Open Journal of Applied Sciences, 2013, 3, 345-359
http://dx.doi.org/10.4236/ojapps.2013.36045 Published Online October 2013 (http://www.scirp.org/journal/ojapps)
Closed Form Moment Formulae for the Lognormal SABR
Model and Applications to Calibration Problems
Lorella Fatone1, Francesca Mariani2, Maria Cristina Recchioni3, Francesco Zirilli4
1Dipartimento di Matematica e Informatica Università di Camerino via Madonna delle Carceri 9, Camerino, Italy
2Dipartimento di Scienze Economiche Università degli Studi di Verona Vicolo Campo_ore 2, Verona, Italy
3Dipartimento di Management Università Politecnica delle Marche Piazza Ma rtelli 8, Ancona, Italy
4Dipartimento di Matematica “G. Castelnuovo” Università di Roma “La Sapienza” Piazzale Aldo Moro 2, Roma, Italy
Received August 11, 2013; revised September 17, 2013; accepted September 30, 2013
Copyright © 2013 Lorella Fatone et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
We study two calibration problems for the lognormal SABR model using the moment method and some new formulae
for the moments of the logarithm of the forward prices/rates variable. The lognormal SABR model is a special case of
the SABR model [1]. The acronym “SABR” means “Stochastic-

 ” and comes from the original names of the model
parameters (i.e., ,,

 ) [1]. The SABR model is a system of two stochastic differential equations widely used in
mathematical finance whose independent variable is time and whose dependent variables are the forward prices/rates
and the associated stochastic volatility. The lognormal SABR model corresponds to the choice 1

 and depends on
three quantities: the parameters ,


and the initial stochastic volatility. In fact the initial stochastic volatility cannot
be observed and can be regarded as a parameter. A calibration problem is an inv erse problem that con sists in determine-
ing the values of these three parameters starting from a set of data. We consider two differen t sets of data, that is: i) the
set of the forward prices/rates observed at a given time on multiple independent trajectories of the lognormal SABR
model, ii) the set of the forward prices/rates observed on a discrete set of known time values along a single trajectory of
the lognormal SABR model. The calibration problems corresponding to these two sets of data are formulated as con-
strained nonlinear least-squares problems and are solved numerically. The formulation of these nonlinear least-squares
problems is based on some new formulae for the moments of the logarithm of the forward prices/rates. Note that in the
financial markets the first set of data considered is hardly available while the second set of data is of common use and
corresponds simply to the time series of the observed forward prices/rates. As a consequence the first calibration prob-
lem although realistic in several contexts of science and engineering is of limited interest in finance while the second
calibration problem is of practical use in finance (and elsewhere). The formulation of these calibratio n problems and the
methods used to solve them are tested on synthetic and on real data. The real data studied are the data belonging to a
time series of exchange rates between currencies (euro/U.S. dollar exchange rates).
Keywords: SABR Model; Calibration Problems; FX Data
1. Introduction
We study two calibration problems for the lognormal
SABR model using the moment method and some new
formulae for the moments of the logarithm of the forward
prices/rates variable. The lognormal SABR model is a
special case of the “Stochastic-

 ” model which has
become known under the acronym of SABR model [1].
The SABR model is widely used in the theory and prac-
tice of mathematical finance, for example, it is widely
used to price in terest rates derivatives and options on cu r-
rencies exchange rates.
Let be a real variable that denotes time and t
t
x
, t,
be real stochastic processes that describe, res-
pectively, the forward prices/rates and the associated
stochastic volatility, as a function of time. The SABR
model [1] assumes that the dynamics of the stochastic
processes t
v
0,t
x
, t, , is defined by the following
system of stochastic differential equations:
v0t
dd,
tttt
xxvWt

0,

 (1)
dd,
ttt
vvQt0,


 (2)
with the initial cond itions:
C
opyright © 2013 SciRes. OJAppS

L. FATONE ET AL. 346
00
,
x
x (3)
(4)
00
,vv
where


0,1   is the

-volatility and 0

 is the
volatility of volatility. Note that in the original paper [1]
the volatility of volatility

was called .

The
stochastic processes W , are standard
Wiener processes such that 00, , t,
, are their stochastic differentials and we assume
that:
,
t t
Q0,t0
WQ dt WdQ
0t
ddd, 0,
tt
WQ tt 

 (5)
where  denotes the expected value of

and
is a constant known as correlation coefficient.
The initial conditions 0

1,1  

x
, 0 are random variables that
are assumed to be concentrated in a point with pro-
bability one. For simplicity, we identify these random
variables with the points where they are concentrated.
We assume 0 (with probability one) so that
Equation (2) implies that (with probability one)
for . Note that the initial stochastic volatility 0
and the stochastic volatility t, , cannot be
observed in the financial markets. That is, 0 must be
regarded as a parameter of the model together with
v 
t
v
0v

0
v
0tv

0tv


,

and

.
The value of the parameter


0,1   determines the
forward prices/rates process, that is, it determines
Equation (1). The most common choices of

are:
0

, 12

 and 1

.
Setting 0

 in (1) the forward prices/rates process
reduces to:
. (6) dd,
ttt
xvWt0
The correspond ing model (6), (2), (3), (4) is known as
the normal SABR model. This model has a forward
prices/rates process whose increments are stochastic
normally distributed, that is, the increments are normally
distributed with mean zero and a stochastic standard de-
viation lognormally distributed. This permits to the for-
ward prices/rates t
x
, , to become negative. Usual-
ly this is not a desirable property. In fact, in financial
applications most of the times prices/rates are supposed
to be positive. However, in some anomalous circumstan-
ces negative quan tities such as neg ative interest rates can
be cons idered.
0t
The choice 12

 in (1) gives the following
forward prices/rates process:
dd,
tttt
xxvWt0.
model the volatility , is a constant, that is,
(7)
The model (7), (2), (3), (4) can be seen as a stochastic
volatility version of the CIR model with no drift. The
CIR model is a short term interest rate model introduced
by Cox, Ingersoll and Ross (CIR) in [2]. In the CIR
0t
vv
t
v, 0t

, 0t. Note atmodel (7), (2), (3), (4)
CIR model (with no drift) when 0
th the
reduces to the


.
When 0

 the volatility is governed by (2). I
SABR l (7), (2) when the initial conditions (3), (4)
are positive (with probability one) negative forward
prices/rates can be avoided.
Finally, the choice 1
n the
mode


in (1) produces:
dd,0.xxvWt
tttt

 (8)
(8), (2), (3), (4) is knowThe m
SA odel n as lognormal BR model. It is a stochastic volatility version of the
Black model. The Black model is a special case of the
Black-Scholes model [3] obtained when the drift para-
meter of the Black-Scholes model is equal to zero. In the
Black model the underlying asset price is modeled as a
geometric Brownian motion. Unlike in the Black model,
where the volatility is a constant, in the lognormal SABR
model the volatility is a stochastic process itself (see (2)).
Note that model (8), (2), (3), (4) reduces to the Black
model when 0


. In the lognormal SABR model the
positivity (witbability one) of the forward prices/
rates t x is guaranteed for 0t when the initial condi-
tions (3, (4) are po sitive (wobability o ne). In parti-
cular when the initial conditions (3), (4) are positive
(with probability one) the ab solute value in (8 ) can be re-
moved.
The c
h
hoice m
pro
)ith pr
ade in this paper of studying t
ic
entrate on the study of t
he log-
es/rates
he log-
no
ran
no
rmal SABR model is motivated by the fact that the
lognormal model is the most used SABR model in the
practice of the financial markets. Moreover, after the
normal SABR model (that has been studied in [4]) the
lognormal SABR model is mathematically the simplest
model in the class of the SABR models (1)-(4).
Note that in the SABR model the forward pr
dom variable is represented as a compound random
variable and that the SABR model can be seen as a sto-
chastic state space model [5]. Compound random vari-
ables and state space models are widely used in science
and engineering. This means that the methods and the
results presented here to study the lognormal SABR
model can be extended outside mathematical finance to a
wide class of problems.
In this paper we conc
rmal SABR model (8), (2), (3), (4), i.e., in (1) we
choose 1


