A MULTIVARIATE STOCHASTIC UNIT ROOT MODEL
WITH AN APPLICATION TO DERIVATIVE PRICING
By
Offer Lieberman and Peter C. B. Phillips
December 2014
COWLES FOUNDATION DISCUSSION PAPER NO. 1964
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
YALE UNIVERSITY
Box 208281
New Haven, Connecticut 06520-8281
http://cowles.econ.yale.edu/
A Multivariate Stochastic Unit Root Model
with an Application to Derivative Pricing
O¤er Liebermanyand Peter C. B. Phillipsz
Revised, September 18, 2014
Abstract
This paper extends recent …ndings of Lieberman and Phillips (2014)
on stochastic unit root (SUR) models to a multivariate case including a comprehensive asymptotic theory for estimation of the model’s
parameters. The extensions are useful because they lead to a generalization of the Black-Scholes formula for derivative pricing. In place
of the standard assumption that the price process follows a geometric
Brownian motion, we derive a new form of the Black-Scholes equation that allows for a multivariate time varying coe¢ cient element
in the price equation. The corresponding formula for the value of a
European-type call option is obtained and shown to extend the existing option price formula in a manner that embodies the e¤ect of a
stochastic departure from a unit root. An empirical application reveals
that the new model is consistent with excess skewness and kurtosis in
the price distribution relative to a lognormal distrubution.
Key words and phrases: Autoregression; Derivative; Di¤usion; Options; Similarity; Stochastic unit root; Time-varying coe¢ cients.
JEL Classi…cation: C22
Our thanks to the Editor and three referees for helpful comments and suggestions on
the original manuscript.
y
Bar-Ilan University. Support from Israel Science Foundation grant No. 1082-14 and
from the Sapir Center in Tel Aviv University are gratefully acknowledged. Correspondence
to: Department of Economics and Research Institute for Econometrics (RIE), Bar-Ilan
University, Ramat Gan 52900, Israel. E-mail: o¤[email protected]
z
Yale University, University of Auckland, Southampton University, and Singapore Management University. Support is acknowledged from the NSF under Grant No. SES
1258258.
1
Introduction
Unit root and local unit root time series models have attracted much attention in the last few decades, providing a wellspring of work that has been
found useful in applied research in many disciplines, including …nance. The
prototype model
Y1 = "1;
Yt =
+ Yt
1
+ "t , t = 2; :::; n; "t
iid
0;
2
"
; t = 1; :::; n;
(1)
has substantial ‡exibility and, when the autoregressive parameter is in the
vicinity of unity, data generated from the model take many plausible forms
that include stationary, trend stationary, random wandering, and explosive
possibilities. A key mechanism in determining the large sample limit form
of the process is the invariance principle
Pbnrc for standardized versions of partial
sums of the innovations Sbnrc = t=1 "t ; where bnrc is the integer part of
nr: The simplest case involves the Donsker result
bnrc
1 X
p
"t ) W (r) , r 2 [0; 1] ;
" n t=1
(2)
where W (r) is standard Brownian motion and ) denotes weak convergence,
but much more general results are known to hold (e.g., Phillips, 1987a; see
Giraitis et. al., 2012, for a recent discussion). As is well known, the limit
theory has implications for standardized versions of the output process Yt
when is in the vicinity of unity.
To illustrate, let n be the number of subintervals into which a T -year
period is subdivided, such that n=T is …xed for a given data-frequency, and
let A and ";A denote the mean and standard deviation in annualized terms,
so that
r
n
n
and ";A =
(3)
":
A =
T
T
P
Then, when = 1, we have Ybnrc = bnrc + bnrc
t=1 "t , and for large n this
leads directly to
p
Ybnrc Yn (r) = T A r + T ";A W (r) :
(4)
1
It is emphasized that even though the model (1) is written in terms of any
frequency, the large sample behavior (4) is expressed in common annualized
terms than involve A and ";A . It is di¢ cult to over-emphasize the role
that this last formula plays in the literature. For instance, the celebrated
Black-Scholes formula (Black and Scholes, 1973, henceforth BS) for option
pricing, critically depends on the assumption that stock prices, S (r), follow
a geometric Brownian motion, viz.,
dS (r)
=T
S (r)
A dr
+
p
T
";A dW
(r) = dYn (r) :
(5)
A tacit assumption that leads to the limit theory embedded in (4) and (5)
is that the coe¢ cient of Yt 1 in (1) is …xed and equals unity for all t. For
some data sets and models, this assumption may be reasonable on average,
but it is often likely to be restrictive. Recognition of this limitation has led
to the consideration of local unit root (LUR) models where is …xed (within
an array framework) but lies in the vicinity of unity (Chan and Wei, 1987;
Phillips, 1987b; Phillips and Magdalinos, 2007). A more realistic working
hypothesis might relax the requirement that the coe¢ cient be …xed and allow for some time variation and possible dependencies on other stochastic
variables. Phillips and Yu (2011) explored some time variation in the localizing coe¢ cient to collapse in …nancial markets and bubble migration but
used a deterministic : The stochastic localizing coe¢ cient we use here has
the form
Y1 =
Yt =
+ "1 ;
+ t (a; n) Yt
1
where
t (a; n) = exp
+ "t ; t = 2; :::; n;
a0 ut
p
n
(6)
(7)
and ut is an L 1 vector and is the source of the variation in the autoregressive
coe¢ cient. In applications, ut will typically stand for a vector of excess
returns1 on market indices and/or related stocks, but this need not be the
case. A formal factor
p interpretation of (7) is possible in which the loading
coe¢ cients an = a= n are local to zero and the factors (observable and
1
In other words, in applications ut would typically be a demeaned I (1) process.
2
unobservable) are measured through ut ; while the AR coe¢ cient t (a; n) is
driven by the exponential function so that the model is a nonlinear factor
formulation.
We assume that partial sums of t = (ut ; "t )0 satisfy the invariance principle
bnrc
X
u
u"
1=2
n
BM ( ) ; =
;
(8)
0
2
t ) B (r)
u"
t=1
"
where B = (Bu ; B" )0 is a vector Brownian motion with positive de…nite
and component L L submatrix u > 0 and scalar 2" > 0. The parameters
and are the single-period mean and covariance matrix quantities, which
are …xed for a given frequency and are to be distinguished from their annualand T -year counterparts.
The model (6)-(7) is a multivariate version of a (single regressor) stochastic unit root (STUR) model introduced in Lieberman and Phillips (2014)
and belongs to the general class of time varying coe¢ cients (TVC) models.
That paper explored the connection of the STUR model to recent developments in the literature, including similarity models, speci…cally, Lieberman
(2010, 2012), who investigated autoregressive similarity-based models with
non-stochastic regressors. The term ‘similarity’originated from the theory of
empirical similarity, developed in Gilboa et. al. (2006), and under which the
value of t (a; n) is dictated by the degree of similarity between Yt and Yt 1 ,
as measured by the input ut to the exponent of (7). The main feature of the
STUR model is that for any given t, the coe¢ cient t (a; n) can be less than-,
equal to-, or greater than unity, with a time speci…c value that is determined
by ut . We note that the random coe¢ cient structure is at the heart of the
arguments developed in Meyn and Tweedie (2005) and subsequent work on
GARCH processes.
This paper derives the stochastic limit theory of the STUR model (6)-(7)
and uses this limit theory to generalize the classic stochastic di¤erential equation (sde) (5) so that it embodies the limit of (6)-(7). The special case where
a = 0 produces the limit process (4) and so the new limit theory provides an
extension of (5) to include a TVC feature. Within this framework, a further
contribution of the paper is to derive the BS sde for derivative pricing and the
BS price of a European call option under the new scheme. Furthermore, we
provide a new asymptotic theory for estimation of all the model parameters
and apply these results in the construction of option pricing formulae.
3
The idea of modifying the base model (5) to enhance realism is by no
means new. Two main streams of extension appear in the literature. The
…rst is the stochastic volatility (SV) model (e.g., Hull and White (1987),
Heston (1993)). In that model, if the volatility process is not correlated with
W (r), the process is consistent with a symmetric volatility smile (Renault
and Touzi (1996)), whereas if there is a negative correlation between the
two, the process will be consistent with an asymmetric volatility skew, which
is often claimed to be empirically better suited to stock options (see, for
instance, Hull, 2009).
In the second stream of literature it is suggested to replace the standard Brownian motion driver process in (5) by a fractional Brownian motion
(FBM), B H , with a Hurst parameter H. See, for instance, Hu and Øksendal
(2000) and Biagini et. al. (2008). While the FBM model is reported to …t
certain data sets better than the base model (5), B H has correlated increments and is not a semimartingale so that the model introduces arbitrage
possibilities, classic Itō calculus is inapplicable and new methods of stochastic
integration using Wick algebra are required, see Bjork and Hult (2005). In
contrast, the extension based on a similarity STUR model allows for the use
of standard Itō calculus and is convenient for analysis and empirical work.
In sum, the BS model (5) is simple, tractable and possesses features such
as completeness and no-arbitrage but su¤ers limitations such as no implied
volatility smile and the absence of heavy tails. These features of the BS
model suggest that there is value in an extension of the model that captures
its main advantages while overcoming its main empirical shortcomings.
The plan for the remainder of the paper is as follows. Section 2 develops
the limit theory for the multivariate STUR model and provides the associated
sde. Section 3 provides a comprehensive asymptotic theory of estimation of
the model parameters, in both the = 0 and the 6= 0 cases. Section 4 derives the BS sde corresponding to our model and the price process associated
with it. The value of a European call option under the new model is given in
Section 5. A numerical section is supplied in Section 6, comprised of a simulation sub-section for the results of Section 3, and an empirical application,
showing that our model is consistent with excess skewness and kurtosis, as
compared with the lognormal distribution implied by the BS model. Section
7 concludes and proofs will be given in the Appendix.
4
2
Continuous Limit of the STUR Model
By backward substitution, the model (6) gives the following solution from
initialization at Y1 = + "1 ,
Y2 = (
2
+ 1) +(
2 "1
+ "2 ) ; Y3 = (
and generally for any t
t 1
X
Yt =
=
+
+ 1) +(
3
3 2 "1
+
3 "2
+ "3 ) ;
2,
t
Y
s=1
j=s+1
t 1
X
t
Y
s=1
3 2
j
j
j=s+1
!
