Estimation of Value at Risk and ruin probability for
diffusion processes with jumps
Begoña Fernández
Universidad Nacional Autónoma de México
joint work with Laurent Denis and Ana Meda
PASI, May 2010
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We will consider a process X that satisfies
Z
Xt = m +
t
t
Z
σs dBs +
0
bs ds +
0
Nt
X
γT − Yi , t > 0,
i=1
(1)
i
Xt∗ = sup Xu .
0≤u≤t
• We will obtain upper and lower bounds for
P[Xt∗ > z]
• We will use these bounds for estimations to Ruin
Pobabilities and VaR where
For q = 1 − α in ]0, 1[, the Value at Risk VaR associated
with Xt∗ will be given by
VaR (X ∗ ) = inf{z ∈ IR, P(X ∗ ≤ z) ≥ q}
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Z
Xt = m +
t
Z
σs dBs +
0
t
bs ds +
0
Nt
X
i=1
γT − Yi , t > 0,
i
where B is a one-dimensional Brownian Motion , N is a Poisson
Process independent of B, T1 , T2 , . . . , are the jump times for N, the
random variables Yi , i ≥ 1 are i.i.d. and independent of the Poisson
Process and the Brownian Motion.
We assume the following hypotheses and denote them by (H):
(1) b is an integrable process.
Rt
(2) For all t > 0, E( 0 σs2 ds) < +∞.
(3) The jumps of the compound Poisson process are non-negative,
i.e., Y1 ≥ 0 P-a.e., and we assume that Y1 is not identically equal
to 0.
P t
(4) The process ( N
i=1 γT − Yi , t > 0) is well-defined and integrable.
i
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Hypothesis UBï¿ 12
(1) We assume that the law of the jumps admits a Laplace transform
defined on ] − ∞, c[ where c is a positive constant (or c = +∞)
and we put
L(x) = E[exY1 ], x < c.
(2) There exists γ ∗ > 0 such that γs ≤ γ ∗ , P-a.s. for all s ∈ [0, t].
(3) There exist 0 < δ < (c/γ ∗ ) and a constant Kt (δ) ≥ 0, such that, for
all s ∈ [0, t],
Z
Z s
δ2 s 2
σu du + λs(L(δγ ∗ ) − 1) ≤ Kt (δ) a.e.
(2)
δ
bu du +
2
0
0
Let us remark that Assumption (UB)(3) is fulfilled if we replace it by the
stronger one:
(30 ) There exist constants b∗ (t) ≥ 0, a∗ (t) ≥ 0 such that,
Z t
Z s
bu du ≤ b∗ (t) P-a.e. ∀s ∈ [0, t].
σu2 du ≤ a∗ (t),
0
0
In this case one has, for all 0 < δ < c/γ ∗ ,
a∗ (t)
Begoña Fernández Universidad Nacional Autónoma
Estimation
of Value
Risk with
and ruin
Laurent
probability
Denis and
for
Ana Meda
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The main result to get the upper estimate is:
Lemma
Assume (UB). Let δ ∈ ]0, c/γ ∗ [ be as in (3) of (UB), and let z ≥ m.
Then,
P(Xt∗ ≥ z) ≤ exp{δ(m − z) + Kt (δ)}.
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Proof.
We have
P(Xt∗ ≥ z) = P (δXt∗ ≥ δz)
!
≤ P
sup Ms ≥ exp{δ(z − m) − Kt (δ)} ,
0≤s≤t
where {Ms , s ≥ 0} is the martingale defined by
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"Z
Ms = exp
s
σu dBu +
δ
0
Nt
X
i=1
#
γ ∗ Yi
δ2
−
2
Z
0
s
!
σu2 du − λs(L(γ ∗ δ) − 1) .
{Ms , s ≥ 0} is a martingale since it is the product of a continuous
martingale and a purely discontinuous one. The maximal inequality for
exponential martingales gives the result.
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Proposition
Let us assume (UB), and let
A = {δ ∈ ]0, c/γ ∗ [ | δ satisfies (UB)(3)}.
Then
VaRα (Xt∗ )
≤ inf
δ∈A
Kt (δ) ln α
−
m+
δ
δ
.
Proof. Thanks to Lemma 1.1, P(Xt∗ > z) < α is implied by
z >m+
Kt (δ) ln α
−
,
δ
δ
which yields the result.
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Hypothesis (LB):
There exist constants b∗ (t) ≤ 0, a∗ (t) > 0, and γ∗ ∈ IR, such that
Z t
Z s
2
σu du ≥ a∗ (t) ,
bu du ≥ b∗ (t) and γs ≥ γ∗ P-a.e. ∀s ∈ [0, t].
0
0
The lower bound depends on the sign of γ∗ , so we shall discuss each
case separately.
Lemma
Let us assume (LB), and γ∗ ≤ 0. Then, for all z ∈ IR and t ≥ 0,
P(Xt∗ ≥ z) ≥ P(
Nt
X
p
a∗ (t)|Z | + γ∗
Yi ≥ z − m − b∗ (t)),
i=1
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Sketch of the proof Proof
. Since by hypothesis the compound Poisson and the Brownian motion
are independent, we use the product structure of the measure and
adopt a natural notation, a change of time and the fact that we know
the distribution of the sup of the Brownian Motion.
Z
P 1 (Xt∗ (·, w2 ) ≥ z)dP 2 (w2 ).
