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Adaptive Filtering for Stochastic Volatility by using Exact Sampling
ShinIchi Aihara
1
, Arunabha Bagchi
2
and Saikat Saha
3
1
Department of Mechanical Systems, Tokyo University of Science Suwa, 5000-1 Toyohira, Chino, Nagano, Japan
2
Department of Applied Mathematics, Twente University, P.O.Box 217,7500AE, Ensched, The Netherlands
3
Department of Electrical Engineering, Lik¨oping University, SE-58183 Lik¨oping Sweden
Keywords:
Particle Filter, Stochastic Volatility, Parameter Identification, Adaptive Filter.
Abstract:
We study the sequential identification problem for Bates stochastic volatility model, which is widely used as
the model of a stock in finance. By using the exact simulation method, a particle filter for estimating stochastic
volatility is constructed. The systems parameters are sequentially estimated with the aid of parallel filtering
algorithm. To improve the estimation performance for unknown parameters, the new resampling procedure is
proposed. Simulation studies for checking the feasibility of the developed scheme are demonstrated.
1 INTRODUCTION
In the early 1960s, the linear filtering theory is formu-
lated by Kalman and Bucy (Kalman and Bucy, 1961)
and nonlinear filtering has already been well devel-
oped by many researchers, see Bensoussan (Bensous-
san, 1992) and the bibliography therein. The realiza-
tion problem for the nonlinear filter is still not easy.
The recent development of particle filtering theory
(Doucet et al., 2000) enable us to realize the nonlin-
ear filtering in an easy way with the aid of the digital
computer.
In this paper we consider the Bates model which
is used in the fiance problem. In this model, we ob-
serve the tick value of stock price and need to esti-
mate the movement of the volatility process for trad-
ing the stock and/or options. It is not possible to apply
the nonlinear filtering theory to this volatility estima-
tion problem, because this is out of the usual filter-
ing problem in the continuous stochastic systems (Ai-
hara and Bagchi, 2006). To circumvent this difficulty,
the particle filter theory is usually applied in (Aihara
et al., 2008; Capp´e et al., 2005; Javaheri, 2005). The
Bates model is given by
dS
t
= µ
S
S
t
dt +
√
v
t
S
t
dB
t
+ S
t
dZ
J
t
−λm
J
S
t
dt, (1)
dv
t
= κ(θ−v
t
)dt + ξ
√
v
t
dZ
t
(2)
where B
t
and Z
t
are standard Brownian motion pro-
cesses with correlation ρ and Z
J
t
denotes the pure-
jump process. Noting that the process S
t
denotes
the stock value, the observation data y
t
= logS
t
/S
0
is given by
dy
t
= (µ
S
−λm
J
−
1
2
v
t
)dt +
√
v
t
dB
t
+ dq
J
t
, (3)
where q
J
t
is a compound Poisson process with in-
tensity λ and Gaussian distribution of jump size,i.e.,
N(µ
J
,σ
2
J
), and the mean relative jump size is given
by m
J
= E(exp(U
s
) −1) = exp(µ
J
+ σ
2
J
/2) −1 and
where the λm
J
S
t
term in (1) compensates for the in-
stantaneous change in expected stock introduced by
the pure-jump process Z
J
t
. The particular properties
of this model are
1. The observation mechanism (3) contains the sig-
nal dependent noise.
2. The observation noise has a correlation with the
systems noise.
From the first property, the estimation of stochastic
volatility becomes out of filtering problem. To cir-
cumvent this difficulty, all systems are discretized and
the particle filter is applied in (Aihara et al., 2008;
Johannes and Polson, 2006). However the usual dis-
cretization method transformed the original continu-
ous non-Gaussian system into the conditional Gaus-
sian. Recently, Brodie and Kaya (Broadie and Kaya,
2006; Smith, 2008; van Haastrecht and Pelsser, 2010)
proposed the exact simulation method from the fact
that the original system has a non-central chi-square
distribution and we use this technique to the particle
filtering (Aihara et al., 2012). Introducing the new
Brownian motion
˜
Z
t
=
1
p
1−ρ
2
(Z
t
−ρB
t
), (4)
326
Aihara S., Bagchi A. and Saha S..
Adaptive Filtering for Stochastic Volatility by using Exact Sampling.
DOI: 10.5220/0004454703260335
In Proceedings of the 10th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2013), pages 326-335
ISBN: 978-989-8565-70-9
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)