The use of statistical tests to calibrate the normal SABR model
Lorella
Fatone, Francesca Mariani,
Maria Cristina Recchioni
and Francesco Zirilli
We investigate the idea of solving calibration problems for stochastic dynamical systems using statistical tests. We consider a specific stochastic dynamical system: the normal SABR model. The SABR model has been introduced in mathematical finance in 2002 by Hagan, Kumar, Lesniewski, Woodward [5]. The model is a system of two stochastic differential equations whose independent variable is time and whose dependent variables are the forward prices/rates and the associated stochastic volatility. The normal SABR model is a special case of the SABR model. The calibration problem for the normal SABR model is an inverse problem that consists in determining the values of the parameters of the model from a set of data. We consider as set of data two different sets of forward prices/rates and we study the resulting calibration problems. Ad hoc statistical tests are developed to solve these calibration problems. Estimates with statistical significance of the parameters of the model are obtained. Let T > 0 be a constant, we consider multiple independent trajectories of the normal SABR model associated to given initial conditions assigned at time t = 0. In the first calibration problem studied the set of the forward prices/rates observed at time t = T in this set of trajectories is used as data sample of a statistical test. The statistical test used to solve this calibration problem is based on some new formulae for the moments of the forward prices/rates variable of the normal SABR model. The second calibration problem studied uses as data sample the forward prices/rates observed on a discrete set of given time values along a single trajectory of the normal SABR model. The statistical test used to solve this second calibration problem is based on the numerical evaluation of some high dimensional integrals. The results obtained in the study of the normal SABR model are easily extended from mathematical finance to other contexts in science and engineering where stochastic models involving stochastic volatility or stochastic state space models are used. This website contains some auxiliary material that helps the understanding of [4]. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
Keywords. SABR model, calibration, statistical tests, Montecarlo.
2010 Mathematics Subject Classification. 34K50, 62H15, 37N40, 34K29.
2 The normal SABR model and the moments of the forward prices/rates variable
3 The multiple trajectories statistical test
4 The single trajectory statistical test
5 Numerical experiments (Movie 1, 2, 3)
The SABR model has been introduced
in 2002 by Hagan, Kumar, Lesniewski, Woodward [5] and is widely used in the practice of the financial markets. The model is a system of two stochastic differential equations
whose independent variable
is the time t, t > 0, and whose dependent variables are
the forward prices/rates ξ_{t}, t > 0, and the associated stochastic volatility v_{t}, t > 0.
The SABR model is characterized by the following (real)
parameters: a correlation coefficient ρ
∈ (−1, 1), the β volatility, β ∈ [0, 1], and the volatility of volatility ε > 0. Moreover the initial stochastic
volatility
The parameters ε, ρ,
The first
step in the formulation of a hypotheses testing
problem consists in the
definition of the null hypothesis H_{0} and of the alternative hypothesis H_{1}. When the goal
is to establish an assertion
about the parameters
of a probability distribution on
the basis of support from a data sample, usually the assertion is taken to be the null hypothesis H_{0}
and the negation of the assertion itself is taken to be the alternative hypothesis H_{1 }(or viceversa).
The hypotheses are classified
as “simple” or “composite” depending
on their formulation, see [6]. In [4] we study a “simple”
null hypothesis concerning the parameters of the normal
SABR model, that is we consider the hypothesis:
H_{0} :
(ε, ρ,
where ε^{∗}> 0, ρ^{∗}∈ (−1, 1),
H_{1} :
(ε, ρ,
When we consider
a “simple” null hypothesis, such as (1.1), the associated
decision table is the following one:

retain (do not reject)
H_{0} and conclude
that H_{1 }fails to be substantiated; (1.3)

