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.

 

1  Introduction

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)

6  References

 

1 Introduction

 

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  > 0 assigned at time t = 0 cannot be observed in the financial  markets and must be regarded as a parameter of the model. In the family of SABR models parametrized by β ∈ [0, 1]  the normal (β = 0) and the lognormal  (β = 1) models are the most used ones.  From the mathematical point of view the normal SABR model (β = 0) is the simplest one, see [5]. In [4] we study two calibration  problems for the normal SABR model.

The parameters ε, ρ,  are the unknowns  of the calibration  problems for the normal SABR model. These unknowns must be determined starting from a set of data. The sets of data considered are sets of forward prices/rates. We use statistical tests to solve the calibration problems studied; more precisely to the parameter values obtained as solution  of the calibration  problems we associate  a statistical significance level using a statistical test.

The first step in the formulation of a hypotheses testing problem consists in the definition of the null hypothesis H₀ and of the alternative hypothesis H₁. 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  H₀   and the negation of the assertion itself is taken to be the alternative hypothesis H₁ (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:

 

H₀ :              (ε, ρ, ) = (ε^∗, ρ^∗, ),                             (1.1)

 

where ε^∗> 0, ρ^∗∈ (−1, 1),   > 0 are given.The alternative hypothesis is:

 

H₁ :              (ε, ρ, )  (ε^∗, ρ^∗, ).                             (1.2)

 

When we consider a “simple” null hypothesis, such as (1.1), the associated decision table is the following one:

 

-       retain (do not reject) H₀ and conclude that  H₁ fails to be substantiated;    (1.3)

-        reject H₀ and conclude that H₁ is substantiated.                                       (1.4)

 

In a statistical  test two types of error can occur:

 

Type I error: Rejection of H₀ when H₀ is true;

Type II error: Non-rejection of H₀ when H₀ 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 H₀  and H₁  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 H₀ is a decision rule that specifies when to reject H₀ (and as a consequence when to retain H₀). Usually this rule consists in specifying the set of values of the test statistic for which H₀ 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 H₀ 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 χ² 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 Black-Scholes 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 Black-Scholes model is reduced to the Student’s T and the χ² 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,   of   the normal SABR model with a given statistical significance level α is based on these moment formulae.  The data sample used in this statistical test is a sample of forward prices/rates observed at time t = T > 0 on a set of independent trajectories of the model associated to given initial conditions assigned at time t = 0. The decision resulting from the test is assumed comparing the theoretical values of three moments of the forward prices/rates variable of the normal SABR model when the null hypothesis H₀  is true with the observed values of these moments computed on the data sample. To perform the hypothesis testing the estimators of the moments considered are combined in a vector valued random variable that is used as test statistic.  The probability  density function  of this test statistic is obtained by numerical simulation.

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 H₀  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:

 

₀ = ,           (2.3)

v₀ = .          (2.4)

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 W₀ = Q₀ =    0, dW_t, dQ_t, t > 0, are their stochastic differentials  and we assume that:

 

< dW_tdQ_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 ,  are  random variables that we assume to be concentrated in a point with probability one. For simplicity we identify these random variables with the points where they are concentrated. We assume   > 0. The assumption  > 0 with probability one and equation (2.2) imply that v_t  > 0 with probability one for t > 0. Unlike   the initial stochastic volatility   cannot be observed in the financial markets and must be regarded as a parameter of the model.  Similarly the stochastic volatility v_t, t > 0, cannot be observed in the financial markets. The parameters ε, ρ,  are the unknowns of the calibration problem for the normal SABR model.

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 β-volatility. 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' ) =     g_N (t – t’ , k, v, v’, ε, ρ) ,

          (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) =        ω sinh(πω) (ς (k)v) (ς (k)v’),

          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)=  (1 − ρ²), k ∈ R.           (2.9)

 

In (2.7) when t’ = 0 we must choose ξ’  =   and v’= .

