Estimation of Jump-Diffusion Processes via
Empirical Characteristic Functions
Michael Rockingera
Maria Semenovab∗
February 2005
Abstract
This article proposes an estimation procedure for the affine stochastic volatility models with jumps both in the asset price and variance processes. The
estimation procedure is based on the joint (here bi-variate) unconditional characteristic function for the stochastic process for which we derive a closed form
expression. The estimation of the general model and of various restrictions, on
SP500 data, is performed using the continuous empirical characteristic function method. The estimation suggests that besides a stochastic volatility, jumps
both in the mean and the volatility equation are relevant.
a
Corresponding author. HEC Lausanne and FAME, Institute of Banking
and Finance, Route de Chavannes 33, CH-1007 Lausanne-Vidy, Switzerland.
E-mail: [email protected]
b HEC Lausanne and FAME, Institute of Banking and Finance, Route de
Chavannes 33, CH-1007 Lausanne-Vidy, Switzerland.
E-mail: [email protected]
Keywords: Affine jump-diffusions, Characteristic functions, Stochastic
volatility, Empirical estimation
JEL classification: G12, C22, C52
∗
Michael Rockinger is also CEPR. Both authors acknowledge help from the Swiss National Science
Foundation through NCCR (Financial Valuation and Risk management) as well as through grant on
“Multivariate modelling of asset prices using the information contained in transaction data”. The
usual disclaimer applies.
1
Estimation of Jump-Diffusion Processes via
Empirical Characteristic Functions
Abstract
This article proposes an estimation procedure for the affine stochastic volatility
models with jumps both in the asset price and variance processes. The estimation
procedure is based on the joint (here bi-variate) unconditional characteristic function
for the stochastic process for which we derive a closed form expression. The estimation
of the general model and of various restrictions, on SP500 data, is performed using the
continuous empirical characteristic function method. The estimation suggests that
besides a stochastic volatility, jumps both in the mean and the volatility equation are
relevant.
2
1
Introduction
In this paper we derive the estimation procedure for the affine jump-diffusion process
using the joint unconditional empirical characteristic function. We also provide the
analytical solution for such a characteristic function. Our method allows the estimation of the parameters of a process with unobservable state variables avoiding
the discretization of the process and the simulation of the latent variables. Three
specifications of the jump-diffusion processes are estimated using the continuous
empirical characteristic function method: affine diffusion with diffusive stochastic
volatility model (SV) of Heston (1993); affine jump-diffusion with stochastic volatility and jumps in the asset price (SVJ) of Bates (1996); and eventually the affine
jump-diffusion with jumps both in the asset price and stochastic volatility processes
(SVJJ) described by Duffie, Pan, and Singleton (2000). The last model is of the
major interest to us due to the growing literature on the relevance of jumps in the
volatility for the correct specification of the asset price generating process.
The difficulty with the estimation of the jump-diffusion processes is based on
several issues. First, most of the existing stochastic-process estimation-procedures
require the knowledge of analytical solutions for the density function or its moments.
Second, for some state variables, only a discrete sample of observations can be obtained, while the estimated model is formulated in continuous time. Third, some
state variables are not observable at all. To deal with these unobservable variables,
several inference procedures have been proposed. The most popular are: Simulated
method of moments (Duffie, Singleton, 1993), Efficient method of moments (Gallant,
Tauchen, 1996), Simulated maximum likelihood (Durham, Gallant, 2001; Brandt,
Santa-Clara, 2002 ), Markov Chain Monte Carlo and Sequential Bayesian inference
(Jacquier, Polson, Rossi, 1994; Jones, 1998; Eraker, Johannes, Polson, 2003), as well
3
as Characteristic function methods, (Singleton, 2001; Chacko and Viceira, 2002; Jiang
and Knight, 2002).
We favor the estimation procedures that involve the characteristic function for a
number of reasons. First, the fact that there is a one-to-one relationship between
the distribution function and the characteristic function allows the estimation of the
model parameters using the characteristic function of the process instead of its density
function without any loss of information. This is a convenient feature, considering
that analytical solutions for characteristic functions of processes are available for a
wider set of models then there are solutions that yield an expression for the density
functions. Duffie, Pan, and Singleton (2000), for example, derive a closed form expression of the characteristic function for the affine jump-diffusion. Second, both the
presence of latent variables and the absence of the continuous sample of observations
may result in the need for a discretization of the stochastic process. Simulation based
methods would require the discretization of the process even if there are no latent
variables. The use of an appropriate characteristic function method allows estimation
of the model parameters without simulation or discretization of the process.
For example, the Conditional Characteristic Function (CCF) estimation procedure
proposed by Singleton (2001), and applied to the affine diffusions without latent
variables does not call neither for simulation nor for the discretization of the process.
Moreover, the estimator attains the efficiency of the ML estimator. However, the
procedure carries a significant computation burden and in the presence of the latent
variable in the model one is forced to simulate this variable.
Unconditional Characteristic Function methods (UCF) discussed by Chacko and
Viceira (2002), Jiang and Knight (2002) are based on integrating the latent variable
out, so that there is no need for simulations or discretization of the process at all.
4
Chacko and Viceira (2002) completely integrate out the latent variable obtaining
the marginal characteristic function of the asset log-price. Jiang and Knight (2002)
partially keep the information contained in the data, namely in some block of past
observations, by constructing a joint unconditional CF. Computations required by
the UCF procedure are less demanding then in the case of CCF. This comes at some
cost, however. The smaller is the information set on which we condition the lower
is the efficiency of the estimator. Jiang and Oomen (2004) provide an extension to
Jiang and Knight (2002) by allowing for a stochastic volatility model with jumps in
the mean.
The estimator developed by Jiang and Knight (2002) is a good compromise between computational costs and efficiency. In the following sections we provide a more
rigorous discussion on the characteristic function estimators. For the moment we
would like to note that our estimation procedure can be seen as an extension of the
method proposed by Jiang and Knight (2002), extension that includes jumps both in
the price and the variance process.
