Download Determining the volatility of a price process in the presence of rounding errors

The University of Chicago
Department of Statistics
TECHNICAL REPORT SERIES
DETERMINING THE VOLATILITY OF A PRICE PROCESS
IN THE PRESENCE OF ROUNDING ERRORS ∗
Yingying Li and Per A. Mykland
TECHNICAL REPORT NO. 573
Departments of Statistics
The University of Chicago
Chicago, Illinois 60637
September 2006
∗
Yingying Li is Doctoral Student, Department of Statistics, The University of Chicago, Chicago, IL
60637. E-mail: [email protected]. Per A. Mykland is Professor, Department of Statistics,
The University of Chicago, Chicago, IL 60637. E-mail: [email protected]. We gratefully
acknowledge the support of the National Science Foundation under grant DMS-0204639 and DMS0604758. We thank Professor Michael J. Wichura for very helpful comments related to the Lemma 4
of this paper.
Determining the Volatility of a Price Process in The Presence of
Rounding Errors
∗
Yingying Li and Per A. Mykland
September 2006.
This version: September 22, 2006
Abstract
Let S denote the price process of a security, and suppose that S follow a geometric Brownian
motion with volatility σ 2 . We consider the case when the observations at the discrete time
(α )
points 0, 1/n, 2/n, · · · , 1 are the rounded-off values Si/nn = αn bSi/n /αn c (i = 0, · · · , n),
where αn > 0 is the round-off level corresponding to the sample frequency n. We investigate
Pn
(α )
(αn )
))2 ,
the asymptotic behavior of the “Realized Volatility” V n = i=1 (log(Si/nn ) − log(S(i−1)/n
which is commonly used as an estimator of the volatility σ 2 .
We prove the convergence of V n or scaled V n under different conditions on αn . A bias
corrected estimator of the volatility is proposed and an associated central limit theorem is
shown. Simulation results show that improvement in statistical properties can be substantial.
KEY WORDS: Bias-correction; Diffusion Process; Market Microstructure; Realized Volatility; Rounding Errors.
∗
Yingying Li is Doctoral Student, Department of Statistics, The University of Chicago, Chicago, IL 60637. Email: [email protected]. Per A. Mykland is Professor, Department of Statistics, The University of Chicago,
Chicago, IL 60637. E-mail: [email protected]. We gratefully acknowledge the support of the National
Science Foundation under grant DMS-0204639 and DMS-0604758. We thank Professor Michael J. Wichura for very
helpful comments related to the Lemma 4 of this paper.
1
Determining the Volatility of a Price Process in The Presence of Rounding Errors
1
2
INTRODUCTION
We consider a security price process S, which is the solution to the following stochastic differential
equation
dSt = µSt dt + σSt dBt , t ∈ [0, 1]
(1)
where Bt is a standard Brownian motion, and µ ∈ R and σ ∈ (0, ∞) are constants.
It is a common practice in finance to use the sum of frequently sampled squared returns (the
R1
“Realized Volatility” (RV)) to estimate the integrated volatility 0 σ 2 dt (for process (1), this is
simply the volatility σ 2 ). However, recent empirical studies in finance showed evidence that market
microstructure makes this estimator fail when the prices are sampled at high frequencies, and
sampling sparsely gives more reasonable estimates. We investigate the case when the contamination
due to market microstructure is simply round-off errors, and we are interested in the limiting
behavior of the RV.
More specifically, let αn be a sequence of positive numbers which represents the accuracy of
measurement when one observes the process n times during the time period [0,1]. Suppose at time
i/n (i = 0, · · · n), one observe the value kα when the true value Si/n is in [kα, (k + 1)α) with k ∈ Z.
For every real s we denote by s(αn ) = αn bs/αn c its rounded-off value at level αn . We investigate
the asymptotic behavior of the estimator (the RV)
V
n
n
X
(α )
(αn )
=
(log(Si/nn ) − log(S(i−1)/n
))2 .
