Valuation of options on discretely sampled variance: A general analytic approximation

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1 Valuatio of optios o discretely sampled variace: A geeral aalytic approximatio Gabriel Drimus Walter Farkas, Elise Gourier 3 Previous versio: Jauary 3 This versio: July 4 Abstract The values of optios o realized variace are sigificatly impacted by the discrete samplig of realized variace ad may be substatially higher tha the values of optios o cotiuously sampled variace. Uder geeral stochastic volatility dyamics, we aalyze the discretizatio effect ad obtai a aalytical correctio term to be applied to the value of optios o cotiuously sampled variace. The result allows for a straightforward implemetatio i may of the stadard stochastic volatility models proposed i the literature. Fially, we compare the performace of differet umerical methods for pricig optios o discretely sampled variace ad give recommedatios based o the optio s characteristics. keywords: optios o realized variace, variace swaps, stochastic volatility. jel: G3, C63. Itroductio Early literature o variace derivatives assumed cotiuous samplig of the realized variace. The first cotributio belogs to Dupire993 ad Neuberger994 who itroduced the cocept of the log-cotract ad argued that delta hedgig this cotract leads to the replicatio of the cotiuously Istitute of Bakig ad Fiace, Uiversity of Zürich, Plattestrasse 4, CH-83 Zürich, Switzerlad. gabriel.drimus@bf.uzh.ch ad walter.farkas@bf.uzh.ch. Departmet of Mathematics, ETH Zürich, Rämistrasse, CH-89 Zürich, Switzerlad. 3 ORFE Departmet, Priceto Uiversity, Sherrerd Hall, Priceto NJ 8544, USA. egourier@priceto.edu Fudig from the Natioal Ceter for Competece i Research NCCR FINRISK through Project D Mathematical Methods i Fiacial Risk Maagemet ad from the Swiss Natioal Sciece FoudatioSNSF is gratefully ackowledged. Electroic copy available at:

2 sampled variace. Carr, Mada998 subsequetly exteded the results ad showed how variace swaps as well as corridor variace swaps ca be replicated by a static positio i a cotiuum of vailla optios, dyamically delta-hedged with the uderlyig asset. Broadie, Jai8 further show that volatility derivatives ca be dyamically hedged usig variace swaps ad a fiite umber of Europea optios. For a detailed overview of the various theoretical developmets, we refer the reader to Carr, Lee9. I derivative markets, variace cotracts are specified with discrete samplig; i particular, daily samplig is the most commo covetio. For liear cotracts o realized variace such as variace swaps, the discretizatio effect is usually small, as foud i Broadie, Jai8. The explaatio follows from the fact that liear cotracts o variace do ot deped o the volatility of variace. Itki, Carr used the method of forward characteristic fuctios to show how to price discrete variace swaps i geeral time-chaged Lévy models. Uder the assumptio of a stochastic clock idepedet of the drivig Lévy process, sectio 6 i Carr, Lee ad Wu shows that discrete samplig icreases the value of variace swaps. For o-liear cotracts o variace such as optios o variace the discretizatio effect becomes substatial, especially for shorter maturities. The short-time limit of the discretizatio gap, uder geeral semi-martigale dyamics, has bee derived recetly i Keller-Ressel, Muhle-Karbe; the authors also develop Fourier pricig methods for optios o discrete variace uder expoetial Lévy dyamics. I the cotext of the Hesto993 model, Sepp proposes a approximatio by combiig the distributio of quadratic variatio i the Hesto993 model with that of discrete variace i a idepedet Black, Scholes973 model. The approach leads to a tractable characteristic fuctio for the discretely sampled variace ad provides good accuracy ear the at-the-moey regio, across maturities. I this paper we provide a comprehesive treatmet of the discretizatio effect uder geeral stochastic volatility dyamics. Additioally, we do ot restrict attetio to particular strike rages or to particular maturity rages. We begi by provig that, coditioal o the realizatio of the istataeous variace process, the properly scaled residual radomess arisig from discrete samplig ca be well approximated with a ormally distributed radom variable as formulated i Theorem. of the followig sectio. I the fiacial ecoometrics literature, see Bardorff-Nielse, Shephard, ad more geerally, Bardorff-Nielse et al.6, related results were obtaied ad used i the aalysis of high frequecy data ad the estimatio of stochastic volatility models. I cotrast, we adopt a coditioal approach, which was pioeered i the study of stochastic volatility models by Hull, White987, i the case of zero correlatio betwee volatility ad the uderlyig asset, ad by Romao, Touzi997 ad Willard997 i the case of o-zero correlatio. Additioally, as made precise i the ext sectio, we cosider a differet limitig sequece, which accouts for the correlatio Electroic copy available at:

3 iduced terms i the discrete variace. I a first step, we reduce the dimesioality of the Mote-Carlo pricig scheme by elimiatig the eed to simulate the path of the log-returs. I a secod step, a further simplificatio makes possible to avoid the simulatio of the istataeous variace path, by simulatig directly from the distributio of the itegrated cotiuous variace. Most importatly, a variatio o the latter approach termed hereafter the coditioal Black-Scholes scheme leads to a explicit discretizatio adjustmet term, for which two simple aalytic represetatios are provided i equatios 3 ad 4. This explicit discretizatio adjustmet term is easily computable by stadard Fourier trasform methods, i ay stochastic volatility model with a closed-form expressio for the characteristic fuctio of cotiuously sampled variace, e.g. Hesto993 or the 3/ model i Lewis ad Carr, Su7. The rest of the paper is orgaized as follows. I Sectio, the core sectio, we develop the theoretical results ad also defie the proposed umerical schemes, specifically, the Mote Carlo ad the Fourier based methods. Sectio 3 icludes a overview of the umerical methods aalyzed ad Sectio 4 presets the performace of the differet pricig schemes. The fial sectio summarizes the coclusios. Proofs ot give i the mai text, as well as additioal umerical results, ca be foud i the Appedix. Optios o discretely sampled variace I this sectio we study the magitude of the discretizatio effect i valuig optios o realized variace.. Model We start by settig a geeral stochastic volatility framework. We cosider a filtered probability space Ω, F, {F t }, Q satisfyig the usual coditios ad let B t, W t t be a two-dimesioal Browia motio with correlatio ρ. Additioally, let N t t be a idepedet Poisso process with costat itesity λ. Assume that the stock price ad its istataeous variace S t, v t t satisfy the followig dyamics uder a pricig measure Q: ds t S t = r δ λee Zt dt + v t db t + e Zt dn t dv t = av t dt + bv t dw t where a : R + R ad b : R + R are Borel measurable fuctios such that the two-dimesioal SDE,, admits a uique ad o-explodig solutio S t, v t t. The jump size i the log-retur at time t is deoted by Z t ; jump sizes are assumed to be idepedet ad idetically distributed ad to follow a ormal distributio N µ Z, σz. The risk free iterest rate, divided 3