!
+1
+1
!
!
+
t 1
X
t
Y
s=1
+
= : ht ( ) + Yt ;
t 1
X
j=s+1
e
a0
p
n
s=1
Pt
j
!
j=s+1
"s + "t
uj
"s + "t
(9)
(10)
say. In what follows it is convenient to expand the probability space as
necessary to ensure that the convergence in (8) is in probability so that
P
n 1=2 bnrc
t=1 t !p B (r) : This procedure, which is standard in modern asymptotic theory, enables limit variates such as the vector Brownian motion
B (r) (which typically escape from the underlying probability space through
the action of the asymptotics) to be included in the same space as the random
sequence. The space augmentation also allows for the convenient replacement
of weak convergence by almost sure convergence or, as here, convergence in
probability, which is another well-known device in modern asymptotic theory. For further information, readers are referred to Shorack and Wellner
(1986, Theorem 4, pp. 47-48). Standardizing Yt we then have the following
result.
Lemma 1 In a suitably expanded probability space as n ! 1
Z r
Z r
0
a0 Bu (r)
a0 Bu (p)
0
1=2
n
Ybnrc !p e
e
dB" (p) a u"
e a Bu (p) dp
0
:= Ga (r) ;
0
(11)
and
0
1@
n
bnrc 1
X
s=1
0
@
bnrc
Y
j=s+1
1
1
A + 1A
j
!p e
5
a0 Bu (r)
Z
0
r
e
a0 Bu (p)
dp := Ha (r) : (12)
0
0
Using the di¤erentials d ea Bu (r) = ea Bu (r) a0 dBu (r) + 12 a0
d
= e
Z
r
e
0
a0 Bu (r)
a0 Bu (p)
dB" (p)
a
0
u"
Z
r
a0 Bu (p)
e
u adr
and
dr;
(13)
dp
0
(dB" (r)
a0
u" dr) ;
we …nd that Ga (r) follows the sde
Z r
0
a0 Bu (r)
e a Bu (p) dB" (p)
dGa (r) = e
a
0
u"
0
1
a0 dBu (r) + a0
2
u adr
r
e
a0 Bu (p)
dp
0
+ dB" (r)
= Ga (r) a0 dBu (r) + dB" (r) +
Z
a0
ua
2
a0
u" dr
Ga (r)
a0
u"
which has the form of a nonlinear di¤usion driven by vector Brownian motion
(Bu ; B" ) : Observe that when a = 0; Ga (r) reduces simply to the Brownian
motion B" (r) :
It follows from (10) - (12) that for large n,
Z r
p
0
a0 Bu (r)
e a Bu (p) dp + nGa (r) : (14)
hbnrc ( ) + Ybnrc Yn (r) = n e
0
The …rst term in (14) contributes an additional drift (n dr) to the di¤erential
equation (13), leading to the following approximate continuous time law of
motion for Yn (r)
p
(15)
dYn (r) = n dr + n (Ga (r) a0 dBu (r) + dB" (r)
0
a ua
Ga (r) a0 u" dr :
+
2
In this system the nonlinear di¤usion process Ga a¤ects both the martingale
component
p and the drift. When a = 0; the system reduces to dYn (r) =
n dr + ndB" (r) ; which corresponds in form to the classic equation (5).
6
3
Estimation of the Model Parameters
3.1
=0
The case
This section develops asymptotic theory for the estimation of the model
parameters. Let a
^n denote the least squares estimator of a.
Theorem 2 For the model (6)–(8) with
of a
^n is given by:
(1)
R1
Ga (r) dr
(^
an a) ) R01
G2a (r) dr
0
(2)
R1
a
^n ) R 01
B" (r) dr
B"2
0
(3)
p
n (^
an
a) ) R 1
1
(r) dr
1
1
u
G2a (r) dr
n
oZ
2
0
+E (ut a) ut
0
fE
0
if
u"
= 0:
1
u
if
u" ;
u
= 0, the asymptotic distribution
u" ;
u"
if
u"
6= 0 and a = 0:
("t ut u0t )g a
1
G2a
6= 0:
(r) +
Z
Z
1
Ga (r) dr
(16)
0
1
Ga (r) dBu" (r) ;
0
(4)
p
n^
an ) R 1
0
1
B"2 (r) dr
1
u
Z
1
B" (r) dBu" (r) , if
u"
= 0 and a = 0:
0
The following properties are an immediate consequence of Theorem 2 in
the case where = 0:
Remark 1 The estimate a
^n is not consistent for a unless u" = 0. Thus,
endogeneity in ut plays an important role in in‡uencing the asymptotic behavior of a
^n :
7
Remark 2 Under the hypothesis H0 : a = 0 and when u" 6= 0, we may use
Theorem 2(2). In particular, in the L = 1 subcase, we have
R1
" 0 B" (r) dr
; if u" 6= 0 and a = 0;
a
^n )
R1
2
u 0 B" (r) dr
where
is the correlation coe¢ cient between " and u.
2
",
De…ne the following estimates of
1X
Yt
n t=1
n
^ 2";n =
and
0
p
ea^n ut =
n
2
Yt
1
2
u
and
u"
1X
vech (ut u0t ) ;
; vech ^ u;n =
n t=1
n
X
^ u";n = 1
Yt
n t=1
n
p
0
ea^n ut =
n
Yt
1
ut :
The next result gives the limit theory of these estimates.
Theorem 3 For the model (6)–(8) with = 0, the asymptotic distributions
of ^ 2";n , vech ^ u;n and ^ u";n are given by:
(1)
2
R1
G
(r)
dr
a
0
0
2
1
^ 2";n
R1
u" :
u" u
" )
2 (r) dr
G
a
0
(2)
p
n vech ^ u;n
vech (
u)
) (1) ;
where (r) is the Brownian motion weak limit of p1n
(3)
2
R1
G
(r)
dr
a
0
^ u";n
R1
u" )
G2a (r) dr
0
Remark 3 ^ 2";n is not consistent, unless
8
u"
Pbnrc
t=1
vech (ut u0t
u) :
u" :
= 0. In the a = 0 case, Theo-
rem 3(1) implies
^ 2";n
2
"
)
R1
0
2
B" (r) dr
R1
B"2
0
The restricted estimator ^ 2";n;0 = n
under H0 : a = 0.
1
(r) dr
Pn
t=1
(Yt
0
u"
1
u" :
u
Yt 1 )2 , is consistent for
2
"
Remark 4 In the case L = 1, (1) =d N (0; 4 + 2 4u ), where 4 is the 4th
cumulant of ut : If ut is Gaussian (1) =d N (0; 2 4u ).
Remark 5 ^ u";n is not consistent, unless u" = 0. In the a = 0 case,
Theorem 3(3) reduces to
^ u";n
u"
The case
1
2
0
B" (r) dr
0
B"2 (r) dr
R1
)
The restricted estimate ^ u";n;0 = n
under H0 : a = 0.
3.2
R1
Pn
t=1
(Yt
u" :
Yt 1 ) ut is consistent for
u"
6= 0
Theorem 4 For the model (6)–(8) with 6= 0, the asymptotic distribution
of a
^n is given by:
R1
p
Ha (r) dr 1
n (^
an a) ) R01
u" :
2 (r) dr u
H
a
0
The following remarks apply when
6= 0:
Remark 6 The estimator a
^n is consistent for all a in contrast to the case
where = 0. The result is explained as follows. With
no loss of generality
aut
p
set L = 1. The nonlinear model (6) is Yt = + e n Yt 1 + "t (t = 2; :::; n)
and may be written in linear pseudo-model form as follows
Yt =
=
au
1 a2 u2t
1 a3 u3t
1
pt +
+
+ Op
3=2
6n
n2
n 2 n
ut
a u2t
1 a2 u3t
1
+a p +
+
+ Op
3=2
6n
n2
n 2n
+
9
Yt
Yt
1
1
+ "t
+ "t + Op
1
n
;
or
Yt =
+ aZt + "t ; where
ut Yt 1 a u2t Yt 1 1 a2 u3t Yt 1
Yt 1
+
:
Zt = p
+
+ Op
3=2
2 n
6 n
n2
n
p
When = 0; Yt = Op ( n) : In that case, Zt = Op (1) and is correlated with
the equation error "t when u" 6= 0; which explains the inconsistency of a
^n
when = 0: When 6= 0; we have Yt = Op (n) ; as shown
in the proof of
p
Lemma 1. In that case, the pseudo-regressor Zt = Op ( n) and the stronger
signal ensures that a
^n is consistent in spite of the presence of correlation with
the equation error "t when u" 6= 0: Thus, drift in the generating mechanism
plays an important role in the asymptotic properties of the estimate a
^n :
p
Remark 7pWhen u" = 0, n (^
an a) ) 0 and the convergence rate of
a
^n exceeds n: Again, the presence of endogeneity in ut plays a role in the
asymptotics of a
^n :
Remark 8 When a = 0, Ha (r) =
0r
and
p
3
n^
an )
2 0
1
u
u" :
In the a = 0 and L = 1 subcase, we have
p
n^
an )
3
2
"
(17)
:
0 u
Let ^ n be the least squares estimate of and de…ne
Z 1
Z 1
1 0
0
a ua
Ha (r) dr;
B (a; u ) = Ha (1) a
Ha (r) dBu (r)
2
0
0
where Ha (r) =
1
Ha (r).
Theorem 5 For the model (6)–(8) with
^ n ) B (a;
6= 0,
u) :
Remark 9 By Theorems 4 and 6(2) below, since a
^n ; ^ u !p (a;
10
u)
when
6= 0, the rescaled estimate
^n
n
B a
^n ; ^ u
is consistent for .
Remark 10 When a = 0, Ha (1) = 1, leading to ^ n ) .
