 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 Email: lorella.fatone@unicam.it, francesca.mariani@univr.it, m.c.recchioni@univpm.it, zirilli@mat.uniroma1.it 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 . The acronym “SABR” means “Stochastic-” and comes from the original names of the model parameters (i.e., ,,) . 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 . 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 ttx, 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  assumes that the dynamics of the stochastic processes tv0,tx, t, , is defined by the following system of stochastic differential equations: v0t dd,ttttxxvWt0, (1) dd,tttvvQt0, (2) with the initial cond itions: Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 346 00,xx (3) (4) 00,vvwhere 0,1 is the -volatility and 0 is the volatility of volatility. Note that in the original paper  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 tQ0,t0WQ dtWdQ0t ddd, 0,ttWQ 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tv0v0v0t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,tttxvWt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 tx, , 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,ttttxxvWt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 . In the CIR 0tvvtv, 0t, 0t. Note atmodel (7), (2), (3), (4) CIR model (with no drift) when 0th the reduces to the. When 0 the volatility is governed by (2). ISABR l (7), (2) when the initial conditions (3), (4) are positive (with probability one) negative forward prices/rates can be avoided. Finally, the choice 1n the mode in (1) produces: dd,0.xxvWttttt (8) (8), (2), (3), (4) is knowThe mSA 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  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 tx 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 chhoice m pro)ith prade in this paper of studying t ic entrate on the study of t he log-es/rateshe log-norannormal 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 ) 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 prdom variable is represented as a compound random variable and that the SABR model can be seen as a sto- chastic state space model . 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 concrmal 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, , 0v 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 eCopyright © 2013 SciRes. OJAppS L. FATONE ET AL. 347lognormal 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  in the study of the normal SABR model. Note that the data set used in the first calibratio n prob- leproach to study the calibration prob- le least-squares prhat, extending the results presented in , it is posignificance levels in th is paper. of the moments of the lof 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 apm 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 , . 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 nonlinearoblems 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 tssible 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 somegarithm 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 oLet us deduce closed form formulae for the m logarithm of forward prices/ratSABR model. Let us consider the model given by: dd,0,ttttxxvWt (9) dd,tttvvQt0, (10) with ositive initial conditions (3)assumption (5). That is, we assumep, (4) and the 0x (with 0probability one) and 00v (with probability one), so that Equations (9), (10) imply 0tx (witability one), and 0tv (withbility one) for 0t. Let h prob probalnttx, 0t, belogarithm of the forward prtes. Using the variables t the ices/ra, te storeEquations (9), (10) and the initial conditions (3), (4) are rewritten as foow v, 0t, th ll s:chastic diffential 21ddd,0,2ttttvtvWt  (11) 0,dd,tttvvQt (12) 00 0ln ,x (13) Starting from the expression transition robability density funpr00.vv (14) obtained in  for the pction of the stochastic ocesses