The rest of the article is organized as follows. In Section 2 we derive the joint
unconditional characteristic functions for the three data generating processes: SV,
SVJ, and SVJJ. Section 3 presents the continuous empirical characteristic function
estimation procedure for the three models. Section 4 contains estimation results and
their discussion. Major conclusions are provided in Section 5.
5
2
Analytical solutions for characteristic functions
In this section we present the SVJJ model, justify our choice of the data generating
process, and derive the joint unconditional characteristic function for the model. SV
and SVJ models can be seen as special cases of the SVJJ process.
2.1
The SVJJ model
The data generating process of the logarithmic asset price, st = log(St ), in the SVJJ
framework is assumed to have the following dynamics in line with the Duffie, Pan,
and Singleton (2000) specification:
p
dSt
(1)
= (µ − λm)dt + Vt dWt + (ezs − 1)dqt ,
St
(1)
p
1
(1)
dst = (µ − λm − Vt )dt + Vt dWt + zs dqt ,
2
(2)
or equivalently
where
dVt = β (α − Vt ) dt + σ
(1)
and where Wt
(2)
as well as Wt
p
(2)
Vt dWt + zv dqt ,
(3)
are standard Brownian motions with a correlation
coefficient ρ. As equation (3) indicates, the instantaneous variance of log-prices is
modelled as a one-factor square-root process plus a jump term. The one-factor squareroot process was originally proposed by Cox, Ingersoll and Ross (1985). The presence
of jumps in the volatility dynamics changes the long-run mean of the variance process
from α to α + (mv λ)/β, where mv is the expected jump in variance. The meanreversion rate and the volatility of volatility are not affected by the jumps and are
determined by the parameters β and σ.
The jump process in (2) and (3) is the compensated compound Poisson process,
such that the expected value of the asset log-price is independent of the jump parame6
ters. The univariate Poisson counter qt has a constant intensity λ and a compensator
λms , with ms being the expected relative jump in the asset log-price. Jumps in
returns and volatility are simultaneous. The SVJ and SV models are obtained as restrictions of the SVJJ model. In particular, the SVJ process is the result of restricting
zv to zero. The SV model is obtained by fixing the intensity of the jump process, λ,
to be equal to zero.
To further specify the jump part of the process (2-3) one needs to make assumptions on distributions of jump sizes in the asset log-price and variance processes and
their correlation structure. Theoretically, there exist plenty of choices for the distribution of a jump size. In reality, only two distributions are used in modelling
jump-diffusions due to their computational simplicity, tractability and flexibility at
the same time. Merton (1976) and Bates (1996) models assume that jumps in logasset prices are normally distributed, zs ∼ N(µj , σ 2j ). Ramezani and Zeng (1999)
and Kou (2002) propose to use, for the jumps in log-returns, an asymmetric double
exponential distribution with the density
f (y) = p·η 1 e−η1 y 1y>0 + (1 − p)·η 2 e−η2 |y| 1y<0
(4)
where η 1 > 0, η 2 > 0, and p represents the probability of an upward jump. In other
words, this distribution is a mixture of two exponential random variables with means
1/η 1 and 1/η 2 and a mixing parameter p. Jumps in volatility can also be normally
distributed, but most often they are assumed to have an exponential distribution,
zV ∼ exp(µV ), an assumption we also adapt in this research.
Bates (2000) proposed to use time-varying jump frequencies, λ, that would depend
on the current factor level. However, Andersen, Benzoni, and Lund (2002, p.1263)
found that the p-values associated with the overall goodness-of-fit test are marginally
lower than in the models with constant jump intensity. Independent arrival of jumps
7
in the log-price and volatility might have allowed greater flexibility of the model
and more pronounced impact of the stochastic volatility on the asset-price process.
In that case the difficulty of the estimation would have been increased as well as
the variance of the parameter estimates. This trade-off between the flexibility and
precision motivated Eraker, Johannes, and Polson (2003, p.1285) to compare the two
specifications of the jump part of the process. As a result, the authors found that
there is no evidence for the misspecification of the SVJJ model with simultaneous
jumps and that the two models exhibit very similar behavior.
Following Chacko and Viceira (2003) we could have assumed a more general version of the stochastic volatility process, i.e. a non-affine jump-diffusion:
γ/2
dVt = β (α − Vt ) dt + σVt
(2)
dWt
+ zv dqt .
Such a model results in a non-linear PDE for the characteristic function, that does
not have an exact analytical solution, a difficulty we wanted to avoid from the very
beginning. There is also one more parameter to estimate, γ. Chacko and Viceira
(2002) showed, using stochastic volatility, that in the presence of jumps in the logprice, the estimate for γ is not statistically different from the value of 1.0, i.e. the
model boils down to the one specified in (2-3).
2.2
Conditional characteristic function
The empirical characteristic function estimation procedure, that will be described in
the next section, requires the closed-form solution for the joint unconditional characteristic function of a stochastic process. In our derivation of the joint unconditional
characteristic function for the affine jump-diffusion we start from the simpler problem
- finding the solution for the conditional characteristic function of the state variable
vector (s0t , Vt0 ). Then we define the CCF of (s0t ), the asset log-price alone. This last
8
one can be easily transformed to the CCF of the asset log-returns rt = st+1 −st . Next,
we obtain the marginal characteristic function of the asset log-returns unconditional
of the current volatility level. Finally, the joint unconditional characteristic function
of the asset log-returns is derived.
Closed form solutions for the conditional characteristic functions of the affine
jump-diffusion processes have been available for a long time. The general form of
the solution can be found in Duffie, Pan, and Singleton (2000). We specialize the
definition of the conditional characteristic function for the process (2), following Jiang
and Knight (2002), and using obvious notations as
ϕ(u1 , u2 ; sT , VT | st , Vt ) = ϕt (u1 , u2 ; sT , VT ) = E [exp(iu1 sT + iu2 VT ) | st , Vt ] .
(5)
The conditional characteristic function (5), being a function of state variables st and
Vt , should satisfy the following PDE as a result of applying Itô’s lemma (omitting
the arguments of the CF)
¶
µ
∂ϕ
1
1 ∂2ϕ
∂ 2ϕ
1 ∂2ϕ 2
∂ϕ ∂ϕ
+
µ − λm − Vt +
β (α − Vt ) +
V
V
+
σρ
+
σ Vt +
t
t
∂t
∂s
2
∂V
2 ∂s2
∂s∂V
2 ∂V 2
+λEt [ϕ (st + zs , Vt + zV ) − ϕ (st , Vt )] = 0.