(2)
i=1
Jacod (1996) and Delattre and Jacod (1997) have previously studied the problem of inference for
volatility based on a rounded Brownian motion. While this work is the inspiration for the current
paper, these earlier results are not quite as relevant to securities prices, as rounding happens on the
original (dollar, euro, etc) scale and not on the log scale. As we shall see in this paper, the more
realistic type of rounding leads to a bias which is no longer nonrandom (as in section 4 of Delattre
Determining the Volatility of a Price Process in The Presence of Rounding Errors
3
and Jacod (1997)), but instead requires a somewhat more complicated correction. We emphasize,
however, that we owe a lot to the earlier developments by Delattre and Jacod. Rounding has also
been studied by Zeng (2003), who has developed a Bayesian inference algorithm for this problem.
We prove the convergence of V n or scaled V n , under different assumptions on αn . The theorems
show the problems of using realized volatility as an estimator of the volatility in the presence
of rounding errors, and explain why “sampling sparsely ” could be a practically helpful way to
estimate the volatility (but “sampling sparsely ” doesn’t solve all the problems). We then propose
a bias corrected estimator, and prove an associated central limit theorem. Simulation results show
that substantial improvement in statistical properties can be achieved. These main results are
presented in section 2. Section 3 is devoted to prove the theorems. And Section 4 for conclusions
and discussion.
Our main bias correction applies to the case of “small rounding”, as in Delattre and Jacod
(1997). This kind of asymptotics is quite realistic in practice, cf. the findings for additive error in
Zhang, Mykland, and Aı̈t-Sahalia (2005a). Small rounding asymptotics has also been studied in
Kolassa and McCullagh (1990), where it is shown to be related to additive error.
2
2.1
THE MAIN RESULTS
The Theorems
Theorem 1. Let the accuracy of measurement αn ≡ α be independent on the number of observa(α)
(α)
tions n. Redefine Si/n = α if Si/n = 0. Then,
1
1
√ V n →P
σ
n
r
∞
2 X log(kα)
1
L1
(log(1 + ))2 ,
π
k
k=1
Determining the Volatility of a Price Process in The Presence of Rounding Errors
4
where LaT is the local time of the continuous semimartingale Xt = log St , t ∈ [0, T ]; defined as in
Revuz and Yor (1999), page 222.
One sees that the realized volatility V n blows up as the sample frequency n becomes higher,
just as in Jacod (1996), though the form of the limit is different.
√
Now let βn = αn n.
Theorem 2. If αn → 0 as n → ∞ in such a way that βn → β ∈ [0, ∞), then
V
n
β2
→P σ +
6
Z
2
0
1
β2
1
dt
−
π2
St2
Z
0
1
∞
2 2
1 X 1
2 2 σ St
exp{−2π
k
}dt.
k2
β2
St2
k=1
The above limit is a quantity increasing in β, and always bigger than σ 2 when β 6= 0. It blows
up to ∞ as β approaches ∞.
In the case that βn → 0, V n converges to σ 2 at the limit; but before the limit, V n is always
biased as an estimator of σ 2 .
If βn → 0 fast enough, one has the following central limit theorems:
Theorem 3. If
√ 2
nβn → γ < ∞, then
√
β2
n[V n − σ 2 − n
6
Z
1
0
1
dt] →L N (0, 2σ 4 ).
St2
In this case, one can estimate the bias and find a bias-corrected estimator of σ 2 :
Theorem 4. If
√ 2
nβn → γ < ∞, let V0n := V n −
α2n
6
Pn
1
i=1 (S (αn ) )2 ,
i/n
√
n[V0n − σ 2 ] →L N (0, 2σ 4 ).
one has
Determining the Volatility of a Price Process in The Presence of Rounding Errors
5
The above theorems clearly show the problems of using realized volatility as an estimator of
the volatility in the presence of rounding errors. And one can see that in all the cases, when one
uses a sub-sampled frequency nsparse < n to do the estimation, the ill-posedness of the estimation
problems could be weaken. This is consistent with what people have recently found in the context
of additive error (see, for example, Zhang, Mykland, and Aı̈t-Sahalia (2005b)).
When αn → 0 fast enough, the realized volatility converges to σ 2 , but one needs to do a bias
correction to solve the finite-sample estimation problem. This can give rise to substantial improved
confidence intervals, as we shall now see.
2.2
The Simulation Results
Denote by V n CI the nominal 95% confidence interval (CI) based on V n and V0n CI the nominal
95% confidence interval based on V0n , as follows.
