Power-type derivatives for rough volatility with jumps
PPower-type derivatives for rough volatility with jumps
Weixuan Xia * Abstract
In this paper we propose an efficient pricing-hedging framework for volatility derivatives which simul-taneously takes into account path roughness and jumps. Instead of dealing with log-volatility, we directlymodel the instantaneous variance of a risky asset in terms of a fractional Ornstein-Uhlenbeck processdriven by an infinite-activity Lévy subordinator, which is shown to exhibit roughness under suitable con-ditions and also eludes the need for an independent Brownian component. This structure renders thecharacteristic function of forward variance obtainable at least in semi-closed form, subject to a genericintegrable kernel. To analyze financial derivatives, primarily swaps and European-style options, on av-erage forward volatility, we introduce a general class of power-type derivatives on the average forwardvariance, which also provide a way of adjusting the option investor’s risk exposure. Pricing formulae arebased on numerical inverse Fourier transform and, as illustrated by an empirical study on VIX options,permit stable and efficient model calibration once specified.MSC2020 Classifications: 60E10; 60G22; 60J76JEL Classifications: C65; G13K EY W ORDS : Rough volatility; volatility jumps; forward variance; power-type derivatives “Rough volatility” is a relatively new and yet already familiar jargon that has flourishedin the financial world since the pioneering research work of [Gatheral et al, 2018] [13],which provided important empirical evidence suggesting rough sample paths of volatil-ity observed in high-frequency financial time series and thus the presence of short-termdependence, while the idea of introducing frictions into volatility quantities goes back tothe much earlier work of [Alòs et al, 2007] [2] motivated from observations in option price-implied volatility surfaces. Over the past three years, a good number of works have beendevoted into empirical justifications of the presence of rough volatility in various assettypes. To name a few, [Livieri et al, 2018] [23] confirmed the existence of rough volatil-ity by studying implied volatility-based approximations of spot volatility of the S&P500index, [Takaishi, 2020] [34] collected further evidence supporting volatility roughness inthe cryptocurrency (in particular Bitcoin) market, and [Da Franseca and Zhang, 2019] [9]even demonstrated that roughness is also present in the VIX index.Although the introduction of rough volatility has been a successful reproducer of styl-ized features of historical volatility of asset prices, a series of difficulties have arisen in the * Correspondence address: Department of Finance, Boston University Questrom School of Business. Email: [email protected]. a r X i v : . [ q -f i n . P R ] S e p ower-type volatility derivatives meantime because of the loss of Markov and semimartingale properties. As a result, whendeveloping pricing-hedging techniques accounting for rough volatility one will proba-bly sojourn at Monte-Carlo simulation methods, whereas the inaccessibility of infinites-imal generators has disabled methods associated with the Feynman-Kac formula. So far,simulation-based pricing-hedging methods have already been studied in depth; for ex-ample, [Jacquier et al, 2018] [18] adopted a hybrid simulation scheme for the calibrationof the rough Bergomi model initially proposed in [Bayer et al, 2016] [4] on VIX futuresand options. On the other hand, under the so-called “rough Heston model” which is con-structed from a stationary power-type kernel and belongs to the family of affine Volterraprocesses discussed in [Jaber et al, 2019] [17] (see also [Gatheral and Keller-Ressel, 2019][14]), characteristic function-based pricing methods were derived in [El Euch and Rosen-baum, 2019] [11] which depend, partially, on solving a fractional Riccati equation, wheretheir applicability was also demonstrated by a simple calibration exercise on S&P500 im-plied volatility surfaces; the paper [El Euch and Rosenbaum, 2018] [10] by the same au-thors considered from a theoretical standpoint similar hedging problems, after being ableto write the characteristic function of the log asset price in terms of a functional of its cor-responding forward variance curve. We also notice the up-to-date work of [Horvath et al,2020] [16], which highlighted a martingale framework using forward variance curves inthe goal of studying volatility options. It is worth mentioning that all these recent workshave universally emphasized the role of a Brownian motion while paying little attention tojumps in the asset prices and their volatility, which are, of course, thought to complicatethe pricing problems to great extent. For instance, the aforementioned Riccati equationwill turn into an integro-differential equation entailing more computationally expensivenumerical schemes and the resulting model distributions will no longer be stable but sub-ject to substantial changes under integral operations.All relevant models aside, one should however bear in mind that the main idea behindrough volatility is the exhibition of short-term dependence, or more precisely, rapidly de-caying autocorrelation near the origin, rather than an inherent reliance on Brownian sam-ple paths, or path continuity, which characteristic is at bottom an estimation assump-tion imposed in [Gatheral et al, 2018] [13] and deemed nonessential. In fact, extensiveuse of the Brownian motion in the cited literature is understandably an act of simplic-ity, mainly due to log-instantaneous volatility shown to be empirically close to normallydistributed. On the other hand, disregarding the exclusive use of the Brownian motionsheds light upon another important aspect - the presence of volatility jumps. In a semi-martingale setting, this would send us back to the work of [Todorov and Tauchen, 2011][36], which, by inferring from high-frequency VIX index data the activity level of some pre-sumed mean-reverting instantaneous variance model, showed that stock market volatilityshould be most suitably depicted as a purely discontinuous process without a Browniancomponent. Notably, this concern may not exist anymore in a non-semimartingale modelwith frictions, as pointed out in the same paper; in short, the activity level of the processcan be flexibly adjusted according to the controlling fraction parameter. For this reason,inclusion of volatility jumps in a model that is already fractional has seemingly been con-sidered insignificant and neglected for investigation. Nevertheless, since increased activ- eixuan Xia ity levels are an inevitable consequence of rough sample paths, using a fractional Brow-nian motion with a fraction parameter less than 1 only increases the vibrancy of the re-sulting variance process, which somewhat turns aside the empirical findings of [Todorovand Tauchen, 2011] [36]. In connection with this, we expect that replacing the Brownianmotion with a purely discontinuous component, which is less vibrant, is able to strikea balance between these two important aspects (short-term dependence and volatilityjumps) and eventually yield desirable modeling outcomes.These inspire us to take on a new path deviating from the use of a fractional Brown-ian motion in the establishment of rough volatility and switch to purely discontinuoussquare-integrable Lévy processes of infinite activity. In more detail, we want to proposea general framework for the instantaneous variance based on a generalized fractionalOrnstein-Uhlenbeck process subject to an integrable and possibly non-stationary ker-nel; a key feature of this formulation is that it is not derived from taking natural logarithmbut yet is capable of simultaneously capturing path roughness and possible jumps in theinstantaneous variance. Needless to say, despite that inclusion of jumps may not resultin significant improvement of the model fit of volatility distributions observed at highfrequencies, as noted in [Gatheral et al, 2018, Sect. 6] [13], it is undoubtedly innocuousand the model distribution in logarithm can also become arbitrarily close to normalityby properly adjusting its scale parameter, thanks to the central limit theorem. On thecontrary, introducing jumps into the instantaneous variance gives rise to an analyticallytractable structure for the characteristic function of the average forward volatility, thus fa-cilitating the pricing and hedging of corresponding financial derivatives. The only inexactmeasure to be taken, however, is a parametric kernel approximation, and can be lookedupon as a replacement of the geometric-averaging approximation adopted in [Horvath etal, 2020, Eq. (16)] [16]. To be more specific, for a certain investment horizon the averageforward volatility can be decently approximated by a forward volatility with an unknownwindow to be treated as a constrained parameter, and this does not pose any problems aslong as the kernel satisfies a certain differentiability condition.Besides, although our main results are given under a general setting, attention will bedrawn to three particular types of stationary kernels, all of which are comfortable to workwith and have their own advantages. While the first type is recognized for its incommen-surable simplicity, the second is compatible with the transformation of the instantaneousvariance dynamics into a usual Ornstein-Uhlenbeck process driven by a fractional Lévyprocess. The third type, being part of our innovation, is arguably a result of reverse en-gineering and designed specifically to ensure a closed-form characteristic function - atruly desirable tool that underlies efficient calibration. Remarkably, our objective is notto compare the overall suitability of different types of kernels, but rather, to seek one thatfacilitates numerical implementation to the greatest degree; in fact, all these three typesare very similar in shape.Our ultimate interest in the present paper lies in analyzing European-style financialderivatives written on the average forward volatility, such as the VIX index, includingswaps and options. Under the proposed rough-volatility framework, we obtain pricing-hedging formulae for a more general class of power-type derivatives, which raise the un- ower-type volatility derivatives derlying volatility or the standard option payoff to a certain nonnegative power, and canthus be applied to conveniently adjust the derivative investor’s risk exposure. While thesederivatives are rather newfangled in the volatility market, their counterparts written onequity prices have been thoroughly studied already and one is referred to [Raible, 2000][30] and [Xia, 2017] [40] on single-asset options and [Blenman and Clark, 2005] [5], [Wang,2016] [37], and [Xia, 2019] [41] on exchange options. The proposed pricing-hedging for-mulae will make extensive use of the incomplete gamma function but only involve onenumerical Fourier-type integral, and thus are very convenient to implement in practice.The remainder of this paper is organized as follows. In Section 2 we establish ourmodel framework starting from a fractional instantaneous variance dynamics and pro-vide a comprehensive analysis of its properties, including covariance function and pathregularity, and then give an integral representation for the characteristic function of theaverage forward variance. Some simulation techniques are discussed in Section 3, withsome pertinent convergence results. Section 4 contains our new pricing-hedging formu-lae for power-type derivatives which naturally generalize standard volatility derivatives,initiating our empirical study based on the VIX index in Section 5. Moreover, we alsoprovide some insight into how the model framework may be further extended to accom-modate the presence of rough volatility of volatility in Section 6 by means of a stochastictime change argument, in catering for the noted finding of [Da Franseca and Zhang, 2019][9]. Conclusions are drawn in Section 7 and all mathematical proofs presented in the end. We begin by synthesizing some crucial ingredients of a non-Gaussian fractional Lévy pro-cess, which are necessary for establishing a model for the instantaneous variance of arisky asset whose sample paths have roughness and jump features. Of course, allowingfor positivity of the instantaneous variance the background-driving Lévy process must benonnegative, i.e., a subordinator.To this end, consider a continuous-time stochastic basis S : = ( Ω , F , (cid:80) ; (cid:70) ≡ { F t } t ≥ ),where the filtration (cid:70) is assumed to satisfy the usual conditions. Let X ≡ ( X t ) be anadapted and square-integrable Lévy subordinator supported on S , which is exclusivelycharacterized by a Poisson random measure N defined on ( R ++ , R + ). According to theLévy-Khintchine representation, X has the characteristic exponentlog φ X ( l ) : = log (cid:69) (cid:163) e i l X (cid:164) = (cid:90) ∞ + ( e i l z − ν (d z ), l ∈ R ,where i denotes the imaginary unit and ν is the intensity measure associated with N . Forpracticality we impose the assumption that ν is non-atomic so that the law of X is ab-solutely continuous with respect to Lebesgue measure and we denote by ξ : = (cid:69) [ X ] > ξ : = (cid:69) (cid:163) X (cid:164) > X has independent and stationary incre- eixuan Xia ments, it has the familiar covariance function, for any u > (cid:163) X t , X t + u (cid:164) = ξ t − ξ t ( t + u ).For a continuously differentiable kernel g ∈ C (1,1) defined in the domain {( t , s ) : t > s ∈ [0, t )} satisfying the integrability condition (cid:90) t g ( t , s )d s < ∞ , ∀ t > X ( g ) t : = (cid:90) t g ( t , s )d X s = (cid:90) t (cid:90) ∞ + g ( t , s ) z N (d z , d s ), t ≥