In the
6= 0 case, we de…ne the following estimates of
1X
=
Yt
n t=1
n
^ 2";n
and
n
p
a
^0n ut = n
e
2
Yt
1
X
^ u";n = 1
Yt
n t=1
2
u
and
u"
1X
; vech ^ u;n =
vech (ut u0t ) ;
n t=1
n
n
n
2
",
0
p
ea^n ut =
n
Yt
1
ut :
Theorem 6 For the model (6)–(8) with 6= 0, the asymptotic distributions
of ^ 2";n , ^ 2u;n and ^ u";n are given by
R
2
( R01 Ha (r)dr) 0
2
1
)
(1) ^ 2";n
1
u" ;
"
u" u
2 (r)dr
H
a
0
p
2
(2)
n 2u;n
u ) (1) ; and
R
2
( R01 Ha (r)dr)
^
(3) u";n
1
u" )
u" :
H 2 (r)dr
0
4
4.1
a
A STUR Extension of the BS Model
The Price Process
A fundamental building block in the BS option price formula involves a random walk with a drift, which in the discrete case amounts to equation (1)
with
= 1; i.e., a STUR process with parameter a = 0. The results of
Section 2 suggest a generalization of the BS formula. Top…x ideas, it is convenient to set the parameters as T = n ; and ";T = n " ; so that when
= 1, equation (4) becomes
Yn (r) =
Tr
+
11
";T W
(r) :
(18)
The subscripts A and T will be used in what follows to distinguish between
annualized and T -year period quantities, respectively.
A key assumption in the BS option price model is that stock prices, S (t),
follow a geometric Brownian motion, viz.,
dS (r)
=T
S (r)
A dr +
p
T
";A dW
(r) =
T dr
+
";T dW
(r) :
(19)
That is, the right side of (19) is just (18), which specializes (15) above in
the case a = 0, thereby suggesting the latter as a suitable extension giving a
geometric nonlinear di¤usion limit process corresponding to a more ‡exible
time varying coe¢ cient discrete model. We use this extension to obtain
derivative pricing formulae under weaker conditions than BS, including the
price of a European call option.
1=2
De…ne B := (Bu ; B" )0 =
B; so that B is vector standard Brownian
motion (SBM). Write the lower triangular square root of as
!
!
1=2
1=2
0
0
u
1=2
1;1
:=
:=
1=2
1=2
1=2
1=2
0
0
1
( 2"
u" )
2;1
2;2
u" u
u" u
1=2
:=
1=2
1
;
2
where
u";T
1=2
u
is the positive de…nite square root of u . Let an = (T =n)1=2 a,
= n u" = T u";A , and T = n = T A . We show in the Appendix
that
Yn (r) = T
1=2
u;A Bu (r)
a0n
Ae
Z
r
e
a0n
1=2
u;A Bu (p)
dp +
p
T Gan ;A (r) ;
(20)
0
where we use
p
nGa (r) =
Gan ;A (r) = e
a0n
p
T Gan ;A (r) and
1=2
u;A Bu (r)
a0n
u";A
Z
r
h
e
1=2
A
a0n
0
12
i Z
2
r
e
a0n
1=2
u;A Bu (p)
dB (p)
0
1=2
u;A Bu (p)
dp :
(21)
Let b (r) = (Ga (r) a0 ; 1)0 and bA (r) = (Gan ;A (r) a0n ; 1)0 . Because
p
p
p
1=2
nb (r)0 dB (r) = nb (r)0 1=2 dB (r) = T b (r)0 A dB (r) ;
and
Ga (r) a =
p
we get
p
p
p
nGa (r) a= n = T Gan ;A (r) a= n = Gan ;A (r) an ;
p
p
nb (r)0 dB (r) = T bA (r)0
1=2
A dB
(r) :
Thus, we may rewrite (15) as
p
T
h
p
+ T bA (r)0
dYn (r) = T
a0n
A+
1=2
A
i
u;A an
2
Gan ;A (r)
a0n
u";A
dr
(22)
dB (r) :
We replace the standard formula (19) with (22), which leads to the following
geometric nonlinear di¤usion that is based on the STUR model
dS (r)
=
S (r)
T
A+
p
T
a0n
u;A an
a0n
Gan ;A (r)
2
i
h
p
1=2
0
dB (r) :
+ T bA (r)
A
u";A
dr
(23)
Whereas (19) is a geometric Brownian motion, the system (23) is a geometric
price process that involves the nonlinear di¤usion Gan ;A (r) and Brownian
motion driver process B : The system collapses to (19) when a = 0. So,
(23) may be regarded as a process that is parametrically local to geometric
Brownian motion.
Next, consider the process G (r) = log (S (r)) and let [S]r denote the
quadratic variation process of S (r). We show in the Appendix that
dG (r) =
T
2
";A
A
2
p
T a0n
p
a0n u;A an p 0
T
T an
2
p
1=2
+ T bA (r)0 A dB (r) :
+
u";A
u";A
13
(24)
dr
Gan ;A (r)
T
a0n
u;A an
2
G2an ;A (r) dr
When a = 0 and
u"
= 0, we retain the classic formula
2
";T
dG (r) =
T
dr +
2
2
";A
=T
A
p
dr +
2
ndB" (r)
p
T
";A dB"
(25)
(r) :
In this case, (25) implies that
log (S (r))
log (S (0))
2
";T
N
T
so that
2
2
";T
S (r) = S (0) exp N
T
2
r;
r;
2
";T r
2
";T r
:
(26)
In the Appendix we show that when a 6= 0, S (r) satis…es
S (r) = S (0) exp
T
Z r p
a0n
T
+
0
T
a0n
p
+ T a0n
u;A an
2
Z
2
";A
A
2
p 0
T an
p 0
T an
u;A an
2
u";A
u";A
r
Gan ;A (s)
G2an ;A (s) ds
r
Gan ;A (s)
1=2
u;A
(s) dBu (s) +
p
T
1=2
2;A B
(r) : (27)
0
Equation (27) is the price process under the physical measure. It will be used
in place of (26) when calculating BS option prices, after it is reformulated in
terms of the risk neutral measure, Q, details of which are given in Section 6.
4.2
BS European Option Pricing
Following Hull (2009), let f (r; x) be the price at r of a European-style derivative of a stock, such as a European call option, where x S (r) is the stock
price and Z (r) is the price of the riskless asset. Let ( S (r) ; Z (r)) be the
associated self-…nancing portfolio at time r: The value of the portfolio at time
14
r is
V (r) =
(r) S (r) +
S
Z
(r) (r) ;
where (r) = erf;T r and rf;T = T rf;A is the period-T risk-free rate of interest.
The sde corresponding to the portfolio V (r) is
dV (r) =
S
(r) dS (r) +
Z
(28)
(r) rf;T (r) dr;
which is the self-…nancing condition. We show in the Appendix that
S
(29)
(r) = fx ;
which is the condition in the classic case, and that
Z
(r) =
1
fr +
T rf;A (r)
T
fxx S 2 (r) bA (r)0
2
A bA
(r) :
(30)
When a = 0 the last condition collapses to the well-known condition
Z
(r) =
1
fr +
T rf;A (r)
Since V (r) = f (r; x), it follows that
that using (29) and (30) we obtain
fx S (r) +
1
T rf;A (r)
fr +
S
T
fxx S 2 (r)
2
(r) S (r) +
T
fxx S 2 (r) bA (r)0
2
Z
2
";A
:
(r) (r) = f (r; x) ; so
A bA
(r)
(r) = f;
and, …nally,
T
fxx S 2 (r) bA (r)0 A bA (r) = T rf;A f:
(31)
2
Equation (31) is the generalized BS sde for a European style derivative of a
stock. When a = 0 the formula reduces to the well-known relationship
T rf;A fx S (r) + fr +
T rf;A fx S (r) + fr +
T
fxx S 2 (r)
2
15
2
";A
= T rf;A f:
When K = 1 and a 6= 0, (31) becomes
T
fxx S 2 (r) G2an ;A (r) a2n
2
+2an u";A Gan ;A (r) + 2";A
= T rf;A f:
2
u;A
T rf;A fx S (r) + fr +
While the sde’s are not used in the sequel to price options, their derivation
is of some independent interest, showing how the famous BS sde’s are generalized in our model. Notably, the sde for the SV model in equation (6)
of Heston (1993) includes extra terms (relative to BS) which involve partial
derivatives of the asset with respect to the volatility process, whereas the
above sde’s involve extra terms (relative to BS) which are functionals of the
new limit process Ga (r) re‡ecting the impact of time variation in the discrete
model’s autoregressive coe¢ cient.
4.3
Market Incompleteness
0
In model (6) the time varying coe¢ cient t (a; n) = exp apunt introduces
an additional source of uncertainty in the generating mechanism. The vector ut leads to variation in the autoregressive coe¢ cient t that typically
has unforecastable elements, thereby introducing an additional source of risk
to investors concerning the price generating mechanism. In applications, ut
may, for instance, carry the import of economy-wide common shocks or index movements that a¤ect returns indirectly via the generating mechanism
itself rather than through the equation error shocks "t ; although these two
shocks may well be correlated. If there are shocks to the way price evolves
from the previous price so that the mechanism is not a martingale and the
conditional expectation is not the immediately preceding price, the market is ine¢ cient. This ine¢ ciency may be interpreted as a form of market
incompleteness because the factors and shocks embodied in ut comprise additional unforecastable states of nature that are uncovered in the market.
These shocks involve risk beyond that of the simple equation error. In the
continuous time context, the shocks are manifest in the additional random
p
a0
an
Gan ;A (r) a0n u";A dt; that appears in the drift of
component, T n u;A
2
the log price stochastic di¤erential equation. In e¤ect, the TVC model implies uncertainty and risk in the price process drift, producing an additional
16
source of uncertainty/risk in investment alpha.
