitable numeri al metho d to al ulate V aR?
  • Ma
y w e nd a new metho d, whi h is a urate, an b e applied to general nonlinea r and high dimensional mo dels, and to long-term risk measurement p roblems? W e emphasize that, in the sto hasti mo del as ab
  • ve,
The V aR under interest is the quantile
  • f
the ma rginal distribution
  • f
the solution at ho rizon T to a sto hasti dierential equation. Thus the p roblem is: nd a suitable numeri al metho d to app ro ximate su h quantiles. RiskLab Risk Da y 2000 :6

slide-8
SLIDE 8
  • The
answ er
  • The
Monte Ca rlo metho d.

+

The Euler s heme. RiskLab Risk Da y 2000 :7

slide-9
SLIDE 9
  • What
is the Euler s heme?
  • Let (Xt)
b e a real valued diusion p ro ess, solution to

dXt = b(Xt)dt + σ(Xt)dWt,

where (Wt) is a r
  • dimensional
Bro wnian motion. The Euler s heme is a time dis retization
  • f
the SDE:

Xn

(p+1)T/n = Xn pT/n + b(Xn pT/n)T

n + σ(Xn

pT/n)(W(p+1)T/n − WpT/n).

n

is the numb er
  • f
steps
  • f
the dis retization and p = 0,1,...,n − 1 .

Xn

T

is a go
  • d
app ro ximation
  • f XT
. RiskLab Risk Da y 2000 :8

slide-10
SLIDE 10
  • What
is the Monte Ca rlo metho d?
  • The
la w
  • f Xn

T

is to
  • mplex
to al ulate expli itly . W e use N i.i.d.
  • pies
  • f Xn

T

to app ro ximate E [f(XT )] .

1 N

N

  • i=1

f(Xn,i

T ) → E [f(XT )],

as n,N → ∞. The left hand side an b e easily simulated
  • n
a
  • mputer:
  • ne
simply has to simulate the indep endent Gaussian in rements
  • f
the Bro wnian motion. RiskLab Risk Da y 2000 :9

slide-11
SLIDE 11
  • The
dis retization erro r
  • Under
hyp
  • theses
  • f
unifo rmly hyp
  • ellipti it
y t yp e, XT has a smo
  • th
densit y pXT . Therefo re given a p
  • sitive
real 0 < δ < 1 , there exists a quantile ρ(δ) su h that

P [XT ≤ ρ(δ)] = δ.

F
  • r
the mollied Euler s heme, Xn

T

also has a smo
  • th
densit y , thus there exists a quantile ρn(δ) su h that

P [Xn

T ≤ ρn(δ)] = δ.

RiskLab Risk Da y 2000 :10

slide-12
SLIDE 12
  • The
dis retization erro r ( ont.)
  • W
e have p roved that the dis retization erro r
  • n
the quantile satises

|ρn(δ) − ρ(δ)| ≤ C(T) qT(δ)n,

where

qT(δ) = inf

y∈[ρ(δ)−1,ρ(δ)+1] pXT (y) ≃ pXT (ρ(δ)).

RiskLab Risk Da y 2000 :11

slide-13
SLIDE 13
  • The
statisti al erro r
  • W
e an so rt the simulated
  • Xn,i

T ,i = 1,...,N

  • ,
and thus get the empi- ri al quantile ρn

N(δ)

. The lassi al theo ry tell us that the statisti al erro r is
  • f
  • rder

|ρn(δ) − ρn

N(δ)| ∼

1 qT(δ) √ N .

RiskLab Risk Da y 2000 :12

slide-14
SLIDE 14
  • The
global erro r
  • f
the metho d
  • Thus
the global erro r
  • f
the simulated quantile satises

E |ρ(δ) − ρn

N(δ)| ≤

C(T) qT(δ) √ N + C(T) qT(δ)n.

F
  • r
a p ra ti al use,
  • ne
needs an a urate lo w er b
  • und
  • f
the den- sit y
  • f XT
in
  • rder
to ho
  • se n,N
in terms
  • f
the desired a ura y . F
  • r
stri tly unifo rm ellipti generato rs, see, e.g., Azen ott. In dege- nerate ase, under restri tive assumption
  • n
the drift b , see Kusuok a & Stro
  • k.
RiskLab Risk Da y 2000 :13

slide-15
SLIDE 15
  • The
ase
  • f
high dimension and ma rginal la w
  • Let (Xt)
b e a Rd valued p ro ess, solution to

dXt = b(Xt)dt + σ(Xt)dWt.