0. (1)With this representation, for any t , u >
0, it is easy to see that (cid:69) (cid:163) X ( g ) t (cid:164) = ξ (cid:82) t g ( t , s )d s and,by using the Lévy-Itô isometry (see, e.g., [Lyasoff, 2017, Sect. 16.32] [24]), the covariancefunction of the centered process (cid:161) X ( g ) t − (cid:69) (cid:163) X ( g ) t (cid:164)(cid:162) takes the following form, (cid:69) (cid:163) X ( g ) t X ( g ) t + u (cid:164) = ξ (cid:90) t g ( t , s ) g ( t + u , s )d s .Notably, if lim s (cid:37) t g ( t , s ) = ∞ for every t >
0, then the integral (cid:82) t g (0,1) ( t , s )d s is divergentfor every t >
0. In this case, X ( g ) exhibits short-term dependence with rough sample pathsin the sense that there exists (cid:36) ∈ (0, 1) such that (cid:69) (cid:163) X ( g ) t X ( g ) t + u (cid:164) = (cid:69) (cid:163) X ( g )2 t (cid:164) + ξ C ( t ) u (cid:36) + O ( u ),where (cid:69) (cid:163) X ( g )2 t (cid:164) = ξ (cid:82) t g ( t , s )d s by the dominated convergence theorem and C ( t ) is someconstant depending only on t > g ( t , s ) = ( t − s ) d − F (cid:179) − d , d − d ; − t − ss (cid:180) , (2)for some fraction parameter d ∈ (1/2, 3/2), where F ( · , · ; · ; · ) is the Gauss hypergeometric-(2, 1) function ([Abramowitz and Stegun, 1972, Sect. 15] [1]) and which is non-stationary.We note that the specific form (2) was initially chosen in [Jost, 2006] [19] in an attemptto match the Weyl integral representation of a fractional Brownian motion living in real-valued time used in [Mandelbrot and van Ness, 1968] [25]. It was shown in [Tikanmäkiand Mishura, 2011] [35], however, that such transformation does not necessarily lead tothe same finite-dimensional distribution in the more general case of fractional Lévy pro-cesses. Another popular choice of g is the following obviously stationary and yet struc-turally much simpler Riemann-Liouville kernel , g ( t , s ) ≡ g ( t − s ) = ( t − s ) d − Γ ( d ) , (3) Continuous differentiability is a highly desirable property of the kernel for modeling purposes, hence assumed throughout thispaper. Intuitively, by ruling out kinks and discontinuities it ensures that the frictions brought by g do not have sudden changes.However, it is not required for defining fractional Lévy subordinators and thus not to be comprehended as any implicit assumption forthe ongoing analysis. The same kernel was used in [El Euch, 2018] [10] and [El Euch, 2019] [11] in constructing the (generalized) rough Heston model. ower-type volatility derivatives for d > Γ ( · ) denotes the usual gamma function. In particular, with (3) the frac-tional process X ( g ) can also be understood as a consequence of repeated path integrationof the subordinator X , i.e., for d ∈ N ++ ≡ N \ {0}, X ( g ) t = (cid:90) · · · (cid:90) t (cid:124) (cid:123)(cid:122) (cid:125) d X s d s . . . d s (cid:124) (cid:123)(cid:122) (cid:125) d , (4)which can be extended through Cauchy’s repeated integration formula. Of course, forthis special choice roughness is only present if d ∈ (1/2, 1), while it also has the obviousdrawback, compared to the Molchan-Golosov kernel, that the resulting fractional processfails to have stationary increments.In any case, a well-suited candidate for X , having infinitely many jumps on compacttime intervals, can simply be a one-sided tempered stable process, i.e., a tempered sta-ble subordinator, which has three parameters - a > b > c ∈ (0, 1), leading to thefollowing characteristic exponent,log φ X ( l ) = a Γ ( − c )(( b − i l ) c − b c ), l ∈ R , (5)so that ν (d z ) = ae − bz / z c + (0, ∞ ) ( z )d z , for z >
0, which is clearly an infinite measure.By taking c (cid:38) c = φ X ( l ) =− a log(1 − i l / b ). With (5) it is also straightforward to verify that ξ = a Γ (1 − c )/ b − c and ξ − ξ = a Γ (2 − c )/ b − c . For more concrete properties of the tempered stable process onemay refer to [Rosi ´nski, 2007] [31] and [Küchler and Tappe, 2013] [21]; see also [Schoutens,2003, Sect. 5.3] [32] for an overview. Let us consider, instead of the natural logarithm of the instantaneous volatility of a riskyasset, the instantaneous variance process, denoted V ≡ ( V t ). Intuitively speaking, our ideais to express V as a Volterra-type stochastic integral, up to shifting and positive scaling,analogous to the fractional Lévy process X ( g ) in (1) with a suitable kernel chosen to allowfor short- or long-term dependence as well as long-term mean reversion. This is done byassuming the following fractional Ornstein-Uhlenbeck structure, V t = V e − κ t + ¯ V (1 − e − κ t ) + X ( h ) t , t ≥
0, (6)where κ > V ≥ h isa continuously differentiable kernel having a power-law left tail and a quasi-exponentialright tail, h ( t + u , t ) = (cid:40) O ( u d − ), as u (cid:38) O (cid:161) e − κ u u ( d − + (cid:162) , as u → ∞ , ∀ t >
0, (7)for a fraction parameter d > · ) + denotes the positive part. This condition sub-tly embodies the intuition of introducing frictions into V without jeopardizing its mean- eixuan Xia reverting property, and it automatically ensures that (6) is well defined becausesup t > (cid:90) t h ( t , s )d s < ∞ .Besides, we assume that V > d ∈ (1/2, 1),then V exhibits mean reversion in the long term but is simultaneously allowed to haveshort-term dependence. As before, with the Lévy-Itô isometry the centered process ( V t − (cid:69) [ V t ]) is seen to have the covariance function equal to (cid:69) (cid:163) X ( h ) t X ( h ) t + u (cid:164) = ξ (cid:90) t h ( t , s ) h ( t + u , s )d s ,for any u >
0, which generally depends on t >
0. Under (7), if d < s (cid:37) t h ( t , s ) = ∞ for any t > C ( t ) ∈ R depending only on t such that (cid:69) (cid:163) X ( h ) t X ( h ) t + u (cid:164) = (cid:69) (cid:163) X ( h )2 t (cid:164) + ξ C ( t ) u d − + O ( u ), as u (cid:38)
0, (8)with (cid:69) (cid:163) X ( h )2 t (cid:164) = ξ (cid:82) t h ( t , s )d s . In this case, the covariance function is rough at the ori-gin and V does exhibit short-term dependence. On the other hand, with the right-tailbehavior in (7) V always reverts to a positive mean in the long term, in thatlim t →∞ (cid:69) [ V t ] = ¯ V + ξ lim t →∞ (cid:90) t h ( t , s )d s > h is specialized as the product of a usual exponential kerneland the Riemann-Liouville kernel, h ( t , s ) ≡ h ( t − s ) = e − κ ( t − s ) ( t − s ) d − Γ ( d ) , (9)which is clearly strictly positive and will be referred to as the type-I kernel. A remark-able difference is, however, that we have abandoned negative time by assuming that theinstantaneous variance process is only observed starting from time 0.Indeed, the structure (6) represents a wide range of approaches towards achievingpath roughness and mean reversion at the same time, while it gives rise to an Ornstein-Uhlenbeck process driven by a fractional Lévy process, i.e., the structure used in [Garnierand Sølna, 2018] [12], by choosing h ( t , s ) = g ( t , s ) − κ (cid:90) ts e − κ ( t − v ) g ( v , s )d v , (10)where g is the kernel mentioned in the previous section, esp. (1). Note that with an appli-cation of Itô’s formula and the Fubini-Tonelli theorem (6) is reformatted into V t = V e − κ t + ¯ V (1 − e − κ t ) + (cid:90) t e − κ ( t − s ) d X ( g ) s , (11) ower-type volatility derivatives which is the solution of the fractional stochastic integral equation V t = κ (cid:90) t ( ¯ V − V s )d s + X ( g ) t .Also, we remark that, due to the condition (7), the second term on the left-hand sideof (11) is bounded for every fixed t >
0, so that lim s (cid:37) t h ( t , s )/ g ( t , s ) > s (cid:37) t g ( t , s ) = ∞ for any t >
0. This means that if g is chosen in such a way that thesample paths of X ( g ) exhibit short-term dependence, then those of V must also exhibitshort-term dependence, and in fact, the path roughness of V and X ( g ) must be of thesame degree. If g is further taken to be the stationary Riemann-Liouville kernel, then bystraightforward calculations (10) yields the following type-II kernel which also happensto be stationary , h ( t , s ) ≡ h ( t − s ) = ( t − s ) d − + ( − κ ) − d e − κ ( t − s ) ( Γ ( d ) − Γ ( d , − κ ( t − s ))) Γ ( d ) , (12)where Γ ( · , · ) denotes the upper incomplete gamma function, and the correlation structure(8) can be made more precise with (cid:69) (cid:163) X ( h )2 t (cid:164) = ξ (cid:82) t h ( s )d s and some C ( t ) ∈ Γ (1 − d ) sin( π d ) π Γ ( d ) × ( e − κ t , 1), (13)which only depends on t . This shows that roughness is established if and only if d ∈ (1/2, 1). Also, the long-term mean is given by lim t →∞ (cid:69) [ V t ] = ¯ V + ξ κ − d /2.Of course, with the type-I kernel (9), it is not possible to interpret V as an Ornstein-Uhlenbeck process driven by a fractional Lévy process. Nonetheless, V can still haveshort-term dependence, i.e., for d ∈ (1/2, 1), (cid:69) (cid:163) X ( h ) t X ( h ) t + u (cid:164) ξ = t d − (2 d − Γ ( d ) + C ( t ) u d − + O ( u ), as u (cid:38) t →∞ (cid:69) [ V t ] = ¯ V + ξ κ − d .As an important aspect of our innovation, for d ∈ (1/2, 1) generating short-term de-pendence we propose to construct h by combining the scaled exponential kernel and theRiemann-Liouville kernel in a piecewise fashion, i.e., h ( t , s ) ≡ h ( t − s ) = ( t − s ) d − − τ d − Γ ( d ) + θ e − κτ if t − s < τ , θ e − κ ( t − s ) if t − s ≥ τ , (14)where τ > θ > τ solves the transcendental equation e κτ τ d − = − κθ Γ ( d − Unlike the type-I kernel, the type-II kernel is not necessarily strictly positive, though the resulting process V obviously is. eixuan Xia In order for (15) to be solvable on R ++ , θ has to satisfy the constraint θ ≥ − κ Γ ( d − (cid:179) − de κ (cid:180) d − , (16)under which τ = d − κ W i (cid:179) κ d − − κθ Γ ( d − d − (cid:180) , i ∈ { −
1, 0}, (17)with W · ( · ) being the Lambert W function, a.k.a. the product logarithm (see [Corless et al,1996] [8]). Note that the two solutions in (17) coincide if and only if equality holds in(16). We will refer to (14) as the type-III kernel. Of course, a downside of such piecewiseconstruction is its incompatibility with differentiability at τ if one requires long-term de-pendence ( d > θ to be exactly the lower bound in (16) and specify (14) accordingly, then τ = (2 − d )/ κ andthe type-III kernel is uniquely parameterized by κ and d and reads h ( t − s ) = ( t − s ) d − − ((2 − d )/ κ ) d − Γ ( d ) − e d − κ Γ ( d − (cid:179) − de κ (cid:180) d − if t − s < − d κ , − e − κ ( t − s ) κ Γ ( d − (cid:179) − de κ (cid:180) d − if t − s ≥ − d κ , (18)which will play an important role during our implementation. With (18), we have in (8)that C ( t ) ≡ C = Γ (1 − d ) sin( π d )/( π Γ ( d )) while the long-term mean becomeslim t →∞ (cid:69) [ V t ] = ¯ V + ξ (4 − d ) d (2 − d ) Γ ( d − (cid:179) − d κ (cid:180) d > κ = d = d < − τ = d = Figure 1: Comparison of kernels
On a different note, despite that by the formulation (6) V lacks increment stationar-ity, there is comprehensibly no negative impact placed on characteristic function-basedmodel calibration. ower-type volatility derivatives In the next proposition we give some partial results on the regularity of the samplepaths of V . Indeed, even from its appellation the interpretation of the so-called “pathroughness” is not confined to the involvement of short-term dependence, but should belinked to how irregular the paths may become. Allowing for practicality, we focus on thecase of an infinite Lévy measure ν . Proposition 1.