, and we study the calibration problem for
this modat is, we study the problem of determining
the unknown parameters
el. Th

,

, 0 v
 of the lognormal
SABR model starting from the owldge of a set of data.
The sets of data considered are: i) the set of the forward
prices/rates observed at a given time on multiple inde-
pendent trajectories of the lognormal SABR model, ii)
the set of the forward prices/rates observed on a discrete
set of known time values along a single trajectory of the
kn e

L. FATONE ET AL. 347
lognormal SABR model. The formulation of the cali-
bration problems corresponding to these two sets of data
is based on some new closed form formulae for the mo-
ments of the logarithm of the forward prices/rates vari-
able. Using these formulae the calibration problems con-
sidered are formulated as constrained nonlinear least-
squares problems. The moments formulae are deduced
extending to the lognormal SABR model a method in-
troduced in [4] in the study of the normal SABR model.
Note that the data set used in the first calibratio n prob-
le
proach to study the calibration prob-
le
least-squares
pr
hat, extending the results presented in [4], it is
po
significance levels in th is paper.
of the moments of the
lo
f the
Lognormal SABR Model
oments of
the es of the lognormal
m, that is, a data sample made of observations at a
given time on multiple trajectories, is hardly available in
the financial markets. In fact, in the financial markets
usually it is not possible to repeat the “experiment” as
done routinely in contexts where observations are made
in experiments carried out in a laboratory. This implies
that the first calibration problem although realistic in se-
veral fields of science and engineering has limited appli-
cations in finance. Instead, the second calibration prob-
lem is of practical use in finance since single trajectory
data samples are easily available in the financial markets
and can be identified with time series of observed for-
ward prices/rates.
An alternative ap
m for the lognormal SABR model correspo nding to the
single trajectory data sample consists in extending the
method proposed in [6,7] to study a similar calibration
problem for the Heston model and for some of its varia-
tions. This method is based on the idea of maximizing a
likelihood function. However, the use of closed form
moment formulae (see Formulae (65)-(68)) rather than
the use of a likelihood function involving the transition
probability density fun ction of the differential model and
the solution of a kind of Kushner equation (see [6,7])
gives to the method based on the moment formulae pre-
sented here a substantial computational advantage in
comparison to the method suggested in [6], [7]. A similar
statement holds when the method presented here is com-
pared to methods where averages of quantities implicitly
defined by the differential model (such as the moments)
are computed using statistical simulation .
The numerical solution of the nonlinear
oblems that translate the calibration problems consider-
ed can greatly benefit from the availability of a good ini-
tial guess to initialize the optimization algorithm. In Sec-
tion 3 we discuss briefly how to exploit the first moment
formula obtained in Section 2 to build the initial guesses
needed.
Note t
ssible to define ad hoc statistical tests that can be used
to associate a statistical significance level to the parame-
ter values obtained as solution of the calibration prob-
lems. We do not consider statistical tests and statistical
The remainder of the paper is organized as follows. In
Section 2, new formulae for some
garithm of the forward prices/rates variable of the log-
normal SABR model are derived. In Section 3, the cali-
bration problems for the lognormal SABR model corre-
sponding to the two data sets discussed previously are
formulated as constrained nonlinear least-squares prob-
lems. Finally, in Section 4 we solve numerically the cali-
bration problems presented in Section 3 and we discuss
the results obtained in numerical experiments on synthe-
tic and on real data. The real data studied are time series
of euro/U.S. dollar exchange rates.
2. Formulae for the Moments o
Let us deduce closed form formulae for the m
logarithm of forward prices/rat
SABR model. Let us consider the model given by:
dd,0,
tttt
xxvWt

 (9)
dd,
ttt
vvQt0,


 (10)
with ositive initial conditions (3)
assumption (5). That is, we assume
p, (4) and the
0x
 (with
0
probability one) and 00v
 (with probability one), so
that Equations (9), (10) imply 0
t
x (witability
one), and 0
t
v (withbility one) for 0t.
Let
h prob
proba


ln
tt
x
 , 0t, belogarithm of the
forward prtes. Using the variables t
the
ices/ra

, t
e storeEquations (9), (10) and the
initial conditions (3), (4) are rewritten as foow
v, 0t,
th ll s: chastic diffential
2
1
ddd,0,
2
tttt
vtvWt 

  (11)
0,dd,
ttt
vvQt


 (12)

00 0
ln ,
x
 
 (13)
Starting from the expression
transition robability density fun
pr
00
.vv (14)
obtained in [8] for the
pction of the stochastic
ocesses t

, t v, 0t, we deduce explicit formulae
(eventually involving integrals) for the moments with
respect to zero of t

, t v, 0t, and of t
x
, t v, 0t.
In particular, we derive closed form formulae (that do not
involve integrals) f the momenw rt
to zero of t
ore first fivts ithespec

, 0t.
In [8], using the backward Kolmogorov Equation
associated to (11), (12), the following formula for the
transition probability density function
L
p of the sto-
chastic processes ,
tt
v

, 0t, implicitly defined by
(11)-(14) has been obtained:

L. FATONE ET AL. 348



 
,,,,,
1
L
pvtvt 


de , ,,, ,,
2
,, ,,,0,0,
kL
kg
ttkvv
vv tttt











 
 


(15)
where is the imaginary unit and we have  t



,
t t
vv,



, ,
t
vv

,0 tt

, 0tt

. In (15),
when 0 we must choose 0
t


, 0
vv

n. The functio
L
g
is gi

ven by:

 




2
0
, e
de sinh,
,,,,0, 1,1,
k
vvv
2
8
2
2
,, ,,e
vv
s
Lv
gskv
K
kvKkv
skvv




  

 














 

 

 (16)
where the function
K

of denotes the second type modi- fied Bessel functionorder

(see [9] p. 5). Finally,

2k
 , ,k is defined as


22
2
1
kk
k

 
2
2
i,.k 

 (17)
The function
L
g
can be rewritten as follows [8]:









2
222
22
22
22
2
81
,,,,ee
k
Lv
gskvv vv






22
2
0
2
1
2c osh2
22
0
,
ed sinhsine
2
de ee,
,,,,0,1,1.
vv
s
sus
vv k
vv kyv vvvuvv
yy
u
uus
s
y
sk
vv










 


 
 








 



 
(18)
Formulae (15), (16) or (15), (18) give
L
p
ilit
as a two
dimensional integral of an explicitly known integrand.
Note that in (15) the transition probaby density
function
L
p is written using the variables


ln
tt
x
 ,
t
v, 0t. It is easy to obtain a formula analogous to
formula (15) for the transition probabil
ritten in the original variables t
ity density
function w
x
, t v, 0t.
Formulae (15), (16) and (15), (18) are representation
formulae for
L
p that hold when

  . These 1,1 for-
mulae have been obtained in [8]. Previously when
0

 for
L
pnly series expanwers of osions in po

with base point 0

 were known (see, for example,
] and t references therein).
Let us begin deg some formulae for the moments
,nm
, ,0,1,,nm with respect to zer
[10,11he rivin o of the vari-
ables t
x
, t v, 0t, of the lognor mal SABR model (9),
3), (
 
,,,,ded,,,,, ,
nm
nm tvtvvpvtvt

 
 




(10), (4), namely,
0
,,
,0,0,,0,1,
L
vt
tttmn


 
 

0n

.
(19)
We distinguish two cases, that is: the case and
the case
When 0n.
0n

we have:

,

 
0, 0
,,,d d,,,,
mL
tvtvvp vtv 

 
  


0
,
,,,,,
d,0,,,,,
,,,0,0,0,1,
m
L
mL
t
k ttkvv
vvgttvv
vttttm




 




0
de d
km
kvvg


 









 
 




 (20)
where

is the Dirac’s delta. From (18) we have:




2
2
0
2
0
2
32 cosh
22
00
d,0,,,,
2
ddeee,
,,0,1,1,0,1,.
mL
vv
yy
myu
vv
vvgsv v
s
s
vv y
sv m








 





 

 



