An Analysis of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab *

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1 An Analysis of he Heson Sochasic Volailiy Model: Implemenaion and Calibraion using Malab * Ricardo Crisósomo December 14 Absrac This paper analyses he implemenaion and calibraion of he Heson Sochasic Volailiy Model. We firs explain how characerisic funcions can be used o esimae opion prices. Then we consider he implemenaion of he Heson model, showing ha relaively simple soluions can lead o fas and accurae vanilla opion prices. We also perform several calibraion ess, using boh local and global opimizaion. Our analyses show ha sraighforward seups deliver good calibraion resuls. All calculaions are carried ou in Malab and numerical examples are included in he paper o faciliae he undersanding of mahemaical conceps. Keywords: Sochasic volailiy, Heson, Black-Scholes biases, calibraion, characerisic funcions. JEL Classificaion: G13, C51, C5, C61, C63. * The auhor acknowledges he commens of seminar paricipans a he CNMV. A previous version of his work was circulaed in parial fulfillmen of he requiremens of an MSc degree in mahemaics a he Naional Disance Educaion Universiy (UNED). The opinions in his aricle are he sole responsibiliy of he auhor and hey do no necessarily coincide wih hose of he CNMV. Comisión Nacional del Mercado de Valores (CNMV). c/ Edison 4, 86 Madrid. rcayala@cnmv.es.

2 Lis of Conens 1 Inroducion 3 From Characerisic Funcions o Opion Prices 4.1 The General Valuaion Framework Valuing a European Call Though Characerisic Funcions An applicaion o he Black and Scholes Model The Heson Model Closed-form Soluion of he Heson Model Model Implemenaion Calibraion o Marke Prices Calibraion Procedure in he Heson Model Local Opimizaion Global Opimizaion More Calibraion Exercises Conclusion 1 6 Appendix A Derivaion of 1 and B Heson Equivalence of ln ( w) o he definiion in Gaheral (6) S C Risk Neuraliy in he Heson Model D Daases used for Calibraion References 34

3 1. Inroducion The Black and Scholes (BSM) model provides a coheren framework for pricing European opions. However, his mehod is based on several assumpions ha are no represenaive of he real world. In paricular, he BSM model assumes ha volailiy is deerminisic and remains consan hrough he opion s life, which clearly conradics he behavior observed in financial markes. While he BSM framework can be adaped o obain reasonable prices for plain vanilla opions, he consan volailiy assumpion may lead o significan mispricings when used o evaluae opions wih non-convenional or exoics feaures. During he las decades several alernaives have been proposed o improve volailiy modelling in he conex of derivaives pricing. One of such approaches is o model volailiy as a sochasic quaniy. By inroducing uncerainy in he behavior of volailiy, he evoluion of financial asses can be esimaed more realisically. In addiion, using appropriae parameers, sochasic volailiy models can be calibraed o reproduce he marke prices of liquid opions and oher derivaives conracs. One of he mos widely used sochasic volailiy model was proposed by Heson in The Heson model inroduces a dynamic for he underlying asse which can ake ino accoun he asymmery and excess kurosis ha are ypically observed in financial asses reurns. I also provides a closed-form valuaion formula ha can be used o efficienly price plain vanilla opions. This will be paricularly useful in he calibraion process, where many opion repricings are usually required in order o find he opimal parameers ha reproduce marke prices. In his paper we analyze he valuaion of financial opions using he Heson model. Our aim is o illusrae he use of he model wih an emphasis on he implemenaion and calibraion. Secion presens he valuaion framework and explains how characerisic funcions can be used o esimae opion prices. Secion 3 inroduces he Heson model and discusses he implemenaion of is closed-form soluion. Finally, Secion 4 analyzes he calibraion problem, considering boh local and global opimizaion mehods. For all relevan secions, generic and ready-o-use Malab codes have been developed and numerical examples are provided in order o illusrae he use of he Malab rouines. 3

4 . From Characerisic Funcions o Opion Prices A considerable amoun of research has been recenly devoed o analyze he use of characerisic funcions in opion s valuaion. The raionale is ha when you go beyond he classical BSM framework, he underlying sochasic processes ha are used o calculae opion values have characerisic funcions which are simpler and more racable han heir densiy funcions. Therefore, in many sophisicaed models, i is easier o work wih characerisics funcions insead of using densiy funcions..1 The General Valuaion Framework When markes are complee and arbirage-free, opion values can be calculaed as he presen value of heir expeced payoff under he risk-neural measure rt V e EQ H( S) (.1) where V is he opion value a ime =, r is he risk free rae, T is he ime o mauriy and HS ( ) is he opion payoff. In order o use (.1), we firs need o specify he dynamics of he price process S. In paricular, since we are working wih expecaions, we should consider he probabiliy disribuion of S a (poenially) differen imes, as required by each opion payoff. In he classical framework, he expecaion above is obained by means of he risk-neural densiy. For insance, he payoff of a European call wih srike K and expiraion dae T is given by H( S ) ( S K). Consequenly, is value a ime = is T rt ( ) C ( ) e S T K q ST dst (.) where qs ( T ) is he risk-neural densiy of he underlying asse S a he erminal dae T. The problem wih (.) is ha here are many price processes for which he densiy funcion qs ( T ) is no available in a closed-form or is difficul o obain. However, if we work wih he logarihm of he underlying asse price, here are many of such price processes wih boh simpler and analyically racable characerisic funcions. Characerisic funcions exhibi a one-o-one relaionship wih densiy funcions. In paricular, he characerisic funcion of a given sochasic process X, is he Fourier ransform of is probabiliy densiy funcion iwx iwx ( w) E[ e ] e f ( x) dx (.3) Therefore, by applying he Fourier Inversion heorem, we can recover he densiy funcion of he process X in erms of is characerisic funcion 1 iwx f ( x) e ( w) (.4) 4