t, tv, 0t, we deduce explicit formulae (eventually involving integrals) for the moments with respect to zero of t, tv, 0t, and of tx, tv, 0t. In particular, we derive closed form formulae (that do not involve integrals) f the momenw rt to zero of tore first fivts ithespec, 0t. In , using the backward Kolmogorov Equation associated to (11), (12), the following formula for the transition probability density function Lp of the sto- chastic processes ,ttv, 0t, implicitly defined by (11)-(14) has been obtained: Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 348  ,,,,,1Lpvtvtde , ,,, ,,2,, ,,,0,0,kLkgttkvvvv tttt   (15) where is the imaginary unit and we have t, t tvv, , , tvv,0tt, 0tt. In (15),when 0 we must choose 0t, 0vv n. Thefunctio Lg is given by:  20, ede sinh,,,,,0, 1,1,kvvv2822,, ,,evvsLvgskvKkvKkvskvv      (16) where the function K of denotes the second type modi- fied Bessel functionorder  (see  p. 5). Finally, 2k, ,k is defined as 2221kkk 22i,.k (17) The function Lg can be rewritten as follows : 2222222222281,,,,eekLvgskvv vv2220212cosh2220,ed sinhsine2de ee,,,,,0,1,1.vvssusvv kvv kyv vvvuvvyyuuussyskvv      (18) Formulae (15), (16) or (15), (18) give Lp ilitas a two dimensional integral of an explicitly known integrand. Note that in (15) the transition probaby density function Lp is written using the variables lnttx, tv, 0t. It is easy to obtain a formula analogous to formula (15) for the transition probabilritten in the original variables tity density function wx, tv, 0t. Formulae (15), (16) and (15), (18) are representation formulae for Lp that hold when  . These 1,1 for- mulae have been obtained in . Previously when 0 for Lpnly 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 tx, tv, 0t, of the lognor mal SABR model (9), 3), (  ,,,,ded,,,,, ,nmnm tvtvvpvtvt  (10), (4), namely, 0,,,0,0,,0,1,Lvttttmn   0n.(19) We distinguish two cases, that is: the case and the case When 0n. 0n we have: , 0, 0,,,d d,,,,mLtvtvvp vtv   0,,,,,,d,0,,,,,,,,0,0,0,1,mLmLtk ttkvvvvgttvvvttttm 0de dkmkvvg    (20) where  is the Dirac’s delta. From (18) we have: 22020232 cosh2200d,0,,,,2ddeee,,,0,1,1,0,1,.mLvvyymyuvvvvgsv vssvv ysv m    (21) Using formula (29) on pa ge 1 46 of  we have: 222228eed sinhsinessusvuuu  1232 22 120dee2 ,,,,vyvvyyvvvvvK y  and this implies that: (22) 22222280, 2220cosh1202e,,,e 2d sinhsined()e,,,,0,0,0,1,.smsmusyumvtvt suuusyK yvtttt m      (From Formula (24) on page 197 of  it follows that: 23)  cosh1/20sinh 2de ,sinhyu uyK yuu   . (2on page 92 of  we can conclude4) From Formulaes (23) and (24) and using formula (37) that 0,0 ,,, 1tvt ,, 0,vttt0t , ,and that ,tv0,1 ,,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: , Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. Copyright © 2013 SciRes. OJAppS 349  222220,28220212220ee2ed sinhsine21d,,,,0,0,0,1,.2coshmusssmtk gvv uuussvvvt tttmvv vvu      (25) Formula (25) is equivalent to Formula (23) and shows at in order to have a convergent integral we must choo- 01,,,d dd,,,,,2kmvt vv ttkvv     LWhen using Formulaes (15) and (18) in the de- finition (19) we have: 0n thse 12 32m . 22222,022 2282202222211201,,,de dde,,,,,21ee2ed sinhsine212coshdeknmnm Lsnsusnvvmtvtvvk gttkvvnnvv uuussnnKvvvvvv          u22 ,2cosh,,,0,0,1,2,,0,1,.vv vvuvttttnm   (26) Note that for when with respect to zero of the variables t11,2,,n1111nn  or 1111nn the integrals appearing in th nand the conding mothe moments logno(26) wi 11, 2,,,n ments are 0,1,,m rrespocon- vergent. A similar existence condition for of thermal SABR model has already been derived in , however in  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. .tv, 0t,  ,,,,ddnmnm tvtvvp  0,,,,,,,,,0,0,,0,1,Lvtvtvttttmn   (2) The procedure used here to calculate the moments (27) generalizes the procedure used in  to calculatem7 the oments with respect to zero of the variables tx, tv, 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, ,nmments