- 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
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- 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
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
SLIDE 8
The
answ er
The
Monte Ca rlo metho d.
+
The Euler s heme. RiskLab Risk Da y 2000 :7
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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
Ω
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,