5 Given his relaionship, all he probabiliy evaluaions ha are required o calculae opions values can be also compued using characerisic funcions.. Valuing a European Call hrough Characerisic Funcions Following he reasoning in Heson (1993), he value of a European call opion can be obained by using a probabilisic approach C S e K (.5) rt 1 where 1 and are wo probabiliy-relaed quaniies. Specifically, 1 is he opion dela and is he risk-neural probabiliy of exercise P( ST K). Insead of using densiy funcions, hese probabiliies can be compued via characerisic funcions as follows (proof in Appendix A): iwln( K ) 1 1 e ln S ( w i) T 1 Re iwln S ( i) (.6) T iwln( K ) 1 1 e ln S ( w) T Re iw (.7) Therefore, saring wih he characerisic funcion of he log-price ln S ( w), we can esimae T he price of a European call opion by firs calculaing he probabiliies (.6) and (.7) and hen subsiuing heir values in (.5). This mehod presens wo main advanages: Generaliy: This approach can be applied for any underlying price process S whose characerisic funcion is known. Semi-analyical soluion: The inegrands in (.6) and (.7) should be evaluaed numerically. However, hey are smooh funcions ha decay rapidly and can be evaluaed efficienly using appropriae inegraion rouines 1. This lead o numerical implemenaions ha can value plain vanilla opions in a fracion of a second..3 An Applicaion o he Black and Scholes Model Before moving ino he Heson model, we will apply he characerisic funcion mehod o value a call opion under he BSM framework. The risk-neural dynamics of he underlying asse in BSM are described by a Geomeric Brownian Moion ds rs d S dw (.8) 1 See Kahl and Jäckel (5) or Schmelzle (1). For example, using he Malab s implemenaion proposed in his paper, he compuaional imes required for pricing a European call opion are.387 seconds in he BSM model and.4866 seconds in he Heson model. 5

6 where S is he price of he underlying asse a ime, r is he risk free rae, is he volailiy of he underlying reurns, and W is a Weiner process. Using sochasic calculus, equaion (.8) can be easily solved o yield S 1 ( r ) Z S e (.9) where Z is he sandard normal disribuion. Therefore, he disribuion of S is lognormal, while ln ( S ) is normally disribued. In paricular, he risk-neural evoluion of ln ( S ) is normally disribued wih mean ln( S) ( r.5 ) and variance. This means ha, in pracice, i is easier o work wih he process ln ( S ) raher han using S direcly. Black and Scholes Characerisic Funcion The characerisic funcion of a normal random variable is given by 1 iw(mean) w (variance) ( w) e (.1) Therefore, he characerisic funcion of ln ( S ) can be easily calculaed as iw[ln(s ) ( r.5 ) ].5w BSM ( ) ln( S ) w e (.11) Once we have he characerisic funcion, he nex sep is o esimae 1 and. These probabiliies can be compued by numerical inegraion or, alernaively, Euler s formula ix ( e cos x isin x ) could be applied o furher expand (.6) and (.7), and in order o obain more specific expressions for 1 and under he BSM framework. Since our aim is o gain a beer undersanding of he general characerisic funcion approach, we will compue 1 and direcly using (.6) and (.7). We will repea his procedure in Heson BSM secion 3, where we will use ln( S ) ( w) insead of ln( S )( w) in order o calculae he value of a European call under he Heson model. Funcion 1 below (chfun_norm.m) shows how o compue he characerisic funcion of he BSM model in Malab, while funcion (call_bsm_cf.m) calculaes he call value based on equaions (.5) o (.7). In addiion, example 1 illusraes he pracical use of hese funcions by pricing an individual call opion. As a reference, wihin he BSM framework, he esimaed value of a call opion wih parameers S = 1, K = 1, =.15, r =. and = T = 1 is C = As he example shows, using he characerisic funcion approach, we obain he same call value. Malab Funcion 1: Characerisic funcion of he Black-Scholes model (chfun_norm.m) funcion y = chfun_norm(s, v, r,, w) % Characerisic funcion of BSM. % y = chfun_norm(s, v, r,, w) % Inpus: % s: sock price % v: volailiy 6

7 % r: risk-free rae % : ime o mauriy % w: poins a which o evaluae he funcion mean =log(s)+ (r-v^/)*; var = v^*; y = exp((i.*w*mean)-(w.*w*var*.5)); end % mean % variance % characerisic funcion of log (S) evaluaed a poins w Malab Funcion : Call value in he Black-Scholes model (call_bsm_cf.m) funcion y = call_bsm_cf(s, v, r,, k) % BSM call value calculaed using formulas.5 o.7 % y = call_bsm_cf(s, k, v, r,, w ) % Inpus: % s: sock price % v: volailiy % r: risk-free rae % : ime o mauriy % k: opion srike % chfun_norm: Black-Scholes characerisic funcion % 1s sep: calculae pi1 and pi % Inner inegral 1 in1 real(exp(-i.*w*log(k)).*chfun_norm(s,v,r,,w-i)./(i*w.*chfun_norm(s, v, r,, -i))); in1 = inegral(@(w)in1(w,s,v,r,,k),,1); %numerical inegraion pi1 = in1/pi+.5; % Inner inegral in real(exp(-i.*w*log(k)).*chfun_norm(s, v, r,, w)./(i*w)); in = inegral(@(w)in(w,s, v, r,, k),,1); %numerical inegraion pi = in/pi+.5; % final pi % nd sep: calculae call value y = s*pi1-exp(-r*)*k*pi; end Numerical Example 1: Call opion valuaion using he Black-Scholes model % funcion y = call_bsm_cf(s, v, r,, k) >> call_bsm_cf(1,.,., 1, 1) ans =