reject H_{0} and conclude that H_{1} is substantiated. (1.4)
In a statistical test two types of error can occur:
Type I error: Rejection of H_{0} when H_{0} is true;
Type II error: Nonrejection of H_{0} when H_{0} is false.
The significance level α, 0 < α < 1,
is the maximum probability of making a Type I error. The false alarm
probability λ, 0 < λ
< 1, is the maximum
probability of making a Type II error. Given α, 0 < α < 1, and/or λ, 0 <
λ < 1, performing a statistical test about the hypotheses H_{0} and
H_{1} considered above consists
in assuming the decisions (1.3), (1.4) with statistical significance α
and/or with false alarm probability λ.
Note that taking
a decision based
on statistical inference
starting from a data
sample, the possibility of making Type I and/or Type II errors cannot be avoided. In [4] we limit our attention
to Type I errors.
The analysis developed here
to study Type I errors can be easily
extended to the study
of Type II errors.
A statistical test usually
is done defining a random variable whose value on the data
sample determines the decision to take. This random variable
is called test statistic.
A test of the null hypothesis H_{0} is a decision rule that specifies
when to reject H_{0}
(and as a consequence when to retain H_{0}). Usually
this rule consists
in specifying the set of values of the test statistic for which H_{0} is to be rejected. This set
is called rejection
region R = R_{α} of the test and depends on the significance level α, 0 < α < 1. The hypothesis H_{0} is
to be retained when the test statistic evaluated on the data sample takes value in the complement of the rejection region R = R_{α}.
Given a significance
level α, 0 < α < 1,
a test is completely
determined defining the corresponding test statistic and rejection
region. In the tests considered later
we use a vector
valued random variable
as test statistic and the rejection
region is determined using
numerical methods. Note that
in the more familiar
context of the elementary statistical tests used to determine the parameters of the normal random
variable, that is in the Student’s T and the χ^{2} tests [6], a scalar test statistic is used and the cutoff points which determine
the rejection region are read from tables
that evaluate one dimensional integrals.
The use of statistical tests to solve calibration problems for stochastic dynamical systems has already been considered in [3] where the calibration of the BlackScholes asset price dynamics
model has been studied.
In [3] the data considered are the observations on a discrete
set of time values of the asset price. The resulting
calibration problem for the BlackScholes model is reduced
to the Student’s T and the χ^{2} tests. In the study of the calibration problem of the normal SABR model no elementary statistical tests can be used. That is new ad hoc tests must
be developed using numerical methods.
These statistical tests are examples of the
application of numerical methods to statistics, that is these tests are examples of
computational statistics. New formulae
for the moments of the state variables
of the normal SABR model are presented. The first statistical test developed to estimate the parameters ε, ρ,
The data sample used in this statistical test although realistic
in many contexts of
science and engineering it is hardly available
in the financial markets. In fact in the
financial markets it is not possible
to repeat the “experiment”,
that is observations at time t = T of multiple independent trajectories of the stochastic dynamical system under investigation are usually not available.
This is a serious drawback.
The second statistical test presented overcomes this difficulty. In fact it uses as data of
the calibration problem
the observations of the forward prices/rates made on a discrete set of known time values along a single trajectory of the normal SABR
model. This second type of data is easily available
in the financial markets. In
fact it is simply the
time series of the forward prices/rates observed. The statistical test that
uses this last set of data is computationally
more demanding than the statistical test based on the moment formulae and, in realistic
circumstances, involves the
numerical evaluation
of some high dimensional integrals. Moreover the question of how to choose the hypothesis H_{0} to be tested in the previous calibration problems is discussed. Finally the tests presented
are performed on some samples
of synthetic data and