Assume ν ( R ++ ) = ∞ and (6). For any fixed T > we have the followingthree assertions.(i) If d > , then the sample paths of V are a.s. continuous with a.s. zero quadraticvariation over [0, T ] .(ii) If d = , then the sample paths of V are a.s. discontinuous with a.s. finite quadraticvariation over [0, T ] .(iii) If < d < , then the sample paths of V are a.s. discontinuous and unboundedwith a.s. infinite quadratic variation over [0, T ] . Notably, for d >
1, the sample paths of V are smoothed in a way that all the jumpsgenerated by X are expunged and, as will be seen in the proof in Section 8.1, they areactually Hölder-continuous for certain exponents. On the other hand, in the situation ofassertion (iii), V can have infinitely large jumps, so that its sample paths form maps from[0, T ] to [0, ∞ ]. We stress, however, that this will not be a problem for modeling in practicebecause V t is a.s. finite for any fixed t ≥ d = V only exhibits mean reversion. For the type-I and type-II kernelsit can be easily verified that in the case d = V is exactly the usual Lévy-driven Ornstein-Uhlenbeck process and for the type-III kernel this is also true in the limit as d (cid:37) t ∈ [0, t ] be a fixed time point, we can recast (6) conditional on F t as V t = V e − κ t + ¯ V (1 − e − κ t ) + (cid:90) t h ( t , s )d X s + (cid:90) tt h ( t , s )d X s (19)or equivalently, V t = V t e − κ ( t − t ) + ¯ V (1 − e − κ ( t − t ) ) + (cid:90) t (cid:161) h ( t , s ) − e − κ ( t − t ) h ( t , s ) (cid:162) d X s + (cid:90) tt h ( t , s )d X s . (20)Notably, the first integral on the right-hand side of (20) indicates that V cannot be aMarkov process or a semimartingale in general. In fact, it is so if and only if h is chosensuch that h ( t , s ) − e − κ ( t − t ) h ( t , s ) ≡
0, a clear contradiction with the inclusion of short-term dependence; for instance, with the aforementioned three types of kernels this inte-gral does not vanish. With the loss of the Markov property, it is oftentimes more comfort-able to work directly with (19). The conditional mean and covariance of the instantaneousvariance can be directly written down. For any fixed t > t and u > (cid:69) [ V t | F t ] = V e − κ t + ¯ V (1 − e − κ t ) + (cid:90) t h ( t , s )d X s + ξ (cid:90) tt h ( t , s )d s (21) eixuan Xia andCov (cid:163) X ( h ) t , X ( h ) t + u (cid:175)(cid:175) F t (cid:164) = ξ (cid:90) tt h ( t , s ) h ( t + u , s )d s − ξ (cid:179) (cid:90) tt h ( t , s )d s (cid:180)(cid:179) (cid:90) t + ut h ( t + u , s )d s (cid:180) . After constructing the instantaneous variance model with roughness and jumps, we nowproceed to giving a convenient structure for the forward variance curve, i.e.,˜ V t ( u ) : = (cid:69) [ V t + u | F t ], u > t ≥
0, (22)which in light of (21) admits the following stochastic integral representation,˜ V t ( u ) = V e − κ ( t + u ) + ¯ V (1 − e − κ ( t + u ) ) + X ( H u ) t + ξ (cid:90) t + ut H u ( t , s )d s , (23)where X ( H u ) is associated with the u -shifted kernel H u ( t , s ) : = h ( t + u , s ), u >
0. (24)Apart from the contemporaneous instantaneous variance, frictions in the forward vari-ance curve also result from a new fractional Lévy process X ( H u ) containing additional in-formation over the entire variance history. In consequence, with a general kernel h theMarkov property of ˜ V ( u ) is completely lost. If one prefers to view t + u > t as being time-independent, then (23) really gives a martingale dynamics for the forward variance curveover [0, t + u ], which is similar to the martingale framework developed in [Horvath, 2020,Sect. 3] [16]. However, we intentionally refrain from operating on such a framework asit is primarily beneficial from a simulation-based viewpoint. It is also worth emphasiz-ing that, since lim s (cid:37) t H u ( t , s ) = h ( t + u , t ) = O (1), ∀ t , u >
0, the modified process X ( H u ) deprives the sample paths of ˜ V ( u ) of any degree of roughness.Under continuous monitoring over a fixed window Θ >
0, the average forward volatilityprocess is identified as the square root of the Θ -running average of the forward varianceprocess I ∗ t ( Θ ) : = (cid:115) Θ (cid:90) Θ ˜ V t ( u )d u , t ≥ Θ = = H u in u over a compact interval, hence resulting in significant complication. For this reason,we use the fact that H u and the integrated kernel (cid:82) u H w d w (cid:177) u are both continuously dif-ferentiable and bounded functions with identical tail behaviors for every u ∈ (0, Θ ] andpropose the following approximation of I ∗ ( Θ ) I t ( ∆ ) = (cid:113) ˜ V t ( ∆ ), t ≥
0, (25) ower-type volatility derivatives for some adjusted window ∆ ∈ (0, Θ ] which depends on the shape of h and should mostlikely be treated as an additional parameter . In other words, by (25) we attempt to ap-proximate I ∗ ( Θ ) as the ∆ -forward volatility. It is clear that neither I ∗ ( Θ ) nor I ( ∆ ) can pos-sess rough sample paths and from now on we will refer to I ( ∆ ) as the (adjusted) averageforward volatility, taking it as a replacement of I ∗ ( Θ ). At this point, a comfortable inte-gral formula for the conditional characteristic function of the (adjusted) average forwardvariance is readily available, which we formulate as the following proposition. Proposition 2.
In the setting with (6) and (25), we have for any ≤ t < t ≤ T and ∆ ∈ (0, Θ ] that φ t , t ( l ; ∆ ) : = (cid:69) (cid:163) e i l I t ( ∆ ) (cid:175)(cid:175) F t (cid:164) = exp (cid:179) i l (cid:179) I t ( t − t + ∆ ) − ξ (cid:90) tt H ∆ ( t , s )d s (cid:180) + (cid:90) tt log φ X ( l H ∆ ( t , s ))d s (cid:180) , l ∈ R ,(26) with the modified kernel H ∆ specified in (24). With Proposition 2 one can deduce pricing-hedging formulae for derivatives contractswritten on the average forward variance, e.g., the squared VIX index, which will be ex-plained in detail in the next section. Most importantly, although it is not possible to de-rive a similar formula for the characteristic function of the average forward volatility, wewill demonstrate how this difficulty may be overcome for volatility derivatives by way ofpower-type extensions. Moreover, by forcing ∆ (cid:38) (cid:69) (cid:163) e i lV t (cid:175)(cid:175) F t (cid:164) , l ∈ R .Since I ( u ) is an adapted process for every u > I t ( t − t + ∆ ) in (26) ismeasurable with respect to F t , and is recoverable from an entire realized sample path of I ( ∆ ) up to t . Also, computation of the first Riemann integral is fairly straightforward, andone gets the following explicit expression under the type-I kernel, (cid:90) tt H ∆ ( t , s )d s = Γ ( d , κ∆ ) − Γ ( d , κ ( t − t + ∆ )) κ d Γ ( d ) (27)and similarly, under the type-II kernel, it is (cid:90) tt H ∆ ( t , s )d s = e − κ∆ ( Γ ( d ) − Γ ( d , − κ∆ )) − e − κ ( t − t + ∆ ) ( Γ ( d ) − Γ ( d , − κ ( t − t + ∆ )))( − κ ) d Γ ( d ) . (28)On the other hand, to evaluate the second Riemann integral in (26) analytically is far froman easy task, even with the specialization that X is a tempered stable subordinator havingthe characteristic exponent (5). In this case, for h being the kernel of either type I or typeII, no explicit expression exists for such an integral, the computation of which has to resortto numerical methods such as the Gauss quadrature rule ([Golub and Welsch, 1969] [15]).The reason behind this problem is simple: the type-I and type-II kernels are both formed The determination rule of ∆ will be further discussed in Section 5.1. The expression under the type-III kernel is specifically put into Corollary 1 and one is referred to its proof in Section 8.3. eixuan Xia by multiplying power and exponential functions but no elementary substitution worksfor integrals of the form (cid:82) (1 − e − κ s s d − ) c d s , for d > κ > c ∈ (0, 1). At this point,it is no surprise that with the type-III kernel an explicit formula is accessible due to theaforementioned power-exponential multiplication reflected in the time-invariant thresh-old τ >
0, while by its piecewise nature neither path roughness nor the mean-revertingproperty of the instantaneous variance is sacrificed. This gives rise to the following im-portant corollary.
Corollary 1.
Let h be the type-III kernel in (14), with d ∈ (1/2, 1) , and let X have thecharacteristic exponent (5). Then we have φ t , t ( l ; ∆ ) = exp (cid:181) i l I t ( t − t + ∆ ) + (cid:40) Υ − ( s ) | t − t + ∆ s = ∆ if τ > t − t + ∆ , Υ − ( s ) | max{ τ , ∆ } s = ∆ + Υ + ( s ) | t − t + ∆ s = max{ τ , ∆ } if τ ≤ t − t + ∆ (cid:182) , l ∈ R , (29) where for s ∈ [ ∆ , t − t + ∆ ] Υ + ( s ) : = i l a θ Γ (1 − c ) e − κ s κ b − c + a Γ ( − c ) (cid:179) e κ s ( b − i l θ e − κ s ) c + i l c κθ F (cid:179)
1, 1; 1 − c ; be κ s i l θ (cid:180) − b c s (cid:180) (30) and Υ − ( s ) : = i l a Γ (1 − c ) sb − c (cid:179) d τ d − − s d − Γ ( d + − θ e − κτ (cid:180) + a Γ ( − c ) s (cid:181)(cid:179) b − i l θ e − κτ + i l τ d − Γ ( d ) (cid:180) c × F (cid:179) − c , 1 d − dd − l s d − i l τ d − + ( b − i l θ e − κτ ) Γ ( d ) (cid:180) − b c (cid:182) . (31)As a remark, by forcing τ → ∞ one obtains the characteristic function in the absence ofmean reversion, with κ (cid:38) τ (cid:38) d (cid:37) , in which the Υ − ( s ) and Υ + ( s ) termsin (31) vanish respectively, underlie contrast analysis for our implementation. Despite non-stationarity in general, using (6) we can still simulate the sample paths ofthe instantaneous variance process. To do this, we discretize the generic interval [0, T ] bymeans of the uniform partition T M : = (cid:110) nTM (cid:111) Mn = , M ∈ N ++ , M (cid:192)
1. (32) Noticeably, the latter extremal case is only achievable when θ is strictly larger than the lower bound in (16), and not from (18),where θ vanishes in the limit. ower-type volatility derivatives Then, using the Gauss quadrature rule and the Lévy properties of X , we have the followingestimator of (6) on T M which replaces the Volterra-type stochastic integral with a finiterandom sum.ˇ V nT / M = V e − κ nT / M + ¯ V (cid:161) − e − κ nT / M (cid:162) + n − (cid:88) k = h (cid:179) nTM , kTM (cid:180) ˇ X k , n ≥
1, (33)where ˇ X k ’s are i.i.d. random variables with characteristic function φ ˇ X k ( l ) : = (cid:69) (cid:163) e i l ˇ X k (cid:164) = ( φ X ( l )) T / M , for l ∈ R . If the kernel is stationary, h ( nT / M , kT / M ) = h (( n − k ) T / M ). Thenext proposition describes the convergence rate of the discretized process ˇ V towards V over (0, T ] (with trivial equivalence at time 0). Proposition 3.