(21)
Using formula (29) on pa ge 1 46 of [12] we have:
2
22
22
8e
ed sinhsine
ssus
vu
uu

  




 
12
32 22 12
0dee2 ,
,,,vy



vv
yy
vv
vvvK y





 




and this implies that:
(22)



2
2
2
22
2
8
0, 2
2
2
0
cosh
12
0
2e
,,,e 2
d sinhsine
d()e,
,,,0,
0,0,1,.
s
ms
m
us
yu
m
v
tvt s
u
uus
yK y
vtt
tt m








 
 










   
  





(
From Formula (24) on page 197 of [12] it follows that:
23)
 

cosh
1/2
0
sinh 2
de ,
sinh
yu u
yK yu
u
 
 

. (2
on page 92 of [12] we can conclude
4)
From Formulaes (23) and (24) and using formula (37)
that


0,0 ,,, 1tvt



 ,, 0,vttt


0 t

 , ,




and that


,tv
0,1 ,,tv


 

, ,


,, 0,0tttt

v



. Note that the moments
0, ,2,3,, mm

 diverge. In fact, when 0n

integrating (20) first with respect to k when k

,
and then with respect to

when

we have: ,

L. FATONE ET AL. 349

 


22
2
22
0,
2
82
2
0
2
12
22
0
e
e
2ed sinhsine
2
1
d,,,,0,0,0,1,.
2cosh
m
u
sss
m
tk g
vv u
uus
s
vvvt tttm
vv vvu







 
 

 
 
 






(25)
Formula (25) is equivalent to Formula (23) and shows
at in order to have a convergent integral we must choo-
0
1
,,,d dd,,,,,
2
k m
vt vv ttkvv

 
 

 
 

 
L
When using Formulaes (15) and (18) in the de-
finition (19) we have:
0n
th
se 12 32m .









2
2
222
,0
22 22
8
22
0
2
22
22
1
12
0
1
,,,de dde,,,,,
2
1
ee
2ed sinhsine
2
12cosh
de
k nm
nm L
s
nsus
nvv
m
tvtvvk gttkvv
nn
vv u
uus
s
nn
Kvvvv
vv
 



 





 

 

 
 


 



 

 
 



 




 





u
22 ,
2cosh
,,,0,0,1,2,,0,1,.
vv vvu
vttttnm 


 
   
(26)
Note that for when with respect to zero of the variables t

11,2,,n

11
11nn    or

11
11nn   the
integrals appearing in th n
and the conding mothe moments
logno
(26) wi 1 1, 2,,,n
ments are 0,1,,m rrespocon-
vergent. A similar existence condition for
of thermal SABR model has already been derived
in [13], however in [13] no explicit integral repre-
sentation formula for the moments like Formula (26) is
given.
Note that when 1n and 0m formula (??) gives

1,0 ,,, etvt




, ,,,0,0vtttt 

 
   ,
wh
,
defined by (11)-(14), i.e.

.
t
v, 0t,
 
,,,,dd
nm
nm tvtvvp

 
 0,,,,,,
,,,0,0,,0,1,
Lvtvt
vttttmn






   
 
(2)
The procedure used here to calculate the moments (27)
generalizes the procedure used in [4] to calculate
m
7
the
oments with respect to zero of the variables t
x
, t v,
0t, of the normal SABR model.
Substituting (15) into (27) and using the propeiesf
ourier transform we have:
en

1,1   . Recall that ex


,

.
Let us consider now the mo,,0,1, ,nm ments rt o
the F ,nm


 
  
 
,0
0
0
0
0
1d
d d,,
j
m
vv kkgttk,,,, ,
d
d
d,,,,,
d
,,,0,0,,0,1,.
nnj
nm L
jj
j
j
nnjjm
Lk
j
j
n
tvt vv
jk
nvvgttkv v
jk
vttttnm
j


















 
 


 
 





 (28)
Let us calculate the integrals contained in (28). For
let
 

0,1,j
j
G be the-th order derivative with j
respect to k of the function
L
g
evaluated at 0k

,

that is, let


,,d d,0,,
jj
jL
Gsvvgks vv

,
,,svv


the previ. To simplify the notation we have omitted in ous formula, and we will omit from now on, the
dependence of
L
g
and of
j
G0,1,j, fro, m

and

. Using the functions
j
G, 0,1,j, Formula

L. FATONE ET AL. 350
(28) can be
.

We restrict our attention to
rewritten as follows:

 
,
0
0
,,,
d,,
nm
nnjj
j
tvt
nvv G
j






  





 (29)

,
,,,0,0,,0,1,
m
jttv v
vttttmn 


 
  

,0 ,,, ntvt


,

,




,v

3 in th
n
, 1n
,0, 0,ttt tn


e solution of the ca
,2,3,4 . The choice
0,1,. In fact, in
libration problems
of considering
Section
we will use
0m
,0

in
lems isthe
due to
moments used to solve theob
arkets the variable
t
calibration pr
financial m the fact that in the
x
, 0t, and, as a consequence, the variable t

,
0, can beserved, while the variable t v0,
cannot be observed. That is, the moments ,nm
,
0,1,n, 0m, cannot be easily estimated from
eata, while the moments ,0n
, 0,1,n, c
timated immediately from observed data.
Let us define
 
 
*,0
,, ,,,
,,
nn
nnjj
sv tvt
nDsv



 


t
obs rved
be es
ob
d
, t
an
0
,, ,0,1,
j
jj
sttvn 


.











(30)
where


0
0
0d



,d,,
d,,,,,,0,1,,
d
jj
j
Lk
j
DsvvGsvv
vgskvv sv j
k









(31)
and
 
0
d
,,d,,,,
d
,, ,0,1,.
j
jL
j
Dskv vgskvv
k
skv j


 





(32)
We have
 

0
00
,,,
dd,,, ,
d
,,0,1,.
jj
k
j
L
j
k
DsvDskv
vgskv v
k
sv j







 





(33)
The functions
j
D,
tial value 0,1,j problem
, can be deter
solving the inis deduced belo
function
mined by
w. The
L
g
(see [8]
is the solution of the following initial value
problem):
2 22




0,, ,,, ,.
L
gkvvvvk vv 

 
  (35)
Equation (34) is the Fourier transform of the b
Kolmogorov equation of th e lognor mal SABR model and
the parameter
ackward
k


gate variable in the Fourie
that appears in (34) is the con-
ju r transform of the variable




.
Integrating both sides of (34), (35) with respect to v
when v

 we have:
2 22
222
00 0
0
2
22
DD D k
vvDkv
22
,,
,,
2
2
2
2
22
1
L
LL
L
L
g
gg
k
vvgkv
s
v v
kv

 


g skv



 




(34)



20
1,,
,,
2
v
v

s
kv D skv
















(36)


00,,1,,.Dkvkv 

 

(37)
When 0k

, Equations (36), (37) reduce to:
2
22
00
2,,,
2
DD
vskv
sv







,  (38)


00, 1,.Dv v

  (39)
Equations (36)-(39) define initial value problems satis-
fied by and by, respectively. Recall tha
is given in (33). It is easy
see t
0
D

lation between
hat
0 D
and t the re-
0
D0 D

to


Dsv

0,1

,
this ,sv solutio

,
n usis ing a
la
a solution of (38),
(39). Let us obtain procedure that
ter will be extended to deduce explicit formulae for the
functions
j
D when 0.
Defining Lj
0,

and0 L
 through the relations:




00
,Dsv,L vvs


, , ,sv



ln v 

,
v

 (i.e.


ev




,

) and






00
,,Ls Lsv




, ,s



, problem (38),
(39) can be rewritten as follows:
2
s



)
22
00
0
2,,,
28
LL
Ls



 

 (40


2
00,e.L
  (41) ,




The solution of (40), (41) is given by:
 
2
00
,d,e,,, (42) Lss s

 
 

 



whre e


22
22
8
01
,ee,
s
s
ss





2,.
2s





 (43)
ntegral (42) when The i

212qj and 0j

can be computed using the formula


2
222
0
418
82
ee e ,
,,.
sq sq
sq
 




 
)
d,e
e e
q
qsq
s



 


 

(44
It follows that the solution of problem (40), (41)
0
L


L. FATONE ET AL. 351
is given by:




,,,,,1,
jj
DsvvLsvsvj

 
 2,, (51)

22
28 82
0,eee e, ,
ss
Ls s
 

 


.   (45)
From (45) it follows that
and considering the change of variable


ln v 

,
v

, define the following functions







,, ,
jj j
Lsv LseLs




, , s



,
1, 2,j.