rt othe F,nm     ,000001dd d,,jmvv kkgttk,,,, ,ddd,,,,,d,,,0,0,,0,1,.nnjnm LjjjjnnjjmLkjjntvt vvjknvvgttkv vjkvttttnmj     (28) Let us calculate the integrals contained in (28). For let  0,1,jjG be the-th order derivative with jrespect to k of the function Lg evaluated at 0k, that is, let ,,d d,0,,jjjLGsvvgks vv, ,,svvthe previ. To simplify the notation we have omitted in ous formula, and we will omit from now on, the dependence of Lg and of jG0,1,j, fro, m  and . Using the functions jG, 0,1,j, Formula L. FATONE ET AL. 350 (28) can be.We restrict our attention to rewritten as follows:  ,00,,,d,,nmnnjjjtvtnvv Gj   (29) ,,,,0,0,,0,1,mjttv vvttttmn  ,0 ,,,ntvt, , ,v 3 in thn, 1n,0, 0,ttt tne solution of the ca,2,3,4 . The choice0,1,. In fact, in libration problems of considering Section we will use 0m,0 in lems isthedue to moments used to solve theob arkets the variable t calibration prfinancial m the fact that in the x, 0t, and, as a consequence, the variable t, 0, can beserved, while the variable tv0, 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,, ,,,,,nnnnjjsv tvtnDsv tobs rved be es obd, tan0,, ,0,1,jjjsttvn. (30) where 000d,d,,d,,,,,,0,1,,djjjLkjDsvvGsvvvgskvv sv jk(31) and  0d,,d,,,,d,, ,0,1,.jjLjDskv vgskvvkskv j (32) We have  000,,,dd,,, ,d,,0,1,.jjkjLjkDsvDskvvgskv vksv j (33) The functions jD, tial value 0,1,jproblem, can be detersolving the inis deduced belofunction mined by w. The Lg (see is the solution of the following initial value problem): 2220,, ,,, ,.Lgkvvvvk vv  (35) Equation (34) is the Fourier transform of the bKolmogorov 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: 22222200 00222DD DkvvDkv22,,,,2222221LLLLLgggkvvgkvsvvkv g skv (34) 201,,,,2vvskv D skv (36) 00,,1,,.Dkvkv   (37) When 0k, Equations (36), (37) reduce to: 222002,,,2DDvskvsv , (38) 00, 1,.Dv v  (39) Equations (36)-(39) define initial value problems satis- fied by and by, respectively. Recall thais given in (33). It is easy see t0Dlation between hat 0D and t the re- 0D0D toDsv0,1, this ,svsolutio, n usis ing a laa solution of (38), (39). Let us obtain procedure that ter will be extended to deduce explicit formulae for the functions jD when 0. Defining Lj0,  and0L 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: 2s) 220002,,,28LLLs (40200,e.L (41) ,The solution of (40), (41) is given by:  200,d,e,,, (42) Lss s   whre e2222801,ee,ssss2,.2s (43) ntegral (42) when The i212qj and 0j can be computed using the formula 2222041882eee ,,,.sqsqsq ) d,ee eqqsqs  (44It follows that the solution of problem (40), (41) 0L Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. Copyright © 2013 SciRes. OJAppS 351is given by: ,,,,,1,jjDsvvLsvsvj 2,, (51) 2228 820,eee e, ,ssLs s  . (45) From (45) it follows that and considering the change of variable ln v, v, define the following functions ,, ,jj jLsv LseLs, , s, 1, 2,j.  00ln,,lvDsvvLs v  (46) 2ne1,,.vsvLet us derive the initial value problems thsaIt is easy to see that (47), (48) imply that the function , is the solution of the following initial value problem: 1Lat are tisfied by the functions  0,,,jjDsvDskvk222 02111232 0e281e, ,2LL DLsDsv,  (52) , erentiating times with (34), (35) and sutin,srespect to ,v 1, 2,j. Diffjbstituk problemg 0k in the resulting equation 1j the s, we have that whenfunction 1,Dsv, ,sv, satisfies the following initial value problem: 10, 0,,L (53) 2222102,, ,22DDvDsvsv1v (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 jL, 2,3, ,j satisfy the initial value problems:  e funcfollowing ine problem: 222 32 22123211e282ee2,,2,3,,jj,jjjjLL jLj DsDjjDsv j   (54) jitial valu2222221221=122,jjj2,,2,3,,jjDDjvjvDsvDjvD (49) (50) Let us assume that jvvsv j 0,0,,2,3, ,jLj (55) where ,,jjDs Ds0,0,,2,3, .jDv vj  e, , s, 2,3,j. The solution of (52)-(55) can be written as follows:     0032232 1,dd ,1ee,e,,22,,1,2,,sjjLs sjjjDD Dsj    (56) where we have defin ed 21(, )jjj1,0Ds, s, . Let