P(Xt∗ ≥ z) =
Ω2
For fixed w2 , set v =
PNt
i=1 Yi (w2 ).
P 1 (Xt∗ (·, w2 ) ≥ z) ≥ P 1 ( sup
Then we have
s
Z
σu (·, w2 )dBu ≥ z − m − b∗ (t) − γ∗ v ).
s∈[0,t] 0
As B is a P 1 -Brownian motion,
Z
Rs =
s
σu (·, w2 )dBu
0
is a P 1 -continuous martingale. By the change of time property, we
know that there exists a P 1 -Brownian motion B̃ such that
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For γ∗ ≥ 0 we can give a lower bound which is always valid. For this,
let us introduce the process
Z
Ys = m +
s
Z
bu du +
0
s
σu dBu , s > 0.
0
As γ∗ ≥ 0, the process X dominates Y , so for all α,
VaRα (Xt∗ ) ≥ VaRα (Yt∗ ).
Proposition
If we assume hypotheses (LB), and γ∗ ≥ 0, for all α,
VaRα (Xt∗ ) ≥ VaRα (Yt∗ ) ≥ m + b∗ (t) +
p
a∗ (t)Φ̄−1 (α/2).
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Now, if we make the additional hypothesis (denoted by (LB0 )) that
there exists a constant a∗ (t) such that
Z t
σu2 du ≤ a∗ (t),
0
we can considerably improve the bounds.
Lemma
Under (LB), (LB0 ), and γ∗ ≥ 0, for all z ∈ IR and t ≥ 0,
p
p
P(Xt∗ ≥ z) ≥ P( a∗ (t)Z1 − a∗ (t) − a∗ (t)|Z2 |
+ γ∗
Nt
X
Yi ≥ z − m − b∗ (t)),
i=1
where Z1 and Z2 are N(0, 1) independent random variables that do not
depend on N and (Yi ).
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We will assume in this section that the process Xs , s ∈ [0, t] is given by
Z
s
σu dBu −
Xs = m + bs +
0
Ns
X
i=1
γT − Yi ,
m ≥ 0,
(9)
i
where b is a constant and we assume that there exist constants a∗ and
γ ∗ > 0 such that
∀s ∈ [0, t], σs2 ≤ a∗ , γs ≤ γ ∗ a.e.
(10)
If γ ∗ = 1, we have the classical risk process (see [Asm, Gra]). If the
insurer invests in a risky asset we obtain this general model, see for
example [GaGrSc]. We can apply Lemma 1.1 to estimate the ruin
probability.
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Proposition
Let θ = E(Y1 ). Let X be as in (9), with coefficients bounded as in (10)
above. Assume in addition that a∗ > 0 and c = ∞; or that
limx→c/γ ∗ L(x) = +∞; and that the following safety load condition
holds:
b − λθγ ∗ > 0.
(11)
Denote by δ ∗ the greatest positive root in ]0, c/γ ∗ [ of
h(δ) = −bδ +
δ2 ∗
a + λ(L(γ ∗ δ) − 1) = 0.
2
Then,
P( sup −Xs ≥ 0) ≤ e−δ
∗m
.
0≤s≤t
As a consequence, we have the following upper bound for the ruin
probability:
P(Xs < 0 for some s in ]0, ∞[) ≤ e−δ
∗m
.
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The jumps have exponential law with parameter ν > 0.
Corollary
Suppose (UB), (LB), and (LB0 ) hold. Assume that for all s ∈ [0, t],
γ∗ ≤ γs ≤ γ ∗ , a.e., with γ ∗ > 0.
If γ∗ > 0 then
VaRα (Xt∗ )
VaRα (Xt∗ )
γ∗
γ∗
≤ lim inf
≤ lim sup
≤
.
ν
| ln α|
| ln α|
ν
α→0
α→0
If γ∗ ≤ 0 then
p
VaRα (Xt∗ )
VaRα (Xt∗ )
γ∗
2a∗ (t) ≤ lim inf p
and lim sup
≤
.
| ln α|
ν
α→0
| ln α|
α→0
The geometric Brownian motion model.
Now we consider a case widely used in finance: imagine X satisfies
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So for all t ≥ 0,
Z
Yt = ln m +
t
Z
σs dBs +
0
t
b̃s ds +
0
Nt
X
γ̃T − Yi ,
i=1
i
where the processes b̃ and γ̃ are defined by
1
b̃s = bs − σs2 , s ≥ 0
2
and
+∞
X
ln(1 + γs Yi )
γ̃s =
1{s∈]Ti ,Ti+1 ]} ,
Yi
i=0
where T0 = 0 and we adopt the convention that ln(1 + 0)/0 = 1. If Y is
integrable, we can apply all the results of the previous sections to get
an estimate for VaRα (Yt∗ ) and hence for VaRα (Xt∗ ), thanks to the
following result, whose proof is now obvious:
Proposition
Let X be as in (12). Under (GBM), for all α ∈ ]0, 1[,
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The work with heavy tails is in process and it will appear soon!!!!!!!!!!!
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Thanks
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S. Asmussen; Ruin Probabilities, World Scientific, Singapore,
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D. Talay, Z. Zheng; “Quantiles of the Euler Scheme for Diffusion
Processes and Financial Applications”, Conference on
Applications of Malliavin Calculus in Finance (Rocquencourt,
2001). Math. Finance 13 no. 1, pp 187-199, 2003.
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