 


00
ln
,,l
v
DsvvLs v


 

(46)
2
n
e1,,.vsv



Let us derive the initial value problems th
sa
It is easy to see that (47), (48) imply that the function
, is the solution of the following initial value problem:
1
L

at are
tisfied by the functions
 
0
,,,
jj
DsvDskvk


2
22 0
2
11
1
2
32 0
e
28
1e, ,
2
LL D
L
s
Dsv




,



 










(52)
,
erentiating times with
(34), (35) and sutin
,s
respect to
,v
 1, 2,j. Diffj
bstituk problemg 0k

in the resulting equation 1j

the
s, we have that when
function

1,Dsv
, ,sv

, satisfies the following
initial value problem:


10, 0,,L
 
 (53)
2
222
1
0
2,, ,
22
DD
vDsv
sv

1
v



 
 (47)

10, 0,,Dvv 

 (48)
and when 2,3,j thtion

,Dsv
,
,,sv 
 satisfies the
where




00
,,e1Ds Ds 


, , s 


.
Moreover, from (49), (50) it follows that the functions
j
L
, 2,3, ,j

 satisfy the initial value problems:


e func
following ine problem:

2
22 32 2
2
1
232
1
1e
282
ee
2
,,2,3,,
jj
,
j
j
jj
LL j
Lj D
s
Dj
jD
sv j




 









 



 (54)
j
itial valu

2
22
22
2
1
22
1
=1
22
,
jj
j
2
,,
2,3,,
jj
DD
j
vjvD
svD
jvD














 (49)
(50)
Let us assume that
jv
v
sv j




 


0,0,,2,3, ,
j
Lj  
 (55)
where




,,
jj
Ds Ds

0,0,,2,3, .
j
Dv vj


   e, , s


 


,
2,3,j

.
The solution of (52)-(55) can be written as follows:
 
   
0
0
32232 1
,dd ,
1ee,e,,
22
,,1,2,,
s
j
j
Ls s
jj
jDD D
sj

 
 









 











(56)
where we have defin ed
21
(, )
jj
j






1,0Ds

, s
,


.
Let us give the explicit expressio ns of
 

,,ln,,
jj
DsvvLsv sv
 



*00
,, ,
jtv


 ,t
 when 1,2,
, and of
. Recall that
3, 4j
  
,=Dsv ,l
jj
vLs

n ,v
 and
that
, ,sv

 j1,2,,


0,1Dsv


,
we have:

,.sv 
 Using Formulae (44), (56)

22
,e 1e,,,
ss
vsv





  


 (57)
2
11
2
Dsv







2
22 2
22
2
2
23
2
56
46
1e 1e
,e1e 23
1e
1e,,,
256
s
s s
ss
s
Ds v
vsv

 







2
3
3
2e
1e
s
s
vv








 




 













 


 
 




(58)

L. FATONE ET AL. 352

22 22
2 2
22
2
34
23 356
326
3
456
6
1e1e1e 1e1e
,6 e18e
2318 1018
1e 1e
3e 56
ss ss
ss
ss
s
vv
Dsv
v
 






 

2
s 


 

 
 



    
 
 
 




  





 
 
 


222 2
10
es
 



2
22 2
2
5479
10
691415
15
1e 1e 1e 1
3e 60421015
11e1e1e
e3
4907030
sss
s
sss
s
v
v












 
  

 
  

 

 

 

 
  
  

,, ,sv





(59)
and finally
,
 
4123
,,,,,,D svI svIsvIsvsv



1
I
, 2
I
and 3
I
(60)
where are the fol lowing function s:

 
2
222
222 2
2
2
2
45
26 36
1
547910
310
54
10
1e
,4e 1e91e
5
1e 1e 11e
42 s
 
 

  
 
e
4e4
5755
1e
4e
s
ss s
sss
s
s
s
v
Isv
v
v

 
















 
 
 





 
 
 
 






22
22222
2
910
65 9121415
215
72
1e1e
4
10155
1e 1e1e1e1e
2e3532
70 18841415
e
ss
sssss
s
v
v

 




 

 


  
 

 



 
 
 

 
 






22 22
26152021
11e1e1e1e
9,,,
630 225700105
ss ss
ssv
 

 

 
 

   
 

 


(61)

22 22
2 2
22
2
45
56 7910
610
2
6914
15
1e1e1e1e 1e
,6e6 e
56 21915
31e1e
e
10 27
ss ss
ss
s
s
vv
Isv
v







 


 

 
 

   
 
 
 

 




 
 
 
2
s 








2
15
1e ,, ,
79
ss
sv
 




 





(62)
and

22 2
2
22 22
2
57910
10
3
6912 1415
215
1e 1e 1e
,12 e351830
1e 1e1e1e
2e 9
45 127015
ss s
s
ss ss
s
v
Isv
v


 




 





 
  
 




 
 
 
 
 

222
2
2222
2
691415
15
711 1518202
21
1e 1e1e
6e 270 7090
1e 1e1e1e1e
e33 2
770 150126100
sss
s
sss s
s
v
v


 



 
 


  
 
 
   
   
 
  


 
 
 
 
 

2
2222
2
1
81322 2728
28
105
11e1e1e1e
e,
20819 1986384
s
ssss
s
vsv




 











 
  
 
 


 ,. 
(63)

L. FATONE ET AL. 353
Recall that
s
tt


, 0
. From (30), (46), (57)-(60),
choosing 0t


, 0
vv


 we have that
s
t

of t and threspect to zero e first five moments with

,
t , are
(64)
,

(65)
,

(66)

6


(68)
The momentsdepend on the pa-
rameters of the logd el
given by:


*
000 0000
,,, 1,,,,tv Dtvtv 






 
*
100 00010
00
,,,, ,
,,
tvDtvDtv
tv











 
 
*2
200000010
01 020
00
,,, 2,
3, ,
,,,
tv DtvDtv
Dtv Dtv
tv
 






 






 
*3 2
300 000010
02 0
,,,3,
3,
tv DtvDtv
Dtv
 



 



 
3 0
00
,,
,, ,
Dtv
tv 

 



( 7)

 
 

*4 4
400 000010
2
020030
400 0
,,, 4,
6,4,
,,, ,
tv DtvDtv
Dtv Dtv
Dtv tv
 






 
 







* ** *
.


1 ,
nor 2
,
ma 3
,
l SABR m
4

o

,

,
s on0
v

and on the time depe t. In particular, *
1
Lnd

, 0 v

and , while on
t*
2
, 
*
3, 
*
4 depend

, 0, v


and
t. T mo me ohent *
0
 does not dependn

, 0 v
,

, t
ems
for
new
lve
alo-
for
in
and
fo
and
th
go
m
ca
r the l
o
are
e cal
fo
course as
re an
nnot cali
ogno rm
the mme m
closed formun
the fnext
ibratiod pr
us rmuleast
all the remadefi
o d that, as al
Sectiorm formulae of
t
l 8oe
nc
be
rm
nts of t
mo
orm
n
ae can
inin
nm
o
use
pr
re in
d i
al SABR
he lo
me
ul
obl
be
g m
incr
vo
n
nt fo
the
m
gnorm
rm
ae used
ems di
ded
omen
lved.
stud
odel. F
ul
in t
ced
ts 
Note
y
h
scusse
of
o
al SABR
ae anno
e
(at l
,nm
brat
ulaes (6
ced
sect
evio
in
ne
i
i
on
5)
odel are
in
ons t
usly
prin
n (
,nm

ready sai
probl
-(68
the
o s
. An
ple)
27)
be
)
o
d
Section 1
u ci
d i. Of
come
,
mease the formulae for
n 1, closed foobservable quantities
implicitly defined by (9), (10), (3), (4) such as (65)-(68)
are very useful to build computationally efficien me-
thods to solve calibration problems.
In [4], Formuaeanalogous to (65)-(6) fr th mo-
ments of the forward prices/rates variable of the normal
SABR model (6), (2), (3), (4) are derived and used to
solve calibration problems.
3. Two Calibration Problems for the
Lognormal SABR Model
Let us study the calibration problems of the lognormal
SABR model (11)-(14) annoued in Section 1. Recall
that the parameters