(6)
The usual practice in solving this kind of PDEs is to guess the general form of the
solution first. Inspired by works of Heston (1993) and Duffie, Pan, Singleton (2000)
we guess
ϕt (u1 , u2 ; sT , VT ) = exp [C (τ ; u1 , u2 ) + J (τ ; u1 , u2 ) + D (τ ; u1 , u2 ) Vt + iu1 st ]
(7)
where τ = T − t. Then, the solution (7) is inserted into the PDE (6) and the terms
in Vt , the ones related to the diffusion part of the process, and the ones related to
9
jumps are grouped together to obtain three ordinary differential equations (ODEs):
∂C
= iu1 µ + αβD,
∂t
1
1
∂D
= iu1 σρD + D2 σ 2 − βD − iu1 (1 − iu1 ),
∂t
2
2
∂J
= −iu1 λm + λ [θ (iu1 , D) − 1] .
∂t
(8)
Boundary conditions are C (0, u1 , u2 ) = 0, J (0, u1 , u2 ) = 0, and D (0, u1 , u2 ) = iu2 ,
so that ϕT (u1 , u2 ; sT , VT ) = exp(iu1 sT + iu2 VT ).
The connection between the jump part of the PDE, see equation (6), and the ODE
for J, see equation (8), deserves some clarification. Let us define the new function,
jump transform for the SVJJ process (2), as
θ(c1 , c2 ) = E [exp {c1 zs + c2 zv } | st , Vt ]
(9)
It then follows that the expectation term in the PDE (6) can be rewritten as
E [ϕ (u1 , u2 ; sT , VT | st + zs , Vt + zV ) − ϕ (u1 , u2 ; sT , VT | st , Vt )]
= Et [exp {C + J + D (Vt + zv ) + iu1 (st + zs )} − exp {C + J + DVt + iu1 st }]
= exp {C + J + DVt + iu1 st } Et [exp {Dzv + iu1 zs } − 1]
= ϕt (u1 , u2 ; sT , VT ) [θ (iu1 , D) − 1] .
The resulting ODE is obvious. In the estimation section of this work we will consider
a specific example of the jump part of the process (2). Hence, the closed form of
the jump transform θ(c1 , c2 ) and the compensator, λms , for the jump process we be
derived.1
1
The specification of the jump transform for the SVJ, SV with-jumps-only-in-volatility and SVJJ
processes, with normal asset log-retun jumps and exponential jumps in variance process, can be found
in Duffie, Pan, and Singleton (2000).
10
The closed form of the conditional characteristic function (5) is obtained by finding
solutions to ODEs (8):
· µ
¶
¸
h2 (1 + g 2 )
αβ
− bτ ,
C (τ ; u1 , u2 ) = iu1 mτ + 2 ln
σ
σ 2 (2biu2 − σ 2 u22 + iu1 (iu1 − 1))
gh − b
D (τ ; u1 , u2 ) =
,
σ2
Z
τ
J (τ ; u1 , u2 ) = −λiu1 τ m + λ
0
[θ (iu1 , D(y; u1 , u2 )) − 1] dy,
(10)
where
b(u1 , u2 ) = iu1 σρ − β,
p
σ 2 iu1 (iu1 − 1) − b2 ,
µ
¶¶
µ
b + iu2 σ 2
hτ
+ arctan
.
g(u1 , u2 ) = tan
2
h
h(u1 , u2 ) =
It turns out that the solution we obtain is also related to Lévy processes. Indeed,
the Lévy-Khintchine representation theorem states that the characteristic function of
a general compensated jump process is
· Z
ψ(u) = exp τ
Rn
¸
¢
¡ iux
e − 1 − iu K)v(dx)
(11)
where v(dx) is a Lévy measure, and K is the compensator of the process. Assuming
that u is a vector, exp [J (τ ; u1 , u2 )] turns out to be the special case of the characteristic function in (11). Thus, the CCF of the SVJJ process is simply a product
of the CCF of Heston’s stochastic volatility model and the characteristic function
of the bivariate compensated jump process. This last observation is not something
unexpected. Jumps in the asset log-prices and the variance process, the way they
are defined in (2), are independent of the current levels of the state variables and the
past jumps, since the Poisson process is memoryless, and the characteristic function
of two independent processes is equal to the product of the characteristic functions
of each process.
11
2.3
Unconditional characteristic function
Finally, we have all the necessary tools to derive the unconditional joint characteristic function of the affine jump-diffusion process. This derivation consists of several
steps first outlined in Jiang and Knight, that are generally applicable to any stochastic process of interest. First, define the conditional characteristic function of the
observable variable, asset log-price, as
ψ(w; st+1 | st , Vt ) = E [exp(iwst+1 ) | st , Vt ] = ϕ(w, 0; st+1 , Vt+1 | st , Vt )
= exp [C (1; w, 0) + J (1; w, 0) + D (1; w, 0) Vt + iwst ] (12)
assuming T − t = 1. Note, that this function is a univariate version of the general
case described in (5). Second, construct the conditional characteristic function of the
asset log-returns:
ψ(w; rt+1 | st , Vt ) = ψ(w; (st+1 − st ) | st , Vt )
= E [exp(iwst+1 ) | st , Vt ] − E [exp(iwst ) | st , Vt ]
(13)
= exp [C (1; w, 0) + J (1; w, 0) + D (1; w, 0) Vt + iwst ] − exp [iwst ]
= exp [C (1; w, 0) + J (1; w, 0) + D (1; w, 0) Vt ] .
(14)
= ψ(w; rt+1 | Vt ).