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’  ) =  p_N (ξ, v, 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 

(t-t’, ξ’, v’ ) = M_n,0 (t, ξ’, v’, t’  ),  (ξ’, v’) ∈ R × R⁺, t, t’ ≥ 0, t – t’> 0, n = 0, 1, . . . ..          (2.11)

Note  that the moments depend, in general, on the unknowns ε, ρ,  of the calibration problems considered in the following Sections and on the time t - t’.  In [4] when t’=0, ξ’ =   and v’=  we derive the following formulae for the moments _,  of the normal SABR model:

 

  (t, , ) = 1, ( , ) ∈ R × R⁺,   t ∈ R⁺,          (2.12)

 

  (t, , ) = , ( , ) ∈ R × R⁺,   t ∈ R⁺,         (2.13)

 

(t, , ) =  +   -1),   ( , ) ∈ R × R⁺,   t ∈ R⁺,          (2.14)

 

(t, , ) =  + 3     -1) +     ),

 

          ( , ) ∈ R × R⁺,   t ∈ R⁺,          (2.15)

 

(t, , )=  + 6     -1) -4     )+

               ,

         

           ( , ) ∈ R × R⁺,   t ∈ R⁺,          (2.16)

 

The moments    depend on the unknowns of the calibration problems considered ε, ρ,  and on the time t.  In particular  depends on ε and     while    depend on ε,   and ρ. The moments _,  do not depend on ε,  and ρ and cannot be used in the solution of calibration problems. The formulae (2.14), (2.15), (2.16) for the moments  have been deduced for the first time in [4].Similar formulae  can be deduced (at least in principle) for the moments M_n,m, n, m = 0, 1, . . ..  These formulae become more and more involved  when n, m increase.

 

 

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, , ρ, and that ξ_T can be observed while v_T cannot be observed.

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   , i = 1, 2, . . . , n, of the random variable ξ_T . Let   be a realization of , i = 1, 2, . . . , n, the set D = { , i = 1, 2, . . . , n} is the set of “observations” used as data sample in  the calibration problem considered, see Figure 1.

 

Figure 1. Data sample 1: the observations    i = 1, 2, . . . , n, of ξ_t  made at time t=T on n independent trajectories of the normal SABR model.

 

 

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 , i = 1, 2, . . . , n, (i.e. given the data set D), determine the values of the parameters ε, ρ and  of the model (2.1), (2.2), (2.3), (2.4) with significance level α.

Chosen the null hypothesis H₀  (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:

 

=( )²   =( )³   ,    =( )⁴   ,           i = 1, 2, . . . , n.          (3.1)

 

It is easy to see that the random variables:

 

,   ,              (3.2)

 

are unbiased estimators of   respectively.  The random variables , are used as components  of the vector valued test statistic L = ( ,  ) of the statistical test used to test the hypothesis H₀ given by (1.1).

Let us consider the realizations ,     in the data sample D of the random variables ,  that is:

 

,   ,   ,          (3.3)

 

where

 

 =( )²  ,      =( )³  ,         =( )⁴  ,          i = 1, 2, . . . , n.          (3.4)

 

Given a statistical significance level α, 0 < α < 1, and the null hypothesis H₀ defined by (1.1), that is given ε^∗  > 0, ρ^∗ ∈ (−1, 1), >0, using the vector valued test statistic L, we want to test the null hypothesis H₀ given by (1.1) against the alternative hypothesis H₁ given by (1.2) with statistical significance level α,   0 < α < 1.

First of all we translate the hypothesis H₀ in a corresponding hypothesis for the moments     associated to the normal SABR model (2.1), (2.2), (2.3), (2.4).