8 3. The Heson Model In 1993, Heson proposed a sochasic volailiy model where he underlying asse behavior was characerized by he following risk-neural dynamics ds rs d V S dw 1 dv a( V V ) d V dw dw dw 1 d (3.1) The parameers used in he model are he following: S is he price of he underlying asse a ime r is he risk free rae V is he variance a ime V is he long-erm variance a is he variance mean-reversion speed is he volailiy of he variance process 1 dw, dw are wo correlaed Weiner processes, wih correlaion coefficien Therefore, under he Heson model, he underlying asse follows an evoluion process which is similar o he BSM model, bu i also inroduces a sochasic behavior for he volailiy process. In paricular, Heson makes he assumpion ha he asse variance V follows a mean revering Cox-Ingersoll-Ross process. Sochasic volailiy models ackle one of he mos resricive hypoheses of he BSM model; namely, he assumpion ha volailiy remains consan during he opion s life. Observing financial markes i can be easily seen ha volailiy is no a consan quaniy. This is also refleced in he differen implied volailiy levels a which opions wih differen srikes and mauriies rade in he marke, which collecively give rise o he so-called volailiy surface. Among volailiy models, Heson s dynamics exhibi several desirable properies. Firs, i models volailiy as a mean-revering process. This assumpion is consisen wih he behavior observed in financial markes. If volailiy were no mean-revering, markes would be characerized by a considerable amoun of asses wih volailiy exploding or going near zero. In pracice, however, hese cases are quie rare and generally shor-lived. Second, i also inroduces correlaed shocks beween asse reurns and volailiy. This assumpion allows modelling he saisical dependence beween he underlying asse and is volailiy, which is a prominen feaure of financial markes. For insance, in equiy markes, volailiy ends o increase when here are high drops in equiy prices, and his relaionship may have a subsanial impac in he price of coningen claims. Consequenly, he Heson model provides a versaile modelling framework ha can accommodae many of he specific characerisics ha are ypically observed in he behavior of financial asses. In paricular, he parameer conrols he kurosis of he underlying asse reurn disribuion, while ses is asymmery. 8

9 However, as expeced, hese benefis come a he expense of higher complexiy. Compared wih BSM, he implemenaion of he Heson model requires more sophisicaed mahemaics and i also involves a more challenging process o calibrae he model o fi marke prices. 3.1 Closed-form Soluion of he Heson Model One of he main advanages of he Heson model is ha he price of European opions can be esimaed using a quasi-closed form valuaion formula. The developmen of he Heson formula follows he general approach ha we explained in secion. As we menioned, he presen value of a European call opion can be esimaed using a probabilisic approach C S e K (3.) rt 1 where 1 and are wo probabiliy-relaed quaniies. Therefore, he call value under he Heson model can compued by firs obaining 1 and using he dynamics described in (3.1) and hen subsiuing heir values in equaion (3.). However, he difficuly arises when we ry o calculae hese probabiliies under he Heson dynamics, since he ransiion densiies for his model are no available in a closed-form. Alernaively, as we showed earlier, 1 and can also be obained using characerisic funcions. Heson Characerisic Funcion In his secion we sar wih he Heson characerisic funcion proposed by Gaheral (6), bu we also inroduce an addiional modificaion. In paricular, he characerisic funcion ha we will use hrough he paper is he following: Heson ln( S ) ( w) e r [ C(, w) V D(, w) Viwln( Se )] 1 ge C(, w) a r ln 1 g D(, w) r r 1 e 1 ge h h h r g r ; h 4 w iw ; a iw ; h (3.3) Our approach differs from Gaheral (6) in ha we apply he characerisic funcion mehod based on he process ln ( S ), insead of ln ( S / K ). Using his approach we obain an Heson expression for ln( S ) ( w) ha can be direcly used wihin he general pricing framework presened in secion. This is in conras wih he formulaion used in Heson (1993) and laer 9

10 in Gaheral (6), where wo disinc funcions are used o calculae 1 and. Appendix B shows he equivalence of our approach o he mehodology provided by Gaheral (6) I should be noed ha he characerisic funcion presened in (3.3) already incorporaes he risk-neural behavior of he process ln ( S ). A discussion of he risk neural paradigm in he Heson model is included in Appendix C. 3. Model Implemenaion Heson Alhough ln( S ) ( w) may have a complicaed appearance, is implemenaion is quie sraighforward. In paricular, once we have esimaed appropriae values for he model parameers V,,,, V a, he Heson characerisic funcion can be easily evaluaed using numerical sofware. Funcion 3 (chfun_heson.m) shows how o compue he Heson characerisic funcion in Malab. Heson Afer obaining ln( S ) ( w), he characerisic funcion can be subsiued in (.6) and (.7) o calculae 1 and. Using hese probabiliies, equaion (3.) will provide he esimaed value of a European call under he Heson Model. Funcion 4 (call_heson_cf.m) performs he calculaions based on such equaions. Example illusraes how o use hese funcions o value a call opion where S = 1, K =, V =.16, V =.16, a = 1, =, = -.8 and = T = 1. Kahl and Jäckel (5) showed ha he esimaed value for his opion under he Heson model is C =.495. As he example shows, our implemenaion yields he same call value. I is also relevan o noe ha some auhors compue he price of vanilla opions in he Heson model using he Fas Fourier Transformaion (FFT). This approach has he advanage ha i can provide simulaneously he prices of opions wih differen srikes and, herefore, i employs lower compuaional ime 3. However, he FFT approach inroduces an addiional parameer and is implemenaion requires modifying he general valuaion formulas presened in secion. Consequenly, since our aim is o develop pracical inuiion on he Heson model, we will no employ his approach. Malab Funcion 3: Characerisic funcion of he Heson model (chfun_heson.m ) funcion y = chfun_heson(s, v, vbar, a, vvol, r, rho,, w); % Heson characerisic funcion. % Inpus: % s: sock price % v: iniial volailiy (v^ iniial variance) % vbar: long-erm variance mean % a: variance mean-reversion speed % vvol: volailiy of he variance process % r : risk-free rae % rho: correlaion beween he Weiner processes for he sock price and is variance % w: poins a which o evaluae he funcion % Oupu: 3 See Carr and Madam (1998). 1