Denote b y (Xi

t)

the
  • rdinate
p ro ess
  • f (Xt)
. Supp
  • se
the generato r
  • f (Xt)
is unifo rmly hyp
  • ellipti ,
then there exists a quantile ρi(δ) su h that

P [Xi

T ≤ ρi(δ)] = δ.

and a quantile ρn,i(δ)
  • f
the Euler s heme su h that

P [Xn,i

T

≤ ρn,i(δ)] = δ.

RiskLab Risk Da y 2000 :14

slide-16
SLIDE 16
  • The
ase
  • f
a ma rginal la w ( ont.)
  • W
e have p roved

|ρn,i(δ) − ρi(δ)| ≤ C(T) qi

T(δ)n.

qi

T(δ) =

inf

y∈[ρi(δ)−1,ρi(δ)+1]

pi

XT (y) ≃ pi XT (ρi(δ)),

where pi

XT

is the i -th ma rginal distribution
  • f XT
. W e re all that the situation here is exa tly the
  • ne
w e gave in the example fo r the p
  • rtfolio
  • f N + M + 1
se urities. RiskLab Risk Da y 2000 :15

slide-17
SLIDE 17
  • Con lusion
  • W
e
  • n lude
b y listing some p rop erties
  • f
  • ur
numeri al app roa h: 1. A desired a ura y an b e a hieved b y ho
  • sing N
and n p rop erly . 2.Our erro r estimates hold fo r hyp
  • ellipti
SDE, whi h is a
  • mmon
situation in nan e. 3. The numeri al
  • st
gro ws
  • nly
linea rly w.r.t the dimension. Mo reo- ver, the numeri al
  • st
an b e redu ed b y using pa rallel a r hite tures. 4. It an b e applied to long term risk measurement p roblems. RiskLab Risk Da y 2000 :16

slide-18
SLIDE 18
  • An
appli ation to Mo del Risk measurement
  • Given
t w
  • maturities: T O < T
. The trader w ants to hedge a Europ ean
  • ption
with maturit y T O written
  • n
a dis ount b
  • nd B(t,T)
. The pa y
  • at
maturit y is denoted b y Φ(B(T O,T)) . The trader uses t w
  • b
  • nds
to hedge the
  • ption:
the b
  • nd
with maturit y T O and the b
  • nd
  • f
maturit y T . RiskLab Risk Da y 2000 :17

slide-19
SLIDE 19
  • The
Heath-Ja rro w-Mo rton mo del
  • Instantaneous
fo rw a rd rate under the sp
  • t
ma rtingale measure:

f(t,T ∗) = f(0,T ∗) +

t

0 σ(s,T ∗)σ∗(s,T ∗)ds +

t

0 σ(s,T ∗)dWs,

with

σ∗(s,T ∗) :=

T ∗

s

σ(s,u)du.

Dis ount b
  • nd
p ri e B(t,T) :

B(t,T) = 1 −

T

t

r(s)B(s,T)ds +

T

t

σ∗(s,T)B(s,T)dWs, 0 ≤ t ≤ T.

RiskLab Risk Da y 2000 :18

slide-20
SLIDE 20
  • The
p
  • rtfolio
  • The
p
  • rtfolio

Vt = HtB(t,T) + HO

t B(t,T O).

Assumption: The p
  • rtfolio
is self-nan ing, i.e.,

Vt = V0 +

t

0 HsdB(s,T) +

t

0 HO s dB(s,T O).

RiskLab Risk Da y 2000 :19

slide-21
SLIDE 21
  • The
fo rw a rd p ri es
  • f
the b
  • nd
and
  • f
the p
  • rtfolio
  • F
  • rw
a rd p ri e
  • f
the b
  • nd:

BF (t,T) := B(t,T) B(t,T O).

F
  • rw
a rd p ri e
  • f
the p
  • rtfolio:

V F

t

:= Vt B(t,T O) = HtB(t,T) + HO

t B(t,T O)

B(t,T O) .