The moments ,  obtained from    given by  (2.14), (2.15), (2.16)  when H₀ is true and t = T , are compared with the moments ,      observed in the data sample. Note that the point =( ,   ) ∈ R³ is the value taken by the test statistic L on the data sample D. In particular to test the null hypothesis H₀ considered in (1.1) we check if the point =( ,   ) ∈ R³ and the point , )∈ R³    are “close”     o r  “far”.  The heuristic decision rule of the statistical test is:

 

-         retain (do not reject) H₀ if the points  and are “close”;          (3.5)

-         reject H₀ if the points  and   are “far”.                                   (3.6)

 

In [4] we determine the relation among α, n, ε^∗, ρ^∗,  that translates  the qualitative expressions “close” and “far” used in (3.5), (3.6) in a quantitative  statement about the norm of the vector     . Recall that the statistical significance level α, 0 < α < 1, is the maximum probability of rejecting the null hypothesis H₀  when the hypothesis is true. We proceed as follows: given α, 0 < α < 1, n > 0, ε^∗ > 0, ρ^∗ ∈ (−1, 1) and > 0 we solve the following inequality for the real unknown A_α,n:

Probability(||L − Pˆ ∗|| ≥ A_α,n)  ≤ α,          (3.7)

 

where  || · || is the Euclidean  norm of ·  in R³ and  L = ( ,  )  is the random variable defined through (3.2) and we determine the infimum  r_α,n of the values A_α,n that satisfy (3.7). The inequality (3.7) is studied in [4] and the infimum   of its solutions is determined numerically using statistical simulation. See  [4] for more details.

Given α, 0 < α < 1, a positive  integer n, the null hypothesis H₀ (given by (1.1)) and the corresponding threshold r_α,n > 0, the decision rule of the statistical test is given by:

 

-        if ||   || ≤ r_α,n retain (do not reject) H₀, with significance level α;             (3.8)

-        if ||   || > r_α,n reject H₀, with significance level α.                                   (3.9)

 

Let us call “moments space” the space where the test statistic L takes values. We can choose R³ (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   and radius  r_α,n (see (3.8)) and the rejection region R_α,n  is its complement.  See Figure 3.

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 χ² tests, see [6]) and of the tests used in the calibration of the Black-Scholes model [3], r_α,n depends on the null hypothesis H₀. 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 Black-Scholes model [3]. In  Section 5 in a study case given n and the null hypothesis H₀ 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    that define the null hypothesis H₀ given by (1.1). The parameters ε^∗, ρ^∗ and    can be chosen as solution of a different formulation of the calibration problem that does not involve statistical significance. For example in [4]  we mention a formulation of the calibration problem for the normal SABR model based on a specific analysis of the moment formulae (2.14), (2.15), (2.16) that gives very good results in numerical experiments. This formulation is based on the knowledge of the random variables defined in (3.3) for two time values T₁, T₂ such that 0 < T₁ < T₂ and is formulated through two nonlinear least squares problems (see [4]).

 

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 t₀, t₁,..., t_M, where t_i > t_i−1, i = 1, 2, ..., M, and t₀ = 0. Note that the time values t_i, i = 0, 1, ...,M, are known. For i = 1, 2, ...,M, we denote with  the forward price/rate observed at time t = t_i  along one trajectory of the stochastic process ξ_t, t > 0. The set D₁ = {  i =1, 2, ..., M } is the data sample used to solve the second calibration problem, see Figure 2.

 

 

 

Figure 2. Data sample 2: the observation  made at t=t_i, i = 1,...,M, on a single trajectory of the normal SABR model.

 

 

 

Let f_k : R^M → R, k = 2, 3, 4, be given functions (or distributions). Let us define:

 

Ƒ_k (M ) = ......  f_k(   ( ,           k = 2, 3, 4,          (4.1)

 

where (  is the joint probability density function of the normal SABR model (2.1), (2.2), (2.3), (2.4) of having   = ξ_i, i = 1, 2, ..., M, conditioned to  =  and  = . That is we have:

 

( = ….. ( (

          ···· ( .           (4.2)

 

The choice of the functions f_k , k = 2, 3, 4, is crucial to build satisfactory statistical tests. In [4] we choose:

 

( ,          k = 2, 3, 4,          (4.3)

 

where , i = 1, 2, ..., M, are positive weights, , i = 1, 2, ..., M − 1, are real numbers and is the Dirac’s delta. Moreover we choose ,  i = 1, 2, ..., M − 1. The weights , i = 1, 2, ..., M, are chosen in [4]. As suggested by (2.14), (2.15), (2.16) we should choose the weight  decreasing when the index i increases.