us give the explicit expressio ns of  ,,ln,,jjDsvvLsv sv *00,, ,jtv ,t when 1,2,, and of . Recall that 3, 4j ,=Dsv ,ljjvLsn ,v and that , ,sv j1,2,,0,1Dsv, we have: ,.sv  Using Formulae (44), (56) 22,e1e,,,ssvsv  (57) 2112Dsv 222 2222223256461e 1e,e1e 231e1e,,,256ss ssssDsvvsv 2332e1essvv       (58) L. FATONE ET AL. 352 22 222 22223423 356326345661e1e1e 1e1e,6 e18e2318 10181e 1e3e 56ss sssssssvvDsvv  2s           222 210es22222547910691415151e 1e 1e 13e 6042101511e1e1ee34907030ssssssssvv            ,, ,sv(59) and finally , 4123,,,,,,D svI svIsvIsvsv1I, 2I and 3I (60) where are the fol lowing function s:  2222222 22224526 36154791031054101e,4e 1e91e51e 1e 11e42 s    e4e457551e4esss sssssssvIsvvv      2222222291065 9121415215721e1e4101551e 1e1e1e1e2e353270 18841415essssssssvv           22 222615202111e1e1e1e9,,,630 225700105ss ssssv         (61) 22 222 22224556 791061026914151e1e1e1e 1e,6e6 e56 2191531e1ee10 27ss ssssssvvIsvv            2s2151e ,, ,79sssv (62) and 22 2222 222579101036912 14152151e 1e 1e,12 e3518301e 1e1e1e2e 945 127015ss ssss sssvIsvv         22222222269141515711 1518202211e 1e1e6e 270 70901e 1e1e1e1ee33 2770 150126100sssssss ssvv                   222222181322 27282810511e1e1e1ee,20819 1986384ssssssvsv     ,. (63) Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 353 Recall that stt, 0. From (30), (46), (57)-(60), choosing 0t, 0vv we have that st of tand 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 0001000,,,, ,,,tvDtvDtvtv   *220000001001 02000,,, 2,3, ,,,,tv DtvDtvDtv Dtvtv    *3 2300 00001002 0,,,3,3,tv DtvDtvDtv   3 000,,,, ,Dtvtv ( 7)   *4 4400 0000102020030400 0,,, 4,6,4,,,, ,tv DtvDtvDtv DtvDtv tv   * ** *. 1, nor 2, ma 3, l SABR m4 o, , s on0v and on the time depet. In particular, *1Lnd, 0v and , while ont*2, *3, *4 depend , 0, v and t. T mo me ohent *0 does not dependn , 0v, , t ems for new lve alo- for in andfoandthgom car the lo aree cal focourse asre annnot caliogno rmthe mme mclosed formun the fnext ibratiod prus rmuleastall the remadefi o d that, as alSectiorm formulae of tl 8oencbe rmnts of t moormnae canininnm ouse prre ind ial SABR he lomeulobl beg mincrvonnt fo themgnormrmae usedems di dedomenlved. studodel. Ful in tced ts Noteyhscusse ofoal SABRae annoe (at l,nm bratulaes (6cedsectevio in neiion 5)odel are in ons tuslyprinn (,nmready saiprobl-(68 the o s. Anple)27)be) odSection 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 , 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 ththat 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 time0v 0T ectories ociated to the in 0tindependent . The set of data of the first calibration prob- leme set of the logarithms of the forward prices/ d at time is thrates observetT in this set of trajectories. In particular, letting be a positive integer, we consider independenpies nt coniT, of variable 1,2,,,inthe rando Tm solution at time tT of (11)-(14). For 1,2,,in let ˆiT be a realization of iT. The set: 1ˆ,1,2,,,iTin (69) is the data sample used in the following calibration prthe 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 0v of the lognormal SABR model (11)-(14To solve this calibration problem we com). pare ththeoretical 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: 11,, 1,2,3,4,njijTinT jn (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: 11ˆˆ,,1,2,3,4.njijTinT jn  (71) The unknown parameters , , 0v of the normal SABR model can be determined as solutions of the fol- lowing constrained nonlinr least-squares problem: 04*00,, 1ˆmin,,,jj jvjTv nT72) sbject to the constea (uraints: 2, 00, 11,0,v (73) where j, 1,2,3,4,j are non negative weights that will be chosen in Section 4. Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 354 Note that roughly speaking when increases the “q nuality” of the moments ˆ,,jnT 1,2,3,4,j estimated from the data sampmer to lve sibrained no n valuesthe 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 constnlinear 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. Tothe moment asymptotic expansion of the moment *1orm0. Let ch that 120TT, we approximate the first- and the send-order Taylor's expansions e point t2t *1oximation o12,TT of bas*1 whenco The first-or0.f 1tT 0vis us. Thed to obtain the seconde initial gappr uess fooximation or th1e param* wheneter 2tT -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 formullution of pand thrat thakes colem (72)e aviailabmput, (73). Th comal guility oonallyis mputationaf the exp very eans that, whl cost, it is for the paramlicit en eter momethe soae mob atiesses efficient necessary, at an affordablepossible to use multiple init. The defined in (69) used in Calibra- tion problem 1 to formulate problem (72), (73) must be codata sample 1 mpleted with the auxiliary data 1ˆiT, 2ˆiT, 1, 2,,,in (observed at tim e 1tTand2tT) 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 1TT 2TT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 0v (when 1TT) or, the initial guess for  (when 2TT). 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.roblethe data ske 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 01teg and le,, ,Mtt t be 1M discrete timues sut 1,iitt e valch tha 1,2,,,iM and 00t. Recall that the times it, 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 ofMtt t For 1, 2,,iM let us denote by ˆi the logarithm of the forward prices/rates observed at time itt 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, 1M discrete time values 01,, , ,Mtt t such that 1,iitt 1,2,,,iM and 00,t and given the data set  defined in (74), determine the values of the parameters 2,  and 0v of the lognormal SABR model (11)-(14). The Calibration problem 2 can be formulated as the following constrained nonlinear least-sq uares problem: 024*,00,, 11ˆmin( ,,),Mjijjiivijtv (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 ˆji, 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 proThe numl expshows t the behavioe conqres problemr the numoptimization orent formin 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 Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 355second-order Taylor’s approximations ofwith base point . From the first-of th (i.e. ng thervations *1 (see (65)) order Taylor's 0te trajectapproximation at 0t of *1 evaluated at the beginning ˆory usie obsi with i small, that is 1,2,,10i in the numerical example ofs Sectioess of econd n 4) -or we obtain Tay the inlor's itial gu0om the der approximation at 0tv. Fr ated at the esing servations ˆiof 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 *1ddt with base point is used to construct the initial gnum imization algorithm used to solvat 0t erical opt1uess 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 0v and the first-order Taylor’s approximation 0 of t*1dd 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 respeare tado hfirsterit is possible to tgnol haal mthe nomore accurate value of the parameters can be obtained increasing the numerousness of the data sample used in ibn probleriman 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’sicportfolios. In these contexts it is not reallknow the exact values of the model paramsense 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 theperiment. Let 0T be given and n, m be positive integers. Let tTm be the time increment and itit, 0,1,, ,im be a discrete set of equispaced time values. Let mtT, mtTvv be the solutions of (11)-(14) at time tT. The n independent realizations ˆiT, 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 ). In the numerical example we choose 1T, 100m, 1000n, 0.1, 0.2 , 001 and 000.5vv. 