F rom the self-nan ing
  • ndition

dV F

t

= HtdBF (t,T).

RiskLab Risk Da y 2000 :20

slide-22
SLIDE 22
  • Exa t
hedging strategy in the ase: σ(t,T) is deterministi
  • A
self-nan ing exa t hedging strategy is a pair (H0

t ,Ht)

Ht = ∂πσ ∂x (t,BF (t,T)),

where the fo rw a rd p ri e
  • f
the
  • ption πσ
is the solution to a Bla k- S holes t yp e equation,

    

∂πσ ∂t (t,x) + 1 2x2(σ∗(t,T O) − σ∗(t,T))2∂2πσ ∂x2 (t,x) = 0, πσ(T,x) = Φ(x).

F rom the self-nan ing
  • ndition,

H0

t = V F 0 +

t

0 HsdBF (s,T) − HtBF(t,T).

RiskLab Risk Da y 2000 :21

slide-23
SLIDE 23
  • Mo
del risk measurement
  • A
bad hoi e
  • r
estimate σ
  • f σ
leads to a wrong hedging strategy

Ht = ∂πσ ∂x (t,BF (t,T)).

The p
  • rtfolio
using the missp e ied hedging strategy:

V F

t = V F 0 +

t

0 HtdBF (t,T).

F
  • rw
a rd p ri e
  • f
Prot & Loss:

P&LF

t := V F t − V F t .

Mo del risk measurement: Quantile, E [U(P&LT O)] , (see Artzner, Del- baen, Eb er and Heath.), et . RiskLab Risk Da y 2000 :22

slide-24
SLIDE 24
  • Quantile
  • f
mo del risk
  • P [P&LT O ≤ ρ(δ)] = δ.
Set

          

u1(t) := σ∗(t,T O), u2(t) := (σ∗(t,T O) − σ∗(t,T)), ϕ(t,x) := ∂πσ ∂x (t,x) − ∂πσ ∂x (t,x).

Thus P&LF

t

is the solution to

  

dBF (t,T) = BF (t,T)u1(t)u2(t)dt + BF(t,T)u2(t)dWt, dP&LF

t = ϕ(t,BF (t,T))dBF (t,T).

RiskLab Risk Da y 2000 :23

slide-25
SLIDE 25
  • Existen e
  • f
a smo
  • th
densit y
  • f P&LT O
.
  • Supp
  • se
that |u2(t)| ≥ a > 0 , fo r any t in [0,T O] , and ϕ(0,BF (0,T)) = , then the la w
  • f P&LT O
has a smo
  • th
densit y pT O (ma rginal la w
  • f
the Ma rk
  • v
p ro ess (BF (t,T),P&LF

t )

). W e also have the stri t p
  • sitivit
y
  • f pT O
  • n
its supp
  • rt.
Mo reover, w e an
  • btain
upp er b
  • unds
  • n
the derivatives
  • f pT O
. Pro
  • f.
W e study the Malliavin va rian e D(P&LF

T O),D(P&LF T O)

. RiskLab Risk Da y 2000 :24

slide-26
SLIDE 26
  • The
lo w er b
  • und.
  • By
adding
  • ne
te hni al assumption, w e
  • btain

pt(y0,y) ≥ 1

T O

u2

2(s)ds

exp(− (ln(Υ−1(T O,y − y0 + Υ(0,x0)) − ln x0) −

T O

(u1(s) − 1 2u2

2(s))ds)2

4

T O

u2

2(s)ds

−C),

where Υ(t,y) :=

y

0 ϕ(t,x)dx.

and Υ−1(t,·) is the inverse fun tion
  • f

Υ(t,·)

. Pro
  • f.
W e apply the te hnique
  • f
Girsanov transfo rmation. RiskLab Risk Da y 2000 :25

slide-27
SLIDE 27
  • Con lusion
  • F
rom
  • ur
study
  • n
the densit y
  • f
the Prot & Loss, w e
  • btain
a p re ise estimate
  • n
the global erro r
  • f
the Monte Ca rlo metho d fo r the app ro ximation
  • f
the quantiles
  • f
the Prot & Loss from missp e ied hedging strategies. Our metho dology an b e applied to wide lass
  • f
p roblems from Risk measurement and V aR analysis... RiskLab Risk Da y 2000 :26