Starting from the data sample , i =1, 2, ...,M, we compute:

 

 (M ) = ,           k = 2, 3, 4.           (4.4)

 

Note that the quantity  is a realization of the random variable:

 

(M)=  ( , ,           k = 2, 3, 4,           (4.5)

 

where

 

 ( , = ,           k = 2, 3, 4,         (4.6)

 

and | , that is  is the random variable conditioned to , i = 1, 2, ..., M. The random variables F₂, F₃, F₄ are used to build the vector valued test statistic F = (F₂, F₃, F₄) of a statistical test used to solve the calibration problem considered.

Given ε^∗> 0, ρ^∗ ∈ (−1, 1), > 0, using the vector valued test statistic F, we want to test the null hypothesis H₀ given by (1.1) against the alternative hypothesis H₁ given by (1.2) with a prescribed statistical significance level α, 0 < α < 1. We proceed as follows: let , k = 2, 3, 4, be given by:

 

(M ) = ......  f_k( ( ,          k = 2, 3, 4,           (4.7)

 

where (  is the joint probability density function  given by (4.2) when H₀ is true. Let =(  be the realization of the test statistic F = (F₂, F₃, F₄) on the data sample D₁. The test of the null hypothesis H₀ consists in verifying if the point  =(  ∈ R³ is “close” or “far” from the point =( , , ) ∈ R³ where , k = 2, 3, 4, are given in (4.7).

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 D₁ the decision rule to test the null hypothesis H₀ given by (1.1) is:

 

-     if ||   || ≤ s_α,M retain (do not reject) H₀, with significance level α;          (4.8)

-     if ||   || > s_α,M reject H₀, with significance level α.                                (4.9)

 

where s_α,M is a positive quantity that depends on α, M and on the null hypothesis H₀.

The value s_α,M is defined as follows:

 

s_α,M = ,                  (4.10)

 

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 by (4.2) when ε = ε^∗, ρ = ρ^∗ and  = .  Recall that the transition probability density functions p_N appearing in the integral (4.2) are one dimensional  integrals of explicit integrands (see [4]  for more details). When M is large (i.e. when M is greater than 3 or 4) the integrals (4.2) and (4.7) are high dimensional integrals that must be computed using the Monte Carlo method. In order to use the Monte Carlo method to evaluate (4.2) and (4.7) we must draw a sample from the probability density functions (  i = 0, 1, ..., M − 1. However this is not easy due to the complexity of the expression of p_N. This difficulty can be overcome using the “importance sampling” method, that allows to draw the sample of the Monte Carlo procedure from auxiliary probability density functions that are similar to the density functions  (  i = 0, 1, ..., M − 1, and are easy to sample. These probability density functions are called sampling distributions. The sampling distribution used to evaluate (4.2), (4.7) are obtained substituting (2.1), (2.2), (2.3), (2.4) with a simplified model that can be solved explicitly (see [4]).

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 F₂, F₃, F₄ defined in (4.5) with the corresponding three-dimensional joint histogram deduced from a (numerically generated) sample of these random variables. We compute the  integrating the joint probability density function of F₂, F₃, F₄ on the spheres of center   and radius A_α,M > 0. These integrals are approximated using the appropriate finite sums of the joint histogram of F₂, F₃, F₄. We determine s_α,M as the infimum  of the A_α,M such that  ≤ α.

Finally we consider the problem of choosing the parameter values that define H₀. 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.