Under T M , for any fixed t ∈ (0, T ] , there exists n ∈ N ∩ [1, M ] such thatthe estimator ˇ V nT / M is conditionally asymptotically unbiased towards V t and (cid:69) (cid:163)(cid:161) ˇ V nT / M − V t (cid:162) (cid:164) = O ( M − ), as M → ∞ .Note that the convergence rate is untrammeled by the fraction index d of h , whichapplies to the three types of kernels discussed. Nonetheless, the above L -convergencefails in the limit as d (cid:38) I ( ∆ ) for a given ∆ >
0, by using the estimatorˇ I nT / M ( ∆ ) = V e − κ ( nT / M + ∆ ) + ¯ V (cid:161) − e − κ ( nT / M + ∆ ) (cid:162) + n − (cid:88) k = H ∆ (cid:179) nTM , kTM (cid:180) ˇ X k + (cid:90) nT / M + ∆ nT / M H ∆ (cid:179) nTM , s (cid:180) d s , n ∈ N ∩ [1, M ], (34)to which the L -convergence criterion in Proposition 3 also applies, and where there isno need to discretize the last deterministic integral which can be evaluated explicitly inmany cases; see, e.g., (25), (26) and Corollary 1 as well.We illustrate (33) by considering X to be a gamma process and an inverse Gaussianprocess, both belonging to the class of tempered stable processes; recall (5). The reasonis that in these two cases the cumulative distribution function of ˇ X can be expressed inclosed form which facilitates the use of inverse transform sampling. In light of the previ-ous discussion based on Figure 1 we focus on the type-I kernel (9) and use the parameters κ = V = ¯ V = ∆ = M = we plot realized sample paths of V over the unit time interval for both d = d = d = d = V are purely discontinuous with probability 1, whereas in the latter case they are contin-uous. These illustrations are to show that the proposed rough stochastic volatility model The values of a and b are chosen for the mean and variance of X in the gamma case to match those in the inverse Gaussian case,which are 1/5 and 1/2500, respectively. eixuan Xia with jumps can be simulated effectively, in spite of increment non-stationarity, thus im-plying potential applications to simulation-based pricing methods, which are howevernot our concentration in the present paper. gamma ( a = b = c (cid:38)
0) inverse Gaussian ( a = (cid:112) π , b = c = V In this section we present the main pricing-hedging formulae for European-style deriva-tives written on the adjusted average forward volatility. The setting of Section 2.2, esp. therepresentation (6), is adopted throughout, and a fixed maturity date T > We begin with swaps written on the average forward volatility. The payoff of a power swapto its investor is at
T S ( p ) T = I pT ( ∆ ), (35)where p ≥ ower-type volatility derivatives p = p = p = t before maturity can besimply computed as the expected value of I pT ( ∆ ) conditional on F t , i.e., as S ( p ) t = (cid:69) (cid:163) I pT ( ∆ ) (cid:175)(cid:175) F t (cid:164) ,and is treatable as a potentially fractional moment of I T ( ∆ ), which is naturally connectedto the so-called “fractional calculus”. Proposition 4.
At time t ∈ [0, T ) , the price of the power volatility swap with payoff(35) satisfies the quasi-recurrence relation S ( p ) t = ( − i) p /2 φ ( p /2) t , T (0; ∆ ), p ∈ N , (cid:32) S ( p ) t = sec π ( p /2 − (cid:98) p /2 (cid:99) )2 p /2 − (cid:98) p /2 (cid:99) Γ (1 − p /2 + (cid:98) p /2 (cid:99) ) × (cid:90) ∞ Re (cid:183) S (2 (cid:98) p /2 (cid:99) ) t − ( − i) (cid:98) p /2 (cid:99) φ ( (cid:98) p /2 (cid:99) ) t , T ( l ; ∆ ) l p /2 −(cid:98) p /2 (cid:99)+ (cid:184) d l , p ∉ N , (36) provided that (cid:69) (cid:163) I pT ( ∆ ) (cid:164) < ∞ . Thanks to the exponential structure (26) the convergence of (36) can be directly linkedto the smoothness of the characteristic function of X . Comprehensibly, it is never exor-bitant to demand that φ t , T ( · ; ∆ ) ∈ C (cid:98) p /2 (cid:99)+ ( R ), which condition remains valid for a wideclass of square-integrable Lévy processes X . For instance, for X a tempered stable pro-cess, its characteristic function (5) immediately renders φ t , T ( · ; ∆ ) ∈ C ∞ ( R ) so that Propo-sition 4 is automatically applicable for all values of p ≥
0. Implementation of (36) is alsoquite straightforward, the specializations in Corollary (1) notwithstanding, by means ofthe Gauss quadrature rule for numerical integration and finite-difference approximationsfor differentiation of integer orders; for example, given the required degree of smoothness,a central approximation reads for (cid:178) > φ ( (cid:98) p /2 (cid:99) ) t , T (0; ∆ ) = (cid:98) p /2 (cid:99) (cid:88) n = (cid:195) (cid:98) p /2 (cid:99) n (cid:33) ( − n φ t , T (cid:179)(cid:179) (cid:98) p /2 (cid:99) − (cid:180) (cid:178) ; ∆ (cid:180) + O ( (cid:178) ).In particular, by taking p = S (1) t = (cid:112) π (cid:90) ∞ Re (cid:104) − φ t , T ( l ; ∆ ) (cid:112) l (cid:105) d l , (37)whilst that for the corresponding variance swap is none but the F t -conditional mean of I T ( ∆ ) and we recall (22) and (25). For simplicity we use φ ( (cid:36) ) t , T ( l ; ∆ ) ≡ ∂ (cid:36) φ t , T ( l ; ∆ )/ ∂ l (cid:36) to denote the (cid:36) th derivative of φ t , T ( l ; ∆ ) with respect to l ∈ R , for (cid:36) ≥ eixuan Xia As we are amidst a non-Markovian setting, it is quixotic to construct a hedge for thesepower volatility swaps based on the contemporaneous average forward variance I t ( ∆ ).In light of the presentation of the characteristic function (26), to hedge such a swap oneshould need at least the forward variance with a prolonged window by the time to ma-turity, whose variable nature with regard to the current date t gives rise to a dynamichedging strategy. In other words, a hedging strategy designated for the time period [ t , T )will require the entire forward variance curve { ˜ V t ( u ) : u ∈ ( ∆ , T − t + ∆ ]}. The good news isthat perfect hedging is possible using only the forward variance curve, with no additionalsources of risk involved in the swap, as the next corollary explains. To that end we firstdefine the time-indexed differential operator (cid:52) t : = ∂∂ I t ( t − t + ∆ ) , t ∈ [ t , T ],for a fixed t ∈ [0, T ). Corollary 2.
In the setting of Proposition 4 we have (cid:52) T (cid:161) S ( p ) t (cid:162) = p ( − i) p /2 − φ ( p /2 − t , T (0; ∆ )2 = pS ( p − t p ∈ N , (cid:32) (cid:52) T (cid:161) S ( p ) t (cid:162) = sec π ( p /2 − (cid:98) p /2 (cid:99) )2 p /2 − (cid:98) p /2 (cid:99) Γ (1 − p /2 + (cid:98) p /2 (cid:99) ) (cid:90) ∞ Re (cid:104) l p /2 −(cid:98) p /2 (cid:99)+ (cid:179) (cid:52) T (cid:161) S (2 (cid:98) p /2 (cid:99) ) t (cid:162) − ( − i) (cid:98) p /2 (cid:99)− (cid:179) l φ ( (cid:98) p /2 (cid:99) ) t , T ( l ; ∆ ) + (cid:106) p (cid:107) φ ( (cid:98) p /2 (cid:99)− t , T ( l ; ∆ ) (cid:180)(cid:180)(cid:105) d l , p ∉ N . (38)The first equation in (38) signifies that if p /2 is an integer, then the size of the hedgeis equal to the spot price of another power variance swap, with the decremented power p /2 −
1. Again, after taking p = (cid:52) T (cid:161) S (1) t (cid:162) = (cid:112) π (cid:90) ∞ Re (cid:104) φ t , T ( l ; ∆ ) (cid:112) l (cid:105) d l .Obviously, for the corresponding variance swap, the hedge is exactly I t ( T − t + ∆ ), with (cid:52) T (cid:161) S (2) t (cid:162) = φ t , T ( · ; ∆ ) ∈ L ( R ). In fact, although this condition is con-siderably benign and realistic, it is only sufficient but not necessary for the results to hold.The only assumption we have made in this regard is that X is a continuous random vari-able, stemming from the non-atomic Lévy measure ν . As mentioned since the introduction, our study for asymmetric power options is moti-vated by the average forward volatility being the square root of the average forward vari-ance. In other words, an option written on the average forward volatility can be effectively Although the notation φ ( − t , T ( · ; ∆ ) can be well understood as an antiderivative, it does not matter here due to multiplication by 0. ower-type volatility derivatives treated as a power-type option on the average forward variance, with power exactly equalto 1/2. For convenience and generality we still conduct our analysis subject to a positivepower coefficient.Let us consider a European-style put option contract on the average forward volatility I T ( ∆ ), having the terminal payoff P ( p , p ,(a)) T = (cid:161) K p − I p T ( ∆ ) (cid:162) + , (39)where K > p , p ≥ I T ( ∆ ) takes val-ues within the unit interval under normal conditions, p > ≤ p < p ≥ P (1, p ,(a)) T corresponds to the terminal payoff of the standard volatility putoption, while P (1, p ,(a)) T represents that of a standard put option on the average forwardvariance. Moreover, in the case of a call option, we have C ( p , p ,(a)) T = (cid:161) I p T ( ∆ ) − K p (cid:162) + . (40)The following proposition is given for arbitrary-time pricing of the asymmetric poweroption. Proposition 5.