,

, e unknowns and are th
that we want to determine these parameters starting from
the knowledge of a set of data. We consider the sets of
data specified previously. The corresponding calibration
problems are formulated using the closed form Formulaes
(65)-(68) for the moments of the logarithm of the for-
ward prices/rates variable and are solved numerically.
Let us begin formulating the first calibration problem.
Let be given. We consider multiple
traj f the lognormal SABR model (11), (12) as-
soitial conditions (13), (14) assigned at
time
0
v

0T
ectories o
ciated to the in
0t
independent

. The set of data of the first calibration prob-
leme set of the logarithms of the forward prices/
d at time
is th
rates observetT

in this set of trajectories.
In particular, letting be a positive integer, we
consider independenpies
n
t co ni
T

, of
variable 1,2,,,in the rando T m

solution at time tT

of
(11)-(14). For 1,2,,in

 let ˆi
T

be a realization of
i
T

. The set:


1ˆ,1,2,,,
i
Tin
  (69)
is the data sample used in the following calibration
pr
the data set defined in
(69), recues of the paraers
oblem:
Calibration problem 1: multiple trajectories calibra-
tion problem.
Given 0T, 0n and 1 
metonstruct the val

,

e
and 0 v
 of the lognormal SABR model (11)-(14
To solve this calibration problem we com
).
pare th
theoretical values of the four moments *
j
, 1,2, 3,4,j

given by (65)-(68) with the estimates of these moments
obtained from the sample 1
 of the observed data.
It is easy to see that the random variables:


1
1
,, 1,2,3,4,
nj
i
jT
i
nT j
n



 (70)
are unbiased estimators of, respectively, *
j
,
1, 2,3, 4j

. For 1,2,3,4j

let us consider the
realization


ˆ,,
jnT in the data sample 1
, of the
random variable


,
jnT, that is:


1
1ˆ
ˆ,,1,2,3,4.
nj
i
jT
i
nT j
n


 
 (71)
The unknown parameters

,

, 0 v
 of the normal
SABR model can be determined as solutions of the fol-
lowing constrained nonlinr least-squares problem:



0
4*00
,, 1
ˆ
min,,,
jj j
vj
Tv nT







72)
sbject to the const
ea
(
uraints:
2,
0
0, 11,0,v

 
 (73)
where
j

, 1,2,3,4,j

are non negative weights that
will be chosen in Section 4.

L. FATONE ET AL. 354
Note that roughly speaking when increases the
“q n
uality” of the moments

ˆ,,
jnT 1,2,3,4,j

estimated from the data samp
mer to lve sib
rained
no n values
the
m construct
good initial guesses of problem (72), (73) we take a
closer look to the explicit expression of
Formulae (65)-(68). In particular, we consider the
(fula (65))
when be su
withder
appr
le increases and this should
ake easisoatisfactorily the Calration prob-
lem 1.
The numerical experience shows that the const
nlinear least-squares problem (72), (73) has many
local minimizers with similar objective functio.
This means that the solution of problem (72), (73) is
sensitive to the choice of the initial guess of
inimization procedure used to solve it. To
the moment
asymptotic expansion of the moment *
1
orm
0. Let ch that 12 0TT, we
approximate the first- and the send-order
Taylor's expansions e point t
2t
 
*
1

oximation o
12
,TT
of bas
*
1 when
co
The first-or 0.
f 1 tT

0
v

is us
. Th
ed to obtain th
e seconde initial g
appr
uess fo
oximation or th
1

e param
* wheneter
2 tT -order f

is used to obtain the
initial guess for the parameter

. To build an initial
guess of the parameter

it is necessary to use
higher-order moments. We prefer to exploit the fact that
11


nt formul
lution of p
and th
r
at th
akes co
lem (72)
e av
i
ailab
mput
, (73). Th
com
al gu
ility o
onally
is m
putationa
f the exp
very
eans that, wh
l cost, it is
for the param
licit
en
eter
mome
the soae m ob ati
esses
efficient
necessary, at an affordable
possible to use multiple init

.
The defined in (69) used in Calibra-
tion problem 1 to formulate problem (72), (73) must be
co
data sample 1 
mpleted with the auxiliary data 1
ˆi
T

, 2
ˆi
T

,
1, 2,,,in (observed at tim e
1
tTand2 tT

)
needed to build the initial guess of the minimization
procedure used to solve problem (72), (73). For sim-
plicity, it is possible to choose  or as it is
do 1
TT 2 TT
ne in the numerical example discussed in Section 4. In
this case, the data contained in (69) are used both to
formulate the nonlinear least-squares problem (see (72),
(73)) and to obtain the initial guess for 0 v
 (when
1
TT) or, the initial guess for

(when 2 TT).
This set of data is realistic in several contexts of
science and engineering where, for example, the obser-
vations are obtained in experiments done in a laboratory.
In fact, repeated experiments are a routine work in a
laboratory. However, most of the times this is not re-
alistic for observations made in the financial markets
where usually it is not possible to repeat the “experi-
ment”. That is, in the financial markets repeated obser-
vations at a given time 0t of independent realizations
of the forward prices/rates random variable t

are
usually not available. This is a serious concern which
implies that the Calibration pm 1 is of limited in-
terest in finance.
The second calibration problem for the lognormal
SABR model (11) -(14) overcomes this difficulty. In fact,
ample considered in the second calibration
problem is the set of the logarithms of the forward pri-
ces/rates observed on a discrete set of known time values
along a single trajectory of the lognormal SABR model.
This data sample is easily available in the financial mar-
ts.
roble
the data s
ke It can be identified with a time series of the log-
arird thms of the forwaprices/rates observed in the finan-
cial market.
Going into details, let
M
be a positive inert
01
teg and le
,, ,
M
tt t be 1
M

discrete timues sut
1, ii
tt e valch tha

 1,2,,,iM

 and 00t. Recall that the
times i t, 0,1, ,iM

, are known. The data of the
second calibration problem are arith the for-
ward prices/rates observed at the times 01 ,, , .
the logms of
M
tt t For
1, 2,,iM

 let us denote by ˆi

the logarithm of the
forward prices/rates observed at time i tt along one
trajectory of the stochastic process ,0.
tt

 T set: he


2ˆ,1,2,, , iiM
  (74)
is the data sample used in the following calibration
problem:
Calibration problem 2: single trajectory calibration
problem.
Given 0M, 1
M

discrete time values
01
,, , ,
M
tt t such that 1,
ii
tt

 1,2,,,iM and
00,t

and given the data set  defined in (74),
determine the values of the parameters
2

,

and 0 v

of the lognormal SABR model (11)-(14).
The Calibration problem 2 can be formulated as the
following constrained nonlinear least-sq uares problem:

0
2
4*
,00
,, 11
ˆ
min( ,,),
Mj
ijjii
vij
tv










 (75)
subject to the constraints (73). The constants ,ij

,
1, 2,,,iM

 1, 2, 3, 4,j

in (75) are non negative
weights that will be chosen in Section 4. Note that when
M
increases the “quality” of the terms


ˆ
j
i

,
1, 2,,,iM

 1, 2, 3,4,j

does not increase, it is only
the number of addenda of (75) that increases. For this
reason we expect Calibration problem 2 to be more
di pro
The numl exp
shows t the behavioe con
qres problem
r the num
optimization or
ent form
in Section 2. In particular, we use the first- and the
fficult than Calibration blem 1.
ericaerience with problem (75), (73)
hatur of thstrained nonlinear
least-sua (75), (73) is similar to the
behaviour of problem (72), (73). This implies that the
availability of a good initial guess foerical
algorithm used to solve (75), (73) is very
helpful to obtain a satisfactory solu tion. Inder to build
this initial guess we exploit the momulae deriv ed