The naive CI based on V n relies on the classical theory of the realized volatility, which says
that when there is no observation error,
√
n[V n − σ 2 ] →L N (0, 2σ 4 ).
The resulting nominal 95% CI is
h
i
p
p
V n CI = V n − 1.96 ∗ 2(V n )2 /n, V n + 1.96 ∗ 2(V n )2 /n .
Since our findings above indicate that there are some problems with using the classical theory
of the realized volatility when the rounding errors are present. We propose a new estimator
V0n = V n −
n
1
αn2 X
.
(α
n) 2
6
i=1 (Si/n )
Determining the Volatility of a Price Process in The Presence of Rounding Errors
6
By Theorem 4,
√
n[V0n − σ 2 ] →L N (0, 2σ 4 ).
Our adjusted nominal 95% CI is then
·
V0n
CI =
V0n
¸
q
q
n
n
n
2
2
− 1.96 ∗ 2(V0 ) /n, V0 + 1.96 ∗ 2(V0 ) /n .
To examine the performance of the new estimator V0n , we did the following simulation:
We simulated sample paths from (1) with µ = 0.02 and σ = 0.2. Assume the observations at the
(α )
time points 0, 1/n, 2/n, · · · , 1 are the rounded-off values Si/nn = αn bSi/n /αn c (i = 0, · · · , n) with
round-off level αn satisfying αn2 n3/2 = 1. For each sample frequency n = 1000, 2000, · · · , 10000,
1000 sample paths were simulated. And correspondingly, 1000 V n CI’s and 1000 V0n CI’s were
worked out. The (actual) coverage probability of V n CI is estimated by the ratio of the total
number of times that the true parameter σ 2 lies inside the V n CI’s to the total number of sample
paths 1000. And by the same way the coverage probability of V0n CI is estimated.
Figure 1 records the estimated coverage probabilities of V n CI and V0n CI. One can read
from Figure 1 that for each n, the nominal 95% CI based on V n only has an estimated coverage
probability of 20% − 30%. While the nominal 95% CI based on V0n seems to be worthy of the name.
1.0
Determining the Volatility of a Price Process in The Presence of Rounding Errors
o
o
o
o
o
o
o
o
o
0.8
o
7
0.4
0.6
o: estimated coverage probability of V0^n_CI
*: estimated coverage probability of V^n_CI
*
*
*
*
*
*
*
*
*
0.0
0.2
*
0
2000
4000
6000
8000
10000
n
Figure 1: The Estimated Coverage Probabilities, V0n CI versus V n CI. The nominal coverage
probability is 95%.
Determining the Volatility of a Price Process in The Presence of Rounding Errors
3
3.1
8
PROOFS OF THE MAIN RESULTS
Proof of Theorem 1
The proof of this theorem is very similar to the proof of Theorem 3 in Li and Mykland (2006)
with T = 1 (except that here we consider the rounding to be “rounding down”; in Li and Mykland
(2006), the rounding is “rounding to the nearest multiple of α” ). The basic framework of the
proofs relies on Jacod (1996). Please refer to the above papers for the details.
3.2
Preparations for proofs of Theorem 2-4:
Notations:
Am := {ω ∈ Ω : St (ω)t∈[0,1] ∈ [
Bm,n := {ω ∈ Ω : max
1≤i≤n
Yi,n :=
1
, m]};
m
√
1
n|Si/n − S(i−1)/n | ≤ 2mσ(2 log n) 2 };
√
(α )
(αn )
n(Si/nn − S(i−1)/n
);
n
U (n, φ) =
1X
(αn )
φ(S(i−1)/n
, Yi,n ) for function φ on R2 ;
n
i=1
h(·) : density of the standard normal law ;
hs (·) : density of the normal law N (0, s2 ).
Lemma 1. ∀m, P (Am
T
c ) → 0 as n → ∞.
Bm,n
(3)
Determining the Volatility of a Price Process in The Presence of Rounding Errors
9
Proof:
dSt = µSt dt + σSt dBt
1
⇒St = S0 exp{(µ − σ 2 )t + σBt }
2
√
√
σ
1
1
⇒ n|Si/n − S(i−1)/n | = S(i−1)/n | n(exp{ √ Zi + (µ − σ 2 ) } − 1)|,
2
n
n
where Zi ∼ N (0, 1), i.i.d., i = 1, 2, · · · , n.