(15)
The past asset log-price does not enter the equation for the conditional characteristic
function of the asset log-returns. Hence, there is no more need for conditioning on the
asset log-price. The consequence is that once we find the way to avoid conditioning on
the volatility level, the solution for the characteristic function will become independent on time and realizations of the state variables, observable or not. The advantage
of obtaining an unconditional solution would be a weaker dependence of parameter
estimates on the current information and, of course, the absence of problems with
12
filtering out the latent variable realizations. The drawback is the loss in an efficiency
of the parameter estimation procedure. As shown by Singleton (2001), already in the
case of an univariate affine jump-diffusion model without latent variables, the characteristic function estimator attains the efficiency of the Maximum likelihood estimator
only if one conditions on the whole sample path.
The aim of stochastic volatility models is to replicate the autocorrelation and
heteroskedasticity features observed in data. Thus, if one wishes to avoid conditioning on latent variables in the presence of dependency in the stochastic process
realizations, and to account for this dependency at the same time, one needs to work
with joint unconditional characteristic functions of asset log-returns. Define the joint
unconditional characteristic function (UCF) of the asset log-returns as
ψ(w1 , ..., wp+1 ; r1 , ..., rp+1 ) = E [exp (iw1 r1 + iw2 r2 + ... + iwp+1 rp+1 )] .
(16)
Working backward with this joint UCF and computing conditional expectations at
each time step (see appendix for details) we arrive at the point where the only random
variable left is the starting value of the stochastic volatility process:
à p+1
!
X
ψ(w1 , ..., wp+1 ; r1 , ..., rp+1 ) = exp
C(1, wk , wk∗ ) + J(1, wk , wk∗ ) E [exp (D (1; w1 , w1∗ ) V0 )] .
k=1
(17)
The expectation in (17) is simply the marginal characteristic function of the stochastic
volatility.
The variance process in the SVJJ model can be divided into two parts: the continuous square-root process with a known stationary distribution, namely the gamma
distribution, and a discrete jump process. Due to the independence of the continuous
and the discrete parts of the stochastic volatility process, its marginal characteristic
function is simply the product of the characteristic functions of the two parts. The
characteristic function of a general compensated jump process is defined in (11), and
13
is equal to 1 at τ = 0. Therefore, the marginal characteristic function of the stochastic
volatility process defined in (2) is equal to the characteristic function of the gamma
distribution
µ
¶−2αβ/σ2
iuσ 2
ϕ(u, Vt ) = 1 −
.
2β
(18)
Finally we obtain the closed form expression of the joint UCF of the asset log-returns
in the framework of the SVJJ model by substituting the solution (18) for the expectation term in equation (17):
ψ(w1 , ...wp+1 ; r1 , ...rp+1 |V0 ) = exp
à p+1
X
!
C(1, wk , wk∗ ) + J(1, wk , wk∗ )
k=1
¶−2αβ/σ2
µ
D (1; w1 , w1∗ ) σ 2
∗ 1−
2β
(19)
¡
¢
∗
∗
= 0, and wk∗ = −iD 1; wk+1 , wk+1
.
with wp+1
The joint UCF defined in (19) has as special cases Heston’s stochastic volatility
model, Bates SV model with jumps in the asset log-price, and the SV with jumpsonly-in-volatility model. The only difference between those models comes from the
definition of the jump process. As we have already pointed out, the jump part of
the joint UCF is independent of its diffusion part. Therefore, the joint UCF for the
stochastic volatility model is obtained by simply setting J(1, u1 , u2 ) equal to zero.
The joint UCF for jump-diffusion models requires only the solution for the jump
transform θ(c1 , c2 ), see equation (9), and the compensator, λms , in addition to the
closed forms of C(1, u1 , u2 ) and D(1, u1 , u2 ) stated in (10).
For example, in the next section we estimate parameters of the SVJJ model with
simultaneous and correlated jumps in asset log-returns and volatility. Assuming that
jumps in volatility have an exponential distribution, zv ∼ exp(1/η v ) and jumps in
log-asset prices are normally distributed conditional on the realization of zv , i.e. zs |
14
zv ∼ N(µj + ρj zv , σ 2j ), the expected relative jump in the asset price is
ms
¶¸
·
µ
1 2
−1
= E[e − 1] = E (E[e | zv ]) − 1 = E exp µj + ρj zv + σ j
2
¡
¢
µ
¶
£
¡
¢¤
exp µj + 12 σ 2j
1 2
= exp µj + σ j E exp ρj zv − 1 =
− 1.
(20)
2
1 − ρj η v
zs
zs
The jump transform for this specific SVJJ process is
¡
¢
exp µj c1 + 12 σ 2j c21
.
θ(c1 , c2 ) = Et [exp {c1 zs + c2 zv }] =
1 − ρj η v c1 − η v c2
Therefore, the jump part of the joint UCF becomes
Ã
! Z Ã
!
¡
¡
¢
¢
τ
exp µj iu1 − 12 σ 2j u21
exp µj + 12 σ 2j
− 1 +λ
− 1 dy.
J (τ ; u1 , u2 ) = −λiu1 τ
1 − ρj η v
1 − ρj η v iu1 − η v D(y; u1 , u2 )
0
(21)
3
Characteristic function based estimators.
3.1
Review
Characteristic function based estimators can be divided into three types: Maximum
likelihood (ML), GMM and empirical characteristic function (ECF) estimators. To
apply the ML-CCF estimator of Singleton (2001) one needs to take several steps:
derive the conditional characteristic function (CCF) of the state vector, obtain the
conditional density function using inverse Fourier transform of the CCF, as the characteristic function is by definition the Fourier transform of the density function, compute the conditional log-likelihood function of the sample and eventually proceed with
an optimization procedure. The conditioning is done on the whole sample path.
The ML-CCF estimator is an efficient estimator. However the efficiency of the
ML-CCF estimator comes at a high computational cost. Calculation of the inverse
Fourier transform requires a choice of a numerical procedure. Usually it is a discrete
15
approximation over an equally spaced grid. The oscillating nature of the function
demands the grid to be sufficiently dense and large. In the multivariate setting like
ours, the number of grid points increases unfortunately at a high rate. One also
needs to control for the truncation error due to the finite integration limit and for the
sampling or, in other words, discretization error, the discussion about which is quite
often omitted in the transform methods for option pricing literature.1
Conditional moments of the transitional density of the affine jump-diffusion process
can be derived in a closed form by computing derivatives of the CCF evaluated at
zero. Thus, the usual GMM estimator can be implemented. One difficulty with this
approach is that there may be many parameters to be estimated. In such a situation,
multidimensionality of moments might lead to numerical instabilities.
