Closed Form Moment Formulae for the Lognormal SABR Model and Applications to Calibration Problems

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 inverse problem that consists in determineing the values of these three parameters starting from a set of data. We consider two different 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 constrained 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 problem 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 calibration 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).


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 practice of mathematical finance, for example, it is widely used to price interest rates derivatives and options on currencies exchange rates.

Let
be a real variable that denotes time and t t x , t , be real stochastic processes that describe, respectively, 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 , is defined by the following system of stochastic differential equations: d d , with the initial conditions:   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 0 0 , , t , , are their stochastic differentials and we assume that: 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 probability 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 The value of the parameter determines the forward prices/rates process, that is, it determines Equation (1).The most common choices of  are: Setting 0   in (1) the forward prices/rates process reduces to: . (6) d d , The corresponding 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 deviation lognormally distributed.This permits to the forward prices/rates t x , , to become negative.Usually 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 circumstances negative quantities such as negative interest rates can be considered.
in (1) gives the following forward prices/rates process: 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 0 t v v t v , 0 t    , 0 t  .Note at model ( 7), ( 2), ( 3), (4) CIR model (with no drift) when 0 th the reduces to the   .
Finally, the choice 1 n the mode   in (1) (8), ( 2), ( 3), ( 4) is know The 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 parameter 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 (wit bability one) of the forward prices/ rates t x is guaranteed for 0 t  when the initial conditions (3 , (4) are positive (w obability one).In particular when the initial conditions (3), (4) are positive (with probability one) the absolute value in (8)  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 stochastic state space model [5].Compound random variables 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 mod at is, we study the problem of determining the unknown parameters el.Th  ,  , 0 v  of the lognormal SABR model starting from the owl dge 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 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 kn e lognormal SABR model.The formulation of the calibration problems corresponding to these two sets of data is based on some new closed form formulae for the moments of the logarithm of the forward prices/rates variable.Using these formulae the calibration problems considered are formulated as constrained nonlinear leastsquares problems.The moments formulae are deduced extending to the lognormal SABR model a method introduced in [4] in the study of the normal SABR model.Note that the data set used in the first calibration proble proach to study the calibration proble least-squares pr hat, extending the results presented in [4], it is po significance levels in this 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 several fields of science and engineering has limited applications in finance.Instead, the second calibration problem 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 forward prices/rates.
An alternative ap m for the lognormal SABR model corresponding 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 variations.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 function 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 presented here a substantial computational advantage in comparison to the method suggested in [6], [7].A similar statement holds when the method presented here is compared 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 considered can greatly benefit from the availability of a good initial guess to initialize the optimization algorithm.In Section 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 parameter values obtained as solution of the calibration problems.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 lognormal SABR model are derived.In Section 3, the calibration problems for the lognormal SABR model corresponding to the two data sets discussed previously are formulated as constrained nonlinear least-squares problems.Finally, in Section 4 we solve numerically the calibration problems presented in Section 3 and we discuss the results obtained in numerical experiments on synthetic and on real data.The real data studied are time series of euro/U.S. dollar exchange rates.

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: d d , 0 , with ositive initial conditions ( 3) assumption (5).That is, we assume p , (4) and the 0 x   (with 0 probability one) and 0 0 v   (with probability one), so that Equations ( 9), (10) 9), (10) and the initial conditions (3), (4) are rewritten as fo ow v , 0 t  , th ll s:   Starting from the expression transition robability density fun pr obtained in [8] for the p ction of the stochastic ocesses t  , t v , 0 t  , we deduce explicit formulae (eventually involving integrals) for the moments with respect to zero of t  , t v , 0 t  , and of t x , t v , 0 t  .In particular, we derive closed form formulae (that do not involve integrals) f th e momen w r t to zero of t or e first fiv ts ith espec In [8], using the backward Kolmogorov Equation associated to (11), (12), the following formula for the transition probability density function L p of the stochastic processes , t t v  , 0 t  , implicitly defined by ( 11)-( 14) has been obtained: where is the imaginary unit and we have ven by: , , e where the function K  of denotes the second type modified Bessel function order  (see [9] p. 5).Finally, The function L g can be rewritten as follows [8]: We distinguish two cases, that is: the case and the case When where  is the Dirac's delta.From (18) we have: .
Using formula (29) on page 146 of [12] we have:  and this implies that: .
From Formula (24) on page 197 of [12] it follows that: on page 92 of [12] we can conclude 4) From Formulaes (23) and (24) and using formula (37) that Formula ( 25) is equivalent to Formula ( 23) and shows at in order to have a convergent integral we must choo- When using Formulaes (15) and ( 18) in the definition (19) we have: .
Note that for when with respect to zero of the variables t   [13], however in [13] no explicit integral representation formula for the moments like Formula (26) is given.
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 , 0 t  , of the normal SABR model.
Let us calculate the integrals contained in (28).For let . To simplify the notation we have omitted in ous formula, and we will omit from now on, the dependence of L g and of can be .