4 yield ad correlatio parameters are deoted by r, δ ad ρ respectively. A large umber of stochastic volatility models proposed i the literature belog to this framework. Importat examples iclude Scott987, Hesto993 ad the 3/ model discussed i Lewis ad Carr, Su7. Cosider a fiite maturity T > ad let = t < t < t <... < t = T be a equally spaced partitio of [, T ] with step-size = T. The aualized discretely sampled variace of logs t over [, T ] is defied as RV = T log Sti. 3 S ti A stadard result i stochastic calculus see, for example, Revuz, Yor 999 establishes that, as, RV coverges i probability to the cotiuously sampled variace or quadratic variatio of logs t over the iterval [, T ] ad scaled by the aualizatio factor T. Specifically, i our setup we have plim RV = T [logs t] T = T T v tdt + t T Z t N t, where plim deotes the limit i probability. I practice, variace cotracts must be specified by usig discrete samplig. For example, a call optio o realized variace with maturity T ad volatility strike σ K delivers, at time T, the payoff: V N T log Sti σk S ti + where V N is a costat kow as the variace otioal; throughout this sectio, we set V N =. Our goal here is to compare the prices of optios o discretely sampled variace to those of optios o quadratic variatio. A applicatio of Itô s lemma to logs t gives d logs t = r δ v t λe[ez ]dt + v t db t + Z t dn t from where, each discrete log-retur ca be writte as ti ti Sti log = r δ λe[e Z ] T S ti v s ds+ vs db s + t i t i N ti k=n ti + for all i {,,..., }. I what follows we deote by Ft W the filtratio geerated by the Browia motio W t drivig the variace diffusio v t ; we recall that the process v t serves to model the stochastic istataeous variace of the asset price. A key observatio is that, coditioal o FT W, the log-returs log S ti S t i, i = {,,..., }, form a sequece of idepedet ormally distributed but ot idetically radom variables. Z k 4

5 Usig that E N ti k=n ti + Z k = λ T µ Z ad V ar N ti k=n ti + Z k = E V ar V ar E N ti k=n ti + N ti k=n ti + Z k N ti N ti + Z k N ti N ti = λ T σ Z + λ T µ Z, we ca coclude that: Sti log S ti F W T N µ,i, σ,i 4 with meas ad variaces µ,i ad σ,i give by µ,i = r δ λe µ Z+ σ Z µz T ti σ,i = ρ v s ds + µ Z + σzλ T t i. ti t i v s ds + ρ ti t i vs dw s for all. The result i 4 follows immediately from the property that the Browia itegral of ay determiistic, locally-bouded fuctio is a Gaussia process see, for example, Revuz, Yor999 ad that jump sizes form a idepedet ormally distributed sequece. We ote that, i the absece T v tdt of jumps, coditioal o FT W, the cotiuously sampled variace T is just a costat, whereas the discretely sampled variace RV still has a residual radomess drive by B t. I practice, for typical parameter values, this residual radomess is ot egligible ad ca lead to substatially higher prices for optios o discretely sampled variace, especially for maturities less tha oe year.. Distributio of RV coditioal o F W t We formulate here the mai theorem which plays a key role i the umerical schemes proposed. 5

6 Theorem.. Coditioal o F W T RV =, the discretely sampled realized variace log Sti S ti coverges i distributio to a ormal radom variable. More precisely, as, we have RV µ,i + σ,i d N, 5 s where s = σ,i 4 + 4µ,i σ,i. Remark: Theorem. shows that the coditioal distributio of RV is asymptotically ormal N M, Σ, which will prove useful i calculatig the price of optios o realized variace. The coditioal momets are give by M = µ,i + σ,i Σ = s. We emphasize that these momets are coditioal o Ft W or alteratively they hold give a kow path of the istataeous variace up to maturity. Proof of Theorem. To establish this result, we use a geeralized versio of the cetral limit theorem CLT for triagular arrays of uequal compoets. To see why this is ecessary, ote that for each, i the expressio of RV, we have a differet sequece of squared log-returs ad the compoets of each sequece have differet variaces. Specifically, we shall use the Lideberg-Feller geeralized CLT see, for example, Ferguso996 as formulated i Theorem 6. i Appedix 6. Defie the triagular sequece Y,i = X,i µ,i + σ,i, where X,i = log Sti S ti is the triagular sequece of log-returs. We seek to apply the Lideberg-Feller Theorem 6. to the sequece Y,i. coditio EY,i = is satisfied. By costructio, the Makig use of the coditioal ormality of the log-returs X,i we obtai, coditioally o F W T : EY,i = σ 4,i + 4µ,iσ,i. 6

7 The computatio of this expectatio follows from the straightforward but tedious use of the higher momets of the ormal distributio. Take ay α 3,. By the local properties of Browia paths see, for example, Revuz, Yor999, the fuctio ht = t vs dw s is Hölder cotiuous with idex α o [, T ]. We coclude that there exists a positive costat K >, idepedet of, such that ti vs dw s t i = h t i h t i K t i t i α = K T α α for all. Similarly, sice the fuctio gt = t v sds is Lipschitz cotiuous o [, T ], there exists a positive costat K >, idepedet of, such that ti v s ds t i = g t i g t i K t i t i = K T for all. From the defiitio of µ,i, we see that for all positive itegers T, the followig boud holds: µ,i r δ λe µ Z+ σ Z µz T + K T + ρ K T α α r δ λe µ Z+ σ Z µz + K + ρ K T α α = C T α α. 6 Similarly, for σ,i we obtai σ,i ρ K T + µ Z + σ Zλ T = C T. 7 We ow show that the sequece Y,i satisfies the sufficiet coditio of Lyapuov 8 for δ =. Specifically, we show that lim s 4 E Y,i 4 = where s = E Y,i = σ,i 4 + 4µ,i σ,i. Similar to the computatio of the secod coditioal momet of Y,i, we ca calculate its fourth momet as follows: E Y 4,i = 6σ 8,i + 4σ 6,i µ,i + 48σ 4,i µ 4,i + 4σ,i µ 6,i. 8 7