The ECF estimation procedure consists of minimizing the weighted difference
between the analytical and the empirical characteristic functions. Depending on the
choice of the weighting rule, the grid of discrete points or a continuous function,
one obtains either the Discrete or Continuous ECF estimator. The discrete ECF
estimation technique, sometimes called Spectral GMM in the literature, is a special
case of Hansen’s (1982) GMM procedure. The continuous ECF estimator can be seen
as a special case of the GMM with a continuum of moment conditions introduced by
Carrasco and Florens (2000a).
There is already a fair amount of research done in the area of ECF estimators.
Parzen (1962) pioneered with an idea of using the ECF for inference purposes. The
ECF estimator for i.i.d. processes has been studied by Csörgö (1981), Feuerverger
and McDunnough (1981b, 1981c), Bryant and Paulson (1983). More specifically, the
discrete ECF was considered by Schmidt (1982) and Tran (1998). Paulson, Holcomb
1
Noticeable contributions where truncation and discretization errors are measured are Davies
(1973), Benhamou (2000), Lee (2003), and Pan (2001).
16
and Leitch (1975), Carrasco and Florens (2002b) explore continuous ECF for i.i.d
processes. Heathcote (1977) studies asymptotic properties of the ECF estimator in
the i.i.d. case.
The ECF estimation procedure can also be used when the process in not i.i.d.
but strictly stationary and with a weak form of dependence. In this situation, either
the conditional or the joint CF can be used, with consequences on the computational burden and estimator efficiency. The relevant literature for the discrete case
is Feuerverger (1990), Knight and Satchell (1996, 1997); and for the continuous case
- Singleton (2001), Jiang and Knight (2002), Carrasco, Chernov, Florens and Ghysels (2002), Yu (2004). Asymptotic properties of the ECF estimators for stationary
processes are established by Knight and Yu (2002).
3.2
Continuous ECF estimator
Our choice of the estimator for SV, SVJ, and SVJJ models is governed by two considerations. First, as the stochastic volatility is modelled to be stationary and the asset
log-price only first-difference stationary, the ECF estimator for non-i.i.d. processes
should be used. Second, implementation of the discrete ECF estimator faces the same
difficulty as GMM, namely the choice of the size of the grid over which the ECF and
the analytical characteristic function are matched. The continuous ECF estimator
requires the choice of a weight function only and is equivalent to matching all the
moments continuously. Moreover, the weight function can be defined in such a way as
to put more weight around the origin, where the characteristic function carries most
of the information. Taking those arguments into account we applied the continuous
ECF to the estimation of the jump-diffusion models.
Let Xt = (4s0t , Vt0 ) be a bivariate stationary process, with a known dynamics,
17
an observable finite realization of the asset log-returns {4s1 , 4s2 , .., 4sT }, an unobservable stochastic volatility process {V1 , V2 , .., VT } , and an unknown parameter
vector θ specifying the distribution of the process. The continuous ECF estimator of a stationary process consists of the following steps. First, split the data into
n = T − p overlapping blocks of size p + 1 defined as yj = (4sj , 4sj+1 , .., 4sj+p )0 ,
j = 1, ..., T −p, and compute the empirical joint unconditional characteristic function
for a given block size as
1X
[exp(iu0 yj )] .
ϕn (u; y) =
n j=1
n
(22)
Then, obtain the exact analytical form of the joint unconditional characteristic function of a p + 1 dimensional vector y
ϕ(u; y, θ) = E [exp(iu0 y)] .
(23)
Closed form solutions for joint unconditional characteristic functions of the SV, SVJ
and SVJJ processes were derived in the previous section. Finally, the parameter vector
θ is estimated by minimizing the weighted difference between the joint analytical and
joint empirical characteristic functions, more specifically by minimizing the integral
In (θ) =
Z
|ϕ(u; y, θ) − ϕn (u; y)|2 g(u)du
(24)
where the integration involves all p + 1 components of u and where g(u) is a p + 1
dimensional continuous real weight function.1
The continuous ECF estimator is the parameter vector θ̂n for which the integrated
squared error (24) reaches its minimum. This estimator is consistent and asymptotically normally distributed for any fixed block p under regularity conditions stated by
1
This integral reveals that the estimation becomes numerically burdensome, even if p is reactively
small.
18
Knight and Yu (2002), i.e.
θ̂n −→ θ0 a.s.,
(25)
√
n(θ̂n − θ0 ) −→ N(0, B −1 (θ0 ) A(θ0 ) B −1 (θ0 )),
(26)
where in equation (25) the convergence is in probability and in (26) in distribution.
The matrices A(θ0 ) and B(θ0 ) are defined in the appendix.
Efficiency of the continuous ECF estimator achieves the one of the maximumlikelihood as p goes to the sample size. However, the numerical implementation of the
minimization problem becomes infeasible as p grows, leaving us with a choice between
large and small block size, efficiency and implementability. One more concern is the
choice of the weight function g(u). In theory, the optimal weight function obtained by
Feuerverger and McDunnough (1981b) is the inverse Fourier transform of the score
function
1
g (u) =
2π
∗
Z
∂ log f (y; θ) −uiy
e
dy,
∂θ
that depends on the density function which is unknown for many processes and specifically for the affine jump-diffusions we are considering in this article. As a consequence,
in an actual implementation, an arbitrary weight function should be used. The arbitrary weight function can be chosen from the set of continuous functions that assign
more weight to an interval around the origin and whose increments vanish outside
some finite interval (Heathcote 1977). It might and might not depend on the unknown
parameters and sample values. The two most common choices for the weight function
p
are exponential g(u) = exp(−u0 u) and normal g(u) = 1/ 2πσ 2w exp [−u0 u/(2σ 2w )].
Here σ 2w is the measure of the width of the weighting function. The continuous ECF
estimation using any of those two functions does not result in an efficient estimator,
however the asymptotic properties are still preserved.