5
The Value of a European Call Option
At time 0 the value of a European call option maturing in T years is given
by
C = e rf;T E Q max fST K; 0g ;
where Q is the risk-neutral measure (de…ned immediately after equation
(27)). To calculate the expectation in the last equation we need to …nd
STQ
S Q (1), the price at time T under Q. To clarify the notation we remark that for any random variable X, by E Q (X) or by E X Q we mean
the expected value of X under Q. Under risk-neutral pricing, S Q (r) must
be a martingale with respect to Q, so that
E Q (S (r)) = E S Q (r) = S (0) erf;T r :
In other words, under Q the price process develops from S (0) at the risk-free
rate rf . To …nd S Q (r), we …rst make the following transformation:
Bu (r)
B" (r)
BuQ (r)
B" Q (r) + r
=
;
2 R;
where the superscript Q indicates that the SBM’s are under Q. As
dB" (r) = dB" Q (r) + dr
(32)
and as
h
1=2
A
i
B (p) =
2
=
h
h
1=2
A
1=2
A
i
2;1
i
Bu (p) +
B Q (p) +
2
h
h
1=2
A
1=2
A
i
2;2
i
B" (p)
r;
(33)
2;2
it follows from (21) that
Gan ;A (r) =
=
e
a0n
Q
1=2
Q
u;A Bu (r)
(r) +
h
GQ
an ;A
1=2
A
i
2;2
(r) ;
17
Z
0
r
e
a0n
1=2
Q
u;A Bu (p)
dp + GQ
an ;A (r)
(34)
Q
Q
say, where GQ
an ;A (r) is Gan ;A (r) with Bu (r) and B" (r) in the former replacing Bu (r) and B" (r) in the latter, respectively. We obtain from (27)
S Q (r) = S (0) exp
p
+ T
T
a0n
p
+ T a0n
= S (0)
A
Q
1=2
u;A
Q
Q
0
1=2
A
(r)
i
B
T a0n
Q
(s) + GQ
an ;A (s)
r
u";A
Q
(s) + GQ
an ;A (s) ds
2
ds
Q
(s) + GQ
an ;A (s) dBu (s)
(r) +
2
Q
u";A
T a0n
Z r
0
Z r
u;A an
2
Z 0r
h
p
2
p
u;A an
2
a0n
p
+ T
2
";A
T
(r) ;
h
1=2
A
i
r
2;2
where
Q
n
(r) = exp T
p
+ T
p
A
a0n
T a0n
u;A an
u";A r
p
T a0n
u";A
Z
2
0
Z r
0
a
a
2
u;A n
Q
2
T n
(s) ds
2
0
Z r
Q
0
T an u;A an
GQ
(s) ds
an ;A (s)
0
p 0 1=2 Z r
+ T an u;A
(s) dBuQ (s)
p h
+ r T
1=2
A
i
0
2;2
18
r
Q
(s) ds
and
Q
2
";A
(r) = exp
Tr
2
Z r p
a0n u;A an p 0
+
T
T an u";A GQ
an ;A (s)
2
0
2
a0 u;A an
T n
GQ
ds
an ;A (s)
2
Z r
i
h
p
1=2
0 1=2
Q
B Q (r)
+ T an u;A
GQ
(s)
dB
(s)
+
u
an ;A
A
= ~ , such that
Q
Q
~
:
2
0
Now, set
(35)
(1) = 1, and let
(1) = e
rf;A T
Q
E
(1) :
The price process under Q at r = 1 is de…ned as
Q
S Q (1) = S (0)
(1)
= S (0) erf;A T
Q
(1)
with
Q
Q
(1) =
Q
(1) ;
(1)
(1)
Q
E
Evidently,
E S Q (1) = S (0) erf;A T ;
as required.
Note that in the a = 0 and
Q
Q
BS
u"
= 0 case,
(1) = exp T
2
";A
(1) = exp
2
A+
T+
p
T
p
T
"
Q
";A B"
and
Q
(1) = e
rf;A T
E
= e
rf;A T
:
19
;
Q
BS
(1)
(1)
(36)
Hence, in this special case,
2
";A
Q
S Q (1) = S (0)
(1)
= S (0) exp
Q
(1)
rf;A
p
T+
2
T
2
Q
";A B"
(1) ;
as is well-known.
A further adjustment can be made such that the mean and variance
of a transformed Q (1) will be the same as of the lognormal distribution
(corresponding to BS pricing). To do so, let
Q
e
1
Q
(1) =
Then E
Q
T
= 1 and V ar
(1)
Q
(1))
Q
T
(1)
Q
(1)
E
Q
E(
!
1=2
2 T
";A
=e
2 T
";A
1
+ 1:
1; giving the mean and variance
Q
BS
of
(1), which is valid under BS pricing. The mean- and variance adjusted
price process under Q is then given by
S
Q
(1) = S (0) exp (rf;A T )
Q
(1) :
(37)
Such a transformation is desirable because the resulting process, S Q (1),
di¤ers from the price process under BS only with respect to the term Q (1),
vs. Q
BS (1), and these two di¤er only in skewness, kurtosis and higher order
moments. Hull (Ch. 18, 2009) argues that the information embodied in the
skewness and kurtosis measures for processes that have the same mean and
variance as BS can be translated into information on volatility smiles.
In order to simulate the value of a European call option maturing in
T years, we calculate GQ
an ;A (r) over the grid fr = 0; 1=n; 2=n; :::; 1g, after
simulating a vector Brownian motion driver process and noting that T corresponds to r = 1. We then calculate the sample mean of max S Q (1) K; 0 ,
denoted by max fS Q (1) K; 0g, using a large number of replications. The
estimated European call option price is
C^ = e
rf;T
max fS Q (1)
C^ = e
rf;T
max fS
K; 0g;
(38)
K; 0g
(39)
or
20
Q
(1)
if the re…nement (37) is preferred.
6
6.1
Numerical Analysis
Simulations
We downloaded from Yahoo Finance daily data on the closing prices of
Google and Nasdaq composite indexes (tickers GOOG and ^IXIC), over the
period 1-2-2009 through to 11-20-2013, giving a total of 1231 observations for
each series. This sample period is post the 2008 market crash, so we avoid
possible issues of structural breaks in the illustrations.
With the obvious notation, we obtained estimates of the following empirical STUR model
\ t = 0:000941
log(Google)
4:9540 ( log (N asdaq)t
p
+ exp
1231
(40)
0:000713)
log (Google)t
1
;
where 0:000713 is the estimated daily return of log (N asdaq)t over the
sample period. We have also obtained the estimates ^ 2u;A = 0:2112, ^ 2";A =
0:0716 and ^";u = 0:6923, over the same sample period.
For a hypothesis test of the form H0 : a = 0, when u" 6= 0, we know
from Remark 8 that
p
3
n^
an )
2
"
0 u
=
3n (
2T
2
u;A
u";A )
:
0;A
With daily data, n=T = 252. Under H0 , the model is a random walk with a
drift and therefore the historical estimates of and are consistent. To verify
the results of Section 3 we generated 250 samples of n = 1000; 2000; :::; 250000
(daily) observations from a random walk model with a drift (i.e., the null
model with a = 0). The value of was taken to be ^ = 0:000941. The
disturbance term was generated from a bivariate normal distribution with
(daily) covariance matrix consistent with the historical estimates mentioned
above, by dividing the annual covariance estimates by 252. For each of the
250 samples we have obtained a
^n and consequently, we computed the sample
21
average (based on the 250 samples) of
p
n^
an
Rn =
; n = 1000; 2000; :::; 250000:
3 " =2 u 0
(41)
We obtained
1 X
Rn = 0:9991;
250
con…rming the result in Remark 8. The graph of Rn against n is provided in
Figure 1.
For the same output, we also obtained the following regression results
Rn =
\
(log
j^
an j) = 7:328
0:508 log (n) , R2 = 0:978:
The implication of both Figure 1 and the last equation is that j^
an j n1=2 ,
corroborating the decay rate stated in Remark 8.
Moving on in a similar way except that the true value of a is a0 = 0:15,
we generated2 250 samples of n = 1000;p
2000; :::; 250000 (daily) observations
from our process, obtaining
an a0 ) against n in Figure 2,
p a plot of n (^
which con…rms the stated n-rate. As a further check, we ran the log-log
regression for this scenario, obtaining
(log \
j^
an a0 j) = 7:642
0:533 log (n) , R2 = 0:973;
again corroborating the result of Theorem 4 that j^
an a0 j n 1=2 .
To complete this subsection, we also estimated the model with
obtaining
\ t = exp
log(Google)
4:9544 ( log (N asdaq)t
p
1231
0:000713)
= 0,
log (Google)t
1
:
Using the same historical estimates for the covariance matrix as in the 6= 0
case, we generated 100; 000 replications of the right side of Theorem 2(2)
with n = 1231 integral points. The resulting kernel density estimate is given
in Figure 3. The distribution is evidently bi-modal with a local minimum at
zero.
2
In this setup the estimates of and of are inconsistent, but are used as if they were
the true values in order to assess the …nite sample implications of Theorem 4.
22
6.2
An Empirical Application
We start with the single regressor STUR model (40), expanding it to a tworegressor application later on. The Akaike (AIC), Schwarz (SC) and sum
of squared errors (SSE) values for the model (40) are 5:976, 5:968, and
0:182, respectively. On the other hand, for the model (1) the …gures are
5:327, 5:319 and 0:349, respectively, and for the model (1) with = 1
the corresponding values are 5:327, 5:323 and 0:349. Thus, in terms of
selection criteria, the STUR model provides a clear improvement over the
basic model.
The estimate of a in (40) is 4:954. Running p
with an increasing sequence
an increases. Thus, using
of subsamples n = 100; 200; :::; 1231 reveals that n^
(17) we would reject the hypothesis H0 : a = 0. This result, together with
the selection criteria …gures, justi…es the use of the TVC model over (1).
For the annual risk-free rate we used the Federal Funds Rate from the
Bloomberg website quoted as rf;A = 0:0007 (0.07% per annum). The 3month treasury yield was identical (in annual terms). Other choices of the
risk-free rate, such as the 12 month treasury yield (0.11%) or the 2-year yield
(0.29%), which may be better suited to use for options with longer expiration
periods might also be considered. But for this empirical illustration we have
used the Federal Funds Rate.
The practice in this illustration is similar to the one described in Christoffersen and Jacobs (2004, p. 1207) in the sense that …rst we obtain consistent
estimates of the model parameters using the stock return data and consequently plug in these estimates instead of the corresponding parameters
under the risk neutral measure.
Google is a non-dividend paying stock and is therefore suited to this
application. We took expirations on 1-3-2014 (36 days), 1-17-2015 (415 days)
and 1-15-2016 (778 days) and considered strike prices K = 950, 1030, 1060,
1100, 1200. The closing price for Goggle stock on 11-27-2013 was 1063.11.
For each scenario, we have calculated the BS classic call option prices (see,
e.g., equation (13.20) of Hull (2009)) in Mathematica. To simulate Gan ;A (r),
1=2
we generated a vector of two standard Brownian motions scaled by ^ A . For
T , we substituted the number of days to expiration divided by 365. The
prices S Q (1) were evaluated with 500 integral points and 2000 replications
over simulated Brownian motions.
