Volatility of prices of financial assets

Interdisciplinary Studies of Complex Systems No. 12 (2018) 5-16 © L. Di Persio, N. Gugole V o l a t il it y o f p r ic e s o f f in a n c ia l a sse t s L u ca D i P e r s i o ,1 N ic o la G u g o le2 1 Introduction W h e n it com es t o analyze a financial tim e series, v o la tility m od ellin g plays an im p o rta n t role. A s an exa m p le, th e varian ce o f financial returns often d isplays a d e p en d en ce o n th e s econ d ord er m om en ts and h eav y-pea k ed and ta iled d istrib u tion s. In ord er to take in to a ccou n t for this p h en om en on , kn ow n at least from th e w ork o f [22] and [14], e co n o m e tric m o d e ls o f ch a n gin g v o la tility have b e e n in trod u ced , such as the A u t o r e g r e s s i v e C o n d i t i o n a l H e t e r o s k e d a s t i c i t y (A R C H ) m o d e l b y E ngle, see [13]. T h e idea b eh in d th e A R C H m o d e l is to m ake v o la tility d ep en d en t on th e va riab ility o f past observation s. T aylor, in [26], stu d ied an altern ative form u lation in w h ich v o la tility was d riven b y u n observ ed co m p o n e n ts, and has com e t o b e k n ow n as th e S t o c h a s t i c V o l a t il i ty (S V ) m od el. B o th th e A R C H and th e S V m od els, cov ered in S ection 2, have b een intensively stu d ied in th e past d ecades, tog e th e r w ith m ore or less sop h istica ted estim ation approach es, see [25], as well as con cern in g c o n c r e t e a p p li c a t i o n s , see, e.g., [9], and references therein. P arallel t o th e stu d y o f d iscrete-tim e e co n o m e tric m od els for financial tim e series, m ore p recisely in th e early 1970’s, th e w orld o f o p tio n p ricin g e x p erien ced a great co n trib u tio n given b y th e w ork o f F isch er B la ck and M y ro n Scholes. T h e B la ck -S ch oles (B S ) m o d e l, see [4], assum es th a t th e price o f the u n derlyin g asset o f an o p tio n co n tra ct follow s a g eom etric B row n ian m o tio n . L a tter ty p e o f a p p roa ch has b een also used w ith in th e fram ew ork o f interest rate d yn am ics, see, e.g., [6], and references therein. O n e o f th e m ost su ccessfu l exten sion s has b een the con tin u ou s-tim e S t o c h a s t i c V o l a t il i ty (S V ) m o d e l, in tro d u ce d w ith the w ork o f H ull and W h ite , see, [19]. A m a jo r co n trib u tio n was su ccessively due t o H eston in [18], in deed he d e v e lo p e d a m o d e l w h ich led t o a q u a si-closed form expression for E u ro p e a n o p tio n prices. D ifferen tly from th e B S m od el, th e v o la tility is n ot longer con sidered con sta n t, b u t it is allow ed t o va ry trou g h tim e in a stoch a stic way. In S ection 3 we will start from a su b-class o f S V m od els, w h ich is th e on e o f L o c a l V o la t il i ty (L V ), b e in g ch a ra cterized b y a d eterm in istic tim e-v a ryin g volatility, and th en we will con sid er th e general S V case, p ro v id in g in form a tion a b o u t the p ricin g eq u a tion as m ade, e.g., in [5] or, from a p o in t o f view m ore cen tred tow ards a pp lica tion s, in [12], and references therein. 1Department of Computer Science, University of Verona. lu ca.dipersio@ u n ivr.it 2Department of Computer Science, University of Verona. 5 6 L . D i P er s io , N . 