19
4
Estimation
In this section we estimate the parameters of the three models, SV, SVJ, and SVJJ,
using the continuous ECF estimator developed above. The analytical characteristic functions for the SV and SVJ are obtained by fixing the relevant jump process
parameters in Equation (21) to be equal to zero.
4.1
Methodological issues
A number of difficulties arise in the estimation of the jump-diffusion models using
continuous ECF. We can logically divide them into two groups: problems related to
the specification of the data generating process, and issues concerning the application
of the continuous ECF method.
Let us start with the estimation difficulties created by the model specification
itself. The probability density generated by the jump-diffusion process is the mixture
of lognormal distributions. Ho, Perraudin and Sorensen (1996) pointed out identification problems in estimating mixtures of lognormals. Substitution of the jump process
compensator from the drift of the asset log-price process, eq. (2), was intended to
tackle this issue.
The stochastic volatility process, the way it is defined, makes it difficult to estimate
the mean reversion parameter, β, and the volatility of volatility, σ, independently.
To see this, consider the conditional and marginal variances of the volatility process:
V ar [Vt | V0 ] =
ασ 2
ασ 2
(1 − e−2β(T −t) ) and V ar [Vt ] =
,
2β
2β
(27)
assuming that V0 = α.
Given conditional volatility implied by the data, the higher estimate of the mean
reversion parameter β will lead to the higher estimate of the volatility of volatility
20
parameter σ, and vice versa. To make this point clear, we note that two sets of
parameters, one with small values for β and σ, and the other with big values, can
generate very similar distributions. This explains the enormous variation in results
obtained by the previous researchers. For example, the mean reversion parameter β
of the SV model estimated on the daily S&P500 returns over the period from 1990 to
1999 has a range from 0.23 in Jiang and Knight (2002) to 16.7 in Chacko and Viceira
(2003). Therefore, the concept of the half-life of the process, though theoretically
nice, becomes practically inapplicable.
Half life, computed as H = ln(2)/β, is the time it would take the process to come
half the way back to its unconditional mean. As long as we do not condition on the
whole sample path, we are unable to pin down the β. The increased estimate of the
mean reversion parameter will make the estimated process to be less persistent than
the true one. However, the diffusion term of the process will be loaded more heavily,
thus increasing the σ estimate and pushing the transitional density of the estimated
variance process closer to the true one.
With regard to the asset log-price process, its probability density is also defined by
parameters that play roles similar to each other. For example, the negative skewness
and excess kurtosis of the return distribution has become a fundamental concern
in modelling financial asset prices. The negative skewness can be generated in the
framework of SVJJ by negative correlation between the diffusion parts of the asset
log-price and the variance processes, by high values of the volatility of volatility, or
by jumps in the state variables processes. Kurtosis as well is affected by the presence
of jumps in the state variables process. Increase in any of the three: an expected
jump size, a variance of the jump size and a frequency of jumps, will lead to the
higher excess kurtosis. As a result, the same models estimated on approximately the
21
same sample can have different parameter estimates leading to more or less similar
distributional characteristics of the model.
Estimation of a stochastic process via continuous ECF method also carries some
difficulties. The first problem arises with the choice of the grid for the integration.
The continuous ECF estimator requires the minimization of the integral (24). The
numerical calculation of this integral is performed using some discrete approximation
procedure. The later, in turn, depends on the weight function g(u). The exponential
weight function has often been used due to the computational convenience. It brings
along the possibility to calculate the integral in (24) by Hermitian quadrature. The
normal density weight function has one more attracting feature then the exponential,
it can be scaled according to the variance of the sample, thus assigning weight to the
center and tails of the distribution proportionally to the importance of the tails. As a
matter of fact, the flexibility of the weight function and the decision about truncation
points of the integral (24) appeared to play a more important role in the estimation
of the jump-diffusion models than the choice of the discretization procedure for the
integral. Thus, we calculated the value of (24) over a fine grid with equally spaced
points1 using the normal pdf as a weight function.
The choice of the block size p + 1, as it has been already stated, affects the
computational burden of the procedure and at the same time its efficiency. Though,
there are no theoretical results concerning the magnitude of the efficiency loss due to
a smaller p, it is easy to notice that a higher p implies overwhelming computational
difficulties. The calculation of the integral (24) for p = 1 involves a double integration
as the block size p + 1 = 2. The estimation of the variance-covariance matrix of
parameters for the same block size requires four dimensional integration. A block
1
One word of caution however, the joint UCF is undefined when all elements of the argument
vector u are equal to zero, so we had to drop these points from the grid.
22
size of 3 would lead to a triple integral in (24). Will this estimator be more efficient
than the one with a block size of 2 ? In practice, the answer is not clear. An increase
in dimensions of the integration would have some impact on the discretization and
truncation errors of the estimators. How big is this impact in comparison to the
efficiency gain is an open question.
For example, Jiang and Knight (2002) present in their Table 2 the SV model
estimation results for p going from 1 to 5. The estimate of the mean reversion
parameter β is the one that varies the most. However, most of the estimates are
within one standard deviation from each other, and it is not obvious that the standard
deviation of parameter estimates decreases with a bigger block size. Moreover, the
volatility of volatility and correlation estimates of Jiang and Knight also show some
variation. Thus, the variability of estimates with different block sizes could be just
a result that confirms our previous discussion: different parameter values offset the
impact of each other and lead to, approximately, the same probability density of the
process. Hence, we perform the estimation on the basis of the joint unconditional
characteristic function derived for the block size of 2.
4.2
Empirical results
The data set consists of the S&P500 daily closing prices for the period between
January 2nd 1980 and December 31st 1999. The chosen time period includes the
market crises of October 1987, year 1997, Fall 1998, i.e. the rare events that had
a huge impact on the asset prices and caused dramatic consequences on the market
volatility see Figure 1. At the same time the sample is big enough to encompass
the periods when the market stabilized after these events. Thus, we have a chance
and a reason to test for jumps in asset returns, jumps in the variance and the mean-
23
reverting feature of the variance. In addition, many researches used approximately
the same sample, therefore we will be able to compare our results with the others.