11 % Characerisic funcion of log (S) in he Heson model % Inerim calculaions alpha = -w.*w/ - i*w/; bea = a - rho*vvol*i*w; gamma = vvol*vvol/; h = sqr(bea.*bea - 4*alpha*gamma); rplus = (bea + h)/vvol/vvol; rminus = (bea - h)/vvol/vvol; g=rminus./rplus; % Required inpus for he characerisic funcion C = a * (rminus * - ( / vvol^).* log((1 - g.* exp(-h*))./(1-g))); D = rminus.* (1 - exp(-h * ))./(1 - g.* exp(-h*)); % Characerisic funcion evaluaed a poins w y = exp(c*vbar + D*v + i*w*log(s*exp(r*))); Malab Funcion 4: Call price in he Heson model (call_heson_cf.m) funcion y = call_heson_cf(s, v, vbar, a, vvol, r, rho,, k) % Heson call value using characerisic funcions. % y = call_heson_cf(s, v, vbar, a, vvol, r, rho,, k) % Inpus: % s: sock price % v: iniial volailiy (v^ iniial variance) % vbar: long-erm variance mean % a: variance mean-reversion speed % vvol: volailiy of he variance process % r: risk-free rae % rho: correlaion beween he Weiner processes of he sock price and is variance % : ime o mauriy % k: opion srike % chfun_heson: Heson characerisic funcion % 1s sep: calculae pi1 and pi % Inner inegral 1 in1 s, v, vbar, a, vvol, r, rho,, k) real(exp(-i.*w*log(k)).*chfun_heson(s, v, vbar, a, vvol, r, rho,, w-i)./(i*w.*chfun_heson(s, v, vbar, a, vvol, r, rho,, -i))); % inner inegral1 in1 = inegral(@(w)in1(w,s, v, vbar, a, vvol, r, rho,, k),,1); % numerical inegraion pi1 = in1/pi+.5; % final pi1 % Inner inegral : in s, v, vbar, a, vvol, r, rho,, k) real(exp(-i.*w*log(k)).*chfun_heson(s, v, vbar, a, vvol, r, rho,, w)./(i*w)); in = inegral(@(w)in(w,s, v, vbar, a, vvol, r, rho,, k),,1);in = real(in); pi = in/pi+.5; % final pi 11

12 % rd sep: calculae call value y = s*pi1-exp(-r*)*k*pi; end Numerical Example : Call valuaion in he Heson model. % funcion y = call_heson_cf(s, v, vbar, a, vvol, r, rho, ); >> call_heson_cf(1,.16,.16, 1,,, -.8, 1, ) ans =.495 1

13 4. Calibraion o Marke Prices Before using a pricing model we should ensure ha i can produce accurae resuls for he opions ha are already raded in he marke. Availabiliy of closed-form soluions is paricularly useful in he calibraion process. Typically, when we seek o obain he opimal model parameers ha are able o reproduce marke prices, we need o perform a subsanial number of plain vanilla opions repricings. Consequenly, accurae and efficien pricing formulas are required in order o obain reliable resuls wihin a reasonable imeframe. 4.1 Calibraion Procedure in he Heson Model The goal of calibraion is o find he parameer se ha minimizes he disance beween model predicions and observed marke prices. In paricular, using he risk-neural measure, he Heson model has five unknown parameers V, V, a,,. Therefore, by calibraing hese parameers values, we seek o obain an evoluion for he underlying asse ha is consisen wih he curren prices of plain vanilla opions. In order o find he opimal parameer se we need o (i) define a measure o quanify he disance beween model and marke prices; and (ii) run an opimizaion scheme o deermine he parameer values ha minimize such disance. A simple and sraighforward approach is o minimize he mean sum of squared differences 1 G( ) C ( K, T ) C ( K, T ) N Mk i i i i i i i1 N (4.1) Mk Where Ci ( Ki, Ti ) are he opion values using he parameer se, and Ci ( Ki, Ti ) are he marke observed opion prices. As shown in Bin (7), he calibraion process presens he problem ha he objecive funcion is no necessarily convex and may exhibi several local minima. This complicaes he esimaion of he opimal he parameer se, since he soluion aained by local opimizaion migh be dependen on he iniial guess. Therefore, a good iniial guess migh be criical and, even hen, in some cases he convergence o he global opimum is no guaraneed. The obvious soluion is o employ global opimizaion. However, global opimizers generally lack he mahemaical racabiliy of local ones, and also require subsanially higher compuaional imes. Since boh mehods have advanages and disadvanages, we will explore boh approaches. 4. Local Opimizaion When a funcion exhibis several minima, local opimizers face he problem ha once a soluion has been found, we canno be sure wheher such soluion is he bes available. In oher words, we canno disinguish if he soluion is a local minimum or a global one, or consequenly, if we have reached a local soluion, here is no easy way o measure how far we are from he global one. 13

14 An alernaive o ackle his problem is o define a crierion for accepable soluions. If we selec a priori which soluions can be deemed accepable, we can a leas ensure ha any acceped soluion will be consisen wih our olerance bounds. Conversely, if we found a nonaccepable soluion, we can run he algorihm wih a differen saring poin and keep searching for soluions ha comply wih our crieria. In our ess, we will require ha he difference beween model and marke prices falls on average wihin he observed bid-ask spreads. Therefore, we will consider he following se of accepable soluions ˆ 1 N 1 N N ˆ Mk Ci ( Ki, Ti ) Ci ( Ki, Ti ) bidi aski (4.) i1 i1 Mk where Ci ( Ki, Ti ) are he model prices wih he opimal parameer se, Ci ( Ki, T i) are he mid-marke opion prices, and bid i / askiare he marke observed bid and ask prices. As a local opimizer we will use he Malab lsqnonlin funcion (leas-squares non-linear), which implemens a rus-region reflecive minimizaion algorihm 4. In addiion, we will also define lower and upper bounds for he opimal parameers. These hresholds are included in he calibraion in order o avoid possible soluions ha, while mahemaically feasible, are no accepable in an economic sense. In paricular, we will use he following bounds: Long-erm variance and iniial variance: Accepable soluions for variance levels should ake a possible value. However, given is mean-reversion, he volailiy of mos financial asse rarely reaches levels beyond 1%. Consequenly, we will use bounds of and 1 for boh forv and V. Correlaion: Saisical correlaion akes values from -1 o 1. As previously menioned, he correlaion beween volailiy and sock prices ends o be negaive. However, posiive correlaions migh also be possible in paricular cases. Therefore, he full range of accepable soluions will be used in he calibraion. Volailiy of variance: Being a volailiy, his parameer should exhibi posiives values. However, he volailiy of financial asses may change dramaically in shor ime periods (i.e. he volailiy iself is very volaile). Consequenly, high upper bounds are required for his parameer. In order o avoid poenial resricions, a broad se of soluions, from o 5, will be used in he calibraion. Mean-reversion speed: To ensure mean-reversion he parameer a should ake posiive values (negaive values will cause mean aversion). However, we have no found clear evidence regarding which upper value could be an appropriae bound. Consequenly, insead of fixing an upper level, maximum values for a will be dynamically se in he calibraion as a by-produc of he non-negaiviy consrain. Non-negaiviy consrain: In addiion o he parameer bounds, anoher condiion is required o ensure ha he variance process in he Heson model does no reach zero or negaive values. In his regard, Feller (1951) shows ha a consrain av N 4 See Yuan (1999) for an overview on he use of rus-region algorihms for solving non-linear problems. 14