8 Usig iequalities 6, 7 we have for all positive itegers T E Y,i 4 6 C 4 T C3 C T 3+α 3+α + 48 C C 4 T +4α +4α + 4 C C 6 T +6α +6α C 3 T +6α +6α. where we have used the fact that + 6α < + 4α < 3 + α < 4 ad T. By a simple applicatio of the classic Cauchy-Schwarz or, alteratively, Jese s iequality, we also obtai: s σ,i 4 σ,i = ρ T where C 4 > does ot deped o. Fially, this gives v t dt = C 4 s 4 E Y,i 4 C 3 T +6α +6α = C4 C 3 T +6α C 4 6α 3 as where we have used that α > 3. By the Lideberg-Feller Theorem 6., we coclude s RV µ,i + σ,i = Y,i s d N,..3 Pricig of optios o discretely sampled variace Theorem. ad basic properties of ormal radom variables give that, coditioal o FT W, the udiscouted value of the discrete variace call ca be approximated by C M σ K σ K = Σ φ + M σ M σ Σ k N K 9 Σ where φ ad N deote the desity ad distributio fuctios of the stadard ormal law. As a result of this formula, give a istataeous variace path, oe ca calculate M ad Σ ad derive the coditioal optio price usig equatio 9. The results derived so far idicate the followig method to price optios o discrete variace by elimiatig the eed to simulate paths of the spot price S t : for each simulated istataeous variace path, compute the coditioal optio price by 9 ad, fially, average over these coditioal 8

9 prices. This will be the basis of two methods we will ivestigate for pricig optios o realized variace, amely the coditioal ormal ad simplified coditioal ormal methods. Later i the sectio, we explore ways to further simplify this coditioal scheme by elimiatig the eed to compute the quatities µ,i ad σ,i for each variace path. We remark that, o the purely theoretical frot, it is possible to draw further o the tools of the geeralized CLT to establish bouds o the approximatio 9. These bouds are available i sectio 7 of the appedix..4 Example of the Black-Scholes model The stadard ad most basic model which fits i our framework is the Black- Scholes973 model, which will prove useful i our simplified coditioal pricig schemes. I the stadard Black-Scholes framework, we set v t = σ a positive costat ad the log-returs ow become i.i.d. ormal: log Sti S ti N r δ σ T, σ T which gives that the distributio of RV satisfies d σ r δ σ RV = T σ + Z i where Z i, with i, here deotes a sequece of idepedet stadard d ormal variables. We obtai that RV = σ χ, λ where χ, λ deotes the o-cetral chi-square distributio with degrees of freedom ad ocetrality parameter λ give by: r δ σ T λ = σ. A well-kow result i mathematical statistics see, for example, the classic treatmet i Muirhead5 establishes the followig covergece i distributio to a stadard ormal: χ, λ + λ + λ d N, as the umber of degrees of freedom. Usig the value of λ from, simple algebraic computatios show that the distributio of RV coverges to a ormal distributio with mea ad variace give by: N σ + r δ σ T, σ4 + 4 r δ σ σ T. 9

10 This is a special case of our more geeral result i Theorem.. It is obtaied usig that, i the stadard Black-Scholes model, µ,i = r δ σ T ad σ,i = σ T which leads to: r δ σ M = σ T + σ 4 4 r δ σ Σ = + σ T as give i. I the Black-Scholes model it is possible to derive a exact closed-form formula for the price of optios o discrete variace. We formulate this result i Lemma. ad ote that it will be used i our coditioal Black-Scholes scheme itroduced later. Lemma.. I the Black-Scholes model with costat volatility σ, we have E RV σk σ + = σ F χ K σ ; λ, + + σ λ σ F χ K σ ; λ, + 4 σ σk F χ K ; λ, where F χ ; λ, deotes the o-cetral chi-square CDF with degrees of freedom ad o-cetrality parameter λ; the value of λ is give by. Proof See Appedix.. σ.5 Optio pricig i a geeral stochastic volatility model We ow retur to the geeral stochastic volatility case ad seek to derive further simplified versios of the coditioal pricig scheme implied by Theorem. ad approximatio 9. We ote that, i a separate study, Sepp explores a differet approach to adjustig for the discretizatio effect. Specifically, the discretizatio effect is treated idepedetly by makig the followig approximatio, i distributio: RV d T T v t dt E Q T T v t dt + RV BS 3 where RV BS is the discretely sampled variace i a idepedet Black- Scholes model with time-depedet volatility σt = E Q v t ; we remark

11 that, i the presece of jumps, equatio 3 ivolves a small modificatio ad we refer the reader to Sepp for details. Give that, i a Black- Scholes model, the Fourier-Laplace trasform of discretely sampled variace is easily obtaied i closed-form ad usig the idepedece assumptio, we ca approximate the trasform of RV : Lλ = E Q e λ RV e λ M L QV λ L RV BS λ where M = E Q T T v tdt ad L QV λ is the trasform of cotiuously sampled variace. As i several stochastic volatility models L QV λ is kow i closed-form, this approach is attractive from the stadpoit of usig semiaalytical trasform techiques ad provides good accuracy for ear the at-the-moey regio. We will propose a alterative, trasform-based approach, which does ot rely o a idepedece assumptio. The advatage will be that it leads to improved accuracy for out-of-the-moey optios. The magitude of the discretizatio effect depeds o the realizatio of cotiuously sampled variace; the smaller larger the latter, the smaller larger the discretizatio effect. Igorig this depedece, will ted to overprice out-of-the-moey variace puts ad uderprice out-of-the-moey variace calls. The followig lemma will prove useful i our calculatios. It was give i Bardorff-Nielse, Shephard, uder the assumptio of a variace process of fiite variatio. This would be usuitable for our dyamics of v t but, fortuately, the assumptio ca be removed ad, hece, we modify its statemet accordigly. Lemma.. Let v t t be a process which is a.s. locally bouded ad has at most a coutable umber of discotiuity poits o every fiite iterval. The, for ay fixed T > ad positive iteger k N\{} we have as. k T k it i T v t dt k T v k t dt a.s. Proof The proof is idetical to Bardorff-Nielse, Shephard except that we do ot require the process v t t to be of locally bouded variatio. The argumet oly requires that v k t be a.s. Riema itegrable o [, T ], a coditio which is satisfied uder our assumptios for Lemma. by, for example, Theorem 6. i Rudi 976. Recall that, by Theorem., the coditioal distributio of RV i.e. give a path of the istataeous variace satisfies RV F W T M /Σ d