For any kn ,
P ( max
1≤i≤n
√
n|Si/n − S(i−1)/n | > kn , St {t∈[0,1]} ∈ [1/m, m])
1
kn
σ
1
≤P ( max | exp{ √ Zi + (µ − σ 2 ) } − 1| > √ , St {t∈[0,1]} ∈ [1/m, m])
1≤i≤n
2
n
n
m n
σ
1
kn
1
≤P ( max | exp{ √ Zi + (µ − σ 2 ) } − 1| > √ )
1≤i≤n
2
n
n
m n
kn
1 2 1
σ
≤P ( max (1 − exp{ √ Zi + (µ − σ ) }) > √ )+
1≤i≤n
2
n
n
m n
1
kn
σ
1
P ( max (exp{ √ Zi + (µ − σ 2 ) } − 1) > √ ).
1≤i≤n
2
n
n
m n
1
1
1
Let Mn = max1≤i≤n Zi , cn = (2 log n) 2 − 12 (2 log n)− 2 (log(4π) + log log n), sn = (2 log n)− 2 . The
extreme value theorem says (c.f. Aldous (1989) page 46 or Ferguson (1996) page 99)
Mn − cn
→D ξ,
sn
where ξ has support (−∞, ∞) and P (ξ ≤ x) = e−e
−x
. It follows that
Determining the Volatility of a Price Process in The Presence of Rounding Errors
10
σ
1
1
kn
P ( max (1 − exp{ √ Zi + (µ − σ 2 ) }) > √ )
1≤i≤n
2
n
n
m n
σ
1
1
kn
=P ( min exp{ √ Zi + (µ − σ 2 ) } < 1 − √ )
1≤i≤n
2
n
n
m n
√
1 2
1
kn
n
√
=P ( min Zi < [( σ − µ) + log(1 −
)]
)
1≤i≤n
2
n
m n σ
√
n
1
1
kn
=P (Mn > [(µ − σ 2 ) − log(1 − √ )]
)
2
n
m n σ
[(µ − 12 σ 2 ) n1 − log(1 −
Mn − cn
>
=P (
sn
sn
≈P (ξ >
[(µ − 21 σ 2 ) n1 − log(1 −
√
n
k√
n
)]
σ
m n
√
k√
n
)] σn
m n
sn
− cn
− cn
)
) for large n.
1
For kn = 2mσ(2 log n) 2 , the above probability goes to 0 as n → ∞.
A parallel argument gives the same conclusion for P (max1≤i≤n (exp{ √σn Zi + (µ − 12 σ 2 ) n1 } − 1) >
k√
n
)
m n
1
when kn = 2mσ(2 log n) 2 .
In summary,
P ( max
1≤i≤n
√
1
n|Si/n − S(i−1)/n | > kn , St {t∈[0,1]} ∈ [1/m, m]) → 0 as n → ∞ for kn = 2mσ(2 log n) 2 ,
i.e. P (Am
Lemma 2. Given
\
c
Bm,n
) → 0 as n → ∞.
√
1
nαn → β ∈ [0, ∞), on Am , there exists N, cm ∈ (0, m
], such that for all
(α )
n ≥ N, i = 0, 1, 2, · · · , n, Si/nn ≥ cm .
Proof:
(α )
∀i = 0, 1, 2, · · · , n, Si/nn ≥ Si/n − αn ;
and
Si/n ≥
1
on Am , and αn → 0 as n → ∞,
m
Determining the Volatility of a Price Process in The Presence of Rounding Errors
11
hence the conclusion.
√
Lemma 3. Given βn = nαn → β ∈ [0, ∞), ∀m > 0, √
³
Yi,n
(αn )
nS(i−1)/n
=O
2 log n
n
´1/2
on Am
T
Bm,n .
Proof: On Bm,n ,
Yi,n =
√
√
(α )
(αn )
n|Si/nn − S(i−1)/n
| ≤ n(|Si/n − S(i−1)/n | + 2αn ) ≤ 2mσ(2 log n)1/2 + 2βn .
1
By lemma 2, one can find a cm ∈ (0, m
] such that for large n, on Am
T
Bm,n ,
Yi,n
2mσ(2 log n)1/2 + 2βn
√
≤
.