We restrict our attention to rewritten as follows: , , , where , d and .
We have .
, can be deter solving the ini s deduced belo function mined by w.The L g (see [8] is the solution of the following initial value problem ): Integrating both sides of (34), (35) with respect to v when v    we have: When 0 k  , Equations (36), (37) reduce to: Equations ( 36 is 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 L j  0 ,  and 0 L  through the relations: , (39) can be rewritten as follows: The solution of (40), (41) is given by: e e e , , , .
It follows that the solution of problem (40), (41 Copyright © 2013 SciRes.OJAppS is given by:   , e e e e , , From (45) it follows that and considering the change of variable 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 , erentiating times with (34), (35) and su tin and when 2,3, Moreover, from (49), (50) it follows that the functions j L  , 2, 3, , j   satisfy the initial value problems: following in e problem: , , Let us assume that where  .The solution of (52)-(55) can be written as follows: , where we have defined 2 1 ( , ) Let us give the explicit expressions of , and of .Recall that 3, 4 , e 1 e , , , , I and 3 I (60 where are the following functions: ,  The moments depend on the parameters of the log del given by: ease the formulae for n 1, closed fo observable quantities implicitly defined by ( 9), ( 10), ( 3), (4) such as (65)-( 68) are very useful to build computationally efficien methods to solve calibration problems.

Two Calibration Problems for the Lognormal SABR Model
Let us study the calibration problems of the lognormal SABR model ( 11)-( 14) annou ed 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 forward 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) asso itial conditions (13) 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: , that is: The unknown parameters  ,  , 0 v  of the normal SABR model can be determined as solutions of the following constrained nonlin r least-squares problem: s bject to the const ea ( u raints: where j  , 1, 2,3, 4, j  are non negative weights that will be chosen in Section 4.

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Note that roughly speaking when increases the "q n uality" of the moments  This set of data is realistic in several contexts of science and engineering where, for example, the observations 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 realistic for observations made in the financial markets where usually it is not possible to repeat the "experiment".That is, in the financial markets repeated observations at a given time 0 t  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 p m 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 prices/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 marts.roble the data s ke It can be identified with a time series of the logari rd thms of the forwa prices/rates observed in the financial market.
Going into details, let M be a positive in er t  11)-( 14).
The Calibration problem 2 can be formulated as the following constrained nonlinear least-squares problem: , subject to the constraints (73).The constants in (75) are non negative weights that will be chosen in Section 4. Note that when M increases the "quality" of the terms   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 num l exp shows t the behavio e con q res problem r the num optimization or ent form in Section 2. In particular, we use the first-and the fficult than Calibration blem 1. erica erience with problem (75), (73) hat ur of th strained nonlinear least-s ua (75), ( 73) is similar to the behaviour of problem (72), (73).This implies that the availability of a good initial guess fo erical algorithm used to solve (75), ( 73) is very helpful to obtain a satisfactory solution.In der to build this initial guess we exploit the mom ulae derived  .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  .