2 G u g o le Discrete-time models D iscrete-tim e m o d e ls for th e volatility, as said in th e in tro d u ctio n , are b o rn in ord er t o analyze and re p ro d u ce th e b eh a vior o f real financial tim e series, w h ich are o ften ch a ra cterized b y a n u m ber o f s t y l i z e d f a c t s , i.e., features o f p articu la r interest. • T h e variance o f returns o f financial p ro d u cts is o ften s u b je ct t o th e so called v o l a t i l i t y c l u s t e r i n g e f f e c t . T h is m eans th a t th e returns sh ow an high serial a u tocorrela tion : p erio d s o f h igh v o la tility are follow ed b y p erio d s w ith th e sam e feature and viceversa. • A s n o te d in the p ion eer w orks b y M a n d e lb ro t, see [22], and Fam a, see [14], th e varian ce o f financial returns o ften displays a d ep en d en ce o n th e secon d ord er m om en ts and h eav y-pea k ed and ta iled distrib u tion s. • S tock returns o ften ex h ib it th e so ca lled le v e r a g e e f f e c t : th e co n d itio n a l variance resp on d s in an a sy m m etric w ay w ith resp ect t o rises or falls o f th e asset price. • T h e c o v a r i a t i o n e f f e c t ca p tu res th e fact th a t th e volatilities o f different financial assets co u ld b e correlated : large changes in th e returns o f an asset ca n in d u ce a sim ilar b e h a v io r in oth er assets. In th e follow in g we w ill b riefly in trod u ce th e A R C H m o d e l, see [13], tryin g to em ph asize its lim its. T h en , we w ill treat th e S V m od el, see [26], and related exten sions, in ord er t o m o d e l the a forem en tion ed stylized facts. It is w orth to m en tion th a t different, m ore n u m erica lly orien ted m e th o d s, ca n b e also fru itfu lly exp lo ite d , as, e.g., su ggested in [10, 11] and references therein. 2.1 A R C H model O n e o f th e m ost p o p u la r d iscrete-tim e m o d e ls for th e stoch a stic v o la tility is th e A R C H m o d e l, w h ich establishes a co n n e ctio n b etw een the past squared returns o f a financial asset and its current co n d itio n a l b e th e return p rocess o f som e ob serva tion m o d e l. variance. W e let { y t }t==1 In th e origin al form u lation o f E n gle, see [13], th e d y n a m ic o f th e A R C H (1 ) was given b y y t| F t-i - N (m ,^ 2), a2 t = w + a y t-1 (1) (2) w here w, a > 0 are real n o n -s to ch a s tic param eters, F t d en otes th e globa l in­ form a tio n u p t o tim e t. N aturally, eq. (2) co u ld b e gen eralized t o th e general A R C H (p ) case p a t = w + ^ 2 a® y 2- 1 , a® > 0, i= 1 in w h ich th e co n d itio n a l variance is given b y a linear co m b in a tio n o f p -lag ged squ ared error term s. A s n o te d b y N elson, see [23], th e A R C H m o d e l presents at least 2 draw backs: • C on strain ts m ust b e im p osed o n th e param eters in ord er t o gu aran ­ tee th e p o s itiv ity o f th e co n d itio n a l variance, h ow ever th e y are often v io la te d in th e classical estim ation p rocedu res. 7 V o la tility o f p r i c e s o f fin a n c ia l a s s e ts • It is n o t possible t o m o d e l th e co n d itio n a l variance as a ra n d o m oscil­ la to ry p rocess, w h ich is a recurrent situ a tion ob served in real data. In th e follow in g we will present th e S t o c h a s t i c V o l a t il i ty (S V ) m o d e l due to T aylor, see [26] and [27], and able t o o v ercom e th e a forem en tion ed difficulties. 2.2 Stochastic volatility (S V ) model T h e p e cu lia rity o f th e S V m o d e l b y T aylor is th a t th e varian ce o f th e returns is m o d e le d as an u n observ ed p rocess. In [27] T aylor show s th a t this m o d e l ca n b e tra n sp osed in to a con tin u ou s tim e version, useful w hen it com es t o p rice o p tio n s and oth er m o d e rn financial instrum ents. D en otin g again { y t } t = 1 as th e return p rocess o f som e o b serva tion m od el, th e S V p aram etriza tion sets yt = e x p (h t /2 )e t , ht = w + a h t- 1 £ t — N (0 ,1 ) + nt, 2 nt — N ( 0 ,a ? ) w here th e £t ’s and th e nt ’s are in d ep en d en t. (3) N otice th a t { h t }t==1 represents n o th in g b u t th e log a rith m o f th e v o la tility o f th e return p rocess this way, th e p o s itiv ity o f th e related varian ce is gu aran teed. In { y t }t==1. a ca n b e seen as a p ersisten ce param eter. N otice th a t { h t }t==1 is a stan d ard autoregressive A R (1 ) p rocess o n ly w h en |a| < 1, case in w h ich it is s trictly sta tio n a ry w ith m ean an variance 2 Mh = E[ht] = W , 1 —a ah = V a r(h t) = -— -r 4 t . 