√ (αn )
ncm
nS(i−1)/n
Since βn → β < ∞, the above inequality implies that
√
³
Yi,n
(α )
n
nS(i−1)/n
is O
2 log n
n
´1/2
or smaller.
Lemma 4.
Z
0
1Z
∞
2 2
1 β2 β2 X 1
βbu + yσx/βc 2
2
2 2σ x
) dydu = σ + 2 ( − 2
h(y)(
exp{−2π
k
}).
x
x 6
π
k2
β2
k=1
Determining the Volatility of a Price Process in The Presence of Rounding Errors
Proof:
Z
1Z
βbu + yσx/βc 2
) dydu
x
0
µ
¶
βbU + Y σx/βc 2
=E
, U ∼ unif [0, 1], Y ∼ N (0, 1)
x
β2
= 2 E(bU + Y σx/βc)2
x
β2
σ 2 x2
= 2 E(bU + Zc)2 , Z ∼ N (0, 2 )
x
β
Z
l+1
2
β X
= 2
h σ2 x2 [l2 (l + 1 − z) + (l + 1)2 (z − l)]dz
x
β2
l
l
Z
β 2 X l+1
= 2
h σ2 x2 (z 2 + {z} − {z}2 )dz
x
β2
l
h(y)(
l
β2
σ 2 x2
= 2 [EZ 2 + E({Z}(1 − {Z}))], Z ∼ N (0, 2 )
x
β
∞
2
2
2
2
X
β
1
1 β
σ x
=σ 2 + 2 ( − 2
exp{−2π 2 k 2 2 }),
2
x 6
π
k
β
k=1
where {z} = z − bzc, is the fractional part of z.
The last equality above is proved by using the Fourier expansion:
Let f (z) = {z} − {z}2 for z ∈ R. Let L = 1/2, f (z) has Fourier coefficients
bk (f ) = 0;
1
a0 (f ) =
L
Z
L
1
f (z)dz = ;
3
−L
12
Determining the Volatility of a Price Process in The Presence of Rounding Errors
Z
1 L
kπz
f (z) cos(
)dz
ak (f ) =
L −L
L
Z 1
2
=2
z(1 − z) cos(2kπz)dz
− 12
Z
1
=2
z(1 − z) cos(2kπz)dz
0
=−
Therefore,
1
k2 π2
.
∞
1
1 X 1
f (z) = − 2
cos2kπz
6 π
k2
k=1
and
E(f (Z)) =
∞
1 X 1
1
− 2
RφZ (2kπ),
6 π
k2
k=1
where RφZ (·) represents the real part of the characteristic function of Z.
2 2
For Z ∼ N (0, σβx2 ), one has,
E(f (Z)) = E({Z}(1 − {Z})) =
∞
2 2
1 X 1
1
2 2σ x
− 2
exp{−2π
k
},
6 π
k2
β2
k=1
which finishes the proof of lemma 4.
13
Determining the Volatility of a Price Process in The Presence of Rounding Errors
3.3
14
Proof of Theorem 2
Recall that V n is defined in (2). For large n,
V n IAm ∩Bm,n
n
X
(α )
(αn )
=
(log Si/nn − log S(i−1)/n
)2 IAm ∩Bm,n
i=1
(αn )
n
(αn )
Si/n − S(i−1)/n
1X√
=
[ n log(
+ 1)]2 IAm ∩Bm,n
(α )
n
S n
i=1
n
X
(4)
(i−1)/n
√
Yi,n
[ n log( √ (α ) + 1)]2 IAm ∩Bm,n
n
nS(i−1)/n
i=1
=
1
n
=
Yi,n
1X√
[ n( √ (α )
n
nS n
n
i=1
(i−1)/n
Yi,n
Yi,n
1
1
− ( √ (α ) )2 + θ3 )]2 IAm ∩Bm,n , θ ∈ (0, √ (α ) ).
n
n
2 nS
3
nS(i−1)/n
(i−1)/n
By lemma 2, one can find
cm ∈ (0,
Define
1
(α )
] such that for large n, Si/nn ≥ cm for all i = 0, 1, 2, · · · , n.
m

y


 ( )2 , when x ≥ cm ;
x
φcm (x, y) =
3
8
6


 ( 4 x2 − 3 x + 2 )y 2 ,
cm
cm
cm
(6)
when x < cm .