Some Numerical Experiments
In this section we discuss three numerical experiments.
In the first numerical experiment we solve the Calibration 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 exc ange rates).
The numerical experiments presented in this section can be "interpreted" as follows.As already said, the num ical experiment can be seen as a "physical experiment" 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 pr ing and to hedge y important to eters.In some setting of the first numerical ex "physical model" (i.e. in this case the lo rmal SABR model) where the parameters of the mode ve a precise physic eaning (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 possible way.N te that in this kind of experiments usually a the cal ratio em.The second and third numerical exp ents c e seen as experiments in finance or in a different context where it is not possible to make repeated 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 lognormal SABR model) are used to do option's ic 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 0 T  be given and n , m be positive integers.Let t T m   be the time increment and be the solutions of ( 11)-( 14  L -error of the tion (81) of the least-squares problem with respect to the "true" solution ( ) is 0.167.
Let u at the da ration problem 2 are pposed to be prices/ s observed in the financial markets, that is, these data are not affecte by noise as the da erved in physic H dy the f the solution of Calib tion problem 2 found with the numerical proc ure proposed with respect to the presence of noise in the data.We add noise to the synthetic data contained in 2  and we study the quality of the solution of Calibration problems 2 found as a function of the noise properties.In particular given where rand is a random entry taken from a uniform distribution on the interval     we compute the relative 2 L -error between the "true" solution (76) and the solution of Calibration 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 , N E  be this mean relative error.The mean relative errors  3.
Table 3 shows mean relative errors greater than the corresponding mea elative errors of the first numerical experiment.The calibration problem studied in the second experiment is more difficult than th observations.There is no a similar effect in   parameters is to acquire at e observation 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 th t exploits more deeply than Calibratio problem 2 the information contained in the prices.We do not consider this problem he .F ample, w e odel is used instead of th ormal SABR m roblem 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 ad c minimization algorithms to solve problems (72), ( 73) and ( 75 cast exchange rates and to compute option prices on exchange rates. We solve Calibration problem 2 using the real data of Figure 1 associated to a time window made of 1 20 M   consecutive observation times, that is the observations corresponding to 20 consecutive trading days, and we move this window across the data set discarding the datum corresponding to the first observation time of the window and inserting the datum corresponding to the next observation time after the window.The calibration problem (75), ( 73) is solved for each choice of the data window.We choose The initial guess of the numerical method used to solve the nonlinear least-squares problem (75), (73) has been chosen as follows:     0 , , 0.05, 0.05, 0.05 . Note that a data window made of twenty data has too few points to implement satisfactorily the asymptotic analysis of the moment formulae discussed in Section 3. Note that to make possible the effective numerical solution of problem (75), (73) the independent variables in (75), ( 73) have been rescaled.
The reconstructions of the parameters obtained moving the data window along the data set of 35) Equation (34) is the Fourier transform of the b Kolmogorov equation of the lognormal SABR model and the parameter ackward k   gate variable in the Fourie that appears in (34) is the conju r transform of the variable       .
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 value of the parameters can be obtained increasing the numerousness of the data sample used in ib n probl erim an b u   let us consider the following "noisy" data sample: enteri in the nonlinear least-squar when compared, for example, to the quantity of data contain in the data sample are of high quality due to the average over a sample of 1000 = n dollars and are the closing value of the day (in New year is m tr ing days and that a month is made of about 21 trading days.

Figure 1 Figure 1 .
shows the euro/U.S. dollar currency's exchange rate as a function of time.T own in Figu s available at [15].We use lognormal SABR mo interpret the data show In order to u model in the he logarithm of the data shown ormulating a ne rsion of Calibration problem 2 (i.e. a "single trajectory" calibration probl m) that defines more accurat model ely the

Figure 1 are shown in Figure 2 .
In Figure2the abscissa corresponds to the data window used to reconstruct the model parameters.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, Figure2shows that the parameters ,   reconstructed remain essentially stable when the ndow is moved along the data time series.Occasi wi onally  and  have spikes that probably indicate that the numerical procedure used to solve problem (75), (73) has failed.

Table 2 .
The comparison of Tables1 and 2shows a

Table 3 . Calibration problem 2: the mean relative error
1