Table 1 provides the summary statistics for the sample.
Insert Figure 1 somewhere here.
Insert Table 1 somewhere here.
Some scaling of the data and parameters was found to be essential for the estimation algorithm to converge. The choice of the scaling procedure is, however,
absolutely arbitrary. We found it to be more comfortable to work with log-returns
expressed in percent. The ECF for log-returns expressed in decimal points has an
enormous dispersion. While the ECF for our sample returns, expressed in percent,
approaches zero for the bounds of the support u = 6 and −6, the ECF for the returns
in decimal points has a value around 0.9 for the same arguments u = 6 and −6. Thus,
scaling the returns allows us to work with smaller bounds of integration.
Table 2 presents our estimation results for the SV, SVJ, and SVJJ models. The
stochastic volatility parameter estimates are in line with those obtained in a number of
recent studies that performed the estimation under the physical measure and included
only asset prices into the sample (Andersen et al. (2001), Chacko and Viceira (2003),
Eraker et al. (2003), Jiang and Knight (2002)). The long-run mean, α, of the variance
process is equal to 0.5496 and it is within the range of the smallest value of 0.0255
reported by Chacko and Viceira and the largest value of 0.9052 found by Eraker et al.
The speed of mean-reversion estimate, β = 7.8803, lies between 0.0231 of Eraker et al.
and 16.6997 of Chacko and Viceira. The volatility of volatility parameter, σ = 2.1446
is within the bounds of 0.1434 and 4.7148 obtained by the same researchers. Our
estimate for the mean of the asset log-return process is higher than the one found in
24
the literature.1 The correlation between the asset log-return process and the variance
process is negative in support of the stylized fact of the growing volatility in the
periods of low returns, the so called leverage effect.
Insert Table 2 somewhere here.
The inclusion of jumps in the SV model has some effect on the parameter estimates. The allowance for jumps in the asset log-returns reduces the value of the
correlation coefficient ρ that is required to generate the return distribution similar
to the observed one. The modelling of jumps in volatility reduces this correlation
coefficient even further. Moreover, the long-run mean of the variance process and the
volatility of volatility coefficient drop down significantly in the presence of jumps in
the variance. Note, that the unrealistically high values of the volatility of volatility
coefficient are considered to be one of the major drawbacks of the SV model.
While jumps in returns have negative mean, jumps in volatility are positive on
average. This last result, combined with the negative correlation between the jump
sizes, is in line with the leverage effect of the stochastic volatility observed in the
diffusion part of the model. The variance of the return jump distribution is reduced
when we allow for jumps in the volatility process. In other words, the information
contained in the jump part of the variance process and not accounted for in the SVJ
model is accumulated in the variance of the return jump distribution estimate. The
intensity of the Poisson process is higher in the SVJJ model then in the SVJ model.
The explanation is that by including volatility jumps in the model we are able to
detect more discontinuities in the process then by just estimating the SVJ model.
1
The estimation algorithm is still a work in progress that requires some fine-tuning. We hope to
arrive to a more reasonable mean return parameter, which is famous for its estimation difficulty.
25
The jump size correlation coefficient, ρj , was found to be the hardest one to
estimate. As it has been pointed out before, the SVJ model is obtained from the
SVJJ model by setting the variance jump size zv = 0. Once there are no more jumps
in the variance process, the parameter ρj has no reason to exist anymore. In this
case, as recognized by Davies (1977), the usual t-statistics is no longer distributed as
normal. Notice also, that the test of zv = 0 involves a boundary. Again, as noticed
by Chernoff (1954), the usual t-statistics does not apply.
5
Conclusions
This paper provides the analytical solution for the unconditional joint characteristic
function of the affine jump-diffusion process. The unconditional characteristic function allows us to estimate parameters of a process with unobservable state variables
avoiding the discretization of the process and the simulation of the latent variables.
Three specifications of the jump-diffusion processes are estimated using the continuous empirical characteristic function method: Heston’s (1993) stochastic volatility
model, a stochastic volatility model with jumps in the asset price, and a stochastic
volatility model with jumps in both the price and the unobservable variance process.
The first goal was to test the estimation procedure in itself. The second goal was
to find the data-generating process that fits the best the historical distribution of the
asset prices with a specific emphasis on jumps. We found that the continuous ECF
procedure proved to be a good compromise between the efficiency and computational
feasibility.
26
Appendix
Asymptotic properties of the continuous ECF estimator, based on the findings of
Feuerverger and McDunnough (1981b), Besbeas and Morgan (2001), Knight and Yu
(2002), are the following. Keeping in mind that
n
n
p
1X
1X
0
2
2
|z| = Re(z) + Im(z) , Re ϕn (u; y) =
cos u yj , Im ϕn (u; y) =
sin u0 yj ,
n j
n j
and simplifying the notation, the integral (24) can be written in the following form
Z
In (θ) =
...
Z
=
...
Z
Z
|ϕ(u; y, θ) − ϕn (u; y)|2 g(u)du
©
ª
[Re ϕ(u; θ) − Re ϕn (u)]2 + [Im ϕ(u; θ) − Im ϕn (u)]2 g(u)du
and the continuous ECF estimator θ̂n is consistent,
θ̂n −→ θ0
a.s.,
and asymptotically normally distributed
√
n(θ̂n − θ0 ) −→ N(0, B −1 (θ0 ) A(θ0 ) B −1 (θ0 )),
where θ0 is a true parameter vector and
Z
Z
∂ϕ(u; θ0 ) ∂ϕ(u; θ0 )
g(u)du
∂θ
∂θ0
·
¸
Z
Z
∂ Re ϕ(u; θ0 ) ∂ Re ϕ(u; θ0 ) ∂ Im ϕ(u; θ0 ) ∂ Im ϕ(u; θ0 )
g(u)du,
+
=
...
∂θ
∂θ
∂θ0
∂θ0
B(θ0 ) =
A(θ) = lim
n→∞
Z
...
...