For n large enough, by lemma 2 and lemma 3, (4) can be further written as
n
V n IAm ∩Bm,n =
1X
(2 log n)3/2
(αn )
φcm (S(i−1)/n
, Yi,n )IAm ∩Bm,n + O(
)IAm ∩Bm,n
n
n1/2
i=1
=U (n, φcm )IAm ∩Bm,n + O(
where U (·, ·) is defined in (3).
(5)
(2 log n)3/2
)IAm ∩Bm,n ,
n1/2
Determining the Volatility of a Price Process in The Presence of Rounding Errors
Furthermore,
c
V n IAm = V n IAm ∩Bm,n + V n IAm ∩Bm,n
(2 log n)3/2
c
)IAm ∩Bm,n + V n IAm ∩Bm,n
1/2
n
(2 log n)3/2
c
+ (V n − U (n, φcm ))IAm ∩Bm,n
+ O(
)IAm ∩Bm,n
n1/2
= U (n, φcm )IAm ∩Bm,n + O(
= U (n, φcm )IAm
= U (n, φcm )IAm + op (1)
(by Lemma 1).
By Delattre and Jacod (1997),
Z




U (n, φcm ) →P
1Z 1Z
0




Z
h(y)φcm (St , βbu + yσSt /βc)dydudt, if β > 0;
0
1Z
0
h(y)φcm (St , yσSt )dydt, if β = 0.
Note that cm ≤ 1/m,
µ
φcm (S(i−1)/n , Y ) =
Y
S(i−1)/n
¶2
IAm + φcm (S(i−1)/n , Y )IAcm .
Lemma 4 gives, when β > 0,
Z
U (n, φcm )IAm →P
0
1
∞
1 2 2 β2 β2 X 1
σ2S 2
(σ
S
+
−
exp{−2π 2 k 2 2 t })dtIAm
t
2
2
2
6
π
k
β
St
k=1
It is easy to check that the above convergence is also true when β = 0.
15
Determining the Volatility of a Price Process in The Presence of Rounding Errors
Therefore, for β ∈ [0, ∞),
V n IAm = U (n, φcm )IAm + op (1)
Z 1
∞
2 2
1 2 2 β2 β2 X 1
2 2 σ St
(σ
S
+
→P
−
exp{−2π
k
})dtIAm .
t
2
6
π2
k2
β2
0 St
k=1
That is to say, for any δ > 0, ² > 0, there exists N, such that for all n > N,
Z
n
P (|V IAm −
1
0
∞
2 2
1 2 2 β2 β2 X 1
2 2 σ St
(σ
S
+
−
exp{−2π
k
})dtIAm | > δ) < ².
t
6
π2
k2
β2
St2
k=1
On the other hand, since Am % Ω, there exists M large, such that
P (AcM ) < ².
Therefore, for n > N,
Z
∞
2 2
1 2 2 β2 β2 X 1
2 2 σ St
P (|V −
(σ St +
− 2
exp{−2π
k
})dt| > δ)
2
6
π
k2
β2
0 St
k=1
Z 1
∞
2 2
1 2 2 β2 β2 X 1
c
n
2 2 σ St
≤P (AM ) + P (|V IAM −
(σ
S
+
−
exp{−2π
k
})dtIAM | > δ)
t
2
6
π2
k2
β2
0 St
n
1
k=1
<2².
This proves Theorem 2.
16
Determining the Volatility of a Price Process in The Presence of Rounding Errors
3.4
17
Proof of Theorem 3 and Theorem 4
By (4), for large n,
√ n
nV IAm ∩Bm,n
n
√ 1X
√
Yi,n
= n
[ n( √ (α )
n
nS n
i=1
(i−1)/n
Yi,n
1
− ( √ (α )
2 nS n
(i−1)/n
Yi,n
1
)2 + θ3 )]2 IAm ∩Bm,n , θ ∈ (0, √ (α )
3
nS n
).