the numerical results obtained are shown.
2 The normal SABR model and the moments of the forward prices/rates variable
Let R,
R^{+} be respectively
the set of real and of
positive real numbers and
t be a real variable
that denotes time. The real stochastic processes
ξ_{t}, v_{t}, t > 0,
describe respectively the forward
prices/rates and the associated stochastic volatility as a
function of time. The normal
SABR model is given by the following system
of stochastic differential equations:
dξ_{t} = v_{t} dW_{t }, t > 0, (2.1)
dv_{t} = ε v_{t} dQ_{t }, t > 0,
(2.2)
with the initial
conditions:
v_{0} =
The quantity ε > 0
is a parameter known as volatility of volatility. The stochastic
processes W_{t}, Q_{t}, t > 0,
in (2.1), (2.2) are standard
Wiener processes such that W_{0} = Q_{0} = 0,
dW_{t}, dQ_{t}, t > 0, are their stochastic differentials and
we assume that:
< dW_{t}dQ_{t}> = ρ
dt, t > 0, (2.5)
where <·> denotes the expected
value of · and ρ ∈ (−1, 1) is a constant
known as correlation coefficient.
The initial conditions
In the SABR model,
introduced in 2002 by Hagan, Kumar, Lesniewski, Woodward [5], equation (2.1) is replaced
by equation:
dξ_{t} = ξ_{t}^{β} v_{t }dW_{t}, t > 0,
(2.6)
where β ∈ [0, 1] is a parameter known as βvolatility. The normal SABR model corresponds to the choice β
= 0 in the SABR model of [5] defined
by the equations (2.6), (2.2), (2.3), (2.4).
Starting from the expression obtained in [2] for the transition probability density
function of the variables
ξ_{t}, v_{t}, t > 0, of the normal SABR model we derive some
new
formulae for the moments with respect to zero of these variables.
In particular we derive explicit formulae
for the first five moments with
respect to zero of the forward prices/rates ξ_{t}, t > 0.
In [2] using the results of [7] on the Kontorovich Lebedev transform and some
standard methods of mathematical analysis the following formula
for the transition probability density function
p_{N} of the variables
ξ_{t}, v_{t}, t > 0,
defined by (2.1), (2.2), (2.3), (2.4) has been derived:
p_{N} (ξ, v, t, ξ’,
v’, t' ) =
(ξ, v), (ξ’, v’) ∈ R × R^{+},
t, t’ ≥ 0, t – t’>
0, (2.7)
where we
have: ξ_{t} = ξ, v_{t} = v, ξ_{t’ } = ξ’ , v_{t’} = v’ , t, t’
≥ 0, t
– t’ >
0. The function
g_{N} is given by:
g_{N} (s, k, v, v’, ε,
ρ) =
s ∈ R^{+}, k ∈ R, v, v’∈ R^{+}, ε > 0, ρ ∈ (−1, 1), (2.8)
where ι is the imaginary unit and the functions sinh, K_{η} denote respectively the
hyperbolic sine and the second type modified
Bessel function of order η (see [1] p. 5). Finally ς
^{2}(k), k ∈ R, is defined
as follows:
ς ^{2}(k)=
In (2.7) when t’ = 0 we must choose ξ’ =
Starting from the previous formula in [4] we derive the formulae for the moments M_{n,m }, n, m = 0, 1, . . . , with
respect to zero of the transition probability density function
p_{N}, that is:
M_{n,m}(t,
ξ’, v’, t’ ) =
(ξ’, v’) ∈ R × R^{+},
t, t’ ≥ 0, t – t’>
0, n, m = 0, 1, . . . .. (2.10)
The moments M_{n,m }, n, m = 0, 1, . . . ,
defined in (2.10) do not depend
from the variables t, t’ separately, but they depend only from t 
t’, t, t’>0.
Let
Note
that the moments
(
(
The moments
3 The multiple trajectories statistical test
Let T > 0 be given. Recall that the probability
distributions of the random variables
ξ_{T} , v_{T} solutions of (2.1), (2.2), (2.3), (2.4) when t
= T depend on ε,
Let us formulate the first calibration problem
for the normal
SABR model (2.1), (2.2), (2.3), (2.4) that we study. The data of this problem are the forward
prices/rates observed at
time t = T on a set of independent trajectories of the normal SABR model (2.1),
(2.2), (2.3), (2.4). Let n be a positive integer, we consider n independent copies
Figure 1. Data sample 1: the observations^{ }
More precisely we study the following problem: given T > 0, a statistical
significance level α, 0 < α
< 1, a positive integer n and n independent observations
at time t = T of the forward prices/rates ξ_{T} , that
is given
Chosen the null hypothesis H_{0} (i.e.
given (1.1))
this calibration problem
is solved with an ad hoc statistical test that uses D as
data sample.
Let us define the random variables:
It is easy to see that the random variables:
are unbiased estimators of
Let us consider
the realizations
where
Given a statistical significance level α, 0 < α
< 1, and the null hypothesis H_{0} defined
by (1.1), that is given ε^{∗} > 0, ρ^{∗}^{ }∈ (−1, 1),
First of all we translate
the hypothesis H_{0} in a corresponding hypothesis for the moments
The moments