Z 



∂ Re ϕ(r;θ) ∂ Re ϕ(u;θ)
∂θ
∂θ0
· E(θ) +
ϕ(r;θ) Im ϕ(u;θ)
2 ∂ Re∂θ
∂θ0
ϕ(r;θ) ∂ Im ϕ(u;θ)
+ ∂ Im∂θ
· G(θ)
∂θ0
27


· F (θ) 


g(r)g(u)drdu,
with
1 XX
cov (cos(r0 yj ), cos(u0 yk ))
E(θ) =
n j k
n
=
1
{Re ϕ(r + u; θ) + Re ϕ(r − u; θ)} − Re ϕ(r; θ) Re ϕ(u; θ)
2
n−1
1 X
+
(n − k) {Re Ψk (r, u) + Re Ψk (r, −u) + Re Ψk (u, r) + Re Ψk (u, −r)} ,
2n k=1
1 XX
cov (cos(r0 yj ), sin(u0 yk ))
n j k
n
F (θ) =
=
n
n
1
{Im ϕ(r + u; θ) − Im ϕ(r − u; θ)} − Re ϕ(r; θ) Im ϕ(u; θ)
2
n−1
1 X
+
(n − k) {Im Ψk (r, u) − Im Ψk (r, −u) + Im Ψk (u, r) + Im Ψk (u, −r)}
2n k=1
and
1 XX
cov (sin(r0 yj ), sin(u0 yk ))
G(θ) =
n j k
n
=
n
1
{Re ϕ(r − u; θ) − Re ϕ(r + u; θ)} − Im ϕ(r; θ) Im ϕ(u; θ)
2
n−1
1 X
+
(n − k) {Re Ψk (r, −u) − Re Ψk (r, u) + Re Ψk (u, −r) − Re Ψk (u, r)}
2n k=1
where
Ψk (r, u) = E [exp (ir0 y1 + iu0 yk+1 )] .
B(θ0 ) and A(θ0 ) can be consistently estimated: B(θ0 ) by B(θ̂n ) and A(θ0 ) by applying
the Newey and West (1994) procedure.
28
The joint unconditional CF of returns
ψ(w1 , w2 , ..wp+1 ; r1 , r2 , ..rp+1 )
!#
"
à p+1
X
iwk rk
= E exp
k=1
" "
= E E exp
à p
X
iwk rk
!
eiwp+1 rp+1 |sp, Vp
##
#
à p k=1 !
X
£ iwp+1 rp+1 +i0Vp+1
¤
iwk rk E e
|sp, Vp
= E exp
"
"
= E exp
à k=1
p
X
iwk rk
k=1
!
eC(1;wp+1 ,0)+J(1;wp+1 ,0)+D(1;wp+1 ,0)Vp
"
= eC(1;wp+1 ,0)+J(1;wp+1 ,0) E exp
"
= eC(1;wp+1 ,0)+J(1;wp+1 ,0) E exp

à p−1
X
k=1
à p−1
X
 exp
= eC(1;wp+1 ,0)+J(1;wp+1 ,0) E 
iwk rk
iwk rk
k=1
¡Pp−1
k=1
!
!
#
eiwp rp +D(1;wp+1 ,0)Vp
#
£
¤
E eiwp rp +D(1;wp+1 ,0)Vp |sp−1, Vp−1
¢
#
C(1,wp ,−iD(1;wp+1 ,0))+J(1,wp ,−iD(1;wp+1 ,0))
iwk rk e
eD(1,wp ,−iD(1;wp+1 ,0))Vp−1



= exp [C (1; wp+1 , 0) + C (1; wp , −iD (1; wp+1 , 0)) + J (1; wp+1 , 0) + J (1; wp , −iD (1; wp+1 , 0))] ∗
!
##
" "
à p−2
X
iwk rk exp (iwp−1 rp−1 + D (1; wp , −iD (1; wp+1 , 0)) Vp−1 ) | sp−2, Vp−2
∗ E E exp
k=1
...
= exp
à p+1
X
k=1
!
C(1, wk , wk∗ ) + J(1, wk , wk∗ ) E [exp (D (1; w1 , w1∗ ) V0 )]
¡
¢
∗
∗
= 0, and wk∗ = −iD 1; wk+1 , wk+1
.
with wp+1
29
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34
Captions
Table 1 contains descriptive statistics for the S&P500 index daily log-returns.
The sample covers the period between January 2nd 1980 and December 31st 1999.
The returns are expressed in percent.
Table 2 presents parameter estimates for the three models: SV, SVJ, and SVJJ.
The sample is the one described above. Note, that the estimates are daily parameters.
The estimation was done using the continuous ECF estimator with a block size p+1 =
2 and a normal weight function. The variance of the weight function was set to be
equal to 1.5.
Figure 1 plots time series of the S&P500 index daily log-returns. Horizontal
lines are +/ − 3 standard deviations around the mean of the S&P500 log-return
distribution. One can clearly see the presence of sudden changes in the return size.
Also some periods appear tranquil others agitated, justifying the use of a model with
stochastic volatility.
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Table 1. Descriptive statistics .
for the S&P 500 daily log-returns expressed in percent.
mean
standard deviation
0.050037
0.98389
min
max
-22.833 8.7089
skewness kurtosis
-2.6455
64.238
Table 2. Parameter estimates for the jump-diffusion models.
using S&P 500 daily log-returns expressed in percent
µ
α
β
σ
ρ
λ
µj
σj
SV
0 .2 4 6 2
0 .5 49 6
7 .8 8 0 3
2 .1 4 4 6
-0 .7 4 4 9
SV J
0 .2 3 9 2
0 .5 1 9 5
7 .7 1 0 4
2 .1 0 4 8
-0 .5 7 8 7
0 .0 2 3 1
-5 .3 7 2 7
1 0 .8 3 3
SV JJ
0 .1 6 1 5
0 .3 99 5
7 .8 6 3 3
1 .4 8 3 5
-0 .5 8 3 9
0 .0 6 6 1
-5 .0 1 5 6
5 .9 2 2 8
36
µv
ρj
0 .5 4 6 8
-0 .2 4 9 2
Figure 1: S&P 500 daily log-returns and detection of jumps.
37