15 (generally known as he Feller condiion) guaranees ha he variance in a CIR process is always sricly posiive 5. The opion daase ha we use in he calibraion are shown in Appendix D. Using he bounds described above, he implemenaion of he local calibraion algorihm is shown in scrip 1 (Heson_calibraion_local.m). In addiion, funcion 5 (cosf.m) provides he objecive funcion required for scrip 1. For daase D1, he resuls obained wih local opimizaion are he following: V V a Using hese resuls, he model prediced values and is comparison wih he marke prices are shown below: Opion id. Mid price Model price Difference(abs) Wihin bid-ask? YES YES YES YES NO YES NO NO YES YES YES YES YES YES YES As he able shows, he calibraed Heson model provides a good mach for mos raded opions. 1 ou of 15 opions have a prediced value ha falls wihin he observed bid-ask spread. In addiion, when evaluaed in erms of our accepance crierion, he model s average disance from he mid-marke price is.3369, which is lower han he average deviaion in he bid-ask spreads (.6933). The compuaional ime required for he local calibraion is 6.5 seconds. However, he able also highlighs a limiaion of sochasic volailiy models: hese models may have problems o mach he prices of ou-of-he-money (OTM) opions wih shor 5 This condiion is paricularly useful in cerain Mone Carlo discreizaion schemes. In he calibraion, he non-negaiviy consrain has been implemened by inroducing an upper bound in he accepable values of av. Since V and have heir own range of accepable values, his condiion implicily resrics he accepable values of a o hose ha comply wih he non-negaiviy consrain. 15

16 mauriies (see, in paricular, opion n. 5 6 ). More ofen han no, diffusion processes canno generae he subsanial underlying asse movemens ha are rouinely implied by he prices of shor-daed OTM opions. Price jumps are generally perceived as one of he main drivers behind he high quoes for his ype of opions. Consequenly, adding jumps o he underlying price process may be seen as a possible way forward which may improve he overall fi o marke prices. 4.3 Global Opimizaion The main advanage of global opimizaion is ha i does no exhaus is search on he firs minimum aained. Generally, global opimizers include sochasic movemens in heir search paern, which make i possible o overcome local minimums and coninue searching even if a poenial soluion has already been found. However, he use of sochasic mehods also enails cerain drawbacks. The mahemaical properies of hese algorihms are less racable han hose of local (deerminisic) ones. In addiion, despie is name, heir convergence o he global minimum is no guaraneed. In fac, since he exi sequence is deermined sochasically, he algorihm migh decide o erminae early and, in some cases, he soluion aained migh underperform a local search. All in all, even if global opimizaion is heoreically more powerful, when working wih funcions of unknown shape, i is no easy o esablish ex ane which calibraion mehod will perform beer. In order o es he resuls of global opimizaion we employ he Simulaed Annealing framework (SA). This algorihm conducs a guided search, where new ieraions are generaed by aking ino accoun he previous informaion bu also inroducing randomizaion. Iniially, he algorihm sars wih high olerance for random shocks, and differen regions are surveyed during he firs phase. As a consequence, even if a minimum is found, he algorihm keeps searching for beer soluions. As ime evolves, he algorihm decreases is olerance unil i evenually seles in he bes opimum aained. In paricular, we will use he Malab funcion asamin, which was developed by Prof. Shinichi Sakaa. This funcion implemens an Adapive Simulaed Annealing (ASA), dynamically adjusing he olerance for random shocks. The ASA framework has been shown by Goel and Sander (9) o provide good resuls among a range of differen global opimizers. For comparabiliy, we will use he same parameer bounds ha we defined in secion 4.. The implemenaion of he asamin funcion is shown in scrip (Heson_calibraion_global.m), while he required cos funcion is implemened in funcion 6 (cosf_.m). Running scrip, he opimal resuls obained for daase D1 are shown below: V V a Individual conrac deails are included in Appendix D. 16

17 Opion id. Mid price Model price Difference(abs) Wihin bid-ask? YES YES YES YES NO YES NO NO YES YES YES YES YES YES YES As can be seen, he opimal parameers values under ASA are slighly differen o hose of local calibraion. However, here are no significan divergences in he overall resuls. Under global calibraion 1 ou of 15 model values are wihin he observed bid-ask spreads, and he average disance o he mid-marke price is Therefore, he ASA soluion is also accepable according o our crierion and is qualiy is similar o he resuls obained hrough Malab s lsqnonlin. The main drawback of ASA is is subsanially higher compuaional ime (45.1 seconds in ASA vs 6.5 seconds in Malab s lsqnonlin). 4.4 More Calibraion Exercises Based on daase D1 boh ASA and Malab s lsqnonlin yield similar soluions. However, he complexiy of mulidimensional non-linear opimizaion makes i difficul o draw conclusions from a single comparison. In order o obain furher evidence, we carried ou wo addiional calibraion exercises. Firs, we applied boh mehods o an opion daase which, a priori, should be easier o calibrae. In paricular, all he opions in daase D have relaively broad bid-ask spreads and heir implied volailiies are also relaively sable. Second, we also esed a poenially more challenging daase (D3). In his case, he number of opions was doubled and insrumens wih shorer mauriies and divergen implied volailiies were included in he calibraion. The nex able summarizes he calibraion resuls for hese daases. Daase N. of opions Elapsed ime Malab s lsqnonlin Wihin bid-ask Average disance Elapsed ime ASA (asamin) Wihin bid-ask Average disance D sec 15 of sec 15 of D sec 4 of sec 4 of 3. 17