The price of the asymmetric power put option with terminal payoff(39) at time t ∈ [0, T ) is given by P ( p , p ,(a)) t = K p − π (cid:90) ∞ Re (cid:104)(cid:179) K p e − i K p p l + Γ ( p /2 + − Γ ( p /2 +
1, i K p / p l )(i l ) p /2 (cid:180) × φ t , T ( l ; ∆ )i l (cid:105) d l . (41) The price of the asymmetric power call option with terminal payoff (40) at time t ∈ [0, T ) is given by C ( p , p ,(a)) t = P ( p , p ,(a)) t − K p + S ( p ) t , (42) where S ( p ) t is the contemporaneous price of a power swap on I T ( ∆ ) specified in Proposi-tion 4. By taking p = p = P (1,1,(a)) t = K − π (cid:90) ∞ Re (cid:104)(cid:179) K e − i K l + (cid:112) π /2 − Γ (3/2, i K l ) (cid:112) i l (cid:180) φ t , T ( l ; ∆ )i l (cid:105) d l . (43)and, recalling (37), C (1,1,(a)) t = π (cid:90) ∞ Re (cid:104)(cid:114) π l − (cid:179) K e − i K l + i (cid:112) π /2 − Γ (3/2, i K l ) (cid:112) i l (cid:180) φ t , T ( l ; ∆ )i l (cid:105) d l − K eixuan Xia On the other hand, for the standard put option on the average forward variance with p =
2, there is a significant reduction, P (2,1,(a)) t = K − π (cid:90) ∞ Re (cid:104) ( e − i K l − φ t , T ( l ; ∆ ) l (cid:105) d l .The formulae (43) and (44) for the standard volatility options can be implemented withhigh efficiency provided that the conditional characteristic function takes the form of (26)and facilitate calibration of the model on standard option prices.Hedging of the asymmetric power options resembles that of the corresponding powerswap, with complete reliance on the forward variance curve. For the following we adoptthe differential operator (cid:52) t for t ∈ [ t , T ]. Corollary 3.
In the setting of Proposition 5, hedges can be constructed as (cid:52) T (cid:161) P ( p , p ,(a)) t (cid:162) = − π (cid:90) ∞ Re (cid:104)(cid:179) K p e − i K p p l + Γ ( p /2 + − Γ ( p /2 +
1, i K p / p l )(i l ) p /2 (cid:180) × φ t , T ( l ; ∆ ) (cid:105) d l (45) and (cid:52) T (cid:161) C ( p , p ,(a)) t (cid:162) = (cid:52) T (cid:161) P ( p , p ,(a)) t (cid:162) + (cid:52) T (cid:161) S ( p ) t (cid:162) , (46) where (cid:52) T (cid:161) S ( p ) t (cid:162) is as specified in Corollary 2. Once again, with the choice p = p =
1, the standard volatility options can be hedgedin terms of (cid:52) T (cid:161) P (1,1,(a)) t (cid:162) = − π (cid:90) ∞ Re (cid:104)(cid:179) K e − i K l + (cid:112) π /2 − Γ (3/2, i K l ) (cid:112) i l (cid:180) φ t , T ( l ; ∆ ) (cid:105) d l and (cid:52) T (cid:161) C (1,1,(a)) t (cid:162) = π (cid:90) ∞ Re (cid:104)(cid:179) i (cid:112) π /2 − Γ (3/2, i K l ) (cid:112) i l − K e − i K l (cid:180) φ t , T ( l ; ∆ ) (cid:105) d l .As always, one should never neglect the fact that hedging with the underlying aver-age forward variance I ( ∆ ) is possible if V is a conventional Ornstein-Uhlenbeck processwith h ( t , s ) = e κ ( t − s ) . In this respect, Corollary 2 and Corollary 3 provide alternative hedg-ing strategies based on a prolonged window, which remain valid with or without roughvolatility. On the other hand, these newfound hedges, albeit nonexclusive, are certainlyvery convenient since the investor only needs to look at one single forward variance, in-stead of the entire curve, at any given date before maturity. As in the case of equity options, the volatility option investor’s risk exposure can also beadjusted by directly forcing a mutual power effect on the standard option payoff (similarto [Raible, 2000, Sect. 3.4] [30] and [Xia, 2019, pp. 120] [41]). This way of generalization ower-type volatility derivatives understandably does not build any useful connection between options on the averageforward volatility and the corresponding forward variance and is hence considered lessimportant from the viewpoint of this paper’s motivation. Nonetheless, for the sake ofcompleteness and our interest we still provide a comprehensive analysis of the pricing-hedging methods for such so-called “symmetric power options”.In this connection let a European-style put option contract on I T ( ∆ ) have the followingterminal payoff, P ( p ,(s)) T = (cid:161) ( K − I T ( ∆ )) + (cid:162) p , (47)where K > p ≥
0. In this structure both the strike price and the average forwardvolatility undergo the same power impact, and with binomial expansion we can rewrite P ( p ,(s)) T = ∞ (cid:88) k = (cid:195) pk (cid:33) ( − k K p − k I kT ( ∆ ) { I T ( ∆ ) < K } = ∞ (cid:88) k = (cid:195) pk (cid:33) ( − k K p − k S ( k ) T { I T ( ∆ ) < K } , (48)which shows that, conditional on { I T ( ∆ ) < K }, the symmetric put power option can belooked upon as a weighted sum of power volatility swaps, each associated with an integerpower coefficient in N , which at k = K p . Clearly, (48) is afinite sum if and only if p ∈ N .Besides, we observe that the plots of the payoff functions P ( p , p ,(a)) T and P ( p ,(s)) T against I T ( ∆ ) are symmetric with respect to the line segment joining the points (0, K p ) and ( K , 0)over the interval [0, K ]. For 0 ≤ p <
1, the symmetric power option provides a convextransformation of the standard option payoff whereas its asymmetric power counterpartprovides a concave one; for p > Figure 3: Comparison of leverage effects of power put options
As we write the terminal payoff C ( p ,(s)) T = (cid:161) ( I T ( ∆ ) − K ) + (cid:162) p = p (cid:88) k = (cid:195) pk (cid:33) ( − K ) k S ( p − k ) T { I T ( ∆ ) > K } , p ∈ N , (cid:181) (cid:98) p (cid:99) (cid:88) k = (cid:195) pk (cid:33) ( − K ) k S ( p − k ) T + ∞ (cid:88) k =(cid:98) p (cid:99)+ (cid:195) pk (cid:33) ( − K ) k I p − kT ( ∆ ) (cid:182) { I T ( ∆ ) > K } , p ∉ N , (49) eixuan Xia a similar symmetric power call option can be decomposed into exactly (cid:98) p (cid:99) weighted powervolatility swaps incremented by an infinite sequence of power-type derivatives on the re-ciprocal average forward volatility I − T ( ∆ ) conditioned to stay below 1/ K , which vanishesif and only if p is an integer. Also, note that in this case the payoff functions C ( p , p ,(a)) T and C ( p ,(s)) T are both strictly concave resp. convex in I T ( ∆ ) for 0 ≤ p < p > K , ∞ ),with lim x →∞ ( x p − K p ) + /(( x − K ) + ) p = Proposition 6.
The price of the symmetric power put option with terminal payoff (47)at time t ∈ [0, T ) is given by P ( p ,(s)) t = π ∞ (cid:88) k = (cid:195) pk (cid:33) ( − k K p − k (cid:90) ∞ Re (cid:104) ( Γ ( k /2 + − Γ ( k /2 +
1, i K l )) φ t , T ( l ; ∆ )(i l ) k /2 + (cid:105) d l , (50) while that of the similar symmetric power call option with (49) is C p ,(s) t = (cid:98) p (cid:99) (cid:88) k = (cid:195) pk (cid:33) ( − K ) k (cid:179) S ( p − k ) t − π (cid:90) ∞ Re (cid:104) ( Γ (( p − k )/2 + − Γ (( p − k )/2 +
1, i K l )) φ t , T ( l ; ∆ )(i l ) ( p − k )/2 + (cid:105) d l (cid:180) + Σ ( p ) t , (51) where S ( p − k ) t ’s, for ≤ k ≤ p , are the contemporaneous power swap prices as specified inProposition 4 and Σ ( p ) t = if p ∈ N ,1 π ∞ (cid:88) k =(cid:98) p (cid:99)+ (cid:195) pk (cid:33) ( − K ) k (cid:90) ∞ Re (cid:104) Γ (1 − ( k − p )/2, i K l ) φ t , T ( l ; ∆ )(i l ) − ( k − p )/2 (cid:105) d l , if p ∉ N .Hedges of these symmetric power options using I t ( T − t + ∆ ) also come in similarforms, which yield the next result. Corollary 4.