We remark that the estimate a
^n was obtained from daily data. In order to
simulate S Q (1) for any T and n, and thereby for any implied data frequency,
23
we need to take account of the time dimensionality of the parameter. To
this end, we recall that an = (T =n)1=2 a. Moreover, had we estimated (40)
with data of any other frequency, with nf observations over the same sample
period, least squares estimation would have yielded
a
^n;d =
n
a
^n;f ;
nf
where a
^n;d and a
^n;f are the least squares estimates of a based on the daily and
general-frequency data, respectively. With the consistency result of Theorem
4, this means that with our daily data, for any T and n we can replace an by
s
p
T nf
T nf
a
^n;d =
a
^n;d .
nf n
n
For the illustration, we calculated (38) and (39), each with the historical
, denoted by ^, but also with preset values = 0 and = 0:95, in order
to investigate the e¤ect of the correlation on the results. Finally, we also
evaluated (38) and (39) after …tting a multivariate extension to (40), viz.,
\ t = 0:000944
log(Google)
4:5903 ( log (N asdaq)t 0:000713)
p
+ exp
1231
0:3921 ( log (AAP L)t 0:001411)
p
+
log (Google)t
1231
1
; (42)
where Apple inc. (ticker AAPL) had an average return of 0:001411 over the
sample period.
The results are presented in Table 1. A few comments are in order.
As the pricing schemes only di¤er from each other by the terms Q
BS (1),
Q
Q
(1) and
(1), an investigation of their behavior will be su¢ cient for the
explanation of the price di¤erences. First, C^ is mostly larger than BS and
the converse holds for C^ . The reason for the former is that the simulated
Q
standard deviation of
(1) is larger than that of Q
BS (1). We remark that
the standard deviation of Q (1) is equal to that of Q
BS (1) by construction.
Q
Second, the skewness and kurtosis coe¢ cients of both
(1) and Q (1)
(which are equal to each other by construction) are greater than those of
24
Q
BS
(1). Hull (Ch 18, 2009) argues that these results are consistent with
a volatility skew. Thirdly, there is no noticeable pattern of the results with
respect to . Finally, there is not much di¤erence between the single-regressor
and two-regressor results for small T , and a small di¤erence for large T .
Q
In Table 2 we investigate the moments of Q
(1) and Q (1),
BS (1),
Q
where clearly, excess skewness and kurtosis of
(1) and Q (1) relative to
Q
BS (1) are visible. Kernel density estimates of these terms are provided in
Figure 4-5, emphasizing the peakedness of Q (1) relative to Q
BS (1).
Overall, it appears that the TVC feature of the model results in a superior
performance over the basic model in terms of the usual AIC, SC and SSE
criteria, as well as a signi…cant a
^n estimate and peaked distribution which is
more consistent with empirical …ndings.
7
Conclusions
The time-varying coe¢ cient model is a natural extension of the simple AR(1)
model in which, at any given time period, the coe¢ cient of the lag dependent
variable can be less than-, equal to- or greater than unity depending on a
vector of unobserved factors with local to zero loading coe¢ cients. Unlike
the local to unit root model in which the coe¢ cient converges to unity as
the sample size tends to in…nity, in our model the e¤ect of the stochastic
coe¢ cient does not vanish as the sample size increases. As a result, the limit
process is not geometric Brownian motion but a nonlinear di¤usion. The
new model and limit theory provides a generalization to Black-Scholes call
option pricing.
We have established asymptotic theory for estimates of the model parameters. As expected, in the driftless case with endogeneity in the factors
( u" 6= 0), the estimate of the localizing coe¢ cient a
^n is inconsistent. As an
empirical illustration, the results were applied in the pricing of Google’s call
options. In this application the results show that the price process under the
new model (and under the risk neutral measure) has excess skewness and
kurtosis compared with the lognormal distribution.
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Appendix
Proof Lemma 1. For t = bnsc for any s > 0; we have
n
1=2
t
X
j
= B (t=n) + op (1) ;
j=1
so that
n
1=2
Yt
= n
1=2
t 1
X
e
a0
p
n
s=1
= n
= n
1=2
e
a0
p
n
Pt
Pt
j=1
j=s+1
uj
t 1
X
uj
e
a0
p
n
s=1
1=2 fa0 Bu (t=n)+op (1)g
e
"s + Op n
t 1
X
s=1
27
Ps
j=1
n
e
uj
a0
p
n
1=2
"s + Op n
1=2
Ps
o
1
j=0
uj
a0
p
u
n s
"s + Op n
1=2
= n
1=2 a0 Bu (t=n)
e
t 1
X
e
a0 u s
p + Op n 1 "s + op (1)
n
Ps
Ps 1 !
"
j
j=1
j=1 "j
p
p
n
n
fa0 Bu ((s 1)=n)+op (1)g
1
s=1
0
= ea Bu (t=n)
e
t 1
X
a0 Bu ((s 1)=n)
e
s=1
t 1
X
a0 Bu (t=n)
e
a0 Bu ((s 1)=n)
s=1
Setting t = bnrc and noting that E e
right hand side (rhs) of (43) has limit
a0 Bu ( nt )
e
t 1
X
e
a0 Bu ( s n 1 )
dB"
s=1
a0 u s " s
n
a0 Bu (p) 2
s
0
!p ea Bu (r)
n
Z
(43)
+ op (1) :
< 1; the …rst term on the
r
e
a0 Bu (p)
dB" (p) := Ga (r) ;
0
(44)
and the second term is
a0 Bu (t=n)
e
t 1
X
e
a0 us "s
n
a0 Bu ( s n 1 )
s=1
u s "s
u"
n
a0
=
!p
+
a0 Bu (t=n) 1
u" e
t 1
X
0 a0 Bu (t=n)
=
ae
e
a0 Bu ( s n 1 )
s=1
u"
n
t 1
X
a0 Bu ((s 1)=n)
e
n
Z rs=1
0
0
a0 u" ea Bu (r)
e a Bu (p) dp:
+ Op n
1=2
0
Hence,
n
1=2
Ybnrc !p
Ga (r)
= e
a0 Bu (r)
0
a
Z
u" e
a0 Bu (r)
r
e
a0 Bu (p)
dp
0
r
e
a0 Bu (p)
0
= : Ga (r) ;
Z
dB" (p)
a
0
u"
Z
r
e
a0 Bu (p)
dp
0
28
(45)
as required for (11). Next,
(t 1
!
)
t
1 X Y
+1
n s=1 j=s+1 j
t 1
X
1
n
=
e
s=1
!p e
a0 Bu (r)
a0
p
n
Z
Pt
j=s+1
r
e
uj
+1
a0 Bu (p)
!
dp;
0
giving (12).
Proof Theorem 2. For the case
Qn (a) =
n
X
t=2
= 0, the objective function is
fYt
t
In this model, letting t = bnrc,
Z r
Yt
0
a0 Bu (r)
p )e
e a Bu (p) dB" (p)
n
0
a
0
(a) Yt 1 g2 :
u"
Z
r
(46)
a0 Bu (p)
e
dp
=: Ga (r) :
0
(47)
Minimizing (46) with respect to a yields
Q_ n (^
an ) =
2
=)
n
X
t=2
n
X
t=2
=)
n
X
fYt
t
fYt
Yt u t
t
t
(^
an ) Yt 1 g _ t (^
an ) Y t
(^
an ) Yt 1 g ut
(^
an ) Y t
t=2
=)
n
X
t=2
=
n
X
t=2
1
=
n
X
t
1
(^
an ) Yt
ut
2
t
1
=0
=0
(^
an ) Yt2 1
t=2
a
^0n ut
ut 1 + p
n
+ Op n
2^
a0 ut
ut 1 + pn + Op n
n
29
1
1
Yt Yt
Yt2 1 :
1
(48)
The …rst term on the lhs of (48) is
n
X
t=2
u t Yt Yt
1
=
n
X
ut f
t=2
n
X
=n
(a) Yt
t
+ "t g Yt
1
Yt2 1 p X
Yt 1
+ n
u t "t p
n
n
t=2
n
ut
t (a)
t=2
n
X
1
Yt2 1 p X
Yt 1
n
ut
+ n
u t "t p
n
n
t=2
t=2
Z 1
Z 1
n3=2
G2a (r) dBu (r) + n3=2 u a
G2a (r) dr
0
0
Z 1
Z 1
+ n3=2 u"
Ga (r) dr + n
Ga (r) dBu" (r) 1 f
n
a0 ut
1+ p
n
0
u"
= 0g :
0
(49)
The second term on the left hand side (lhs) of (48) is
1 X
p
ut u0t a
^n Yt Yt
n t=2
n
1 X
ut u0t a
=p
^n f t (a) Yt 1 + "t g Yt 1
n t=2
( n
)
n
X
X
1
a0 ut Yt2 1
p
n
1+ p
ut u0t +
"t Yt 1 ut u0t a
^n
n
n
n
t=2
( t=2 n
!
n
2
p X Yt2 1 0
1 X Yt 1
1
p
n2
+
n
(ut a) ut u0t
u
n t=2 n
n
n
t=2
!)