(7)
(i−1)/n
1
Using the cm ∈ (0, m
] as in (5), we define

y


 ( )3 , when x ≥ cm ;
x
ψcm (x, y) =
3x
4


 ( 3 − 4 )y 3 , when x < cm .
cm
cm
(7) can be further written as
√ n
√
(2 log n)2
nV IAm ∩Bm,n = nU (n, φcm )IAm ∩Bm,n − U (n, ψcm )IAm ∩Bm,n + O(
)IAm ∩Bm,n ;
n1/2
and
√ n
nV IAm
√
√
c
= nV n IAm ∩Bm,n + nV n IAm ∩Bm,n
√
=( nU (n, φcm ) − U (n, ψcm ))IAm
√
√
(2 log n)2
c
+ ( nV n − nU (n, φcm ) + U (n, ψcm ))IAm ∩Bm,n
)IAm ∩Bm,n
+ O(
n1/2
√
= nU (n, φcm )IAm − U (n, ψcm )IAm + op (1),
where φcm is defined in (6), ψcm in (8) and U (·, ·) in (3).
(8)
Determining the Volatility of a Price Process in The Presence of Rounding Errors
18
Note that ψcm (St , σSt y) is an odd function of y, and β = 0; by Delattre and Jacod (1997),
Z
U (n, ψcm ) →P
0
1Z
h(y)ψcm (St , σSt y)dydt = 0.
Therefore,
U (n, ψcm )IAm →P 0.
As a consequence,
√ n
√
nV IAm = nU (n, φcm )IAm + op (1).
(9)
Also by Delattre and Jacod (1997),
√
n[U (n, φcm ) −
Z
0
Z
1
Γφcm (St , βn )dt] →stably in law
0
1
∆(φcm , φcm )(St , 0)1/2 dBs ,
(10)
where
Γφcm (St , βn )
Z 1Z
h(y)φcm (St , βn bu + yσSt /βn c)dydu
=
0
µ
¶
Z 1Z
βn bu + yσSt /βn c 2
=
h(y) (
) IAm + φcm (St , βn bu + yσSt /βn c)IAcm dydu
St
0
∞
β2 1
β2 1 X 1
σ2S 2
=(σ 2 + n 2 − n2 2
exp{−2π 2 k 2 2 t })IAm +
2
6 St
π St
k
βn
k=1
Z 1Z
h(y)φcm (St , βn bu + yσSt /βn c)dyduIAcm (by Lemma 4) ;
0
(11)
Determining the Volatility of a Price Process in The Presence of Rounding Errors
19
and
∆(φcm , φcm )(St , 0)
Z
Z
= hσSt (y)φ2cm dy − ( hσSt (y)φcm dy)2
Z
Z
y
y
= hσSt (y)[( )4 IAm + φ2cm (St , y)IAcm ]dy − ( hσSt (y)[( )2 IAm + φcm (St , y)IAcm ]dy)2
St
St
Z
Z
y 4
y
=[ hσSt (y)( ) dy − ( hσSt (y)( )2 dy)2 ]IAm +
St
St
Z
Z
[ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]IAcm ;
hence
∆(φcm , φcm )(St , 0)1/2
Z
Z
y 4
y
=[ hσSt (y)( ) dy − ( hσSt (y)( )2 dy)2 ]1/2 IAm +
St
St
Z
Z
[ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 IAcm
Z
Z
=(2σ 4 )1/2 IAm + [ hσSt (y)φcm (St , y)2 dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 IAcm .
Plugging (11) and (12) into (10), and note that
√ βn2 R 1
n π2 0
1
St2
P∞
1
k=1 k2
exp{−2π 2 k 2
σ 2 St2
2 }dt
βn
(12)
point-
wise goes to zero on the set Am as n → ∞, one has,
Z 1
√
1
Γφcm (St , βn )dt]IAcm
2 dt)]IAm + n[U (n, φcm ) −
0 St
0
Z 1Z
Z
4
2
→ stably in law N (0, 2σ )IAm +
[ hσSt (y)φcm (St , y) dy − ( hσSt (y)φcm (St , y)dy)2 ]1/2 dBs IAcm .
√
β2
n[U (n, φcm ) − (σ 2 + n
6
Z
1
0
For any continuous function g that vanishes outside a compact set, the above stable convergence
implies
√
β2
E[g( n[U (n, φcm ) − (σ 2 + n
6
Z
0
1
1
dt)]IAm )IAm ] → E[g(N (0, 2σ 4 )IAm )IAm ].