1 —a 2 E q u a tio n (3) is n o t th e u n iqu e w ay t o w rite th e d y n a m ic o f th e m o d e l, see [24] for equivalent form u lation s. In p articu lar, th e S V m o d e l ca n b e e x ten d ed in ord er t o take in to a ccou n t the follow in g stylized facts, see [21] for further details: • In som e cases, th e ku rtosis o f a financial tim e series is greater th an 3. T h is co rresp on d s t o fatter tails w ith resp ect to a n orm al distrib u tion . T h e p ro b le m ca n b e solv ed b y allow in g e t in eq u a tion (3) to have a S tudent t-d istrib u tio n . • A financial asset ca n e xh ibit th e so ca lled l e v e r a g e e f f e c t , th a t is, th e v o la tility resp on ds in an asy m m etric w ay t o rises or falls in th e returns. T h is fact can b e in co rp o ra te d in th e S V m o d e l b y in tro d u cin g a n egative instan tan eou s correla tion betw een e t and nt in eq u a tion (3). 2 .2.1 Estim ation procedures D ifferen tly from th e A R C H -ty p e m od els, we d o n o t k n ow the co n d itio n a l d istrib u tio n o f yt in closed form , see eq u a tion (1 ). F or this reason, th e stan ­ d ard M a x im u m L ik e lih o o d (M L ) a p p roa ch is h ard t o im plem en t. In deed, if we d e n o te b y y = ( y 1 , . . . , y N ) th e v e cto r o f N con secu tiv e ob serva tion s o f th e p ro ­ cess y t, b y h = ( h 1, . . . , h N ) the co rre sp o n d in g v e cto r for th e log-v ola tilities, and b y в = (w ,a ,a 1 ) th e v e cto r o f param eters, th en th e lik elih ood ca n be w ritten as L ( y ; в ) = J p ( y , h |e) d h = J p ( y |h , e ) p ( h |e) d h , (4) 8 L . D i P er s io , N . G u g o le w here we integrate w ith resp ect t o th e jo in t p ro b a b ility d istrib u tion o f the d ata. T h e N -dim en sion al integral in eq u a tion (4) requires the use o f co m p u ta ­ tio n a lly in v olv ed nu m erical m e th o d s and for this reason the estim a tion o f the param eters is h ard. F ollow in g [24], we b riefly cite som e altern ative estim ation p rocedu res: • G e n e r a l i z e d M e t h o d o f M o m e n t s (G M M ): this m e th o d was in trod u ced b y T aylor, see [26]. T h e b asic idea is t o m a tch th e em pirical m om en ts o f th e ob served v e cto r y w ith th e co rre sp o n d in g th eoretica l ones, w hich ca n b e co m p u te d explicitly, h en ce th e k ey advan tage is th a t th e c o n ­ d ition a l d istrib u tion o f y t is n o t required. M ore precisely, we n eed to m in im ize th e o b je c tiv e fu n ction Q = g 'W g w ith resp ect t o th e v ector o f param eters 9, w here т > 1, and W is a p o sitiv e definite, sy m m etric w eigh ting m a trix o f d im en sion ( т + 2) x ( т + 2 ). It is p ossible t o m in im ize Q usin g stan d ard num erical routines. • Q u a s i - M a x i m u m L i k e l ih o o d e s t i m a t i o n (Q M L ): this a pp roa ch is based on th e lin earization o f th e S V m o d e l in eq u a tion (3 ). A ssu m in g є t ~ N ( 0 ,1 ) and defin in g w t = log y ^ , it is possible t o p rove th at wt = —1.2704 + ht + £t, ht = w + a h t -1 + nt,nt ~ N ( 0 , ^ ) , (5) w here £t = log є 2 —E [log є)2], V a r(£ t ) = n 2/ 2 . E ven if th e errors £t d o not have a n orm al d istrib u tion , th e u n derlyin g idea o f th e Q M L app roach is t o su p p ose £t ~ N (0 , n 2/ 2 ) i.i.d., and t o a p p ly th e K a lm a n filter to eq u a tion (5) in ord er t o p ro d u ce o n e-step ahead forecasts o f w t as well as h t . D e co m p o s in g th e p re d ictio n error, it is p ossible t o co n s tru ct the G aussian lik elih ood o f th e data , t o b e m in im ized in ord e r t o estim ate th e v e cto r o f param eters 9, see [17]. 