Assume the setting of Proposition 6. Then we have (cid:52) T (cid:161) P ( p ,(s)) t (cid:162) = π ∞ (cid:88) k = (cid:195) pk (cid:33) ( − k K p − k (cid:90) ∞ Re (cid:104) ( Γ ( k /2 + − Γ ( k /2 +
1, i K l )) φ t , T ( l ; ∆ )(i l ) k /2 (cid:105) d l , and (cid:52) T (cid:161) C p ,(s) t (cid:162) = (cid:52) T (cid:161) Σ ( p ) t (cid:162) + (cid:98) p (cid:99) (cid:88) k = (cid:195) pk (cid:33) ( − K ) k (cid:179) (cid:52) T (cid:161) S ( p − k ) t (cid:162) − π (cid:90) ∞ Re (cid:104) ( Γ (( p − k )/2 + − Γ (( p − k )/2 +
1, i K l )) φ t , T ( l ; ∆ )(i l ) ( p − k )/2 (cid:105) d l (cid:180) , ower-type volatility derivatives where (cid:52) T (cid:161) S ( p − k ) t (cid:162) ’s, for ≤ k ≤ p , are the contemporaneous power swap hedges as speci-fied in Corollary 2 and (cid:52) T (cid:161) Σ ( p ) t (cid:162) = if p ∈ N ,1 π ∞ (cid:88) k =(cid:98) p (cid:99)+ (cid:195) pk (cid:33) ( − K ) k (cid:90) ∞ Re (cid:104) Γ (1 − ( k − p )/2, i K l ) φ t , T ( l ; ∆ )(i l ) ( p − k )/2 (cid:105) d l , if p ∉ N . For this empirical study we illustrate the performance of our model framework estab-lished in Section 2 as well as the pricing-hedging formulae presented in Section 4. Allow-ing for overall efficiency, we focus on the closed-form characteristic function presentedin Corollary 1, specializing h according to (18) and taking X to belong to the class of tem-pered stable subordinators. Our data set consists of standard VIX option prices quoted on June 1, 2016, in the unit ofUS$100 (data source: [Cboe Global Markets, Inc., 2016] [7]). On this date, the VIX indexclosed at I ( ∆ ) = . These are four different maturities T =
7, 28, 112, 168 days and the strike price K goes from 0.1 to as high as 0.7, and t = Θ = ∆ . As aforementioned, we can of course take it as a parameter andput it into our calibration scheme. However, since in this case by the definition of thetype-III kernel both H ∆ ( T − · ) and (cid:82) Θ H u ( T − · )d u (cid:177) Θ are stationary, strictly increasing andcontinuous differentiable functions in [0, T ) which possess the same tail behaviors, it isalso reasonable to employ the mean value theorem and impose that they coincide at timelag T , i.e., H ∆ ( T ) = Θ (cid:90) Θ H u ( T )d u , (52)which always has a unique solution in (0, Θ ]. In other words, with (52) ∆ can be impliedfrom the kernel parameters κ and d and the time-to-maturity T . In particular, because thequotient Θ H ∆ (cid:177) (cid:82) Θ H u d u is bounded and converges to 1 towards infinity it is understoodthat a larger maturity T enhances the accuracy of (52). Although (52) can oftentimes besolved numerically with little effort, its solution is easily determined by inverting (14) and The reason behind choosing call options for this empirical study is that the call pricing formula (44) is more complicated thanthose of volatility swaps and put options (37) and (43). eixuan Xia we obtain the following preferable functional form, ∆ = ( Γ ( d )( Q − θ e − κτ ) + τ d − ) d − − T if − log( Q / θ ) κ < τ , − log( Q / θ ) κ − T if − log( Q / θ ) κ ≥ τ , (53)where Q = ( T + Θ ) d − T d − d τ d − Θ Γ ( d + Θ + θ e − κτ if τ > T + Θ ,1 Θ (cid:179) d τ d − ( T − τ ) − T d + τ d Γ ( d + + ( τ − T ) θ e − κτ − θ (cid:161) e − κτ − e − κ ( T + Θ ) (cid:162) κ (cid:180) if T < τ ≤ T + Θ , θ (cid:161) e − κ T − e − κ ( T + Θ ) (cid:162) κΘ if τ ≤ T . (54)Here θ = − ((2 − d )/( e κ )) d − /( κ Γ ( d − τ = (2 − d )/ κ .Since V can only have upward jumps under (6) which already necessarily generate astrictly positive long-term mean, we will fix ¯ V = . Anotherbenefit from this assumption is that, for the prolonged average forward volatility I ( T + ∆ ), by referring to (22) we have a one-to-one correspondence to the observed forwardvariance I ( ∆ ), I ( T + ∆ ) = e − κ T (cid:179) I ( ∆ ) − ξ (cid:179) s (cid:179) s d − − d τ d − Γ ( d + + θ e − κτ (cid:180)(cid:175)(cid:175)(cid:175) min{ τ , ∆ } s = + θ e − κ s κ (cid:175)(cid:175)(cid:175) ∆ s = min{ τ , ∆ } (cid:180)(cid:180) + ξ (cid:179) s (cid:179) s d − − d τ d − Γ ( d + + θ e − κτ (cid:180)(cid:175)(cid:175)(cid:175) min{ τ , T + ∆ } s = − θ e − κ s κ (cid:175)(cid:175)(cid:175) T + ∆ s = min{ τ , T + ∆ } (cid:180) , (55)where the equality follows directly from (64) and ξ = a Γ (1 − c )/ b − c , and which involvesno additional parameters.We recognize the difficulty of reliably calibrating the family parameter c within theunit interval due to the variability of the hypergeometric functions near the endpoints(see Corollary 1). For this reason, we only focus on three particular values: c = c = c = a > b > κ > d ∈ (1/2, 1). More specifically, for each fixed c we minimize the mean absolute error (MAE) between the observed market prices ofthe call options (denoted ˇ C ’s) and the corresponding model prices, so that the optimalparameter set is given by( ˆ a , ˆ b , ˆ κ , ˆ d ) = arg min a > b > κ > d ∈ (1/2,1) (cid:88) K , T (cid:175)(cid:175) ˇ C − C (1,1,(a))0 (cid:175)(cid:175) ,where the sum acts over all available strike prices and maturities. With Q : = (cid:82) Θ H u ( T )d u (cid:177) Θ , derivation of (54) is substantially no different from that of Corollary 1 in Section 8.3, hence omitted. As will be seen in the next section this assumption indeed does not seriously affect the calibration quality. On the other hand,without this assumption ¯ V will show up in (55) as a parameter to be considered for calibration. ower-type volatility derivatives Calibration is done jointly taking into account all four different maturities for the entiredata set. For contrast analysis we also consider two reduced models without mean rever-sion or path roughness, i.e., with κ (cid:38) d (cid:37)
1, respectively. Table 1 and Table 2 belowpresent the calibration results, where the CPU time measured in seconds are displayedas well. Table 1: Calibration results (rounded to 6 significant figures) c Subgroup ˆ a ˆ b ˆ κ ˆ d MAE CPU time0.3 full model 0.462174 2.21264 4.18619 0.824694 0.16747% 4770.73 κ (cid:38) d (cid:37) (cid:63) κ (cid:38) d (cid:37) κ (cid:38) d (cid:37) It is seen that inclusion of mean reversion and path roughness has led to significant im-provement of the model fit compared to the traditional Lévy-driven Ornstein-Uhlenbeckmodel with d (cid:37)
1. If one disregards mean reversion, the model fit becomes highly labile,largely altering the calibrated fraction parameter ˆ d , whereas the fit is still not too muchworse because the selected VIX options do not have very long maturities. Also, the cali-brated basic parameters ˆ a and ˆ b are quite different if the family parameter c has shifted,which can be easily told from the expression of the first moment ξ . These results in gen-eral confirm the validity of the type-III kernel, in association with the desirable closed-form characteristic function. The model with the best fit (marked “ (cid:63) ”) also agrees withthe rapid mean-reverting nature and roughness of the volatility of S&P500 returns in re-ality. Under this model, the formula (53) implies ∆ = T =
168 days, which is only about 15 days.For the subgroup c = T . For the in-the-money options and short maturities, the tworeduced models without mean reversion or roughness seem to fit quite well, as volatilityjumps are already captured by X , while by incorporating mean reversion and roughnesssimultaneously the full model outperforms the two reduced models mainly for the deeplyout-of-the-money options and long maturities. We also note the existent discrepancy be-tween the model prices and the market prices, which we ascribe further to the presence ofclustering in the instantaneous volatility process (see Section 6). We also remark that thisempirical exercise only demonstrates one extremal version of the type-III kernel definedin (14), and in general one need not take θ to be the lower bound in (16) but can incre- The constrained optimization program is written in Mathematica (cid:114) ([Wolfram Research, Inc., 2015] [38]) and run on a personallaptop computer with an Intel(R) Core(TM) i5-7200 CPU @ 2.50GHz 2.71GHz. eixuan Xia ment it by any arbitrary positive number, and τ takes its original form in (17) which hastwo distinct values. In this case, calibration will be roughly twice more computationallyintense due to increased formula complexity, which in turn reflects the elegance of (18). (full model)( κ (cid:38) d (cid:37) In this section we investigate the sensitivity of the pricing and hedging of power-typevolatility derivatives for the VIX index with respect to varying power coefficients, by usingthe general formulae proposed in Section 4. Based on the previous data set, for simplic- ower-type volatility derivatives ity we look at only one strike price K = T = c = p = p = p ∈ [0.5, 1.5] and plot the price changes( C ( p , p ,(a))0 , P ( p , p ,(a))0 , C ( p ,(s))0 and P ( p ,(s))0 ) of corresponding power-type derivatives in Figure5 and the same thing is done for the hedge changes ( (cid:52) (cid:161) C ( p , p ,(a))0 (cid:162) , (cid:52) (cid:161) P ( p , p ,(a))0 (cid:162) , (cid:52) (cid:161) C ( p ,(s))0 (cid:162) and (cid:52) (cid:161) P ( p ,(s))0 (cid:162) ). For the infinite series in Proposition 6 and Corollary 4 we use the approx-imation (cid:80) k = which universally leads to a global error less than 1%. Plots for power swapsare excluded as they are already embodied in the call option pricing formulae. It is seenthat, in terms of risk adjustment, the symmetric power options are able to provide muchseverer leverage effect for the VIX index compared to the asymmetric power options (withidentical power coefficients), despite that the latter are much easier to handle in gen-eral. Also, hedges for the symmetric power options are in comparison more sensitive tochanges in the power coefficient. Figure 5: Power impact on VIX option prices and hedges
Of course, for an asymmetric power option, by letting the two power coefficients varyindependently we can generate a power surface for its price ( C ( p , p ,(a))0 and P ( p , p ,(a))0 ) andhedge ( (cid:52) (cid:161) C ( p , p ,(a))0 (cid:162) and (cid:52) (cid:161) P ( p , p ,(a))0 (cid:162) ) which cannot be realized for symmetric powerones. For a better illustration we further restrict p , p ∈ [0.9, 1.1] which ensures that thepowered spot VIX and strike price are not too distant from each other. Apart from showingthe magnificent impact of powers on the VIX option price and hedges, these are also areliable indicator that the general pricing-hedging formulae can be implemented fairlyefficiently. eixuan Xia Figure 6: Asymmetric power surfaces for VIX option prices and hedges
Needless to say, the discovery of [Da Franseca and Zhang, 2019] [9] provides yet anothervery interesting implication, that the volatility of the average forward volatility, such asthe VVIX index, also possesses rough sample paths aside from being stochastic. Althoughit is noticeably challenging to establish a comfortable framework coalescing both aspectsof roughness, we will briefly discuss how the foregoing pricing problems may be tackledinheriting the structure of (25). For that purpose we recall the setting of Section 2.3 anddefine the composite process ˜ I t ( ∆ ) : = I T t ( ∆ ), t ≥
0, (56)with T t : = (cid:90) t Y ( η ) s d s , (57)where T = Y ≡ ( Y t ) is an (cid:70) -adapted square-integrable Lévy subordinator and η akernel which respectively resemble X and h up to different parameters. The construction(56) emulates the initiative work of [Carr and Wu, 2004] [6] on stochastic time change,which has a fundamental root in the famous Dambis-Dubins-Schwartz theorem. Clearly, Y ( η ) = (cid:82) · η ( · , s )d Y s is a mean-reverting process that introduces frictions into the volatilityof the average forward volatility and, if η is associated with a fraction parameter less than1, say d (cid:48) ∈ (1/2, 1], then Proposition 1 informs us that (cid:112) Y ( η ) also captures volatility-of-volatility jumps. By convention we assume independence between X and Y .Under (56), we are able to at least write the unconditional characteristic function ofthe time-changed average forward variance in terms of nested integrals. Evaluating the conditional characteristic function on F t for some t ∈ [0, t ) in the presence of time change can be cumbersome. ower-type volatility derivatives Proposition 7.