18 In daase D, boh calibraion mehods produce good resuls. All he model prediced values are wihin he observed bid-ask spread. In erms of he disance from he mid-marke prices, Malab s lsqnonlin performs slighly beer, wih an average disance of.393, agains.435 in ASA. In addiion, as expeced, he ASA algorihm akes subsanially longer o reach he opimum. Calibraion ges more difficul in daase D3. Alhough boh mehods provide accepable soluions 7, he number of opions wihin heir observed bid-ask spread falls o 4 ou of 3. However, even in hese challenging condiions, he comparison beween boh mehods exhibis a similar paern, wih Malab s lsqnonlin reaching slighly beer soluion (average disance.197 vs.) and ASA requiring significanly longer compuing imes. Based on hese exercises, we can conclude ha Malab s lsqnonlin provides beer calibraion resuls, and i also employs lower compuaional imes. However, hese resuls could be condiioned by an objecive funcion ha may no be complex enough o exploi he ASA srenghs. In paricular, since ypically we do no know wheher he objecive funcion may exhibis several local minima, a conservaive approach will be o run boh calibraion approaches. The drawback is, of course, ha a global search migh no necessarily improve he resuls provided by a local one. However, he advances in compuing power and numerical mehods keep reducing he ime required for global calibraion. In our exercises, he running ime of ASA was lower han 1 minues, which for many pracical applicaions makes i worh esing for poenially beer soluions. Scrip 1: Heson local calibraion using Malab s lsqnonlin (Heson_calibraion_local.m) % Heson calibraion, local opimizaion (Malab's lsqnonlin) % Inpu on daa.x % Daa = [So,, k, r, mid price, bid, ask] clear all global daa; global cos; global finalcos; load daa.x % Iniial parameers and parameer bounds % Bounds [v, Vbar, vvol, rho, *a*vbar - vvol^] % Las bound include non-negaiviy consrain and bounds for mean-reversion x = [.5,.5,1,-.5,1]; lb = [,,, -1, ]; ub = [1, 1, 5, 1, ]; % Opimizaion: calls funcion cosf.m: ic; x = lsqnonlin(@cosf,x,lb,ub); oc; % Soluion: 7 The average observed deviaion in he marke bid-ask spreads is

19 Heson_sol = [x(1), x(), x(3), x(4), (x(5)+x(3)^)/(*x())] x min = finalcos Malab Funcion 5: Cos funcion for local calibraion (cosf.m) funcion [cos] = cosf(x) global daa; global finalcos; % Compue individual differences % Sum of squares performed by Malab's lsqnonlin for i=1:lengh(daa) cos(i)= daa(i,5) - call_heson_cf(daa(i,1),x(1), x(), (x(5)+x(3)^)/(*x()), x(3), daa(i,4), x(4), daa(i, ), daa(i,3)); end % Show final cos finalcos =sum(cos)^ end Scrip : Heson global calibraion using ASA (Heson_calibraion_global.m) % Heson calibraion, global opimizaion (asamin) % Inpu on daa.x % Daa = [So,, k, r, mid price, bid, ask] clear all global daa; global cos; global finalcos; load daa.x % Iniial parameers and parameer bounds % Bounds [v, Vbar, vvol, rho, *a*vbar - vvol^] % Las bound include non-negaiviy consrain and bounds for mean-reversion x = [.5,.5,1,-.5,5]; lb = [,,, -1, ]; ub = [1, 1, 6, 1, ]; % Opimizaion: calls funcion cosf_.m: asamin('se', 'es_in_cos_func', ); xype = [-1;-1;-1;-1;-1]; ic; [f, x_op, grad, hessian, sae] = asamin ('minimize','cosf_',x',lb',ub', xype) oc; % Soluion: Heson_sol = [x(1), x(), x(3), x(4), (x(5)+x(3)^)/(*x())] x min = finalcos 19

20 Malab Funcion 6: Cos funcion for global calibraion (cosf_.m) funcion [cos flag] = cosf_(x) global daa; global finalcos; global cos; global cos_i; % Compue individual differences for i=1:lengh(daa) cos_i(i)= daa(i,5) - call_heson_cf(daa(i,1),x(1), x(), (x(5)+x(3)^)/(*x()), x(3), daa(i,4), x(4), daa(i, ), daa(i,3)); end % Compue sum of squared differences cos = sum(cos_i.^); % Show final cos and curren soluion finalcos =sum(cos) flag = 1; Heson_sol = [x(1), x(), x(3), x(4), (x(5)+x(3)^)/(*x())] end

21 5. Conclusion Sochasic volailiy models ackle one of he mos resricive hypoheses of he BSM framework, which assumes ha volailiy remains consan during he opion s life. However, by observing financial markes i becomes apparen ha volailiy may change dramaically in shor-ime periods and is behavior is clearly no deerminisic. Among sochasic volailiy models, he Heson model presens wo main advanages. Firs, i models an evoluion of he underlying asse which can ake ino accoun he asymmery and excess kurosis ha are ypically observed (and expeced) in financial asse reurns. Second, i provides closed-form soluions for he pricing of European opions. Availabiliy of closed-form valuaion formulas is paricularly imporan for he calibraion process. In our ess, alhough he objecive funcion is no necessarily convex, boh local and global opimizaion mehods provide reasonable resuls wihin a relaively shor imeframe. However, in cases where he objecive funcion may exhibi several local minima, local opimizaion may underperform a global search. Once he model parameers have been calibraed o fi marke prices, he Heson dynamics can be used o price oher producs ha are no acively raded in he marke. Following hese resuls here are also wo possible areas of furher work. Firs, before using he calibraed model o price exoic producs, a discreizaion scheme will be ypically required in order o obain more granular informaion regarding he underlying asse dynamics during he produc s life. This can be achieved, in mos pracical cases, by implemening a Mone Carlo simulaion scheme. Second, a sep furher will be o include disconinuous jumps in he underlying asse evoluion. Adding jumps o sochasic volailiy enails higher complexiy, bu also provides a poenially more realisic framework. Mos jump models follow a characerisic funcion approach whose implemenaion is similar o he one described here. Therefore, for ineresed readers, we hope ha he explanaions provided in his paper may help hem o connec he dos in heir nex mahemaical journey. 1