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 001). These trajectories are ap- proximated integrating numerically using the explicit Euler method the model (11)-(14). In particular, when 1000n let us denote by 1ˆˆ,iT 1, 2,,,in the approximations of 1ˆ,1,2,,,iTin 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 001). The set: Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 356 1,10001,1,2,,1000,iTnT i  (77) is the sample oftic data used to solve Calibration problem 1. In a similar way when we choose 100T, 1000nˆˆˆ synthe (leaving the other parats unchanged) we generate the data set100, 1000100ˆ,1,2, ,1000iTnTi . 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 samˆmeer t-squaple ed in thnumericalUsing and the first- and second-oor’s appoint ugg io ˆˆ1, 1000Tn. The data sets 1,1000ˆTn and 100,ˆTnuse experiare avai1, 1000ˆn and 100,ˆT1000 lable at . ment Trder Tayl0, as s1000n proximations oested in Sectf 1 with base n 3, we find *t00.515inv and 0.099in as initial guesses of, respectively, 0v and . Note that in the notation of Section 3 we have chosen 11TT and 2100T. The initial guess in of  is chosen as 0.05in . Given 0,,0.099,0.05,0.515inin inv as initial guess, the nonlinear least-squares problem ) is sing 1, 1000ˆTn 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 1j(7der of ma2), (73solved 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 jast-squares prob increasolved ustion foun (78) The relative -error of the initial guess 0,,0.099, 0.05,0.515inin inv is: **00,,,,0.076,0.222,0.508 .vv *2L) is 0e -error of the so lem withe sensitivity of the solution procedure respect to the presence of noise in the0,,0.099, 0.05,0.515inin inv with respect to the “true” soluti2on (76.275. The relativLlution (78) of the least-squares probh respect to the “true” solution (76) is 0.062. To study tproposed with data we add noise of known statistical properties to the synthetic data contained in 1, 1000ˆTn, 100, 1000ˆTn 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 follooisy” data samˆˆˆ2i1,100111 ,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, 1000ˆTn. Given 100 1Trandrandtion (0,ˆTn comute the relatiwepve -error bee” solution (76) and the ibed ab2Ltween solution of Calibration problem 1 found with the numerical procedure descove. We repeat the entire procedure 1N times and we compute the mean of the 2L-relative errors of the 1N solutions found. We denote by the “trur11,,NnE this mean relative erroelative errors r. The mean r11,1000, 1000NnE obtained when 11000N, 1000n 1and 0.01,0.05,0.1 are show in Table 1. nAs 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 invothe experiment. Wensider and we con- stthe correspondingin cople “nolved in 10000n00ata druct the data sams 1,10 0ˆTn, 100, 10000ˆTn and samples 1,10000ˆ,Tn 100, 10000ˆTn. The data sets 1, 10000ˆTn an100, 10000ˆTn used in the numerical experiment are available at . Given 0sy” 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 NnE11,=1000,=1000 as a function of 1. 1 11,1000, 1000NnE 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, =1E0000 as a function of 11.  11,1000, 10000NnE 0.01 0.104 0.05 0.141 0.1 0.170 Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 357and a time increment 0t, let ,itit 0,1, ,,iM be a discrete set otion times. Let ˆˆif observa be the approximat of ,1,2,,,itiM ion of a realization obtained integrating with tt d one trajectolognormodel (11)-(14). Let us choose 100,M 1,the explici ofmal SABR Euler methory the  0.1,0.2, That is, we wan 001 and 000.5.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,iti M along one trajectory of the lognormal SABR model. The set 2ˆˆˆ,1,2,,100,ii (80) e unknownarametersis the sample of synthetic data used to solve Calibration problem 2. The data set 2ˆ used in the numerical xperiment is available at . eProceeding as discussed in Section 3 using the data set ˆ, the least-squares fit of the first-order Taylor’s n of * with b2apprat tof secopointinoximatio ase point evaluated ves itial guess set fit of th e ap with base d at gives 10, giata lor'suateitial gu 0t as inares *1,i, 0vnd 1,2 ,1i. Using the d-or aytval0.114, 00.533inv2ˆ, the