2 .2.2 T he multivariate case A stylized fact w h ich ca n n ot b e ca p tu re d b y the stan d ard u nivariate S V m o d e l is th e so ca lled c o v a r i a t i o n e f f e c t , th a t is, ro u g h ly speakin g, the presen ce o f a correla tion b etw een th e volatilities o f different financial series. O ften , large changes in th e returns o f an asset are follow ed b y large changes in oth er ones. T h is ca n b e due to th e presen ce o f co m m o n u n observ ed factors in fluen cin g th e d yn a m ics o f a set o f assets. V ola tilities are also s u b je c t t o the co m in g o f new in form ation , such as trad in g volu m e, q u o te arrivals, g o v e rm e n t’s health, d iv id e n d an n ou n cem en ts and so on . A ll these p h en om en a suggest th a t a m u ltivariate m o d e l co u ld b e b e tte r th an an univariate one in term o f adherence to real data. V o la tility o f p r i c e s o f fin a n c ia l a s s e ts T h e first m u ltivariate S V m o d e l was p ro p o se d in [16]. 9 W e d en ote b y y t = ( y 1,t , . . . , y N,t ) T th e v e cto r o f returns related t o N different assets at tim e t. T h e d y n a m ic o f th e i-th co m p o n e n t is assum ed t o be ( y i ,t = e x p (h M / 2 ^ i , t , \ h i,t = w® + a . i h i , t - 1 + ni,t, w here є t = (є 1,t , . . . , є N,t) and nt = (n 1,t, . . . , nN,t ) are m u tu a lly ind ep en d en t and n orm a lly d istrib u ted . M oreover V ar(n t) = V a r(є t) = , 1 P 1,2 P 1,2 1 P1,n \ P2,N \ = 1. ( w here |P®,j | < 1, so th a t P1,N P2, N is a correlation m a trix. T h e w eakness o f th e m od el is th a t it d oes n o t allow th e covarian ces o f th e assets to e volve in an in d ep en d en t m an ner o f th e variances. I f i = j , C o v (y i,t,y j,t| h t) е [у м у Ы ^ ] P i,j exp and since Var(y®,t|h t) = ex p ( h i ,t ) , it follow s th a t the m o d e l has con sta n t correlation s, w h ich ca n b e a lim itin g fact in som e situ ation s, see, e.g., [25]. A s in th e univariate case, it is possible to estim ate th e param eters th rou g h a Q M L a pp roa ch , see [16], b y linearizing the co rre s p o n d in g equations. T h e m u ltivariate S V m o d e l a dm its also oth er represen tation s, e.g., th e factorial one, see [20]. T h e m ain advantage w ith resp ect t o th e p reviou s m ul­ tivariate m o d e l, is th e red u ction o f th e d im en sion a lity o f th e p aram eter space: th e returns v e cto r y t = ( y 1 t , . . . , y N,t ) T is a linear co m b in a tio n o f u n obser­ ved and co m m o n factors follow in g a univariate S V d y n a m ic. I f we d e n o te b y f t = ( f 1,t, . . . , f K ,t) T th e set o f co m m o n factors at tim e t, th en yt = B ft + wt , f i t = e x p (h M / 2 ^ i , t , h i,t = Mi + . i = 1, . . . , K , Фі ^-і ^ - 1 + Пі,Ь w here B is a co n sta n t m a trix o f dim en sion N x K , K < N , w t ~ N (0 , П) is th e error v e c to r and it is assum ed in d ep en d en t o f all th e o th e r term . T h e ra n d o m variables є®^ and n®,t are serially and m u tu a lly in d ep en d en t and n or­ m a lly d istrib u ted . W e assum e also th a t |ф®| < 1 so th a t th e fa cto r lo g -v o la tility processes hi t are station ary. F or m ore details a b o u t th e m od el, see [20], [24]. 