22 Appendix A: Derivaion of 1 and The proof is divided in wo pars. In he firs one we derive 1 and based on he relaionship beween he cumulaive densiy funcion (CDF) of a random variable X and is characerisic funcion The second par is devoed o prove (A.1). 1 1 e ( w) F( x) Re iw iwx X (A.1) *** For he firs par we follow he reasoning in Chourdakis (8). We sar wih he value of European call wih mauriy dae T and srike K. In a risk-neural conex, he call value a = is given by rt C e EQ max( ST K,) (A.) Using x = ln ( S T ) and expanding (A.) we ge an expression for he European call value ha is similar o he definiion in erms of 1 and ha we used in (.5) rt x C e ( e K) f ( x) dx log K (A.3) rt x e e f ( x) dx K f ( x) dx log K e I e KI rt rt 1 For a given call opion, by comparing equaions (.5) and (A.3) i can be seen ha should rt be equal o I, while 1 should be equal o I1e / S. The second inegral I is simply he probabiliy of he log-sock price finishing above he log-srike. Therefore, by applying he relaionship in (A.1), his probabiliy can be obained in erms of he characerisic funcion of ln ( ST ) as follows I P(ln S ln K) T log K 1 P(ln S ln K) T 1 1 Re iw which is he definiion of ha we presened in (.7). iwln( K ) e ln S ( w) T x To derive 1, we muliply and divide he firs inegral I 1 by he erm e f ( x) dx rt also equal, in a risk-neural conex, o he capialized spo price (i.e. e S ), which is

23 x e f ( x) dx log K x x I1 e f ( x) dx e f ( x) dx log K x e f ( x) dx rt g( x) e S Working on he fracion above, we obain an alernaive inegral expression for gx ( ) as follows x e f ( x) dx x log K e f ( x) * x log K x log K g( x) dx f ( x) dx e f ( x) dx e f ( x) dx Therefore, he firs inegral I1 can be also expressed as rt * 1 ( ) log K I e S f x dx Since f * ( x ) is, by consrucion, beween and 1, is Fourier ransforms is given by * iwx * ( wi) ( w) e f ( x) dx ( i) Consequenly, using again he relaionship in (A.1) iwln( K ) 1 1 e ln S ( w i) rt T I1 e S Re iwln S ( i) T rt Finally, since 1 I1e / S, he expression for 1 simplifies o: iwln( K ) 1 1 e ln S ( w i) T 1 Re iwln S ( i) T which is he definiion of 1 ha we used in (.6). *** The second par follows he reasoning in Kendall, Suar and Ord (1994) and Wu (7). Firs we sar wih he inegral iwx iwx e X( w) e X( w) I iw Replacing each characerisic funcion by is inegral form, he expression above becomes 3

24 I iwx iwz iwx iwz e e df( z) e e df( z) iw iwx iwz iwx iwz e e e e df( z) iw iw( xz) iw( xz) e e df( z) iw i i Nex, considering Euler s equaliy sin( ) ( e e ) / i, and using w( x z), i can be iw( xz) iw( xz) seen ha sin w( x z) ( e e ) / i. Therefore, applying Fubini s heorem and he fac ha inegral simplifies o lim sin ( ) / d / sgn( ) he n n I sin w( x z) df( z) w sin w( x z) df ( z ) w sgn( x z) df( z) ( F( x) (1 F( x)) F( x) 1 Consequenly, solving for Fx ( ) and hen subsiuing I by is original definiion yields 1 1 F( x) I 1 1 iwx iwx e X( w) e X( w) iw Finally, since he densiy of X is a real-valued funcion, using he properies of Fourier ransforms, ( w ) ( w) ( w) / Re[ ( w)]. has conjugae symmery and X Therefore, he CDF of X can also be expressed as iwx iwx 1 1 e X( w) e X( w) F( x) iw iwx 1 1 e X ( w) Re iw which is he definiion of Fx ( ) ha we used in (A.1). X X X 4

25 Appendix B: Equivalence of our approach o Gaheral (6) The analysis of he Heson call value in Gaheral (6) is based on he process xt ln( FT / K), where FT is he forward price of he underlying asse a he mauriy dae T. Consequenly, is derivaion focus on he fuure value of he European call a ime T Ga CT x K( e T ) (B.1) 1 rt raher han is value oday. However, aking ino accoun ha he forward price is FT Se, equaion (B.1) becomes C Ga T K e ln( Se ( rt / K ) 1 ) S e K rt 1 and calculaing he presen value of he expression above (i.e. muliplying by rt e in a risk neural conex) yields he probabilisic definiion of he European call value ha we used hrough he paper C S e K rt 1 Nex, we need o show ha he definiions ha we used for 1 and are equivalens o hose provided by Gaheral (6). Regarding (i.e. probabiliy of he final log-sock price being greaer han he log-srike), he resul provided in Gaheral s is given by 1 1 e iw (B.) C( T, w) v D( T, w) v iwx Ga Re where x ln( FT / K), and C( T, w), D( T, w ) are defined in he same erms ha we used in (3.3). Expanding Gaheral s resul we obain 1 1 e e iw iwln( FT / K ) C( T, w) v D( T, w) v Ga Re iwln( FT ) iwln( K ) C( T, w) v D( T, w) v 1 1 e e e Re iw iwln( K ) C( T, w) v D( T, w) v iwln( FT ) 1 1 e e Re iw And recalling ha, a ime T in (3.3) is precisely, he characerisic funcion of he Heson model ha we used ( ) rt Heson C T w v D T w viw Se ln( ST ) w e [ (, ) (, ) ln( )] Ga he expression for becomes 5

26 iwln( K ) ln S ( ) 1 1 e w Ga T Re iw which is he definiion of ha we have used hrough he paper. A similar approach can be used o show he equivalence of (.6) o he expression for 1 Ga provided in Gaheral (6). 6