St2
(13)
Determining the Volatility of a Price Process in The Presence of Rounding Errors
20
And by defining ηcm (·, ·) to be

1


 ( )2 , when x ≥ cm ;
x
ηcm (x, y) =
3
8
6


 ( 4 x2 − 3 x + 2 ), when x < cm ,
cm
cm
cm
one has,
V0n IAm = V n IAm −
βn2
U (n, ηcm )IAm .
6
(14)
Again, by Delattre and Jacod (1997),
Z
U (n, ηcm ) →P
1Z
0
h(y)ηcm (St , σSt y)dydt.
As a consequence,
n
U (n, ηcm )IAm =
1X
1
I
→P
(αn ) 2 Am
n
(S
)
i=1
i/n
Z
0
1
1
dtIAm .
St2
By the assumption that
√ 2
nβn → γ < ∞,
one has
√ βn2
β2
n( U (n, ηcm ) − n
6
6
Z
1
0
1
dt)IAm →P 0.
St2
By (9), (14) and (15),
√ n
√
β2
nV0 IAm = n(U (n, φcm ) − n
6
Z
0
1
1
dt)IAm + op (1).
St2
(15)
Determining the Volatility of a Price Process in The Presence of Rounding Errors
21
Also since that g is uniformly continuous,
√
lim E[g( n(V0n − σ 2 )IAm )IAm ]
n→∞
√
β2
= lim E[g( n[U (n, φcm ) − (σ 2 + n
n→∞
6
Z
0
1
1
dt)]IAm )IAm ]
St2
=E[g(N (0, 2σ 4 )IAm )IAm ] (by (13)),
which implies, for any ² > 0, there exists N , such that ∀n ≥ N ,
√
|E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]| < ².
Note also that g is bounded, suppose |g| ≤ Mg . Recall that P (AcM ) → 0 , one can choose M such
that P (AcM ) < ²/Mg .
So for n ≥ N,
√
|E[g( n[V0n − σ 2 ])] − E[g(N (0, 2σ 4 ))]|
√
≤|E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]|+
√
|E[g( n[V0n − σ 2 ]IAcM )IAcM ]| + |E[g(N (0, 2σ 4 )IAcM )IAcM ]|
√
≤|E[g( n[V0n − σ 2 ]IAM )IAM ] − E[g(N (0, 2σ 4 )IAM )IAM ]| + 2M g ∗ P (AcM )
≤3²
Hence we’ve proved that for all continuous g that vanishes outside a compact set,
√
lim E[g( n[V0n − σ 2 ])] = E[g(N (0, 2σ 4 ))],
n→∞
Determining the Volatility of a Price Process in The Presence of Rounding Errors
22
i.e.,
√
n[V0n − σ 2 ] →L N (0, 2σ 4 ).
This finishes the proof of Theorem 4. The proof of Theorem 3 is basically contained in the
proof above.
4
CONCLUSIONS AND DISCUSSION
In summary, we have proved the following convergence in probability:
• If αn ≡ α ∈ (0, ∞):
1
1
√ V n →P
σ
n
• If
r
∞
1
2 X log(kα)
L1
(log(1 + ))2 .
π
k
k=1
√
nαn → β ∈ [0, ∞):
V
n
β2
→P σ +
6
Z
2
0
1
1
β2
dt
−
π2
St2
Z
1
0
∞
2 2
1 X 1
2 2 σ St
exp{−2π
k
}dt.
k2
β2
St2
k=1
And we have proved the central limit theorem that if
•
√
n[V n − σ 2 −
2
βn
6
R1
1
0 St2 dt]
→L N (0, 2σ 4 ) and
√ 2
nβn → γ < ∞,
√
n[V n −
α2n
6
Pn
1
i=1 (S (αn ) )2
i/n
− σ 2 ] →L N (0, 2σ 4 ).
We have used the later result to create a bias-correction that works for “small rounding”.
Note that while we work with observations on a time interval [0, 1], results for the more general
case of time interval [0, T ] is obtained by rescaling. The case of time varying volatility and/or
unequal observation times can be studied using the methods of Jacod and Protter (1998) and
Mykland and Zhang (2006).
Determining the Volatility of a Price Process in The Presence of Rounding Errors
23
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24
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