 

 

5 Numerical experiments

 

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 = ,  =  be the solutions of (2.1), (2.2), (2.3), (2.4) at time t = T . We approximate n independent realizations , i = 1, 2, . . . , n, of the random variable  integrating numerically n times (2.1), (2.2), (2.3), (2.4) in the time interval [0, T ] using the explicit Euler method and a suitable random numbers generator.

We choose T = 1, m = 10000, n = 100, ε = 0.1, ρ = −0.2, ₀ = = 0 and v₀ = = 0.5. That is:

 

(ε, ρ, ) = (0.1, −0.2, 0.5),           (5.1)

 

are the unknown parameters of the normal SABR model that we want to recover as solution of the calibration problem. The synthetic data , i = 1, 2, ..., n, are obtained approximating with the explicit Euler method multiple independent trajectories of (2.1), (2.2), (2.3), (2.4) with the parameter values given in (5.1) and looking at the computed trajectories at time t = T = 1. That  is for n = 100 and i = 1, 2, . . . , n, let  be the approximation of   obtained in this way. The set   = { , i = 1, 2, ...,100} is the data  sample of the statistical test used to solve the calibration problem of the normal SABR model. In a similar way when we choose T = 100 we generate the data set  = { , i = 1, 2, ...,100}.

The data sets  and   used in the numerical experiment that follows can be downloaded here: ,  .

 

We consider the following calibration problem: given  and   and the significance level α, 0 < α < 1, determine the values of the parameters (ε, ρ, ) of the model (2.1), (2.2), (2.3), (2.4) with significance level α.

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:

 

   :           (ε, ρ, ) = (ε^∗, ρ^∗, ) =  (0.1261, −0.3356, 0.515),           (5.2)

 

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 . To perform this test the corresponding threshold r_α,100 must be determined. For this purpose we build a sample of N = 1000 (approximate) realizations of the random variables , defined in (3.2) when n = 100, T = 1 and    is true integrating numerically (100000 times) with the explicit Euler method and the choice previously described of the discretization parameters the normal SABR model (2.1), (2.2), (2.3), (2.4) (when    is true) in the time interval [0, 1]. Moreover we approximate the joint probability density function of the previously defined random variables   , with the corresponding three-dimensional joint histogram associated to the sample of numerousness N = 1000 of the random variables , that has been generated. Proceeding as suggested in Section 3 we determine an approximation of r_α,100 denoted with _α,100. For simplicity we identify r_α,100 and _α,100, that is we assume r_α,100 = _α,100. Given n = 100 and the null hypothesis (5.2) Table 1 shows the values of the threshold r_α,100 assuming r_α,100 = _α,100 as a function of α determined with the previous procedure.

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     made of n = 100 observations of the random variable  compute the point =( ,   )  ∈ R³ associated to the data sample   and compute the point , )∈ R³,   where the quantities ,  are the moments    given by (2.14), (2.15), (2.16)  calculated when t = T = 1, ε = 0.1261, ρ = −0.3356, = 0.515, that is ,  are the moments    evaluated when t = T = 1 and the hypothesis (5.2) is true. We have = (0.2674, −0.0177, 0.2277) and  = (0.2674, −0.0177, 0.2197).

Let α = 0.01, 0.05, 0.1, and _α,100 be the corresponding thresholds when  is true shown in Table 1, the decision rule of the statistical test that has statistical significance α is given by:

 

-      if ||  −  || ≤ _α,100 retain (do not reject)   , with significance level α;          (5.3)

-      if ||  −  || > _α,100 reject   , with significance level α.          (5.4)

 

In this specific experiment given the data sample   the hypothesis (5.2) is retained for the values of α considered in Table 1.

 

:  (ε, ρ, ) =  (0.1261, −0.3356, 0.515), n = 100

α α,100 0.01 0.05 0.1 0.29 0.19 0.16

 

Table 1. The threshold r_α,n = _α,100 as a function of α for the null hypothesis .