27 Appendix C: Risk Neuraliy in he Heson model In order o undersand he use of risk neuraliy we firs sae he main resul and hen we prove i. Main resul We sar wih he Heson dynamics under he physical measure P ds S d V S dw P,1 dv a ( V V ) d V dw P P P, dw dw d P,1 P, Q and we seek o obain a risk-neural evoluion where E ( ds / S ) rd. As we show below, using he mulidimensional Girsanov's heorem and making appropriae choices, he Heson dynamics under he risk-neural measure Q can be expressed as ds rs d V S dw Q,1 dv a ( V V ) d V dw Q Q Q, dw dw d Q,1 Q, where Q P a a, V Q and is a parameer linked o he price of volailiy risk. P av P a Therefore, he Heson dynamics under he risk-neural measure exhibi a similar paern o ha of he physical measure, bu wih a variance process ha is defined by he parameers Q P P Q Q and V insead of a and V. A remarkable feaure is ha a and V already incorporae he impac of he volailiy risk premium. Consequenly, when calibraing he risk-neural model o marke prices, we can direcly solve for esimae explicily. Q a and Q V Q a, and we will no need o In secion 3, for simpliciy, we omied he Q superscrips. However, i should be noed ha Q he values for a and V ha we used hrough he paper are he risk-neural ones (i.e. a and ), and no hose under he physical measure. The use of risk-neural dynamics is jusified Q V when all he risks relaed o holding opions can be hedged away. Wihin he Heson model, here are wo sources of uncerainy: he underlying asse movemens and he volailiy movemens. The firs risk source can be hedged away implemening a dela-hedging sraegy in similar erms o hose of he BSM framework. However, in order o hedge he volailiy risk, a liquid marke for volailiy relaed conracs is needed. Consequenly, he use of risk-neural pricing is condiioned by he assumpion of perfec hedging. If hedging is no possible, we migh need o go back o he dynamics under he physical measure, which requires differen models and hypohesis in order o esimae he appropriae risk premiums and he corresponding real-world disribuion. 7

28 Proof We sar again wih he Heson dynamics under he physical measure ds S d V S dw P,1 dv a ( V V ) d V dw P P P, dw dw d P,1 P, (C.1) where he discouned underlying price is a maringale under P. To obain he risk-neural dynamics we should find an equivalen maringale measure (EMM) where he process ds / S has a drif of rd. To achieve his we perform a change of probabiliy measure using Girsanov s heorem. In paricular, we define a new EMM hrough he Radon-Nikodym derivaive: dq M dp where M is an exponenial maringale of he form T,1 1 T T, 1 T P P M exp C sdws Cs ds DsdWs Ds ds and i is he soluion of he SDE wih iniial value M 1. dm C P,1 P, dw D dw M Since we are working wih EMMs, he expecaion of a given sochasic process Z under he new measure Q can be compued as Q P E ( Z) E ( M Z) Therefore, if we consider he expecaion of infiniesimal incremens Q P M dm P dm P P P,1 P, E ( dz) E dz E 1 dz E dz E ( C dw D dw ) dz M M Using he equaion above, we can compue he drif and volailiy for he process ds / S under Q 8

29 Q ds P ds P P,1 P, ds E E E ( CdW D dw ) S S S E d V dw E C dw D dw d V dw P P,1 P P,1 P, P,1 ( ) ( )( ) P,1 P P P,1 P P P,1 E ( d) E ( V dw ) E ( C dw d) E ( C V ( dw ) ) E D dw d E D V dw P P, P ( ) ( d C V d D V d ( C V D V ) d P, P dw,1 ) Q ds P P P P E E CdW D dw d V dw S,1,,1 1 P P,1 E V ( dw ) V d where we expanded he iniial expressions and we used he fac ha Weiner processes are P,1 P, disribued as N(, ) and, consequenly, E( dw dw ) d. We also used he basic rules of sochasic calculus E( dw) ; E( dwd ) ; E( d ) and E ( dw ) d. Similarly, he drif and volailiy for dv can be compued as,1, ( ) E dv E dv E C dw D dw dv Q P P P P E ( a ( V V ) d) E ( C dw D dw ) a ( V V ) d V dw P P P P P,1 P, P P P, P P P P,1 P, P P, a ( V V ) d E C V dw dw E D V ( dw ) P P a ( V V ) C V D V d 1,1, ( ), E dv E C dw D dw a V V d V dw Q P P P P P P P P, E V ( dw ) V d Now, in order selec he desired EMM, we impose he resricion E Q ds S rd which gives us he equaion ( C V D V ) d rd. Rearranging erms we obain he following relaionship, which defines he marke price of risk r C D V 9

30 Addiionally, we need o se he drif for he volailiy process. In his case, an appropriae choice is ( ) Q P P E dv a V V V d where is a parameer relaed o he price of volailiy risk. This consrain gives us he P P P P equaion [ a ( V V ) C V D V ] d [ a ( V V ) V ] d, which defines he price of volailiy risk V C D Considering he properies of EMMs, he mulidimensional Girsanov s heorem ells us ha he Weiner processes under he new measure Q are Q,1 P,1 r W W V Q, P, V W W Therefore, rearranging erms and subsiuing on he iniial dynamics we ge ds S d V S dw P,1 Q,1 r Sd V Sd W V Q,1 r Sd V SdW V S d V rs d V S dw Q,1 dv a ( V V ) d V dw P P P, P P Q, V a ( V V ) d V d W P P Q, V a ( V V ) d V dw V d a V a V V d V dw P P P Q, and if we inroduce he noaion Q P a a and V Q, he process dv becomes P av P a dv a ( V V ) d V dw Q Q Q, 3

31 P,1 P, Finally, he correlaion condiion dw dw d is equivalen o require P,1 P, E( dw dw ) d. And considering he relaionship beween he Weiner processes under he physical and he risk-neural measure we ge d E dw dw P,1 P, ( ) Q,1 r Q, V E d W d W V E dw dw Q,1 Q, where we have used again he sochasic calculus rules E( dw) ; E( dwd ) and E( d ). 31

32 Appendix D: Daases used for Calibraion Daase D1: 15 opions (3 mauriies, 5 srikes). Spo Mauriy Srike Ineres rae Mid Bid Ask Call opions wrien on Biogen Idec (Nasdaq: BIIB). Marke daa observed on February 14, 14 Daase D: 15 opions (3 mauriies, 5 srikes) Spo Mauriy Srike Ineres rae Mid Bid Ask Call opions wrien on The Priceline Group (Nasdaq: PCLN). Marke daa observed on February 4, 14 3

33 Daase D3: 3 opions (6 mauriies, 5 srikes) Spo Mauriy Srike Ineres rae Mid Bid Ask Call opions wrien on Yahoo (Nasdaq: YHOO). Marke daa observed on March 4,