12 N, where M = Σ = µ,i + σ,i s = σ4,i + 4µ,i σ,i. The simplified coditioal schemes will set the variace-spot correlatio ρ to zero. It turs out that settig ρ = has little material impact o the prices of discrete variace optios. To see this ituitively, ote that, i the limit of cotiuous samplig, the correlatio parameter plays o role i the price of variace optios. I the umerical examples, we will see that for typical market parameters with strogly egative correlatio, the simplified schemes perform very well. We ext wat to apply Lemma. to obtai a approximatio for M ad Σ. Fix the followig otatios for real sequeces a ad b : write a = ob iff lim a /b = ad a = Ob iff lim sup a /b <. Firstly, observe that our istataeous variace process v t t [,T ], havig a.s. cotiuous paths, clearly satisfies the assumptios of Lemma.. Hece, lettig I k = T ρ =, σ,i = t i t i v t dt, we obtai by Lemma.: T T T vk t dt ad recallig that, for σ,i 4 = T I + o 4 σ,i 6 = T I 3 + o. 5 Also, we ote that T σ,i = I. Expadig µ,i, we have: µ,i = r δ T = r δ T r δt ad, writig T for, we obtai ti t i v t dt + 4 r δt σ,i + 4 σ4,i ti t i v t dt M = T µ,i + T σ,i = r δ T r δt I + 4 = I + O T I + o + I 6

13 We ext apply a similar approach to Σ. Usig agai the relatios 4, 5 obtaied by Lemma., simple algebra gives: Σ = T σ,i T µ,i σ,i = I + o + 4 = I + o r δ T I r δ T I + 4 T I 3 + o From 6 ad 7, we have obtaied the followig result for the coditioal distributio of RV : F W RV T N I + O, I + o 8 The Coditioal Normal Scheme. Keepig the leadig order terms i 8, we obtai the approximatio, i distributio: RV F W T N T T v t dt, T T vt dt. 9 Remark: We ote that by keepig the leadig order terms, the expectatio of the discrete realized variace i equatio 9 is approximated with the itegrated variace. However, it is kow see, e.g., Broadie, Jai8 that the fair strike of a variace swap o the discrete realized variace becomes larger whe the samplig period icreases, which implies that the discrete realized variace has a larger expected value tha the itegrated variace. We aalyze this effect i the umerical sectio. By virtue of relatio 9, we formulate the coditioal ormal pricig scheme as follows: a simulate a variace path v t, t [, T ] ad compute I ad I, b price the coditioal variace call by settig M = I ad Σ = I i formula 9 ad c average coditioal prices by repeatig a, b. Note that this approach, while o loger requirig to compute the quatities µ,i ad σ,i, still requires the simulatio of the etire variace path v t o [, T ] i order to allow us to extract both I ad I. We otice that by Jese s iequality I I ad hece the alterative approximatio F W T RV T N v t dt, T v t dt T T which uderestimates the variace term may cause some uderpricig of optios o realized variace at least, relative to 9. O the other had, 7 3

14 this approximatio will make it possible to simulate just from the law of itegrated cotiuous variace I = T T v tdt, without the eed to geerate the etire variace path v t o [, T ]. We call this scheme the Simplified Coditioal Normal scheme. We shall observe, i the umerical examples, that the uderpricig is usually small. The Simplified Coditioal Black-Scholes Scheme. It ca be show that approximatio is asymptotically equivalet to assumig that the coditioal pricig model is Black-Scholes. Specifically, coditioal o a realizatio of the itegrated cotiuous variace T T v tdt, suppose the model for the uderlyig price is Black-Scholes with variace parameter: σ = T T v t dt. We have see that, i a Black-Scholes model, the discretely sampled variace is o-cetral chi-square distributed χ, λ. I tur, keepig oly the leadig order terms i, we have that χ, λ is approximately N σ, σ4. Replacig the value of σ from, we see that the coditioal Black- Scholes approach leads, i fact, to approximatio. By virtue of relatio, we formulate the simplified coditioal Black- Scholes SCBS scheme as follows: a simulate from the law of itegrated T v tdt, b price the coditioal variace call cotiuous variace I = T by settig σ = I i the exact Black-Scholes formula of Lemma., c average coditioal prices by repeatig a, b. We ote that the SCBS scheme does ot require to simulate the etire path of v t, t [, T ]. Pursuig further the SCBS scheme, it is possible to derive a simple discretizatio adjustmet requirig o Mote-Carlo simulatio. Specifically, uder the assumptio ad provided the cotiuously sampled variace I posses a Fourier trasform i closed-form, we ext derive a leadig-order discretizatio adjustmet based o a simple Fourier iversio. I the followig, we regard the udiscouted prices of optios o realized variace as fuctios of the variace strike V = σk ad defie C, C : R R by C V = V E RV V C V = V E I V where I = T T v tdt ad, uder the SCBS scheme, RV FT W N I, I Assumig ERV < ad EI <, we first check that both fuctios C ad C L R, i.e. are itegrable o R. For example, we have for