n
1 X Yt 1
p
+n3=2 E (ut u0t "t )
a
^n
n t=2 n
Z 1
Z 1
1
2
2
3=2
0
0
p
n u
Ga (r) dr + n E f(ut a) ut ut g
G2a (r)
n
0
0
Z 1
+n3=2 E ("t ut u0t )
Ga (r) dr a
^n
n
1
0
30
3=2
Z
1
G2a
f(u0t a) ut u0t g) a
^n
(r) dr + n (E
0
Z 1
0
+ n fE ("t ut ut )g a
^n
Ga (r) dr:
n
^n
ua
Z
1
G2a (r)
(50)
0
0
Combining (49) and (50), the lhs of (48) behaves as
Z 1
Z 1
2
3=2
Ga (r) dBu (r) + u (^
an + a)
G2a (r) dr +
n
0
Z
u"
0
Z
1
+ n fE ("t ut u0t )g a
^n
Ga (r) dr + (E f(u0t a) ut u0t g) a
^n
0
Z 1
+
Ga (r) dBu" (r) 1 f u" = 0g :
1
Ga (r) dr
0
Z
(51)
1
G2a (r)
0
0
The …rst term on the rhs of (48) is
n
X
ut Yt2 1
3=2
n
t=2
Z
1
G2a (r) dBu (r)
(52)
Z
G2a (r) dr:
(53)
G2a (r) dr :
(54)
0
and the second term on the rhs of (48) is
2 X
p
ut u0t a
^n Yt2 1
n t=2
n
2
^n n
ua
3=2
1
0
Using (52) and (53), the rhs of (48) behaves as
Z 1
Z
2
3=2
n
Ga (r) dBu (r) + 2 u a
^n
0
0
31
1
Equating (51) to (54), we obtain
Z 1
Z 1
Z 1
2
2
3=2
Ga (r) dr
Ga (r) dr + u"
an + a)
Ga (r) dBu (r) + u (^
n
0
0
0
Z 1
Z 1
0
0
0
G2a (r)
Ga (r) dr + (E f(ut a) ut ut g) a
^n
+n fE ("t ut ut )g a
^n
0
0
Z 1
+
Ga (r) dBu" (r) 1 f u" = 0g
0
Z 1
Z 1
3=2
2
= n
Ga (r) dBu (r) + 2 u a
^n
G2a (r) dr :
0
In the
0
6= 0 case, the limit is seen to be
R1
Ga (r) dr 1
(^
an a) ) R01
u
G2a (r) dr
0
u"
whereas, in the
p
n (^
an
a) ) R 1
u"
u"
6= 0;
= 0 case, we obtain
u"
1
1
u
G2a (r) dr
0
n
oZ
2
0
+E (ut a) ut
fE
0
if
u" ; if
= 0:
Furthermore, in the
("t ut u0t )g a
1
G2a
(r) +
Z
Z
1
Ga (r) dr
0
1
Ga (r) dBu" (r) ;
0
= 0 and a = 0 case, the limit is
Z 1
p
1
1
n^
an ) R 1
Ga (r) dBu" (r) , if u" = 0 and a = 0
2 (r) dr u
G
0
a
0
u"
which reduces to
p
n^
an ) R 1
0
1
B"2 (r) dr
1
u
Z
1
B" (r) dBu" (r) , if
0
32
u"
= 0 and a = 0:
(55)
Proof Theorem 3. For part (1), we have
1X
=
Yt
n t=1
ea^n ut =
=
ea ut =
n
^ ";n
p
0
1X
Yt
n t=1
n
1X
=
Yt
n t=1
0
p
0
p
n
(^
an
+
1X
"t
n t=1
n
=
+
(^
an
a)0 ut
2n
0
p
ea ut =
a)0 ut
2n
2
Yt
1
p
n (^
an a)0 ut = n
e
n
2
+ op
1+
1
n
(^
an
!
Yt
2
Yt
1
a)0 ut
p
n
!2
1
a)0 ut
p
n
!
!2
(^
an
n
2
1
n
+ op
1X 2
=
"
n t=1 t
ea ut =
n
Yt
1
n
n
2 X a0 ut =pn
"t e
n t=1
1 X 2a0 ut =pn
+
e
n t=1
n
(^
an
(^
an
p
0
(^
an
a) ut
+
n
2
a)0 ut
2n
(^
an a)0 ut
a)0 ut
p
+
2n
n
+ op
2
+ op
1
n
1
n
The …rst term on the rhs of (56) converges in probability to
second term becomes
2X
a0 ut
"t 1 + p + op
n t=1
n
n
+op
1
n
1
p
n
Yt 1 :
33
(^
an
p
!
!2
2
",
Yt
1
Yt2 1 :
(56)
whereas the
(^
an a)0 ut
a)0 ut
+
2n
n
2
The dominant terms in the last expression are equal to
2
n
=
n
X
n
X
"t
(^
an
t=1
a)0 ut
p
Yt
n
1
(^
an
+ "t
2
a)0 ut
Yt
2n
a0 ut (^
an a)0 ut
a0 ut (^
an a)0 ut
"t
Y t 1 + "t
n
2n3=2
t=1
Z 1
2
0
p
=
(^
an a)
Ga (r) dBu" (r)
n
0
Z 1
1
0
0
+ (^
an a)
Ga (r) dr
an a) E ("t ut ut ) (^
2
0
Z 1
0
0
+a E ("t ut ut ) (^
an a)
Ga (r) dr + op (1)
2
1
!
!
Yt
1
!
0
! p 0:
The leading term in the third term on the rhs of (56) is seen to be
!
n
n
2
X
X
1
1
Yt 1
2
(^
an a)0 ut Yt2 1 = (^
an a)0
ut u0t p
(^
an a)
2
n t=1
n t=1
n
Z 1
0
= (^
an a) u (^
an a)
G2a (r) dr + op (1) :
0
(57)
Using Theorem 2(1), (57) becomes
!
R1
G
(r)
dr
a
0
1
R01
u
u" u
2
Ga (r) dr
0
2
R1
G
(r)
dr
a
0
0
1
)
R1
u" :
u" u
2 (r) dr
G
a
0
1
u
R1
0
u" R 1
Ga (r) dr
G2a
0
(r) dr
!Z
1
0
It follows that
^ 2";n
2
"
)
2
u"
R1
0
2
u
2
Ga (r) dr
R1
G2a
0
34
(r) dr
0
u"
1
u
u" :
G2a (r) dr + op (1)
For part (2), we have
1X
vech (ut u0t )
=
n t=1
n
vech ^ u;n
1X
vech (ut u0t
n t=1
n
=
u
+
u)
1X
= vech ( u ) +
vech (ut u0t
n t=1
n
u) ;
so that
1 X
vech (ut u0t
vech ( u ) = p
n t=1
n
p
n vech ^ u;n
u)
) (1) :
Let et be the least squares residual from the regression (6). For part (3), we
have
X
^ u";n = 1
et ut
n t=1
n
1X
=
Yt
n t=1
ea^n ut =
=
ea ut =
n
1X
Yt
n t=1
0
n
1X
=
Yt
n t=1
1+
Yt
1
ut
0
p
p
n (^
an a)0 ut = n
0
p
n
n
(^
an
p
n
ea ut =
e
(^
an a)0 ut
a)0 ut
p
+
2n
n
35
Yt
1
2
+ op
ut
1
n
!
Yt
1
!
ut :
(58)
The dominant terms in the last expression become
1X
"t
n t=1
n
a0 ut
1+ p
n
(^
an
(^
an a)0 ut
a)0 ut
p
+
2n
n
1X
n t=1
n
2
!
Yt
1
!
ut
2
(^
an a)0 ut
a)0 ut
p
= u"
Yt 1 ut
Yt 1 u t +
2n
n
!
2
an a)0 ut
a0 ut (^
a0 ut (^
an a)0 ut
+
Yt 1 u t +
Yt 1 ut + op (1) :
n
2n3=2
(^
an
(59)
By [ u ]j we denote the jth row of u , which is also equal to the jth column
of u . The …rst term in the brackets of (59) behaves as
K
X
j=1
Yt 1
1X
uj;t ut p
a)j
n t=1
n
K
X
1 X Yt 1
p + op (1)
(^
an a)j [ u ]j
n t=1 n
j=1
Z 1
Ga (r)dr + op (1)
an a)
u (^
0
R1
Z 1
Ga (r) dr
1
0
Ga (r)dr
u u
u" R 1
G2a (r) dr
0
0
2
R1
G
(r)
dr
a
0
R1
u" :
2 (r) dr
G
a
0
n
(^
an
=
=
)
=
n
All other terms in the brackets of (59) are negligible and therefore
^ u";n
u"
)
R1
2
0
Ga (r) dr
0
G2a (r) dr
R1
36
u" :
Proof Theorem 4. In the
6= 0 case, the ols estimator of
1 X a^0n ut =pn
e
Yt
n t=2
is equal to
n
^ n = Yn
1
and the plugged-in score function is
2 Xn
p
Yt
n t=2
n
Q_ n (^
an ; ^ n ) =
1 X a^0n ut =pn
e
Yt
n t=2
n
1
!)
p
0
ea^n ut =
Yn
0
p
ut ea^n ut =
n
n
Yt
Yt
1
1
= 0:
This implies that
n
X
p
0
Yn ut ea^n ut =
Yt
n
Yt
(60)
1
t=2
=
n
X
e
p
a
^0n ut = n
1 X a^0n ut =pn
Yt
e
n t=2
n
Yt
1
t=2
1
!
0
p
ut ea^n ut =
n
Yt 1 :
It follows from Lieberman and Phillips (2014) that in this case
Z r
Yt
0
a0 Bu (r)
) 0e
e a Bu (p) dp Ha (r) :
n
0
Now,
n
X
t=2
Yt ut e
p
a
^0n ut = n
Yt
1
=
n
X
0
p
a ut =
0+e
t=2
37
n
Yt
1
0
p
+ "t ut ea^n ut =
n
Yt 1 :
We have
0
n
X
p
a
^0n ut = n
ut e
Yt
1
t=2
n
X
a
^0n ut
p
u
1
+
Yt 1
t
0
n
t=2
!