10 L . D i P e r s io , N . 3 G u g o le Continuous-time models In th e ea rly 1970’s th e w orld o f o p tio n p ricin g ex p erien ced a great con tri­ b u tio n given b y th e w ork o f F isch er B la ck and M y ro n Scholes. T h e y d e v elop ed a n ew m a th em a tica l m o d e l t o treat certa in financial q uan tities p u b lish in g th e related results in th e article T h e P r i c i n g o f O p t i o n s a n d C o r p o r a t e L i a b i li t ie s , see [4]. T h e latter w ork b eca m e s o o n a reference p o in t in th e financial scenario. N ow adays, m a n y traders still use th e B la ck and Scholes (B S ) m o d e l t o price as well as t o h edge various ty p e s o f con tin gen t claim s. A n im p o rta n t p ro p e rty o f th e B S m o d e l is th a t all th e in volved param eters are n o t in fluen ced b y the risk preferen ces o f investors. In p articu lar, the B S a p p roa ch is b ased o n th e so-ca lled risk-neutral p ricin g assu m p tion w h ich grea tly sim plifies th e associated derivatives analysis. In p articu lar, in th e classical B S -m o d e l, th e v o la tility param eter, let us in d ica te it w ith a , is assum ed to b e con sta n t. L a tter h y p oth esis ca n n o t be con sid ered realistic, as sim ple em p irical analyses ca n easily show . In particu lar it is rath er sim ple to sh ow th a t th e im plied v o la tility o f a financial asset is not co n sta n t b u t varies w ith tim e t o m a tu rity T > 0, and w ith resp ect to th e strike p rice K . Such a fact has sta rted to b e co m e m ore and m ore eviden t since the general m arket crash in 1987. A s a con seq u en ce, th e real values o f th e v o la tility p aram eter th a t can b e ob served in th e m arket d o n o t give rise t o a flat shape as th e B S -m o d e l forecasts. In fact, if we fix the strike p rice value and we lo o k at th e co rre sp o n d in g im plied v o la tility section , e.g., w ith respect t o a plain vanilla o p tio n , th e ty p ica l figure th a t appears ju stifies th e d efin ition o f the soca lled s m i l e / s m i r k e f f e c t . T h e latter b eca u se, esp e cia lly for sh ort m atu rities, the im p lied v o la tility section s assum e a shape w h ich resem bles a s m i l e o r a s m ir k . A s a con seq u en ce o f th e B S -m o d e l lack o f d escrip tio n accu racy, new m o ­ dels have b een d e v e lo p e d t o o v ercom e issues o f th e ty p e m en tion ed so far. T h is has b een also p ro d u ce d app roa ch es able t o treat the in creasin gly co m p le x ity ch a ra cterizin g m o d e rn financial instrum ents. B etw een such alternatives t o the B S analysis, we fo cu s ou r a tten tion o n th e so ca lled lo c a l v o l a t i l i t y (L V ) and s t o c h a s t i c v o l a t i l i t y (S V ) m od els. 3.1 Local volatility models T h e L V m od els ca n b e seen as the sim plest exten sion o f th e classical B S case, in ord er t o achieve an e x a ct re p ro d u ctio n o f th e v o la tility sm ile, th rou g h ca lib ra tio n t o m arket data. T h e m ain differen ce is th a t in L V m od els, the instan tan eou s v o la tility is, in general, a fu n ction o f th e current tim e and the cu rren t asset price. I f we d en ote b y S t th e p rice o f th e asset at tim e t, we can w rite th e related S D E as d S t = M(t, S t)S t dt + a (t , S t)S t d W t , w here So > 0, ^ (t, S t ) is the instan tn eous drift, a (t , S t ) is the instan tan eou s v o la tility at tim e t, and W t a B row n ian m o tio n . I f a (t , S t ) = a > 0 th en we tu rn b ack t o the B S case. V o la tility o f p r ic e s o f fin a n c ia l a s s e ts 11 T h e first L V m o d e l a p p eared