15 C V: C V dv = E RV V + dv = E RV RV V dv = E RV < where we iterchaged itegratio ad expectatio as the itegrad is oegative; a idetical argumet holds for CV. Therefore, both fuctios will have well defied Fourier trasforms, hereafter deoted by Ĉu ad Ĉu, respectively. The followig formula, which first appeared i Carr et al. 5, ca be established for the Fourier trasforms: Ĉ u = Ĉu = e iuv C VdV = ϕ u u i ERV u e iuv CVdV = ϕu u i EI u where ϕ u = E e iurv ad ϕu = E e iui deote the Fourier trasforms of RV ad I, respectively. The key idea is to ow cosider the differece betwee the price of optios o discrete variace ad the price of optios o cotiuous variace by defiig the ew fuctio Λ V = C V CV L R. Usig that F E RV = E E RV W T = E I ad by the liearity of the Fourier trasform, we obtai Λu = e iuv ΛVdV = Ĉu Ĉu = ϕu ϕ u u. The resultig discretizatio adjustmet is give by Λu = E e iui u I e. Expadig the secod term of the product uder the expectatio ad keepig the term of order O, we ca write Λu = E e iui I + O u := Λ u + O. Makig use agai of the assumptio EI <, we have that ϕu, the characteristic fuctio of I, is twice cotiuously differetiable with respect to u ad ϕu = E e u iui I. This gives that the leadig term Λ u i ca be writte as Λ u = ϕu u 5

16 We ow proceed to determie the discretizatio adjustmet term that results from cosiderig oly the term Λ u i the expasio. More precisely, we compute Λ V := π = π = π e iuv Λ udu = e iuv π ϕu u du [ e iuv ϕu ] + iv e iuv ϕu u u du [ e iuv ϕu + iv e iuv ϕu u ] V e iuv ϕudu = V e iuv ϕudu π where both boudary terms, resultig from the itegratio by parts, will vaish by a simple applicatio of the classical Riema-Lebesgue lemma see, for example, Feller 99. Fially, we obtai Λ V = V e iuv ϕudu. 3 π We otice that the computatio of the discretizatio adjustmet term Λ V ivolves a simple Fourier iversio of ϕu. Alteratively, the discretizatio adjustmet ca be writte more compactly as Λ V = V q V 4 where q V deotes the desity of the cotiuously sampled variace I = T T v tdt. Both represetatios 3 ad 4 provide a remarkably simple formula for the leadig order discretizatio adjustmet term. For completeess, we ote that uder the SCBS scheme the Fourier trasforms of RV ad I ca also be liked as follows: ϕ u = E e iurv = E E e iurv F W T = E [ = E e iui u I [ = ϕu u E e iui I = ϕu + u ϕu u + O ] + O + O 6 ] e iui u I. 5

17 Formula 5 ca be used to approximate the price of optios o discrete realized variace by usig it directly i a stadard Fourier optio pricer as i Sepp8,. We ote that, as expected, the discretizatio adjustmet i 3-4 is o-egative reflectig that optios o discrete variace are more expesive tha optios o cotiuous variace. Therefore, if we work i a stochastic volatility model which admits a closed-form solutio for ϕu e.g. Hesto993 or the 3/ model i Carr, Su7, we first price optios o cotiuously sampled variace usig stadard Fourier methods from the literature ad the add the positive adjustmet term 3-4, which is also computable by simple Fourier iversio. 3 Summary of methods We summarize here the differet umerical methods which will be used to compute the price of optios o discretely sampled variace. The umerical performace of these methods will be detailed i Sectio 4. Cotiuous Samplig: Optios o quadratic variatio are priced usig Fourier trasform methods. The Fourier-Laplace Trasform of the optio price is iferred from the trasform of the itegrated variace ad we use a Gamma distributed radom variable as cotrol variate as i Drimus; the umerical iversio is performed usig the techique of Iseger6. As the discretizatio effect is eglected, the prices of optios o discrete realized variace are expected to be higher tha those obtaied with this method. This pheomeo was previously discussed i Bühler9, Gatheral8, Keller-Ressel, Muhle-Karbe, Sepp, ad Keller-Ressel, Griessler. Mote Carlo samplig: The prices of optios o variace are obtaied by Mote-Carlo simulatio of the SDE, by the techique i Aderse8: the simulatio techique combies the exact simulatio scheme of Broadie, Kaya6 with a umerically efficiet local momet-matchig method. We use the prices obtaied with this method as referece prices; the umber of Mote Carlo paths used is 5, ad the step size is oe busiess day dt = 5. Sepp approximatio: Optios are priced usig a approximatio i distributio of the discretely sampled variace as a fuctio of the quadratic variatio i a jump-diffusio stochastic volatility model ad the discrete variace i a idepedet Black-Scholes model. The Fourier-Laplace trasform of the discrete realized variace is obtaied usig equatio i Sepp. 7

18 Coditioal Normal method: Optios o discrete realized variace are priced by simulatig paths of the variace process ad computig the average of the coditioal optio prices over all paths. This method is based o the result that the law of realized variace, coditioal o the variace path, is asymptotically ormal by Theorem.. The coditioal momets of realized variace, M ad Σ, are approximated by applyig Lemma.; coditioal optio prices are fially obtaied usig formula 9. The mai drawback of this method is that it requires to simulate the etire path of the variace process. Simplified Coditioal Normal method: Prices of optios are obtaied by simulatig from the law of the itegrated variace. The Laplace trasform of the itegrated variace i a CIR model is kow i closedform, see for example Cox et al.985 or Dufrese. Fourier iversio allows recoverig the cumulative distributio fuctio of the itegrated variace which, applied to uiformly distributed radom variables, geerates a sample from the law of itegrated variace. The coditioal optio prices are calculated usig formula 9. Compared to the coditioal ormal approach, this method does ot require to simulate the etire path of the variace process. Simplified Coditioal Black-Scholes SCBS: Prices of optios o discrete realized variace are computed by simulatig from the law of itegrated variace ad calculatig the coditioal optio prices usig the exact formula i Lemma.. This approach is equivalet to assumig that the coditioal model is Black-Scholes with variace parameter equal to the realizatio of the itegrated cotiuous variace. As i the simplified coditioal ormal method, this approach does ot require to simulate the etire path of the variace process. Fourier-based discretizatio adjustmet: Prices of optios o realized variace are represeted as the sum of the price of a optio o quadratic variatio ad a discretizatio adjustmet term, both computed usig Fourier techiques. The discretizatio adjustmet term is computed usig equatio 3. 4 Numerical Examples I this sectio, we cosider a stadard Hesto993 model. We examie two differet parameter sets. The referece set P results from the estimatio i the study of Bakshi, Cao, Che997 for the S&P5 idex. The estimated parameters are v, k, θ, ɛ, ρ P = 8.65%,.5, 8.65%,.39,.64. The secod set P is a slight modificatio of the first oe ad aims to aalyze the effect of a strogly egative correlatio coefficiet: v, k, θ, ɛ, ρ P = 8