Let φ Y ( l ) : = (cid:69) (cid:163) e i lY (cid:164) , l ∈ R , denote the characteristic function of Y .Then, for any t > , ˜ φ t ( l ; ∆ ) : = (cid:69) (cid:163) e i l ˜ I t ( ∆ ) (cid:164) = π (cid:90) ∞ φ y ( l ; ∆ ) (cid:90) ∞ Re (cid:104) exp (cid:179) − i λ y + (cid:90) t log φ Y (cid:179) λ (cid:90) ts η ( v , s )d v (cid:180) d s (cid:180)(cid:105) d λ d y , l ∈ R ,(58) where φ y ( l ; ∆ ) is as given in (26). Note that the innermost integral in (58) can be expressed explicitly if η is any of thethree types of kernels specified before, while the other three integrals remain numericalin nature. In particular, the outer two integrals are generally not interchangeable, i.e., onecannot apply the Fubini theorem and has to compute them in proper order.Regardless, we can put (58) into the pricing formulae proposed in Section 4 to com-pute the prices of power volatility derivatives at time 0. However, since there are at leastfour numerical integrals involved, coming up with a robust calibration scheme will be anarduous task. As a means of reducing calibration burden in this connection, one shouldperhaps conduct characteristic function-based estimation (see [Yu, 2004] [42]) based onvolatility-of-volatility index data for the parameters of Y beforehand. The modeling of rough volatility is not inherently confined to Brownian sample paths andcan be alternatively incorporated into a purely discontinuous Lévy process, which is alsoable to capture volatility jumps. Importantly, the latter approach aims to balance betweenthe empirical findings of [Gatheral et al, 2018] [13] and [Todorov and Tauchen, 2011] [36].The pricing-hedging framework presented in this paper is expressly tailored for volatil-ity derivatives and built upon an Ornstein-Uhlenbeck structure with a generalized inte-grable kernel to establish short-term dependence. The advantage of this framework liesin the unconditional positivity of the instantaneous variance, which nullifies any consid-eration of the logarithm of volatility, so that integration is facilitated leading eventuallyto a semi-closed characteristic function for the conditional forward variance. By furtherinspecting the structure of this characteristic function, we discover a family of station-ary kernels, classified as type-III, under which full explicitness can be achieved, and thisbeyond doubt lays the foundation for efficient model calibration in practice.Since a volatility derivative is equivalent to a similar derivative written on the corre-sponding variance raised to the power half, it is natural to think of a wider class of power-type volatility derivatives as also a means of introducing leverage effect into the optionpayoffs. This is very much analogous to the original invention of power options written
Even if both η and h are exponential kernels posing no roughness, the time-changed process ˜ I ( ∆ ) cannot be Markovian with respectto the filtration (cid:70) jointly generated by X and Y . eixuan Xia on equity prices. The general pricing-hedging formulae proposed in Section 4 should beinterpreted as model-independent which require nothing more than a semi-closed modelcharacteristic function, and will certainly work without problems for familiar models withpath roughness unaccounted for. In addition, compared to the existing equity-optionpricing formulae that appeared in [Bakshi and Madan, 2000] [3], these formulae are in-evitably more complicated where exponential functions are mostly replaced by incom-plete gamma functions. However, their applicability and efficiency remain much unaf-fected as demonstrated by the empirical study on VIX options.Needless to say, by the lack of distributional stability the use of a jump model circum-vents consideration of the correlation between the instantaneous volatility and the priceprocess it is attached to. For this reason, if one’s ultimate goal is to deal with traditionalequity options with rough volatility, the framework established in this paper may not bea convenient one. On the other hand, if one’s interests specifically start with volatilityquantities, this framework then provides some new insights into how pricing and hedg-ing may be done more efficaciously as well as, intriguingly, the potential of introducingrough volatility of volatility through a time composition process. In this connection, itis still part of our ongoing research to explore the relative importance of volatility jumpsand price-volatility correlation under rough volatility models for various classes of assetprices. A future research could also be devoted to an all-around comparison of the em-pirical performances with the three different types of kernels, despite that their similarityhas been briefly discussed in Section 2.2; of course, for the first two types model calibra-tion will be understandably much slower because of more than one numerical integral tocarry out. For V defined in (6) we focus on the stochastic integral part, namely X ( h ) , the rest beingobviously continuous and of finite variation. Since h is continuously differentiable and h ( t + u , t ) = O (cid:161) e − κ u u ( d − + (cid:162) = o ( u d − ) as u → ∞ for any t ≥
0, by the extreme value theo-rem there exist two positive constants b h ≥ a h >
0, which depend only on the parametersof h including κ and d , such that, for any u > t ∈ [0, T ], (cid:69) (cid:163)(cid:161) X ( h ) t + u − X ( h ) t (cid:162) (cid:164) ∈ (cid:69) (cid:163)(cid:161) X ( g ) t + u − X ( g ) t (cid:162) (cid:164) × [ a h , b h ], (59)where g is the Riemann-Liouville kernel (3) with the same fraction parameter d . Thisrelation enables us to restrict our analysis to the Riemann-Liouville fractional Lévy sub-ordinator X ( g ) without mean reversion. The expectation on the right-hand side of (59) isthen by the Lévy-Itô isometry E ( g ) t , u : = (cid:69) (cid:163)(cid:161) X ( g ) t + u − X ( g ) t (cid:162) (cid:164) = ξ Γ ( d ) (cid:179) (cid:90) t (( t + u − s ) d − − ( t − s ) d − ) d s + (cid:90) t + ut ( t + u − s ) d − d s (cid:180) . ower-type volatility derivatives In particular, for d > X ( g ) t = (cid:82) t (cid:161) (cid:82) s ( s − v ) d − / Γ ( d − X v (cid:162) d s due to (4) and so we only need to consider d ≤ d ∈ (1, 2]. We observe that the obviously uniformly continuous map R ++ (cid:51) u (cid:55)→ Γ ( d ) (cid:90) t (( t + u − s ) d − − ( t − s ) d − ) d s ∈ R ++ (60)is increasing and convex for every t >
0, hence the only need to compare its tail behaviorsagainst power-law tails. It is not difficult to see (cid:90) t (( t + u − s ) d − − ( t − s ) d − ) d s = O ( u d − ) = o ( u d − ), as u → ∞ ,and (cid:90) t (( t + u − s ) d − − ( t − s ) d − ) d s = O (cid:161) u min{2 d − (cid:162) , as u (cid:38) (cid:90) t (( t + u − s ) d − − ( t − s ) d − ) d s ≤ c d u min{2 d − , ∀ u > t ∈ [0, T ],for some constant c d > d . On the other hand, (cid:82) t + ut h ( t + u , s )d s = u d − /((2 d − Γ ( d )). Combining things we obtain E ( g ) t , u ≤ ξ Γ ( d ) (cid:179) c d + d − (cid:180) u min{2 d − . (61)Since the power of u in (61) strictly exceeds 1, we apply the Kolmogorov- ˇCentsov theorem(see, e.g., [Karatzas and Shreve, 1991, Sect. 2.2.B] [20]) [20] to conclude that X ( g ) , andhence V due to (59), admits an a.s. continuous modification over R + ; in particular, themodification is a.s. locally Hölder-continuous for every exponent in (0, min{ d −
1, 1/2}).Furthermore, using the uniform time partition T M (with t =
0) in (32) of the interval[0, T ], we define the T M -quadratic variation Q M ([0, T ]) : = M (cid:88) n = (cid:161) X ( g ) nT / M − X ( g )( n − T / M (cid:162) ≥ M ∈ N ++ , M (cid:192) d ∈ (1, 2], by the convexity of (60) and the upper bound (61) we have (cid:69) [ Q M ([0, T ])] ≤ M E ( g )( M − T / M , T / M ≤ ξ Γ ( d ) (cid:179) c d + d − (cid:180)(cid:179) MT (cid:180) max{2(1 − d ), − →
0, as M → ∞ ,which by nonnegativity and the relation (59) implies that V has a.s. zero quadratic varia-tion over [0, T ] and completes the proof of assertion (i). eixuan Xia Now suppose d ∈ (1/2, 1], and then in the same vein we see that the map (60) is in-creasing and concave and behaves like O ( u d − ) both as u → ∞ and as u (cid:38)
0. Therefore,there exists c d > E ( g ) t , u ≥ ξ Γ ( d ) (cid:179) c d + d − (cid:180) u d − .Note that in this case there is no way to apply the Kolmogorov- ˇCentsov theorem. Let X − denote the càglàd modification of X and set E : = { ω ∈ Ω : ( X t − X t − )( ω ) > ∃ t ∈ [0, T ]}.With ν ( R ++ ) = ∞ , it is a familiar result (see again [Lyasoff, 2017, Sect. 16] [24]) that (cid:80) E = X ( h ) t − X ( h ) t − = (cid:90) t − ( h ( t , s ) − h ( t − , s ))d X s + (cid:90) tt − h ( t , s )d X s , t ≥
0. (62)Since, for any s ∈ [0, t ), R ++ (cid:51) t (cid:55)→ h ( t , s ) ∈ R + is a continuous map with d ∈ (1/2, 1], bythe dominated convergence theorem the first integral in (62) is naught (in the sense of L -convergence) and (cid:90) tt − h ( t , s )d X s = h ( t , t − )( X t − X t − ).In consequence, there must exist some t > X ( h ) t − X ( h ) t − ∝ h ( t , t − ). To put itanother way, E ⊆ (cid:169) ω ∈ Ω : (cid:161) X ( h ) t − X ( h ) t − (cid:162) ( ω ) ∝ h ( t , t − ), ∃ t ∈ [0, T ] (cid:170) . (63)By (7) further, if d = h is uniformly bounded so that h ( t , t − ) > t >
0, andhence (63) proves the a.s. discontinuity of the sample paths of X ( h ) . In this case, since X ( h ) has no Brownian part, its quadratic variation is given by the sum of its squared jumps, (cid:88) t ∈ [0, T ] (cid:161) X ( h ) t − X ( h ) t − (cid:162) = (cid:88) t ∈ [0, T ] h ( t , t − )( X t − X t − ) > (cid:80) -a.s.,where the inequality follows from the finiteness of (cid:80) t ∈ [0, T ] ( X t − X t − ) >
0. On the otherhand, if d <
1, then h ( t , t − ) = ∞ for any t >