in the literatu re is th e so called C o n s t a n t E l a s t i c i t y o f V a r ia n c e (C E V ) m od el, see [7]. T h e latter is ch a ra cterized b y a v o la tility defin ed as a (t , S t ) = a S ^ 1, a > 0, 7 =1 7 = 0 leads t o n orm a lly d istrib u te d returns. w here y m ust b e d eterm in ed w ith a ca lib ra tio n t o m arket d ata. W ith we fin d th e B S m o d e l, w hile 3 .1.1 T he pricing equation D en otin g b y C = C (t , S t ; T , K ) th e tim e-t p rice o f a vanilla o p tio n havin g as u n derly in g th e asset p rice S t , m a tu rity T and strike K > 0, th en it is possible t o show , assum ing existen ce and uniqueness o f th e risk-neutral m easure, th at C solves th e follow in g P D E : d C + rS dC + at + as + w here r > 1„ , t s ) 2 ! f l — = rC 2a ( i ' S ) S a S 2 = r C (6) (6) ' 0 is th e con sta n t instan tan eou s sp o t rate, to b e p riate b o u n d a r y co n d itio n s, d e p e n d in g o n th e n ature o f th e o p tio n co u p le d w ith a p p ro­ o f interest. In p articu lar, settin g C (T , S T ) t o th e o p t io n ’s p a y o ff and solv in g th e eq u a tion b ackw ards from T t o t, it is p ossible t o fin d C (t , S t ). 3 .1 .2 T he Dupire formula S u p pose t o have a set o f vanilla o p t io n ’s prices related t o tim e t. Is there a w ay to set a (t , S ) in such a w a y t o p e rfe ctly fit these p rices? T h e answ er is yes, and com es from th e well k n ow n D u p i r e f o r m u l a , see [3], [15], or [8]: a (T ,K )2 = a ( T , K ; t , S t )2 = 2 dC dC -------+ r K ----dT dK . K (7) 2— dK2 In p articu lar, if eq u a tion (7) h old s at tim e t = 0, th en th e m o d e l is a u tom a ti­ ca lly ca lib ra ted to th e initial m arket v o la tility sm ile. M oreover, it is possible t o sh ow th a t the right h an d side o f eq u a tion (7) is alw ays well defin ed if the real m arket is arbitrage free. M a n ip u la tin g a little b it the D u pire form ula, we ca n rew rite it in th e follow in g way: § + rK § — 2 a ( T - K )2 K 2ё = » . <8 > E q u a tio n ( 8) is sim ilar t o ( 6) in m a n y aspects, h ow ever m ust b e solv ed forw ard in ord er t o find o p t io n ’s prices for all th e values o f K and T , fixin g t and St. S u ppose, for sim p licity o f e x p o sitio n , th a t r = 0. T h e n th e D u p ire form ula (7) tu rn s into dC a (T ,K )2 = ^ ^ d . _ . K K 2— — dK2 ( 9) 12 L . D i P e r s io , N . G u g o le U sually, vanilla o p tio n prices are q u o te d in term s o f th e B S im p lied v o la tility a BS = a B S (t, S t; T , K ), i.e., th a t value o f th e v o la tility w hich, o n ce inserted in to th e B S p ricin g form ula, gives th e m arket price: C (t , St; K , T ) = C b s (t, St; K , T , a B - ). B y usin g th e chain differen tiation rules and th e form ulas o f th e B S greeks, it is p ossible t o w rite eq u a tion in term s o f a B S , instead if C , see [15], i.e., 2d a g s + a B s K )2 = ___________________________ д т ______ т _________________________ а (т ' k 2 - d 1v r (^ y2 + aBs (* ^ + ^ ) 2) ' w here т = T — t and d1 = d v ? ln (K ) + 2aBs ^ . A s a particu la r case, su p p ose th a t a BS is in d ep en d en t o f K , i.e., the v o la tility sm ile has n o skew , so ст(т, K ) = ст(т), w here ^ о d a BS , 2 _ д / 2 a ( T )2 = 2 r a B s ~ d r " - + a BS = д т ( тстВ s ) from w h ich / a ( u ) 2du = r a g - . Jo 3.2 Stochastic volatility models T h e S V m o d e ls represent a natural exten sion o f th e L V m od els. W e will con sid er th e follow in g co u p le o f SD E s: dSt = ^ S t)S t dt + V * St dW t ї dvt = a (t , St, vt) dt + ^ ^ (t, St, vt) V t d Z t , E [dW tdZt] = p d t , (10) w here n is th e v o la tility o f volatility, p represents th e instan