 

For example let us fix the attention on _α,100 when α=0.1 that is on _0.1,100=0.16. The threshold _0.1,100 divides the moments space (i.e. R³) into two regions: the rejection region R_α,n=R_0.1,100      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   and radius   _0.1,100=0.16  and the rejection region R_α,n=R_0.1,100    is its complement.  The retain region and the rejection region are shown in Figure 3.

 

 

Figure 3. The retain (i.e. the sphere) and the rejection (i.e. the complement of the sphere) regions of the multiple trajectory statistical test.      

 

In the following animations we show a cloud of points   when the numerousness k of the sample  used to generate the points of the cloud varies from 10 to 1000 with step 10, that is when k=10, 20, ….,1000. In these animations the red diamond is the point . Animations 1, Animations 2, Animations 3   show  a cloud made of 1 point, 50 points and 500 points respectively. It is easy to see that when the numerousness of the sample k increases the cloud concentrates around the point .

 

Animation 1.

Animation 2.

Animation 3.

 

 

The second numerical experiment presented consists in testing the null hypothesis   given by (5.2) with the statistical test described in Section 4 using a sample of synthetic data.

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   be the approximation of a realization of , i = 1, 2, ..., M, obtained integrating with the explicit Euler method one trajectory of (2.1), (2.2), (2.3), (2.4). Let us choose M = 10, ∆t = 20, ε = 0.1, ρ = −0.2, ₀ = = 0 and v₀ = = 0.5. That is the unknown parameters of the normal SABR model that we want recover as solution of the calibration problem are given in (5.1).  The set   =  is the data sample used to solve the calibration problem.

The data set used in the numerical experiment that follows can be downloaded here: .

In [4] we choose the weights , i = 1, 2, ..., M, decreasing exponentially in time since  the moments (2.14), (2.15), (2.16)  increase exponentially in time. The choice of the constants  ,  i = 1, 2, ..., M − 1, is due to the necessity of “keeping memory" of the path of observations made and coincide with the choice done in Section 4.

We want to test the null hypothesis (5.2) with statistical significance α. First we determine the threshold s_α,10 = _α,10 corresponding to the null hypothesis as a function of α. Note that the meaning of the notation s_{α,10, α,10} and of the assumption s_α,10 = _α,10 is analogous to the notation and the assumption made on r_α,100 and _α,100. For this purpose we build a sample of N = 1000 realizations of the random variables F₂(M), F₃(M),  F₄(M) defined in (4.5) when   is true integrating numerically with the explicit Euler method the normal SABR model (2.1), (2.2), (2.3), (2.4) when is true, on the time interval [0, 200]. We compute (M) = ( (M), (M), (M)), where  (M ), k = 2, 3, 4, are given by (4.5), when is true, using the importance sample Monte Carlo procedure described in [4]. We have (M) = (0.607, 0.117, 16.449). The values of s_α,10 = _α,10  obtained using the procedure described in Section 4 when  is true and α = 0.01, 0.05, 0.1 are shown in Table 2.

Finally, given the statistical significance level α and the data sample  made of M = 10 observations we compute the point =(  ∈ R³ associated to the data sample . In this specific experiment from the data sample  we have  = (1.684, 6.435, 24.604). Table 2 implies that the hypothesis (5.2) tested on  is retained for the values of α considered, that is for α = 0.01, 0.05, 0.1.

 

:  (ε, ρ, ) =  (0.1261, −0.3356, 0.515), M = 10

α sα,10 0.01 0.05 0.1 571 179 50

 

Table 2. The threshold s_α,M =   _α,10 as a function of α for the null hypothesis  .

 

 

 

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.

 

 

6  References

  

[1]   A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher trascendental functions, 2, McGraw-Hill 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 Ill-Posed Problems 21 (2013), no. 1, 59-84.

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[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 Kontorovich-Lebedev transform, Communications on Pure and Applied Analysis 10 (2011), no. 2, 745–760.