L. FATONE ET AL. 355
second-order Taylor’s approximations of
with base point . From the first-
of th (i.e. ng thervations
*
1
 (see (65))
order Taylor's 0t
e traject
approximation at 0t of *
1
 evaluated at the
beginning
ˆory usie obs
i

with i small, that is 1,2,,10i in the numerical
example of
s Sectioess of econd n 4) -or we obtain Tay the in lor's itial gu0 om the der approximation at 0tv
.
Fr

ated at the esing
servations ˆi
of
the
*
1

obevalu (i.e.nd of te trajecthory u

with i close to
M
, that is
91,92, ,100i in the numerical examp oection
we obtain the initial guess of lef S4)

. Sometimes also the
first-order Taylor’s approximation of *
1
d
dt
 with base
point is used to construct the initial g
num imization algorithm used to solv
at
0t
erical opt
1
uess of the
e (75), (73).
In this last case the first-order Taylor’s approximation at
0t of *  is used to obtain the initial guess of the
parameter 0 v  and the first-order Taylor’s approximation
0 of t*
1
d
d
 is used to obtain the initial guf
the parameter tess o

. These approximations are evaluated at
the beginning of the trajectory. As explained more in
detail in Section 4, in financial applications a priori
information about

is available. That is, due to the
financial meaning of the variables, we must expect
0

. In Section 4 we exploit this information to
choose an initial guess for the parameter

.
4. Some Numerical Experiments
In this section we discuss three numerical experiments.
In the first numerical experiment we solve the Calibra-
tion problem 1 using synthetic data. In the second and
third numerical experiments we solve the Calibration
problem 2 using, ctively, synthetic and real data.
The real data studied the da belonging to a time
series of exchange rates between currencies (euro/U.S.
llar excange rates).
The numerical experiments presented in this section
can be “interpreted” as follows. As already said, the
numical experiment can be seen as a “physical experi-
ment” done in the context of a scientific laboratory where
make repeated observations of the same
quantity. This type of experimen usually is based on a
respe
are ta
do h
first
er
it is possible to tgno
l ha
al m
the no
more accurate value of the parameters can be obtained
increasing the numerousness of the data sample used in
ibn probl
eriman b
pring and to hedge
y important to
eters. In some
setting of the first numerical
ex
“physical model” (i.e. in this case the lormal SABR
model) where the parameters of the modeve a precise
physiceaning (i.e. they are masses, charges, ...). In
these circumstances the main scope of a calibration
problem (such as Calibration problem 1) is to determine
umerical values of these parameters in the best pos-
sible way. Nte that in this kind of experiments usually a
the calratioem. The second and third numerical
expents ce seen as experiments in finance or in a
different context where it is not possible to make re-
peated observations of the same quantity. Note that in
mathematical finance the model and its parameters are
mainly an auxiliary tool. In fact, the model is simply an
instrument to interpret the data or to forecast future data.
In the practice of the financial markets the calibrated
financial models (such as the calibrated logno rmal SABR
model) are used to do option’sic
portfolios. In these contexts it is not reall
know the exact values of the model param
sense even the existence of “exact” values for the model
parameters can be debated. In mathematical finance the
key fact is to show that the calibrated model is able to
interpret the observations, that is, for example, to show
the consistency of the option prices computed using the
calibrated model with the option prices observed in the
market.
Let us describe the
periment. Let 0T be given and n, m be positive
integers. Let tTm

 be the time increment and
i
tit

, 0,1,, ,im

 be a discrete set of equispaced
time values. Let m
tT



, m
tT
vv be the solutions of
(11)-(14) at time tT

. The n independent
realizations ˆi
T

, 1,2,,,in

 of the random variable
T

used as data in Calibration problem 1 are ap-
proximated integrating numerically n times (in cor-
respondence of different realizations of the Wiener
processes) the lognormal SABR model (11)-(14) in the
time interval


0,T using the explicit Euler method (see
[14]).
In the numerical example we choose 1T, 100m

,
1000n

, 0.1


, 0.2

 , 00 1  
 and
00
0.5vv


. The parameters




0
,,0.1,0.2,0.5 ,v    (76)
are the “true” values of the unknowns of the calibration
problem considered (i.e. they are the values of the
unknowns used to generate the data). We reconstruct
these unknown parameters solving Calibration prob lem 1
using as data sample the set of the logarithms of the
forward prices/rates observed at time 1tT

 in
1000n

independent trajectories of the lognormal
SABR model (11)-(14) (with the parameter values given
in (76) and 00 1 


). These trajectories are ap-
proximated integrating numerically using the explicit
Euler method the model (11)-(14). In particular, when
1000n

let us denote by 1
ˆ
ˆ,
i
T

 1, 2,,,in the
approximations of 1
ˆ,1,2,,,
i
Tin

 obtained at time
1tT

 integrating with the explicit Euler method
1000n

independent trajectories of the model (11)-(14)
(with the parameter values given in (76) and
00
1 


). The set:

L. FATONE ET AL. 356

1,10001,1,2,,1000,
i
TnT i

 
 (77)
is the sample oftic data used to solve Calibration
problem 1.
In a similar way when we choose 100T, 1000n
ˆ
ˆˆ
synthe

(leaving the other parats unchanged) we generate
the data set

100, 1000100
ˆ,1,2, ,1000
i
TnTi  . As
explained in Section 3, this second datasample is used in
the construction of the initial guess of the numerical
optimization algorithm used to solve the constrained
nonlinear leasres problem (72), (73) corresponding
to the data samˆ
meer
t-squa
ple ed in th
numerical
Using and the first- and
second-oor’s ap
point ugg io

ˆ
ˆ
 
1, 1000Tn
.
The data sets 1,1000
ˆTn
 and 100,
ˆTn
use
experiare avai
1, 1000
ˆn and 100, ˆT

1000
lable at [15]. ment
T

rder Tayl
0, as s
1000n
proximations o
ested in Sectf 1
 with base
n 3, we find
*
t
00.515
in
v
 and 0.099
in
  as initial guesses of,
respectively, 0 v  and

. Note that in the notation of
Section 3 we have chosen 11TT and 2100T

.
The initial guess in

of

is chosen as 0.05
in
  .
Given


0
,,0.099,0.05,0.515
inin in
v  
 as initial
guess, the nonlinear least-squares problem ) is
sing 1, 1000
ˆTn
 as data sample. In this
numerical example the moments considered in (72) are
all of the same orgnitude so that it is possible to
choose in (72) the weights 1
j
(7
der of ma
2), (73
solved u

, 1, 2,3, 4j. Note
that Formulae (65)-(68) suggest that in general the
weight
j

must decrease whenses.
The nonlinear lelem (72), (73) is
ing the FMINCON routine of Matlab. Th e solu-
d starting from the initial guess

the index j
ast-squares prob increa
solved us
tion foun

(78)
The relative -error of the initial guess
0
,,0.099, 0.05,0.515
inin in
v  
 is:
**
 
00
,,,,0.076,0.222,0.508 .vv   


*
2
L
) is 0


e -error of the
so lem wit
he sensitivity of the solution procedure
respect to the presence of noise in the
0
,,0.099, 0.05,0.515
inin in
v  
 with respect to the
“true” soluti2 on (76.275. The relativL
lution (78) of the least-squares probh respect to
the “true” solution (76) is 0.062.
To study t
proposed with data
we add noise of known statistical properties to the
synthetic data contained in 1, 1000
ˆTn
, 100, 1000
ˆTn
 and
we study the quality of the solutions of Calibration
problem 1 found as a function of the noise properties. In
particular given 10


ple: wing “n
distribu interval
let considuser the follo
oisy” data sam


ˆ
ˆˆ2
i



1,1001
11 ,1,2,,1000,
Tn i  
(79)
where is a random entry taken from a uniform
on the1) . In a similar way we
add noise to 100, 1000 to obtain the “noisy” data sam-
ple 100, 1000
ˆTn
.
Given 10
0 1 Trand 

rand
tion (0,
ˆTn


 comute the relatiwepve -error
bee” solution (76) and the
ibed ab
2
L
tween solution of
Calibration problem 1 found with the numerical
procedure descove. We repeat the entire
procedure 1 N times and we compute the mean of the
2
L-relative errors of the 1 N solutions found. We denote
by
the “tru
r
11
,,
N
n
E

this mean relative erroelative
errors r. The mean r
11
,1000, 1000Nn
E
  obtained when 11000N