least-squoximationit, 91iof pr ess der T of 0 e,92,100, as in. To obtain an initial guess of tharae pmeter we take ae prices go doploiting this dvantagundewn the of thrstande vofact 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 usfinagoethat ually nncs uegatiial markets whp and i11enceversa. thExv 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, 0.5,0.533inin v that uses as ˆ,iin  the Calibration problem 2e oˆ 20,30inonFMINCON routincorresponding to the central part , ,80, of 2ˆ 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 2ˆ (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 ,ija subset of datatrajectory) to, 20,30,,80,i 1, 2, 3,4,j are chosen such that the addenda of (e order of magnitude. Using 2ˆ as data and 75) are of0,,0.114, 0.5,0.533inin invsolution of problem (75), (73) found t has i is: e samnitial guess the e i***00,,,,0.019,0.168,0.472 .vv  (81) The relative 2L-error of thnitial guess 3“trsolu 76s point out thta ofsu rateta obss. owsensitivity oraed0,,0.114, 0.5,0.53inin inv with respect to the Calibeves stuue” solution (76) is 0.551. The relative 2L-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 2ˆ and we study the quality of the solution of Calibration problems 2 found as a function of the noise properties. In particular given 20d r, let u let us consider the following “noisy” data sam- ple: 1,100 22ˆ112 ,irand iwhere rand is a random entry taken from a uniform distribution on the interval ,1 . Given 20ˆ0ˆ2,,, (82)  we compute the relative 2L-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 2N times and the mean of the -relative errors of the 2N solutions found is calculated. Let 2L22,NE be this mean relative error. The mean relative errors 22, 1000NE ob- tained when 1000N2 and 0.01,0.05,0.12 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 thn rproblem studied in the first onequantity of the data and abovqu ngdeduced from the data sample 2ˆe calibration . This is due e all to the quality oes problem eda 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 1,1000ˆTn and 100, 1000ˆTn. That is, there are 100 data in 2ˆ and 2000ˆ dat 1, 1000ˆTn100, 1000Tn and the moments estimated from the data contained in 1, 1000ˆTn, 100, 1000ˆTn 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 NE22,=1000 as a function of 2. 1 22,1000NE 0.01 0.239 0.05 0.265 0.1 0.369 Copyright © 2013 SciRes. OJAppS L. FATONE ET AL. 358 Calibration problem 2. In the context of finance the natural way of fw veethan reor exhn the Heston me lognodel this p hour purposis paper. der a timseries of enge raten the prates 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 . 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 . 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 UYork) of one euro expressed in U.S. dollars. Recall that a ade of about 252adhe data set shre 1 i thedel ton inse the form (11)-(14) we take tin 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, ,1ij, 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,,0.05,0.05,0.05inininv. 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. Occasiwionally  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. recCopyright © 2013 SciRes. OJAppS L. FATONE ET AL. Copyright © 2013 SciRes. OJAppS 359 is moved along the data time sThe parameter reconstructed changes when the windoweries. This is cor- rect since 0v 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,, ,v0v obtained calibrating the lognormal SABR modreleasttiREFERENCES  P. S. Hagan, D. Kumar, A. S. Lesniewski and D. E. Wood-ward, “Managing Smile Risk,” Wilmott Magazine, 2002pp. 84-108. http://www.wilmott.com/pdfs/021118_smile.pdf  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.  F. Black and M. Scholes, “c,” Journal of Political Econo-my, Vol. 81, No. 3, 1973, pp. 637-659.  L. Fatone, F. Mariani, M. C. Recchioni and F. 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