Estadistica Aplicada

MONTE CARLO SIMULATION OF STOCHASTIC
PROCESSES IN FINANCE AND INSURANCE:
JUMP-DIFFUSION MODELS
N. N. LEONENKO1
1

Cardi¤ School of Mathematics, Cardi¤ University, Senghennydd Road, Cardi¤ CF24 4AG, UK
LeonenkoN@Cardi¤.ac.uk

1

Model

We consider below a "risk process" given by a jump-di¤usion process with
negative jumps given by the following stochastic di¤erential equation (SDE):dXt = c(Xt)dt + (Xt)dB (t)
N (t)
X
S (t) =
Zi ;

dS (t); t

0;

i=1

where c(x) is a state-dependent premium rate, B (t) is standard Brownian
motion, (Xt)dB (t) introduces a source of volatility in the premiums accrual,
N (t) is an independent Poisson process of intensity , and S (t) is a compound
Poisson process modeling the cumulative claims. We will denote by b(x) the
densityof the i.i.d. components Zi ("the claims").
The jump di¤usion process Xt is Markovian, with a¢ ne in…nitesimal generator :
h(x) := dh(x) + j h(x)
where
and

(x)2 @ 2h(x)
@h(x)
+
d h(x) = c(x)
@x
2
@x2
Z1
[h(x z ) h(x)] (dz )
j h(x) =
0

where (dz ) = b(z )dz is the Lévy measure.
Example 1: (dz ) = 0 (no jumps). In this case when
c(Xt) = rXt; (Xt) = Xt;
the SDE has the form
dXt= rXtdt + XtdB (t); t

0;

and its Ito solution is classical geometric Brownian motion:
2

Xt = X0 exp

r

2

t + B (t) ; t

0;

the main process in …nancial modelling. The pricing of options is a problem
of interest.
Example 2: (dz ) = 0 (no jumps). One popular model in this class is the
generalized Ornstein-Uhlenbeck process with parameters c(x) = p + rx:
dXt = (p + rXt)dt+ dB (t); t

0;

which is famous Vasicek model for interest rate. The usual parametrization
is
dXt = k (a Xt)dt + dB (t); t 0;
where k is the speed of mean reversing (k > 0) and a is a long term target
for X: For X < a; the drift is positive. When X > a; the drift is negative.
In both cases, X is pulled back by the drift term toward the long-term target
a:
This model an be written withthe second parametrization ( > 0; ; 2
R) as
dXt =
(Xt
)dt + dB (t); t 0; X0 = x0
with solution
Xt =

+ (x 0

)e

t

+

Z

t

e

(t u)

dB (t); t

0:

0

For > 0; the process is also ergodic with invariant Gaussian density
N ( ; 2=2 2)and covariance function
Cov (Xt; Xs) = V arX0e

jt sj

; t; s 2 R:

This model is used in Physical applications.
Example 3:p(x) = p and = 0: In this case we obtain a classical risk
process for insurance when X0 = u (the initial capital of insurance company)
and p is the premium rate:
Xt = u + pt

N (t)
X
i=1

Zi ; t

0;

We consider now the ruin probability or the …nite-time …rst-passage probability:
(t; x) = P f
tjX0 = xg ; x 0:
where

= inf ft

or survival probability
(x; t) = 1

(t; x) = P fXs0 : Xt < 0g ;
0; 0

s

tjX0 = xg ; x

0:

For the in…nite horizon problem t ! 1; the survival probability
(x) = P fXt

0; t

0jX0 = xg ;

and probability of ruin
(t) = P fXt < 0 for some t
2

0 jX0 = xg ; x

0:

Simulation

2.1 Simulation of Brownian motion: B (t); 0
Method A)
Divine the interval [0; T ] a grid such that
0 = t1 < t2

< tN

1

t

T:

=Twith ti+1 ti = t. Set i = 1 and B (0) = B (t1) = 0 and iterate the following
algorithm:
1) Generate a (new) r.v. Z N (0:1)
2) i = i + 1;
p
3) Set B (ti) = B (ti 1) = Z
t;
4) If i N , iterate from step 1.
Between any two point ti and ti+1 the trajectory is approximated by linear
interpolation.
Method B)
By using the Karhunen-Loéve expansion
p
1
X
2 2T
(2i + 1) t
B (t) =
Zi i(t);i(t) =
sin
; 0 t T;
(2i + 1)
2T
i=0

where Zi; i = 0; 1; : : : are iidrv. N (0; 1):
Method C) One can simulate Brownian motion as Gaussian processes
with zero mean and given covariance function:
Cov (B (t); B (s)) = min(t; s):
2.2 Simulation of geometric Brownian motion:
2

Xt = X0 exp

r

t + B (t) ; t

2

0;

One can use
2

Xt+

t

= Xt exp (r

2

)(t + t...