tan eou s correlation b etw een the tw o B row n ian m o tio n s W t and Z t , and 7 > 0. In th e lim it n ^ 0, we retrieve th e S V case. T h e H eston m o d e l is, now adays, th e m ost kn ow S V m od el; it was in tro­ d u ce d for th e first tim e in [18]. S tartin g from eq u a tion (1 0 ), th e H eston m od el corre sp o n d s t o th e ch oice a (t , S t , v t ) = e (v — v t), v > 0, в > 0, ^ (t, St, vt) = 1. In oth er w ords, v t is a C o x -In g e rso ll-R o ss (C I R ) p rocess, w here У is th e so ca lled lo n g t e r m m e a n and в represents th e s p e e d o f r e v e r s i o n . T h is te rm in o lo g y reflects the fact th a t, for su fficien tly large tim es, v t w ill m ove a rou n d th e value v w ith an in ten sity d e p e n d in g o n th e m a gn itu d e o f y. A n im p o rta n t feature o f th e C IR p rocess is, u n der som e co n d itio n s on param eters, th e p o sitiv ity : in p articu lar, we have to im p ose 2в-У > n 2. 13 V o la tility o f p r ic e s o f fin a n c ia l a s s e ts 3 .2.1 T he pricing equation In th e B S case, as well as in the S V case, there is o n ly one sou rce o f ran dom ness, m ore p recisely th e p rocess W t , b u t in th e S V case we have also ra n d om changes in th e v o la tility t o b e h edged. T h e idea is t o set u p a p o r tfo lio con ta in in g th e o p tio n o f interest, a q u a n tity —Д 1 o f th e u n derly in g asset and a q u a n tity —Д 2 o f an oth er asset d e p e n d in g o n th e v o la tility value v t . D ifferen­ tiatin g th e p o rtfo lio value and im p o sin g th e usual risk-free co n d itio n s (ra n d om term s equal to ze ro and return equal t o r ) , see [15] for further details, we en d u p w ith th e follow in g P D E : 9C S 2 д 2C , S 8 C , e S d 2C . 1 2 д 2C 2 v ' S ' a S 2 + r S ' a S + Pnv' e S a v a S + 2 n v ‘ e a ? , 1 at + a— = r C — (a — ф в ^ і ) a y , ( (11) ) w here ф = ф (^ S t , v t ) is the so ca lled m a r k e t p r i c e o f v o l a t i l i t y r is k , and ca n be seen as th e extra return (requ ired b y the investors) p e r unit o f v o la tility risk. D efining a = a — Ф вл/^: as th e d rift o f th e v o la tility vt p rocess u n der the risk-neutral m easure, we co u ld rew rite eq u a tion (11) in a m ore co m p a ct w a y as a c at + 1 о a 2с а с a 2c 1 2 n2 a 2c 2 " A a S 2 + r S a S + pnv‘ e S ‘ a v a S + 2 n v ‘ e w _ac = r C —a л • (12) E q u a tio n (12) is a g o o d p o in t t o start w ith , if the aim is t o ca lib ra te th e S V m o d e l t o o p tio n prices, w h ich are clo s e ly co n n e cte d t o th e risk-neutral m easure. In p articu lar, we ca n assum e th at th e S V m o d e l o f interest, o n ce fitted the related param eters t o o p tio n prices, gen erates th e risk-neutral m easure such th a t the m arket p rice o f v o la tility risk ф is equal t o zero. T h is a ssu m p tion makes sense w h en we are interested o n ly in th e p ricin g part, n o t in the statistical prop erties, w h ich are d e scrib e d b y th e p h ysical m easure. 3 .2 .2 Calibrating the parameters o f the H eston m odel T h e m ain advan tage o f th e H eston m o d e l w ith resp ect to oth er (p o te n ­ tia lly m ore realistic) stoch a stic v o la tility m od els is th e existen ce o f a fast and easily im plem en ted q u a si-closed form solu tion for E u ro p e a n op tio n s, see [15] for th e d erivation . T h is co m p u ta tio n a l efficien cy in th e va lu a tion o f E u ro p e a n o p tio n s b e co m e s useful w h en ca lib ra tin g th e m o d e l t o real o p tio n prices. H ow ca n we p e rfo rm th e ca lib ra tio n ? T h e sim plest w ay is t o m in im ize th e d istance b etw een the ob served E u ro p e a n call o p tio n