19 Figure : Relative errors i prices of OTM optios o discrete realized variace usig the parameter set P. The samplig frequecy is oe day. The left part of the graphs moeyess smaller tha correspods to OTM puts whereas the right part moeyess larger tha correspods to OTM calls. T= moth ; ρ=.64 ; samplig freq.= 3.5 Cot. sampl. Sepp Discr. Adj. 3 Cod. Norm. Simp. Cod. Norm Cod. BS T=6 moths ; ρ=.64 ; samplig freq.= Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS T= year ; ρ=.64 ; samplig freq.= Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS %,.5, 8.65%,.39,.9. Ideed, Berard, Cui4 show that the leverage coefficiet plays a importat role i the covergece of the fair strike of the discrete variace swap rate. We aalyze the performace of the differet methods summarized i Sectio 3 i terms of out-of-the-moey OTM optio price error across a wide rage of volatility strikes, for three differet maturities moth, 6 moths ad oe year ad for differet samplig frequecies ragig from oe to five days. The relative error of the methods as a fuctio of the moeyess σ K v is displayed i Figures,, 4, 3 which correspod to parameter sets P ad P ad differet samplig frequecies. Note that the part of the figures which is to the left of moeyess correspods to the error made i pricig OTM put optios, whereas the part to the right of moeyess represets the error made i pricig OTM call optios. This justifies the kik at moeyess equal to i some of the figures. The average relative differeces of the prices of each method compared to those of the Mote Carlo method 9

20 Figure : Relative errors i prices of OTM optios o discrete realized variace usig the parameter set P more egative value of ρ. The samplig frequecy is oe day. The left part of the graphs moeyess smaller tha correspods to OTM puts whereas the right part moeyess larger tha correspods to OTM calls T= moth ; ρ=.9 ; samplig freq.= Cot. sampl. Sepp 9 Discr. Adj. Cod. Norm. 8 Simp. Cod. Norm Cod. BS T=6 moths ; ρ=.9 ; samplig freq.= Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS T= year ; ρ=.9 ; samplig freq.=.5. Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS for differet levels of moeyesses are listed i Tables ad. The upper part of the tables describes the results for a daily samplig frequecy of the realized variace, whereas the lower part uses a samplig period of five days. Results for itermediate samplig frequecies are ot displayed but available upo request. Tables ad also report the computatioal time eeded for each method to calculate the price for all strikes i secods, o a Itel Core i7-8qm CPU.3GHz. As expected, we observe that the cotiuous samplig method uderprices optios for all strikes ad parameter sets. Whereas the bias is rather small for loger maturities, it becomes o egligible for shorter maturities. This method is computatioally much less itesive tha Mote Carlo simulatios, however, its performace is almost systematically outperformed by the Simplified Coditioal Normal method except for out-of-the-moey puts with short times-to-maturities, whose computatioal time is also less tha a secod per optio.

21 Relative errors i prices of OTM optios o discrete realized variace usig the parameter set P. The samplig frequecy is five days. The left part of the graphs moeyess smaller tha correspods to OTM puts whereas the right part moeyess larger tha correspods to OTM 3.5 T= moth ; ρ=.64 ; samplig freq.=5 Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS calls. T=6 moths ; ρ=.64 ; samplig freq.= Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS.3... T= year ; ρ=.64 ; samplig freq.=5 Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS Sepp s approximatio performs well i the at-the-moey ATM regio but overprices deep out-of-the-moey OTM put optios ad uderprices deep out-the-moey OTM call optios. It is appealig ear the ATM level ad appears to be amog the fastest methods cosidered. For short maturities, we fid that the Coditioal Black-Scholes method is the oe which performs the best. The Fourier-based discretizatio adjustmet uderprices OTM put optios ad the Coditioal Normal methods overprices them. Both uderprice OTM calls. The errors are magified whe the leverage coefficiet is larger i absolute value, however the Coditioal Black-Scholes method remais the preferred choice for short maturities, with very small relative errors. Whe the maturity icreases, we fid that the best method becomes the Fourier-based discretizatio adjustmet which is both fast ad accurate. We remark that i the right part of the smile all methods ted to slightly uderprice optios. As explaied i Sectio, oe of the reasos is that the simplified coditioal methods approximate the expected value of discrete realized variace by the expected value of the itegrated variace.

22 Figure 3: Relative errors i prices of OTM optios o the discrete realized variace usig the parameter set P more egative value of ρ. The samplig frequecy is five days. The left part of the graphs moeyess smaller tha correspods to OTM puts whereas the right part moeyess larger tha correspods to OTM calls T= moth ; ρ=.9 ; samplig freq.=5 Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS T=6 moths ; ρ=.9 ; samplig freq.=5 Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS T= year ; ρ=.9 ; samplig freq.=5.3. Cot. sampl. Sepp Discr. Adj. Cod. Norm. Simp. Cod. Norm Cod. BS Comparig the relative errors for two distict values of ρ did ot highlight a sigificat deterioratio of the performace of the approximatios whe the leverage becomes more egative. Additioally, the same umerical experimets were performed also with a differet suite of parameters: i particular we cosidered the case whe θ > v, which is ofte ecoutered i practice. The parameter sets become v, k, θ, ɛ, ρ P3 = 5%,.5, %,.39,.64 ad, combied with a strogly egative correlatio coefficiet: v, k, θ, ɛ, ρ P4 = 5%,.5, %,.39,.9. The results remai qualitatively uchaged; results are reported i Appedix C. I summary, for short-maturity optios, we recommed usig the Coditioal Black-Scholes method. For mid-term to log-maturity optios, we recommed usig the Fourier-based discretizatio adjustmet, which is fast ad accurate.