Z 1
n
X
1
3=2
Ha (r) dBu (r) + p
^ n Yt 1
ut u0t a
0 n
n
0
t=2
Z 1
3=2
Ha (r) dBu (r)
0 n
0
Pn
1
2
t=2 Yt 1
+ p ua
^n n
n2
n
Z 1
Z 1
3=2
Ha (r) dBu (r) + u a
^n
Ha (r) dr :
0n
0
0
(61)
Also,
n
X
p
(^
an +a)0 ut = n
ut e
Yt2 1
t=2
n
X
1 X
+p
ut u0t (^
an + a) Yt2 1
n t=2
n
ut Yt2 1
t=2
5=2
n
Z
1
0
5=2
n
Z
1
Ha2 (r) dBu (r) + p
n
an + a) n3
u (^
1
Ha2
(r) dBu (r) +
0
u
(^
an + a)
n
X
Yt2 1
t=2
Z
1
n3
Ha2 (r) dr
0
(62)
and
n
X
t=2
p
a
^0n ut = n
"t u t e
Yt
1
n
X
1 X
"t ut u0t a
^ n Yt 1
"t u t Y t 1 + p
n t=2
t=2
Z 1
Z
2
0
3=2
Ha (r) dr + fE ("t ut ut )g a
^n n
u" n
n
0
+ n3=2
Z
1
Ha (r) dr
0
(63)
1
Ha (r) dBu" (r) :
(64)
0
It follows from (61)-(63) that the …rst component of the lhs of (60) behaves
38
as
3=2
Z
1
^n
ua
1
Ha (r) dr
Z 1
Z 1
2
5=2
Ha2 (r) dr
Ha (r) dBu (r) + u (^
an + a)
+n
0
0
Z 1
Z 1
3=2
2
0
Ha (r) dr
^n n
+ u" n
Ha (r) dr + fE ("t ut ut )g a
0
0
Z 1
3=2
Ha (r) dBu" (r)
+n
0
Z 1
Z 1
5=2
2
n
Ha (r) dBu (r) + u (^
Ha2 (r) dr
an + a)
0
0
Z 1
+ u" n2
Ha (r) dr:
0n
Ha (r) dBu (r) +
Z
0
0
(65)
0
The second component of the lhs of (60) is
Yn
n
X
t=2
p
a
^0n ut = n
ut e
Yt
1
!
n
1 X
n
Ha (r) dr
ut Yt 1 + p
ut u0t a
^ n Yt 1
n
0
t=2
t=2
Z 1
Z 1
3=2
n
Ha (r) dr n
Ha (r) dBu (r)
0
0
Z 1
3=2
Ha (r) dr
+ ua
^n n
0
Z 1
Z 1
5=2
Ha (r) dBu (r)
(66)
=n
Ha (r) dr
0
0
Z 1
(67)
+ ua
^n
Ha (r) dr :
Z
n
X
1
0
39
Combining (65) with (66) the lhs of (60) behaves as
Z 1
Z 1
2
5=2
Ha2 (r) dr
an + a)
Ha (r) dBu (r) + u (^
n
0
0
Z 1
Z 1
Z 1
Ha (r) dr
Ha (r) dr
Ha (r) dBu (r) + u a
^n
0
0
0
Z 1
2
Ha (r) dr:
+ u" n
(68)
0
The …rst term on the rhs of (60) is
n
X
0
p
ut e2^an ut =
n
n
X
2 X
ut Yt2 1 + p
ut u0t a
^n Yt2 1
n
t=2
t=2
Z 1
Z
5=2
2
n
Ha (r) dBu (r) + 2 u a
^n Ha2 (r) dr : (69)
Yt2 1
t=2
n
0
The second term on the rhs of (60) is
1 X a^0n ut =pn
e
Yt
n t=2
n
1
!
n
X
0
p
ut ea^n ut =
n
Yt 1 :
t=2
We have
1 X a^0n ut =pn
e
Yt
n t=2
n
1
!
n
1 X 0
Yt 1 + p
a
^ ut Yt 1
n t=2 n
t=2
Z 1
Z 1
1
2
0
n
Ha (r) dr + n^
an
Ha (r) dBu (r)
n
0
0
Z 1
Z 1
0
=n
Ha (r) dr + a
^n
Ha (r) dBu (r) :
n
X
1
n
0
0
Using (61) and (70), the second term on the rhs of (60) is
! n
n
X
p
1 X a^0n ut =pn
0
e
Yt 1
ut ea^n ut = n Yt 1
n t=2
t=2
40
(70)
n
Z
1
0
n3=2
a
^0n
Z
1
Ha (r) dr +
Ha (r) dBu (r)
0
Z 1
Z 1
Ha (r) dBu (r) + u a
^n
Ha (r) dr :
0
Using (69) and (71), the rhs of (60) behaves as
Z
Z 1
2
5=2
Ha (r) dBu (r) + 2 u a
^n Ha2 (r) dr
n
0
Z 1
Z 1
Z 1
Ha (r) dr
Ha (r) dBu (r) + u a
^n
Ha (r) dr
0
0
(72)
:
0
Equating (68) to (72), we obtain
Z 1
Z 1
5=2
2
n
Ha (r) dBu (r) + u (^
an + a)
Ha2 (r) dr
0
0
Z 1
Z 1
Z 1
Ha (r) dBu (r) + u a
^n
Ha (r) dr
Ha (r) dr
0
0
0
Z 1
2
+ u" n
Ha (r) dr
0
Z 1
Z
5=2
2
=n
Ha (r) dBu (r) + 2 u a
^n Ha2 (r) dr
0
Z 1
Z 1
Z 1
Ha (r) dBu (r) + u a
^n
Ha (r) dr
Ha (r) dr
0
0
(71)
0
:
0
This leads to the result
p
n (^
an
R1
a) ) R01
Ha (r) dr
Ha2
0
Proof Theorem 5. With Yn = n
1
41
(r) dr
Pn
t=2
1
u
u" :
Yt , the least squares estimator
of
is given by
1 X a^0n ut =pn
e
Yt
n t=2
^ n = Yn
Yn
=
1X
n t=2
Y1
n
Now,
Yn
0
) ea Bu (1)
n
1
a0n ut )2
a
^0n ut (^
p +
+ op
2n
n
Z
1
e
a0 Bu (r)
!
1
n
Yt 1 :
dr = Ha (1)
0
and Y1 =n = Op (n 1 ). Furthermore, using Theorem 4,
1
a
^0
3=2 n
n
X
1 0
a
^
n2 n
1
ut u0t Yt 1
t=2
Hence,
Ha (1)
u t Yt
t=2
and
^n )
X
a
0
Z
) a
!
0
Z
1
Ha (r) dBu (r)
Proof of Theorem 6. By de…nition,
n
^ ";n
1X
Yt
=
n t=1
e
n
p
a
^0n ut = n
n
0
p
ea ut =
2
Yt
Ha (r) dBu (r)
0
a
^n ) a
0
1X
Yt
=
n t=1
1
1
ua
1
Ha (r) dr:
0
1 0
a
2
n
Z
ua
Z
1
Ha (r) dr
e
=
^n
:
B(a)
For part (1),
n
Yt
= B (a) :
0
1X
Yt
=
n t=1
p
n (^
an a)0 ut = n
1X
("t + op (1)
n t=1
0
1
0
ea^n ut =
p
n
2
Yt
1
+ op (1)
2
+ op (1)
n
=
e
p
a0 ut = n
(^
an
(^
an a)0 ut
a)0 ut
p
+
2n
n
42
2
+ op
1
n
!
Yt
1
!2
:
(73)
Now,
2 X a0 ut =pn
"t e
n t=1
n
(^
an
(^
an a)0 ut
a)0 ut
p
+
2n
n
2X
a0 ut
"t 1 + p + op
n t=1
n
n
=
(^
an
1
p
n
(^
an a)0 ut
a)0 ut
p
+
2n
n
By Theorem 4, a
^n
a = Op n
2 X (^
an a)0 ut
p
"t
Yt
n t=1
n
1=2
2
+ op
1
1
n
+ op
!
Yt
2 (^
an
(74)
Yt 1 :
0
a)
Z
1
Ha (r) dBu" (r) + op
0
= Op
1
. Therefore,
n
=
!
1
n
2
1
p
n
;
2
an a)0 ut
2 X (^
"t
Yt
n t=1
2n
n
2 X a0 ut (^
an a)0 ut
p
"t p
Yt
n t=1
n
n
1
= Op
1
n
1
= Op
1
p
n
n
2
an a)0 ut
2 X a0 ut (^
Yt
"t p
n t=1
2n
n
n
1
p
n
1
= Op
;
1
n3=2
;
:
It follows that (74) converges in probability to zero. The last term in (73) is
1 X 2a0 ut =pn
e
n t=1
n
(^
an
(^
an a)0 ut
a)0 ut
p
+
2n
n
43
2
+ op
1
n
!2
Yt2 1 :
(75)
The leading term in the last expression is
1X
n t=1
n
(^
an
2
a)0 ut
p
n
Yt 1
n
t=1
!
Z
a)0 E (ut u0t )
Yt2 1 = (^
an
R1
R01
)
n
X
Ha (r) dr
0
u"
1
!
(^
an
a)
1
Ha2 (r) dr
u
u
Ha2 (r) dr
0
!
R1
H
(r)
dr
a
1
R01
u"
2 (r) dr u
H
a
0
2
R1
H
(r)
dr
a
0
0
1
R1
u" :
u" u
2
Ha (r) dr
0
=
2
0
All other terms in (75) converge in probability to zero. Hence,
^ 2";n
2
"
)
as required.
R1
2
0
Ha (r) dr
0
Ha2 (r) dr
R1
0
u"
The proof of part (2) is identical to the
X
^ u";n = 1
u t Yt
n t=1
n
1X
=
u t Yt
n t=1
n
0
p
ea^n ut =
1X
ut "t + op (1)
n t=1
n
=
(^
an
u" ;
= 0 case. Finally, for part (3),
n
Yt
n
+ op (1)
1
u
1
p
0
ea ut =
p
n (^
an a)0 ut = n
e
a0 ut
1 + p + op
n
(^
an a)0 ut
a)0 ut
p
+
2n
n
44
2
+ op
1
n
1
p
n
!
Yt
Yt
1
1
!
:
(76)
Now, n
1
Pn
t=1
ut "t !p
1 X (^
an a)0 ut
p
ut
Yt
n t=1
n
n
u"
1
and
n
1X
Yt 1 p
=
ut u0t
n (^
an a)
n t=1
n
R1
Z 1
Ha (r) dr
Ha (r) dr
)
R01
u
Ha2 (r) dr
0
0
2
R1
H
(r)
dr
a
0
=
R1
u" :
Ha2 (r) dr
0
1
u
u"
!
All other terms in (76) converge in probability to zero. Hence,
^ u";n
u"
)
R1
2
0
Ha (r) dr
0
Ha2 (r) dr
R1
u"
and the proof of the theorem is completed.
Proof of (20)-(21). We can write (14) as
Z r
p
0
a0 Bu (r)
Yn (r) = T e
e a Bu (p) dp + nGa (r)
Z0 r
0
a0 Bu (r)
= Te
e a Bu (p) dp
0
Z r
p a0 Bu (r)
0
1=2
e a Bu (p) dB (p)
+ ne
2
0
Z r
1 0
0
a (n u" )
e a Bu (p) dp
n
Z 0r
0
0
= T ea Bu (r)
e a Bu (p) dp
0
h
i Z r
1
0
0
1=2
a Bu (r)
+e
e a Bu (p) dB (p) p a0
T
n
2 0
Further, a0 Bu (r) = a0
1=2
u Bu
(r) = (T =n)1=2 a0
45
1=2
u;A Bu
u";T
Z
r
e
a0 Bu (p)
dp :
0
(77)
(r). Hence, (77) be-
comes
Yn (r) =
Te
1=2
u;A Bu (r)
a0n
Z
r
0
1=2
u;A Bu (r)
a0n
+e
1
p a0
n
u";T
Z
h
1=2
T
r
i Z
2
a0n
e
1=2
u;A Bu (p)
a0n
e
r
e
dp
a0n
1=2
u;A Bu (p)
dB (p)
(78)
0
1=2
u;A Bu (r)
dp ;
0
giving the required result.