,
1000n

1 and 0.01,0.05,0.1


are show in Table 1. n
As already explained, it is expected that in this ex-
periment a more accurate value of the parameters can be
obtained by increasing the amount of the
validat data sample used in the calibration problem. Toe this idea, we
increase the number of the data samples invo
the experiment. Wensider and we con-
st
the correspondingi
n
co
ple
“no
lved in
10000n
00
ata d
ruct the data sams 1,10 0
ˆTn
, 100, 10000
ˆTn
 and
samples 1,10000
ˆ,
Tn

100, 10000
ˆTn
. The data sets 1, 10000
ˆTn
 an100, 10000
ˆTn

used in the numerical experiment are available at [15].
Given 0
sy” d
1

 we compute the mean relative errors
11
,1000, 10000Nn   obtained when 11000N, 10000nE

and 10.01,0.05,0.1


. The results obtained are shown
in Table 2. The comparison of Tables 1 and 2 shows a
substantial reduction of thehen
10.05 mean relative errors w


and 10.1


and only a marginal reduction
of the mean relative errors when 10.01

. This sug-
gests that the presof noise even in small quantity
degrades the solution
ence
obtained.
The second numerical experiment presented consists
in solving Calibration problem 2 using a sample of
synthetic data. Given the number of observations
0M
Table 1. Calibration problem 1: the mean relative error
Nn
E

11
,=1000,=1000 as a function of

1.
1

11
,1000, 1000Nn
E
 
0.01 0.124
0.05 0.215
0.1 0.328
Table 2. Calibration problem 1: the mean relative error
N 
11
, =10 n00, =1
E0000 as a function of

1
1
.

11
,1000, 10000Nn
E
 
0.01 0.104
0.05 0.141
0.1 0.170

L. FATONE ET AL. 357
and a time increment 0t

, let ,
i
tit


0,1, ,,iM be a discrete set otion times. Let
ˆ
ˆi
f observa

be the approximat of
,1,2,,,
i
tiM ion of a realization

 obtained integrating with tt
d one trajectolognor
model (11)-(14). Let us choose 100,M 1,t
he explici
ofmal SABR Euler methory the


0.1,

0.2,


That is, we wan
 00 1  
 and 00
0.5.vv

t to reconstruct th p
(76) by solving Calibration problem 2 using as data
sample the set of the logarithms of the forward prices/
rates observed at time ,1,2,,100,
i
ti M along one
trajectory of the lognormal SABR model. The set

2ˆ
ˆˆ,1,2,,100,
ii
  (80)
e unknownarameters
is the sample of synthetic data used to solve Calibration
problem 2. The data set 2
ˆ
 used in the numerical
xperiment is available at [15]. eProceeding as discussed in Section 3 using the data set
ˆ
, the least-squares fit of the first-order Taylor’s
n of * with b
2
appr
at t
of
seco
point
in

oximatio ase point evaluated
ves itial guess
set fit of th e
ap with base
d at gives
1

0, gi
ata
lor's
uate
itial gu
0t
as in
ares
*
1

,
i,
0
v

nd

1,2 ,1i
. Using the d
-or ay
tval
0.114
, 00.533
in
v

2
ˆ, the least-squ
oximation
i
t, 91i
of

pr
ess
der T of
0 e,92,100,
as in

. To obtain an initial
guess of tharae pmeter

we take a
e prices go do
ploiting this
dvantag
unde
wn th
e of th
rstand
e vo
fact and the fact
e “a
in the
latility
priori” information that in finance the correlation be-
tween forward prices/rates and stochastic volatility is
ve. In fact, as it is easy to us
fina
goe
that
ually n
nc
s u
egati
ial markets wh
p and i
11
en
ceversa.
th
Exv

 the initial guess in

of

that we
solution of
data thbservations
choose is in 
g from th0.5 . e in

Startin itial guess


0
,,0.114, 0.5,0.533
inin v


that uses as
ˆ,
i
in  the
Calibration problem 2e o
ˆ

20,30i
non
FMINCON routin
corresponding to the central part
, ,80, of 2
ˆ
 is obtained solving the
linear least-squares problem (75), (73) using the
e of Matlab. Note that in (75) we prefer
to use only a subset of the observations of the data
sample 2
ˆ
 (i.e. 20,30,,80,i
of the
avoid the presence of too many addenda in the objective
function (75). In the numerical computation the weights
,ij
a subset of data
trajectory) to

, 20,30,,80,i 1, 2, 3,4,j are chosen such
that the addenda of (e order of
magnitude. Using 2
ˆ
 as data and
75) are of


0
,,0.114, 0.5,0.533
inin in
v  

solution of problem (75), (73) found
t h
as i
is:
e sam
nitial guess the



e i
***
00
,,,,0.019,0.168,0.472 .vv   
 (81)
The relative 2 L-error of thnitial guess




3
“tr
solu 76
s point out thta of
su rate
ta obss. ow
sensitivity ora
ed
0
,,0.114, 0.5,0.53
inin in
v  
 with respect to the
Calib
eves stu
ue” solution (76) is 0.551. The relative 2
L-error of the
tion (81) of the least-squares problem with respect to
the “true” solution () is 0.167.
Let uat the daration problem 2
are pposed to be prices/s observed in the financial
markets, that is, these data are not affecteby noise as
the daerved in physicHdy the
f the solution of Calibtion problem 2 found
with the numerical procure proposed with respect to
the presence of noise in the data. We add noise to the
synthetic data contained in 2
ˆ
 and we study the quality
of the solution of Calibration problems 2 found as a
function of the noise properties. In particular given
20
d
r, let u

 let us consider the following “noisy” data sam-
ple:




1,100 
22
ˆ11
2 ,
irand i
 
where rand is a random entry taken from a uniform
distribution on the interval

,1 .
Given 20
ˆ
0
ˆ2,,,
(82)

 we compute the relative 2 L-error be-
tween the “true” solution (76) and the solution of Cali-
bration problem 2 found with the numerical procedure
described above. The entire procedure is repeated 2 N
times and the mean of the -relative errors of the 2 N
solutions found is calculated. Let
2
L
22
,
N
E

be this mean
relative error. The mean relative errors 22
, 1000N
E
  ob-
tained when 1000N2

and 0.01,0.05,0.1
2

 are
shown in Table 3.
Table 3 shows mean relative errors greater than the
corresponding meaelative errors of the first numerical
experiment. The calibration problem studied in the
second experiment is more difficult than th
n r
problem studied in the first one
quantity of the data and abov
qu ng
deduced from the data sample 2
ˆ


e calibration
. This is due
e all to the quality o
es problem
ed
a in,
to the
f the
antities enteri in the nonlinear least-squar
when compared, for
example, to the quantity of data contain in the data
sample 1,1000
ˆTn
 and 100, 1000
ˆTn
. That is, there are
100 data in 2
ˆ
 and 2000
ˆ dat 1, 1000
ˆTn

100, 1000Tn and the moments estimated from the data
contained in 1, 1000
ˆTn
, 100, 1000
ˆTn
 using formula (71)
are of high quality due to the average over a sample of
1000=n observations. There is no a similar effect in
Table 3. Calibration problem 2: the mean relative error
N
E