23 Table : OTM optio prices uder parameter sets P3 ad P4. OTM put refers to a optio with moeyess Strike/v.6, ATM to a optio with moeyess ad OTM call to a optio with moeyess.5. Prices are i percetages. The computatioal time is i secods. T = moth T = 6 moths T = year OTM put ATM OTM call Comp. time OTM put ATM OTM call Comp. time OTM put ATM OTM call Comp. time - Daily samplig frequecy Parameter set P Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Parameter set P Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Samplig frequecy every five days Parameter set P Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Parameter set P Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj

24 5 Coclusio Discrete samplig has sigificat impact o the valuatio of optios o realized variace. We start by providig a characterizatio of this effect i Theorem. uder geeral stochastic volatility dyamics ad costruct several methods for pricig optios o discrete variace. We remark amog the various methods, the coditioal Black-Scholes method ad the Fourier-based discretizatio adjustmet; the latter leads to the remarkably simple ad tractable leadig-order discretizatio adjustmet term obtaied i equatio 3. The result ca be implemeted i ay stochastic volatility model which admits a closed-form expressio for the Fourier trasform of cotiuously sampled variace; importat examples of models where our result is directly applicable iclude both affie stochastic volatility models e.g. Hesto 993 as well as tractable o-affie models, such as the 3/ model i Lewis ad Carr, Su 7. Fially, we perform a extesive umerical study which compares the performace of the differet approximatios. Based o accuracy ad computatioal time, we show that the coditioal Black-Scholes scheme performs the best for short-maturity optios, whereas we recommed usig the Fourier-based discretizatio adjustmet for mid- to log-maturity optios. 6 Appedix A As a lemma of Theorem. we shall use the Lideberg-Feller geeralized CLT see, for example, Ferguso996 as formulated i Theorem 6.. Theorem 6. Geeralized CLT: Lideberg-Feller. Let Z,i, =,,... i =,,..., be a triagular sequece of radom variables such that E Z,i =, E Z,i < ad for each fixed =,,... the radom variables Z,, Z,,..., Z, are idepedet. If the Lideberg coditio is satisfied i.e. for all ɛ > lim s E Z,i ; Z,i > ɛ s = 6 where s = Z E,i, the we have the covergece i distributio Z, + Z, Z, s d N,. 7 To verify the Lideberg coditio 6, it is usually easier to check the sufficiet coditio of Lyapuov see, for example, Petrov 995. Specifi- 4

25 cally, if there exists δ > such that lim s +δ E Z,i +δ = 8 the the coclusio of the Lideberg-Feller theorem 7 holds. 7 Appedix B To establish bouds o the approximatio 9, we use the followig geeralizatio of the Berry-Esse iequalities due to Bikelis966, ad which ca also be foud i Petrov7. Theorem 7. Bikelis. Assume Z, Z,..., Z are idepedet radom variables with mea zero ad E Z i 3 <. Let s = E Zi ad L = s 3 E Z i 3. If F x deotes the distributio fuctio defied as F x = P Z i x the, for ay x R, we have s F x Nx A L + x 3 where N is the stadard ormal CDF ad A is a absolute positive costat. By makig use of Theorem 7., we ca provide a boud o the differece RV betwee the coditioal variace call price Cσ K = E[ σk ] F W + T ad the coditioal ormal approximatio C σ K i 9. Propositio 7.. If σk M, the Cσ K C A Σ L σ K. + M σ K Σ If σ K > M, the Cσ K C σ K A Σ L where L = s 3 E Y,i 3 with Y,i as follows: Y,i = log Sti µ,i + σ,i S ti + σ K M Σ. 9 5

26 Proof of Propositio 7. Itegratio by parts shows that for ay radom variable X with E X < ad distributio fuctio F x we have E X σk σ + = EX K σ K + F xdx. 3 For a fixed, we cosider the sequece of idepedet radom variables Y,, Y,,..., Y, defied i 9. We have E Y,i = ad E Y,i 3 < ; the exact formula for E Y,i 3 ca be foud i Lemma 7.. Usig that Y,i s = Σ RV the Theorem 7. of Bikelis implies F RV Σ x + M Nx or, equivaletly x F M RV x N Σ i= µ,i + σ,i A L + x 3 A L x M Σ where F RV deotes the coditioal distributio of the discretely sampled variace ad L = s 3 E Y,i 3. By 3 ad 3 we have σ Cσ K C K x M dx σ K A L σ K F RV x N = Σ 3 dx. + x M Σ Makig the chage of variable x M Σ = y the remaiig calculatios are straightforward. For example, i the case σk M the itegral becomes σ K M Σ A Σ L y 3 dy = A Σ L. + M σ K Σ The case σ K > M is solved similarly. Proof of Lemma. I the mai text, we showed that, i the Blackd Scholes model, RV = σ χ, λ where λ = r δ σ 6 σ T

27 ad χ, λ deotes the o-cetral chi-square distributio with degrees of freedom ad o-cetrality parameter λ. We recall the desity of a χ, λ radom variable: f χ x; λ, = i= e λ λ i f χ x; + i i! where f χ x; deotes the PDF of a chi-square radom variable with degrees of freedom : f χ x; = / Γ/ x/ e x/ x>. It is straightforward to show that x f χ x; = f χ x; +, which i tur allows us to write e λ λ i x f χ x; λ, = + i f χ x; + + i i! i= The expectatio to compute becomes σ E χ, λ σk = = f χ x; λ, + + λ f χ x; λ, σ K σ σ x σ K f χ x; λ, dx Usig property 3 derived above, the result follows immediately. Lemma 7.. If we let Y,i = log Sti S ti µ,i + σ,i we have E Y,i 3 = σ,i 6 Ψd + φd + Ψd φd + 48α + 6 Nd Nd + + 4α + 9 where ad d ± = µ,i ± µ,i + σ,i. σ,i Ψd = d 5 + αd 4 + 4α + 4d 3 + 6α 3 + 4αd + 48α + 8d + 3α 3 + 6α with α = µ,i σ,i. 7