Proof of 24. By stochastic di¤erentiation we have
dS (r)
1
d [S]r
2
S (r)
2S (r)
p
a0n u;A an
1
Gan ;A (r) a0n u";A
=
T A+ T
S (r)
2
h
i
p
1
1=2
0
S (r) T bA (r)
dB (r)
+
A
S (r)
i0
ih
h
1
1=2
1=2
0
2
bA (r) dr
T
S
(r)
b
(r)
A
A
A
2S 2 (r)
dG (r) =
=
T
A+
p a0n
T
T
bA (r)0
2
A bA
u;A an
2
Gan ;A (r)
(r) dr +
p 0
T an
p
T bA (r)0
S (r) dr
u";A
1=2
A dB
(r) :
Now,
bA (r)0
A bA
(r) = G2an ;A (r) a0n
u;A an
+ 2Gan ;A (r) a0n
and thus, because
a0
p
n
u";T
a0
=p T
n
u";A
46
=
p 0
T an
u";A
u";A
+
2
";A
(79)
we get
dG (r) =
T
A+
p
T
a0n
u;A an
2
Gan ;A (r)
a0n
u";A
T
G2an ;A (r) a0n u;A an + 2Gan ;A (r) a0n u";A + 2";A
2
p
1=2
+ T bA (r)0 A dB (r)
2
p 0
";A
=
T
T an u";A dr
A
2
p
a0n u;A an p 0
+
T
T an u";A Gan ;A (r)
2
p
a0 u;A an 2
1=2
Gan ;A (r) dr + T bA (r)0 A dB (r) :
T n
2
Proof of 27. Using (24), we have
G (r)
Z r
G (0) = log (S (r)) log (S (0))
2
p 0
";A
=
T
T an u";A dr
A
2
0
Z r p
a0n u;A an p 0
+
T
T an u";A Gan ;A (s)
2
0
p Z r
a0n u;A an 2
1=2
bA (s)0 A dB (s)
Gan ;A (s) ds + T
T
2
0
2
p
";A
=
T
T a0n u";A r
A
2
Z r p
a0n u;A an p 0
+
T
T an u";A Gan ;A (s)
2
0
a0 u;A an 2
T n
Gan ;A (s) ds
2
p 1=2
p 0 Z r
1=2
+ T an
Gan ;A (s) u;A (s) dBu (s) + T 2;A B (r) ;
0
and (27) immediately follows.
47
dr
Proof of details of Section 4.2. We must have df (r; x) = dV (r) and
by direct calculation
dV (r) =
S
(r)
T
A
+
p
p
+ T S (r) bA (r)0
=
S (r) T
+T rf;A
A+
T
a0n
2
1=2
A dB
p
T
u;A an
a0n
Gan ;A (r)
o
(r) + T rf;A
u;A an
2
p
T
Z (r) (r)g dr +
S
Z
Gan ;A (r)
a0n
u";A
S (r) dr
(r) (r) dr
a0n
(r) S (r) bA (r)0
u";A
1=2
A dB
S (r)
(r) : (80)
Now, in view of (23), we have
(d (S (r)))2 = T S 2 (r) bA (r)0
A bA
(r) dr;
(81)
and since df (r; x) = fr dr + fx dS (r) + 21 fxx (d (S (r)))2 , we deduce that
df (r; S (r)) = fr dr + fx fT A
p
a0n u;A an
+ T
Gan ;A (r) a0n u";A
S (r) dr
2
p
T
1=2
+ fx T S (r) bA (r)0 A dB (r) + fxx S 2 (r) bA (r)0 A bA (r) dr
2
p
a0n u;A an
= fr + fx T A + T
Gan ;A (r) a0n u";A
S (r)
2
p
T
1=2
+ fxx S 2 (r) bA (r)0 A bA (r) dr + T fx S (r) bA (r)0 A dB (r) :
2
(82)
Equating the coe¢ cients of dr and of the stochastic component in (80) and
48
(82) gives
S (r) T
+T rf;A
= fr + fx
A+
p
a0n
T
2
(r) (r)
p
T A+ T
T
S
a0n
Gan ;A (r)
S (r)
u";A
Z
a0n
(r) S (r) bA (r)0
u;A an
2
T
+ fxx S 2 (r) bA (r)0
2
and
p
u;A an
A bA
1=2
A dB
Gan ;A (r)
a0n
S (r)
u";A
(83)
(r)
(r) =
p
T fx S (r) bA (r)0
1=2
A dB
(r) :
(84)
The latter yields
(85)
(r) = fx
S
which is the condition in the classic case. Using this condition in (83) we
have
S
(r) T
+T rf;A
= fr +
Z
A+
p
T
a0n
u;A an
2
a0n
u";A
Gan ;A (r)
a0n
Gan ;A (r)
S (r)
(r) (r)
S (r) T
A+
T
+ fxx S 2 (r) bA (r)0
2
p
T
A bA
a0n
u;A an
2
u";A
(r) :
which implies
T rf;A
Z
(r) (r) = fr +
T
fxx S 2 (r) bA (r)0
2
and (30) follows.
49
A bA
(r)
S (r)
Table 1. Google Call Option Prices
n
K
BS
36
950
116:71
36
1030
54:13
36
1060
37:19
36
1100
20:77
36
1200
3:18
C^ ( = 0) 115:81
C^ (^)
114:58
MV
^
C
(^)
114:78
^
C ( = :95) 113:90
52:51
51:75
51:67
50:23
35:86
36:04
35:66
34:90
20:39
21:37
20:85
21:17
4:41
5:13
4:87
5:98
C^ ( = 0)
C^ (^)
C^ M V (^)
C^ ( = :95)
117:61
115:87
116:18
115:21
56:55
55:81
55:63
55:34
40:23
40:39
39:89
40:41
24:40
25:43
24:77
26:29
6:37
7:24
6:89
8:93
n
K
BS
415
950
179:88
415
1030
136:40
415
1060
122:36
415
1100
105:47
415
1200
71:49
C^ ( = 0) 180:52 137:31 123:30 106:60 72:49
C^ (^)
177:49 134:75 121:00 104:32 71:22
MV
^
C
(^)
173:62 130:73 117:26 101:13 69:48
^
C ( = :95) 176:74 133:99 120:36 104:15 72:17
C^ ( = 0)
C^ (^)
^
C M V (^)
C^ ( = :95)
182:58 139:51 125:51 108:80 74:53
180:46 137:95 124:21 107:52 74:16
180:89 138:63 125:21 109:02 76:71
177:76 135:09 121:47 105:25 73:19
50
Table 1. Google Call Option Prices (Continued)
n
K
BS
778
950
219:53
778
1030
179:85
778
1060
166:63
778
1100
150:32
778
1200
115:58
C^ ( = 0) 219:30 179:44 166:20 149:77 114:94
C^ (^)
211:52 172:02 159:01 143:12 109:87
MV
^
C
(^)
216:59 177:57 164:53 148:29 113:82
^
C ( = :95) 213:42 174:65 162:10 146:64 113:70
C^ ( = 0)
C^ (^)
C^ M V (^)
C^ ( = :95)
217:67 177:75 164:51 148:08 113:33
217:03 177:77 164:78 148:87 115:35
222:94 184:15 171:13 154:88 120:10
221:39 182:99 170:47 154:99 121:72
Note: n is the number of days to expiration as of 11-27-2013; K is the
strike price; S (0) = 1063:11; ‘BS’is the price based on Black and Scholes’s
classic formula; C^ and C^ are based on (38)-(39); ^ is based on Nasdaq’s
and Google’s historical volatility; The superscript MV indicates pricing
under (42).
51
Table 2. Summary Statistics for BS- and STUR Based Simulated Data
T
36
36
36
Statistic SD Skewness Kurtosis
Q
0.084
0.253
3.114
BS (1)
Q
(1) 0.084
0.970
4.543
Q
(1) 0.095
0.970
4.543
415
415
415
Q
BS
Q
0.291
0.291
0.299
0.898
1.124
1.124
4.469
5.332
5.332
778
778
778
Q
BS
Q
0.406
0.406
0.421
1.285
1.988
1.988
6.073
12.103
12.103
(1)
(1)
Q
(1)
(1)
(1)
Q
(1)
52
Rn
1.2
1.1
1.0
0.9
n
50 000
100 000
150 000
200 000
Figure 1: Plot of Rn , given in (41), against n: the
n
an
250 000
6= 0 and a = 0 case.
a0
1700
1600
1500
1400
1300
1200
1100
n
50 000
Figure 2: Plot of
p
n (^
an
100 000
150 000
200 000
250 000
a0 ) against n when a0 = 0:15: the
53
6= 0 case.
0.007
0.006
0.005
0.004
0.003
0.002
0.001
200
100
100
200
Figure 3: Kernel density estimate of the asymptotic distribution of a
^n in the
= 0, u" 6= 0 and a = 0 case, with n = 1231, 100000 replications and
historical estimates of the Google-Nasdaq data.
5
4
3
2
1
0.5
1.0
1.5
2.0
2.5
3.0
Figure 4: Kernel density estimates of the exponent terms: Theoretical Q
BS (1)
Q
Q
Q
(1) (Red),
(1) (orange), based on
(Black), Simulated BS (1) (BLUE),
the multivariate model (42) with T = 36 and = ^.
54
1.2
1.0
0.8
0.6
0.4
0.2
0.5
1.0
1.5
2.0
2.5
3.0
Figure 5: Kernel density estimates of the exponent terms: Theoretical Q
BS (1)
Q
Q
Q
(Black), Simulated BS (1) (BLUE),
(1) (Red),
(1) (orange), based on
the model (40)with T = 778 and = 0:95.
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