prices and the th eoretica l ones. If we d en ote b y в th e set o f param eters o f th e H eston m od el, th en we have to solve th e n on -lin ea r least squares N в = arg m ill £ 2 ( c ° bs — C j ( e ) ) , (13) j=1 w here C ? bs = C obs( K j, T j), i = 1 , . . . , N , is th e set o f ob served o p tio n prices, w hile — (в ) = C j( K j, T j; в ), i = 1 , . . . , N , is th e set o f o p tio n prices p ro d u ce d 14 L . D i P e r s io , N . G u g o le b y th e H eston m o d e l, and 0 d en otes th e p aram eter space. A ltern atively, one co u ld p erfo rm th e m in im iza tion in eq u a tion (13) usin g a dataset o f im plied volatilities instead o f th e co rre sp o n d in g q u o te d o p tio n prices. A different a p p roa ch is a d o p te d , for instan ce, in [1], and it is b ased o n the M a x im u m L ik elih ood m e th o d . W e can im agin e th e sto ck p rice S t at tim e t as a fu n ction o f a v e c to r o f state variables X t follow in g a m ultivariate stoch a stic v o la tility d y n a m ic as in eq u a tion (1 0 ), i.e., S t = f ( X t ) for som e fu n ctio n f . U sually, eith er th e sto ck price itself (o r its log a rith m ) is taken as on e o f the state variables, h en ce we w rite X t = ( S t, Yt) T , w here Yt is th e rem ain in g set o f state variables o f len gth N . In general, part o f th e state v e cto r X t ca n n o t be d ire ctly ob served . In [1], th e idea is t o assum e th a t b o t h a tim e series o f stock prices and a v e cto r o f q u o te d o p tio n prices are ob served . T h e la tter v e c to r at tim e t is d e n o te d b y C t , and m ust b e used in ord er t o infer the tim e series for Y t. I f Yt is m u ltid im en sion al th en a sufficient n u m ber o f different o p tio n prices is n eeded. R o u g h ly speakin g, there are tw o w ays to e x tra ct th e value o f Yt from ob served data: • T h e first m e th o d is t o c o m p u te o p tio n prices as a fu n ction o f S t and Yt, for each p aram eter v e c to r con sid ered d u rin g th e estim ation p roced u re. In this w ay it is p ossible t o iden tify the param eters b o th u n der the ph ysical m easure and th e risk-neutral one. • T h e secon d m e th o d con sists in using th e B S im plied v o la tility as a p ro x y for th e instan tan eou s v o la tility o f th e stock . T h is is a sim p lifyin g p roced u re, and it ca n b e app lied o n ly in the case o f a single stoch a stic v o la tility state variable. Since, in general, th e tran sition lik elih ood fu n ctio n for a s to ch a stic v o la tility m o d e l is n o t k n ow n in closed form , th en an a p p ro x im a tio n m e th o d m ust be used, see [2]. In this w a y it is p ossible to express, in an a p p rox im a te clo se d form , the jo in t lik elih ood o f X t . T h en , in ord er t o fin d th e lik elih ood o f (S t , C t ) T , w h ich is en tirely ob served , it is n ecessary to m u ltip ly th e lik elih ood o f the v e cto r X t b y an a p p rop riate ja co b ia n term . T h e last step is n o t n ecessary w h en a p ro x y for Yt is used. For further details, see [1]. References [1] A it-S a h alia , Y acin e and R o b e r t K im m el. 2007. M a x im u m lik elih ood es­ tim a tio n o f sto ch a stic v o la tility m od els. J o u r n a l o f F i n a n c i a l E c o n o m i c s 8 3 (2 ):4 1 3 -4 5 2 . [2] A it-S a h alia , Y acin e et al. 2008. C lo se d -fo rm lik elih ood exp an sion s for m ul­ tivariate diffusions. T h e A n n a l s o f S t a t i s t i c s , 3 6 (2 ):9 0 6 -9 3 7 . [3] B erg om i, L. 2015. S t o c h a s t i c V o l a t il i ty M o d e li n g . C R C Press. [4] B lack, F . and M . Scholes. 1973. 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