retain (do not reject) H_{0} if the points

reject H_{0} if the points
In [4] we determine the relation among α, n, ε^{∗}, ρ^{∗},
Probability(L −
Pˆ ∗ ≥ A_{α,n}) ≤ α, (3.7)
where 
·  is the Euclidean norm of ·
in R^{3} and L
= (
Given α, 0 < α
< 1, a positive
integer n, the null hypothesis H_{0} (given by
(1.1)) and the corresponding threshold
r_{α,n }> 0,
the decision rule of the statistical
test is given by:

if 

if 
Let us call “moments space” the space where the test statistic
L takes values.
We can choose R^{3 }(the
three dimensional Euclidean space) as “moment space”. Note that the threshold
r_{α,n }divides the moments
space into two regions:
the rejection region R = R_{α,n }and the retain (i.e do not reject) region. In the
moments space the retain region is the sphere of center the vector of the theoretical moments
It is worthwhile to note that the threshold
r_{α,n }depends on α and n; moreover unlike the threshold(s) of the elementary statistical tests of the normal random variable
(i.e. the Student’s T or χ^{2}
tests, see [6]) and of the tests used in the calibration of the BlackScholes model [3], r_{α,n }depends on the null hypothesis H_{0}. This is due to
the fact that in
the study of the normal SABR model we have not reduced the random variables involved in the test to a standard random
variable as it is done in the
elementary statistical tests
for the normal variable [6] and in the tests used in the
calibration of the BlackScholes model [3]. In Section 5 in a study case given n and the null hypothesis H_{0} given by (1.1) we will provide a table of r_{α,n }as a function of α, 0 < α < 1.
Finally in
[4] we consider the question of choosing the parameters ε^{∗}, ρ^{∗} and
4 The single trajectory statistical test
Let us formulate the second calibration
problem for the normal SABR model (2.1), (2.2), (2.3), (2.4) that we study. Let
M be a positive integer and R^{M} be the M dimensional
real Euclidean space. The data of the calibration problem are the forward
prices/rates observed at the discrete times t_{0}, t_{1},...,
t_{M}, where t_{i
}> t_{i−1}, i = 1, 2, ..., M, and
t_{0} = 0. Note that the time values t_{i}, i = 0, 1, ...,M, are known. For i = 1, 2, ...,M, we
denote with
Figure 2. Data sample 2:
the observation
Let f_{k} : R^{M} →
R, k = 2, 3, 4, be given functions (or
distributions). Let us define:
Ƒ_{k} (M ) =
where
····
The choice of the functions f_{k}
, k = 2, 3, 4, is crucial to build satisfactory
statistical tests. In [4] we choose:
where
Starting from the data sample
Note that the quantity
where
and
Given ε^{∗}> 0, ρ^{∗}^{ }∈ (−1, 1),
where
Given the statistical significance level α,
0 < α < 1, the number of observations M, the
observation times t_{i},
i = 1,2,...,M,
and the corresponding data sample D_{1} the decision rule to
test the null hypothesis H_{0 }given by (1.1) is:

if 

if 
where s_{α,M} is a
positive quantity that depends on α, M and on the null hypothesis H_{0}.
The value s_{α,M} is
defined as follows:
s_{α,M }=
where inf{·} stands for infimum of the set of
real numbers {·}. From (4.10) it follows that in order to find s_{α,M
}we must evaluate the integrals (4.7). In particular it is necessary
to compute the joint probability density function
Given α, M and the observation
times t_{i}, i = 1, 2, ..., M, in
order to determine the threshold s_{α,M }we approximate the
joint probability density function of the random variables F_{2},
F_{3}, F_{4} defined in (4.5) with the corresponding
threedimensional joint histogram deduced from a (numerically generated) sample
of these random variables. We compute the
Finally we consider the problem of choosing
the parameter values that define H_{0}. As already said at the
end of Section 3 these parameter values are chosen as the
solution of a different formulation of the calibration problem that does not
involve statistical significance. See, for example, [4] where a formulation
of the calibration problem based on the least squares method is presented.
Let us discuss
some numerical experiments. The first numerical experiment presented
consists in solving
the calibration problem
for the normal SABR model with the statistical test described in Section 3 using a sample of synthetic
data.
Let T > 0 be given and n, m be positive integers. Let ∆t = T /m be a time
increment and t_{i} = i∆t, i = 0, 1, . . . , m, be a discrete
set of equispaced time values.
Let
We choose T = 1, m = 10000, n = 100, ε
= 0.1, ρ
= −0.2,
(ε,
ρ,
are the unknown
parameters of the normal SABR model that we want to recover
as solution of the calibration problem.
The synthetic data
The data sets
We consider the following calibration
problem: given
The first step consists in the formulation of
the null hypothesis (1.1). We proceed as done in [4] to
determine the null hypothesis. That is solving a calibration problem for the
normal SABR model with the least squares method we end up with the null
hypothesis:
that must be tested in the statistical test
procedure described in Section 3.
We want to test the null hypothesis (5.2)
with statistical significance level α using the data sample
Let us perform the test associated to the
calibration problem considered. Given the null hypothesis (5.2), the
significance level α and the data sample
Let α = 0.01, 0.05,
0.1, and

if 

if 
In this specific experiment given the data
sample



Table 1. The threshold r_{α,n}_{ }=
For example let us fix the attention on
In the following animations we show a cloud
of points
The second numerical experiment presented
consists in testing the null hypothesis
Let M > 0 be the number of observations.
Let ∆t be a time increment and t_{i} = i∆t, i = 0, 1, ..., M, be
a discrete set of observation times. Let
The data set
In [4] we choose the weights
We want to test the null hypothesis (5.2)
with statistical significance α. First we determine the threshold s_{α,}_{10
}=
Finally, given the statistical significance
level α and the data sample



Table 2. The threshold s_{α,M}_{
}=
Acknowledgments. The numerical experience reported
in Section 5
has been obtained using the computing grid of ENEA (Roma, Italy). The support
and sponsorship of ENEA are gratefully acknowledged.
[1] A. Erdelyi,
W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher trascendental functions, 2, McGrawHill
Book Company, New York, U.S.A. (1953).
[2] L. Fatone, F. Mariani, M. C. Recchioni, F. Zirilli,
Some explicitly solvable SABR and multiscale SABR models: option pricing and calibration, to appear in Journal of Mathematical Finance (2013), http://www.econ.univpm.it/recchioni/finance/w14.
[3] L. Fatone,
F. Mariani, M. C. Recchioni, F. Zirilli, The use of statistical tests to calibrate the Black–Scholes
asset dynamics model applied
to pricing options with uncertain volatility, Journal
of Probability and Statistics 2012 (2012), article id:10.1155/2012/931609, 20 pages,
http://www.econ.univpm.it/recchioni/finance/w11.
[4] L. Fatone, F.
Mariani, M. C. Recchioni, F. Zirilli, The use of statistical tests to calibrate the normal
SABR model, Journal
of Inverse and IllPosed
Problems 21 (2013), no. 1, 5984.
[5] P. S. Hagan, D. Kumar, A. S. Lesniewski, D. E.
Woodward, Managing
smile risk, Wilmott Magazine, September 2002 (2002),
84–108, http://www.wilmott.com/pdfs/021118smile.pdf.
[6] R. A. Johnson, G. K. Bhattacharyya, Statistics: Principles and Methods, 5th ed., John Wiley & Sons, New York, U.S.A. (2006).
[7] S. B. Yakubovich, The heat kernel
and Heisenberg inequalities related to the KontorovichLebedev transform, Communications on Pure and Applied
Analysis 10 (2011), no. 2, 745–760.