22
,=1000 as a function of

2.
1

22
,1000N
E
 
0.01 0.239
0.05 0.265
0.1 0.369

L. FATONE ET AL. 358
Calibration problem 2.
In the context of finance the natural way of f
w veeth
a
n
reor exhn the Heston m
e lognodel this p
ho
ur purposis paper. der a tim
series of enge raten the p
rates con
.S. dollars and are the closing value of the day (in New
year is m tring days and that a month
is made of about 21 trading days. Figure 1 shows the
euro/U.S. dollar currency’s exchange rate as a function
of time. Town in Figu s available at
[15].
We use lognormal SABR mo interpret the
data show Figure 1. In order to umodel in the
he logarithm of the data shown
ormulat-
ing a nersion of Calibration problem 2 (i.e. a “single
trajectory” calibration problm) that defines more accu-
rat model ely theparameters is to acquire at e observa-
tion times not only the forward prices/rates data, but also
the data relative to the prices of one or of several options
having as underlying the forward prices/rates. This last
problem is a “single trajectory” calibration problem tht
exploits more deeply than Calibratio problem 2 the in-
formation contained in the prices. We donot consider
this problem he. Fample, weodel
is used instead of thormal SABR mrob-
lem has been studied in [6].
Note that the FMINCON routine of Matlab used to
solve problems (72), (73) and (75) , (73) is an elementary
local minimization routine. Higher quality results can be
obtained solving problems (72), (73) and (75), (73) using
global minimization methods. Moreover, the explicit
moment formulae that define the objective functions (72),
(75) can be used to develop adc minimization algo-
rithms to solve problems (72), (73) and (75) , (73). Th is is
beyond oes in th
In the third numerical experiment we consie
xchange rates between currencies (euro/U.S.
dollar exchas) ieriod going from
September 14th, 2010, to July 20th, 2011. The exchange
sidered are daily exchange rates expressed in
U
York) of one euro expressed in U.S. dollars. Recall that a
ade of about 252ad
he data set shre 1 i
thedel to
n inse the
form (11)-(14) we take t
in Figure 1. That is, we solve the Calibration problem 2
using a window of 20 consecutive observations as data
and we study the stability of the solution found with re-
spect to shifts of the data along the time series. The
model resulting from the calibration can be used to fore-
Figure 1. euro/U.S. dollar currency’s exchange rate versus
time.
cast exchange rates and to compute option prices on ex-
change rates.
We solve Calibration problem 2 using the real data of
Figure 1 associated to a time window made of
120M

 consecutive observation times, that is the
observations corresponding to 20 consecutive trading
days, and we move this window across the data set dis-
carding the datum corresponding to the first observation
time of the window and inserting the datum correspond-
ing to the next observation time after the window. The
calibration problem (75), (73) is solved for each choice
of the data window. We choose 1252,t

 00




equal to the first observation (i.e. logarithm of the ex-
change rate observed) of the window considered,
,1 ij


, 1,2, ,19,i

 1, 2,3, 4.j The initial guess
of the numerical method used to solve the nonlinear
least-squares problem (75), (73) has been chosen as fol-
lows:




0
,,0.05,0.05,0.05
ininin
v  
. Note that a data
window made of twenty data has too few points to im-
plement satisfactorily the asymptotic analysis of the
moment formulae discussed in Section 3. Note that to
make possible the effective numerical solution of prob-
lem (75), (73) the independent variables in (75), (73)
have been rescaled.
The reconstructions of the parameters obtained mov-
ing the data window along the data set of Figure 1 are
shown in Figure 2. In Figure 2 the abscissa corresponds
to the data window used to reconstruct the model para-
meters. The data windows are numbered in ascending
order beginning with one according to the order in time
of the first day of the window considered. In particular,
Figure 2 shows that the parameters ,

reconstructed
remain essentially stable when the ndow is moved
along the data time series. Occasiwi onally

and

have spikes that probably indicate that the numerical
procedure used to solve problem (75), (73) has failed.
. rs v


 ,, Figure 2The paramete0onstructed from the
data of Figure 1 versus time.
rec

L. FATONE ET AL. 359
is moved along the data time s
The parameter reconstructed changes when the
windoweries. This is cor-
rect since 0 v
 is the stochastic volatility of the first day
of the window and in a stochastic volatility model (such
as the lognormal SABR model) there is no reason to ex-
pect this value to be constant.
The fact that the values of the parameters 0 ,, ,v
0
v



obtained calibrating the lognormal SABR mod
re
least
ti
REFERENCES
[1] P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Wood-
ward, “Managing Smile Risk,” Wilmott Magazine, 2002
pp. 84-108.
http://www.wilmott.com/pdfs/021118_smile.pdf
[2] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the
Term Structure of Interest Rates,” Econometrica, Vol. 53,
No. 2, 1985, pp. 385-407.
[3] F. Black and M. Scholes, “c,” Journal of Political Econo-
my, Vol. 81, No. 3, 1973, pp. 637-659.
[4] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli
“The Use of Statistical Tests to Calibrate the Normal
SABR Model,” Journal of Inverse and Ill-Posed Prob-
lems, Vol. 21, No. 1, 2013, pp. 59-84.
http://www.econ.univpm.it/recchioni/finance/w15
[5] S. Mergner, “Application of State Space Models in Fi-
ag Göt
22.
http://www.econ.univpm.it/pacelli/mariani/finance/w1.
[7] L. Fatone, F.ioni and F. Zirilli,
al of Inverse and Ill-Posed Problems, Vol.
SABR and Multiscale SABR
[6] F. Mariani, G. Pacelli and F. Zirilli, “Maximum Likeli-
hood Estimation of the Heston Stochastic Volatility Mo-
del Using Asset and Option Prices: An Application of
Nonlinear Filtering Theory,” Optimization Letters, Vol. 2,
No. 2, 2008, pp. 177-2
Mariani, M. C. Recch
“Maximum Likelihood Estimation of the Parameters of a
System of Stochastic Differential Equations That Models
the Returns of the Index of Some Classes of Hedge
Funds,” Journ
15, No. 5, 2007, pp. 329-362.
el on the
al data shown in Figure 1 through the “moving win-
dow” procedure are “stable” (see Figure 2) supports the
idea that the lognormal SABR model is able to interpret
the data considered. In fact, this stability property sug-
gests that the relation between the data and the parame-
ters of the lognormal SABR model established solving
problem (75), (73) is not an artefact of the numerical
methods used to solve the nonlinear -squares prob-
lem. In particular, the stability shown guarantees th at the
lognormal SABR model provides “stable” opon prices
and hedging strategies.
http://www.econ.univpm.it/ recchioni/finance/w5
[8] L. Fatone, F. Mariani, M. C. Recchioni and F. Zirilli,
“Some Explicitly Solvable
Models: Option Pricing and Calibration,” Journal of Ma-
thematical Finance, Vol. 3, No. 1, 2013, pp. 10-32.
http://dx.doi.org/10.4236/jmf.2013.31002
http://www.econ.univpm.it/recchioni/finance/w14
[9] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Trico-
mi, “Higher Trascendental Functions,” McGraw-Hill Book
Company, New York, 1953.
[10] J. Hull and A. White, “The Pricing of Options on Assets
with Stochastic Volatilities,” The Journal of Finance, Vol.
42, No. 2, 1987, pp. 281-300.
http://dx.doi.org/10.1111/j.1540-6261.1987.tb02568.x
[11] B. A. Surya, “Two-Dimensional Hull-White Model for
Stochastic Volatility and Its Nonlinear Filtering Estima-
tion,” Procedia Computer Science, Vol. 4, 2011, pp.
1431-1440. http://dx.doi.org/10.1016/j.procs.2011.04.15
,
4
, Vol. 24, No. 1, 2007,
[12] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Trico-
mi, “Tables of Integral Transforms,” McGraw-Hill Book
Company, New York, 1954.
[13] P. L. Lions and M. Musiela, “Correlations and Bounds for
Stochastic Volatility Models,” Annales de l’Institut Henri
Poincaré (C) Analyse Non Linèaire
pp. 1-16.
,
[14] P. Wilmott, “Paul Wilmott Introduces Quantitative Fi-
nance,” Wiley Desktop Editions, Chichester, 2007.
[15] http://www.econ.univpm.it/recchioni/finance/w16
nance,” Universitätsverltingen, Göttingen, 2004.