28 Proof To simplify otatio we drop the subscripts ad let µ,i = µ ad σ,i = σ. We ote that Y,i d = µ + σ N, µ σ where N, deotes a stadard ormal variable. We thus have to compute µ + σ x µ σ 3 φxdx where φx deotes the stadard ormal desity. By solvig we obtai the roots µ + σ x µ σ = σ x + µσ x σ = d ± = µ ± µ + σ. σ Separatig the positive ad egative regios of the modulus, the itegratio becomes [ d σ 6 x + α x d+ 3 φxdx x + α x 3 φxdx + d d + x + α x 3 φxdx where we put α = µ σ. To compute these itegrals, we ote that by lettig u x = x φxdx ad makig use of the relatioship u x = x φx + u x with u x = Nx ad u x = φx, we obtai u x = xφx + Nx u 3 x = x + φx u 4 x = x 3 + 3x φx + 3 Nx u 5 x = x 4 + 4x + 8 φx u 6 x = x 5 + 5x 3 + 5x φx + 5 Nx. Fially, the remaiig steps ivolve oly algebraic calculatios to arrive at the result stated i the lemma. 8 Appedix C ] 8

29 Table : OTM optio prices uder parameter sets P3 ad P4. OTM put refers to a optio with moeyess Strike/v.6, ATM to a optio with moeyess ad OTM call to a optio with moeyess.5. Prices are i percetages. The computatioal time is i secods. T = moth T = 6 moths T = year OTM put ATM OTM call Comp. time OTM put ATM OTM call Comp. time OTM put ATM OTM call Comp. time - Daily samplig frequecy Parameter set P3 Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Parameter set P4 Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Samplig frequecy every five days Parameter set P3 Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj Parameter set P4 Mote Carlo Cot. samplig Sepp s approx Cod. Normal Simp. Cod. Normal Simp. Cod. BS Discr. Adj

30 Refereces [] Aderse, L., Simple ad efficiet simulatio of the Hesto stochastic volatility model, Joural of Computatioal Fiace, Vol., Issue 3, -4, 8. [] Bakshi, G., Cao, C., Che, Z., Empirical performace of alterative optio pricig models, The Joural of Fiace, Vol. LII, No. 5, 3-49, 997. [3] Bardorff-Nielse, O. E., Shephard, N., Ecoometric aalysis of realized volatility ad its use i estimatig stochastic volatility models, J. R. Statistical Society, 64, 53-8,. [4] Bardorff-Nielse, O. E., Graverse, S. E., Jacod, J., Podolskij, M., Shephard, N., A cetral limit theorem for realised power ad bipower variatios of cotiuous semimartigales, From Stochastic Aalysis to Mathematical Fiace: Festschrift for Albert Shiryaev, 33-68, Spriger 6. [5] Berard, C., Cui, Z., Prices ad Asymptotics for Discrete Variace Swaps, Applied Mathematical Fiace,, 4-73, 4. [6] Bikelis, A., Estimates of the remaider i the cetral limit theorem, Litovsk. Mat. Sb., 6, , 966. [7] Black, F., Scholes, M., The pricig of optios ad corporate liabilities, Joural of Political Ecoomy 8, , 973. [8] Broadie, M., Jai, A., Pricig ad hedgig volatility derivatives, Joural of Derivatives 53, 7-4, 8. [9] Broadie, M., Jai, A., The effect of jumps ad discrete samplig o volatility ad variace swaps, Iteratioal Joural of Theoretical ad Applied Fiace, Vol., No.8, , 8. [] Broadie, M., Kaya, Ö. Exact simulatio of stochastic volatility ad other affie jump diffusio processes, Operatios Research, 54 /, 7-3, 6. [] Buehler, H., Volatility markets: cosistet modelig, hedgig ad practical implemetatio, VDM Verlag, ISBN , 9. [] Carr, P., Gema, H., Mada, D., Yor, M. 5, Pricig optios o realized variace, Fiace ad Stochastics, 94, [3] Carr, P., Lee, R., Volatility Derivatives, Aual Review of Fiacial Ecoomics,, -, 9. 3

31 [4] Carr, P., Lee, R., Wu, L., Variace Swaps o time-chaged Lévy processes, Fiace ad Stochastics, 6, [5] Carr, P., Mada, D., Towards a Theory of Volatility Tradig, Volatility: New Estimatio Techiques for Pricig Derivatives, ed. R. Jarrow, pp , Risk Books, 998. [6] Carr, P., Mada, D., Optio Pricig ad the Fast Fourier Trasform, Joural of Computatioal Fiace, Volume, Number 4, pp. 6-73, 999. [7] Carr, P., Su, J., A ew approach for optio pricig uder stochastic volatility, Review of Derivatives Research,, 87-5, 7. [8] Carr, P., Gema, H., Mada, D. B., Yor, M. Optios o realized variace ad covex orders, Quatitative Fiace, -, ifirst,. [9] Cox, J. C., Igersoll J. E., Ross S. A. 985, A Theory of the Term Structure of Iterest Rates, Ecoometrica, 53, [] Drimus, G., Optios o realized variace by trasform methods: A oaffie stochastic volatility model, Quatitative Fiace,,. [] B. Dupire, Model Art, Risk, September, 8-, 993. [] D. Dufrese, The itegrated square-root process, Research paper o. 9, Uiversity of Melboure,. [3] Feller, W., A Itroductio to Probability Theory ad Its Applicatios, Volume II, Wiley, ISBN , 99. [4] Ferguso, T., A course i large sample theory, Chapma-Hall, ISBN 44378, 996. [5] Gatheral, J., Cosistet modelig of SPX ad VIX optios, Presetatio at the 5th Bachelier Cogress, Lodo 8. [6] Hesto, S., A closed-form solutio for optios with stochastic volatility with applicatios to bod ad currecy optios, The Review of Fiacial Studies 6, , 993. [7] Hull, J., White, A., The pricig of optios o assets with stochastic volatilities, The Joural of Fiace, 4, 8-3, 987. [8] Itki, A., Carr, P., Pricig swaps ad optios o quadratic variatio uder stochastic time chage models - discrete observatios case, Review of Derivatives Research, 3, 4-76,. 3