0, which with (63) gives the a.s. discontinuityand unboundedness of the sample paths of X ( h ) , and hence V , over [0, T ]. An immediateimplication is therefore that the sample paths of V have infinitely large squared jumpsover [0, T ] a.s., which lead to its (a.s.) infinite quadratic variation. Therefore assertions (ii)and (iii) are proved. (cid:228) According to (22) and (25) we can write for 0 ≤ t < t < TI t ( ∆ ) = V e − κ ( t + ∆ ) + ¯ V (1 − e − κ ( t + ∆ ) ) + X ( H t − t + ∆ ) t + ξ (cid:90) t + ∆ t H ∆ ( t , s )d s + (cid:90) tt H ∆ ( t , s )d X s ower-type volatility derivatives = I t ( t − t + ∆ ) − ξ (cid:90) tt H ∆ ( t , s )d s + (cid:90) tt H ∆ ( t , s )d X s , (64)where on the right-hand side (cid:82) tt H ∆ ( t , s )d X s is independent from F t while the otherterms are measurable with respect to F t . Therefore, by the independence lemma, (cid:69) (cid:163) e i l I t ( ∆ ) (cid:175)(cid:175) F t (cid:164) = exp (cid:179) i l (cid:179) I t ( t − t + ∆ ) − ξ (cid:90) tt H ∆ ( t , s )d s (cid:180)(cid:180) (cid:69) (cid:163) e i l (cid:82) tt H ∆ ( t , s )d X s (cid:164) , l ∈ R .For the last expectation we note that the process (cid:82) · t H ∆ ( · , s )d X s has independent incre-ments on ( t , T ) and use the infinite divisibility of the law of X to write (cid:69) (cid:163) e i l (cid:82) tt H ∆ ( t , s )d X s (cid:164) = t (cid:89) t (cid:69) (cid:163) e i l H ∆ ( t , s ) X (cid:164) d s = exp (cid:90) tt log (cid:69) (cid:163) e i l H ∆ ( t , s ) X (cid:164) d s ,where (cid:81) ·· denotes the geometric integral operator (see, e.g., [Slavík, 2007] [33]) and whichyields the desired integral representation (26). (cid:228) Proving the corollary is purely a matter of computation. If h is as in (14), then we can write H ∆ ( t , s ) = h ( t − s + ∆ ) = (cid:40) h − ( t − s + ∆ ), if t − s + ∆ < τ , h + ( t − s + ∆ ), if t − s + ∆ ≥ τ , (65)and h − ( t − s + ∆ ) : = ( t − s + ∆ ) d − − τ d − Γ ( d ) + θ e − κτ and h + ( t − s + ∆ ) = θ e − κ ( t − s + ∆ ) .Straightforward integration of (65) over the interval [ t , t ] thus leads to (cid:90) tt H ∆ ( t − s )d s = (cid:90) tt h − ( t − s + ∆ )d s if τ > t − t + ∆ , (cid:90) t min{ t + ∆ − τ , t } h − ( t − s + ∆ )d s + (cid:90) min{ t + ∆ − τ , t } t h + ( t − s + ∆ )d s if τ ≤ t − t + ∆ = (cid:90) t − t + ∆∆ h − ( s )d s if τ > t − t + ∆ , (cid:90) max{ τ , ∆ } ∆ h − ( s )d s + (cid:90) t − t + ∆ max{ τ , ∆ } h + ( s )d s if τ ≤ t − t + ∆ = s (cid:179) s d − − d τ d − Γ ( d + + θ e − κτ (cid:180)(cid:175)(cid:175)(cid:175) t − t + ∆ s = ∆ if τ > t − t + ∆ , s (cid:179) s d − − d τ d − Γ ( d + + θ e − κτ (cid:180)(cid:175)(cid:175)(cid:175) max{ τ , ∆ } s = ∆ − θ e − κ s κ (cid:175)(cid:175)(cid:175) t − t + ∆ s = max{ τ , ∆ } if τ ≤ t − t + ∆ , eixuan Xia where the second equality uses the substitution s (cid:55)→ t − s + ∆ and (cid:82) max{ τ , ∆ } ∆ ≡ ∆ ≥ τ .Similarly, for the second Riemann integral in (26) we have with the characteristic ex-ponent (5) that ξ = a Γ (1 − c )/ b − c and that (cid:90) tt log φ X ( l H ∆ ( t − s ))d s (66) = (cid:90) t − t + ∆∆ log φ X ( l h − ( s ))d s if τ > t − t + ∆ , (cid:90) max{ τ , ∆ } ∆ log φ X ( l h − ( s ))d s + (cid:90) t − t + ∆ max{ τ , ∆ } log φ X ( l h + ( s ))d s if τ ≤ t − t + ∆ . (67)Since the integrands in (66) are obviously integrable over the designated domains, it onlysuffices to consider the indefinite integrals, I + ( s ) : = (cid:82) ( b − i l e − κ s ) c d s and I − ( s ) : = (cid:82) ( b − i l s d − ) c d s . Note that a Γ ( − c ) is just a scaling factor while the integration of b c is immedi-ate. For I + , we observe by using binomial expansion that I + ( s ) = ( − c + (i l ) c ∞ (cid:88) k = (cid:195) ck (cid:33)(cid:179) − b i l (cid:180) k e − κ ( c − k ) s κ ( c − k ) = ( − c + (i l ) c e − κ cs c κ ∞ (cid:88) k = ( − c ) k (1 − c ) k (cid:179) be κ ks i l (cid:180) k = ( − c + (i l ) c e − κ cs c κ F (cid:179) − c , − c ; 1 − c ; be κ ks i l (cid:180) ,where ( · ) · denotes the Pochhammer symbol, a.k.a. the rising factorial, and which aftersimplification leads to (30). The case of I is slightly more involved but can be proved ina similar fashion, and we obtain I − ( s ) : = s (cid:181)(cid:179) b − i l e − κτ + i l τ d − Γ ( d ) (cid:180) c F (cid:179) − c , 1 d − dd − l s d − i l τ d − + ( b − i l e − κτ ) Γ ( d ) (cid:180)(cid:182) ,thus yielding (31). Putting these together we arrive at (29). (cid:228) First notice that (cid:69) (cid:163) ˇ V nT / M (cid:164) = V e − κ nT / M + ¯ V (cid:161) − e − κ nT / M (cid:162) + ξ n − (cid:88) k = h (cid:179) nTM , kTM (cid:180) TM .For a given t ∈ (0, T ], we choose n ≡ n ( t , M ) = (cid:98) M t / T (cid:99) , so that lim M →∞ ( n ( t , M ) T / M ) = t . Since the Riemann integral (cid:82) t h ( t , s )d s is well-defined for t ∈ [0, T ), (33) constitutes aconventional rectangular Riemann-sum approximation and it is familiar that (cid:69) (cid:163) ˇ V n ( t , M ) T / M − V t (cid:164) = O ( M − ), as M → ∞ . (68) ower-type volatility derivatives This shows asymptotic unbiasedness. In addition to (68), using the relation (cid:69) (cid:163)(cid:161) ˇ V n ( t , M ) T / M − V t (cid:162) (cid:164) = (cid:69) (cid:163) ˇ V n ( t , M ) T / M − V t (cid:164) + Var (cid:163) ˇ V n ( t , M ) T / M (cid:164) ,for proving the L -convergence rate it is sufficient to note that, in the same vein, M Var (cid:163) ˇ V n ( t , M ) T / M (cid:164) = T (cid:161) ξ − ξ (cid:162) n ( t , M ) − (cid:88) k = h (cid:179) n ( t , M ) TM , kTM (cid:180) TM → T (cid:161) ξ − ξ (cid:162) (cid:90) t h ( t , s )d s , as M → ∞ . (cid:228) The relationship between fractional moments and the characteristic function of a real-valued random variable has been well established. In particular, knowing that I T ( ∆ ) isstrictly positive with (cid:69) (cid:163) I pT ( ∆ ) (cid:164) < ∞ for every p >
0, we have ([Pilenis, 2016, Equation (2.19)][29]) S ( p ) t = (cid:69) (cid:163)(cid:161) I T ( ∆ ) (cid:162) p /2 (cid:175)(cid:175) F t (cid:164) = ( − i) p /2 φ ( p /2) t , T (0; ∆ ). (69)If p is even, then (69) is understood as a conventional derivative corresponding to thefirst equation in (36). Otherwise, it represents a fractional derivative and can be written([Laue, 1980, Theorem 2.1] [22]) S ( p ) t = sec π ( p /2 − (cid:98) p /2 (cid:99) )2 p /2 − (cid:98) p /2 (cid:99) Γ (1 − p /2 + (cid:98) p /2 (cid:99) ) × Re (cid:183) ( − i) (cid:98) p /2 (cid:99) (cid:90) λ −∞ φ ( (cid:98) p /2 (cid:99) ) t , T ( λ ; ∆ ) − φ ( (cid:98) p /2 (cid:99) ) t , T ( l ; ∆ )( λ − l ) p /2 −(cid:98) p /2 (cid:99)+ d l (cid:175)(cid:175)(cid:175)(cid:175) λ = (cid:184) . (70)Since φ t , T ( · ; ∆ ) ∈ C (cid:98) p /2 (cid:99) ( R ), we can apply the dominated convergence theorem togetherwith the substitution l (cid:55)→ − l to recast (70) as S ( p ) t = sec π ( p /2 − (cid:98) p /2 (cid:99) )2 p /2 − (cid:98) p /2 (cid:99) Γ (1 − p /2 + (cid:98) p /2 (cid:99) ) × Re (cid:183) ( − i) (cid:98) p /2 (cid:99) (cid:90) ∞ φ ( (cid:98) p /2 (cid:99) ) t , T (0; ∆ ) − φ ( (cid:98) p /2 (cid:99) ) t , T ( − l ; ∆ ) l p /2 −(cid:98) p /2 (cid:99)+ d l (cid:184) .Using that ( − i) (cid:98) p /2 (cid:99) φ ( (cid:98) p /2 (cid:99) ) t , T (0; ∆ ) = S (2 (cid:98) p /2 (cid:99) ) t and the Hermitian property of the characteristicfunction we arrive at the second equation in (36). (cid:228) Based on (26), we have for p /2 ∈ N that (cid:52) T ( φ t , T ( l ; ∆ )) = i l φ t , T ( l ; ∆ ), eixuan Xia so that (cid:161) (cid:52) T ( φ t , T ( l ; ∆ )) (cid:162) ( p /2) = i (cid:179) l φ ( p /2) t , T ( l ; ∆ ) + p φ ( p /2 − t , T ( l ; ∆ )2 (cid:180) .Sending l → (cid:52) T can bedone inside the integral. Interchange with the real part is then simply allowed thanks tothe Hermitian property. Since φ t , T ( · ; ∆ ) ∈ C (cid:98) p /2 (cid:99) ( R ) and φ ( (cid:98) p /2 (cid:99) ) t , T ( l ; ∆ ) = O ( φ t , T ( l ; ∆ )) as l → ∞ , we only need to check integrability of the tails of Re[ φ t , T ( l ; ∆ )]/ l p /2 −(cid:98) p /2 (cid:99) for l ≥ A : = I t ( T − t + ∆ ) − ξ (cid:90) Tt H ∆ ( t , s )d s > F t and allows us to rewrite φ t , T ( l ; ∆ ) = e i l A ϕ t , T ( l ; ∆ ), l ∈ R ,where ϕ t , T ( l ; ∆ ) is the characteristic function of the random variable (cid:82) Tt H ∆ ( T , s )d X s >
0, whose law admits a well-defined density (recall that H ∆ is continuous and ν is non-atomic). Hence, we have (cid:90) ∞ Re (cid:163) e i l A ϕ t , T ( l ; ∆ ) (cid:164) d l = p /2 − (cid:98) p /2 (cid:99) ∈ (0, 1) for any p ∉ N implies the desired tail integrability. (cid:228) Let f t , T ( x ; ∆ ) and F t , T ( x ; ∆ ), for x >
0, respectively denote the density function and thedistribution function of I T ( ∆ ) (cid:175)(cid:175) F t , which exist because the law of X is absolutely contin-uous. For the price of the asymmetric power put option on I T ( ∆ ) at t ∈ [0, T ), we adopt˜ p = p /2 to rewrite its terminal payoff so that P ( p , p ,(a)) t = (cid:69) (cid:163)(cid:161) K p − I pT ( ∆ ) (cid:162) + (cid:175)(cid:175) F t (cid:164) = (cid:90) K p
2/ ˜ p ( K p − x ˜ p ) f t , T ( x )d x = K p F t , T ( K p / ˜ p ; ∆ ) − (cid:90) K p
2/ ˜ p x ˜ p f t , T ( x ; ∆ )d x : = E − E .Further denote ˜ K = K p / ˜ p . Using the Fourier inversion formula we have E = K p (cid:179) − π (cid:90) ∞ Re (cid:104) e − i ˜ K l φ t , T ( l ; ∆ )i l (cid:105) d l (cid:180) ower-type volatility derivatives and E = π (cid:90) ˜ K x ˜ p (cid:90) ∞ Re (cid:163) e − i l x φ t , T ( l ; ∆ ) (cid:164) d l d x = π (cid:90) ∞ Re (cid:104) φ t , T ( l ; ∆ ) (cid:90) ˜ K e − i l x x ˜ p d x (cid:105) d l , (71)where the second equality uses the Fubini theorem since the integral in x is taken over afinite interval. To evaluate the inner integral in (71), we apply the substitution x (cid:55)→ i l x andobserve that (cid:90) ˜ K e − i l x x ˜ p d x = (i l ) − ˜ p − (cid:90) i ˜ K l e − x x ˜ p d x = (i l ) − ˜ p − (cid:179) (cid:90) ∞ − (cid:90) ∞ i ˜ K l (cid:180) e − x x ˜ p d x = (i l ) − ˜ p − ( Γ ( ˜ p + − Γ ( ˜ p +
1, i ˜
K l )),where the second equality follows because the integrand is analytic over the horizontalhalf-strip { x : Re x >
0, Im x ∈ (0, ˜ K l )} with l >
0. This establishes (41) after rearrangement.The pricing formula for the similar asymmetric call option results from a standard par-ity argument that (cid:161) I p T ( ∆ ) − K p (cid:162) + − (cid:161) K p − I p T ( ∆ ) (cid:162) + = I p T ( ∆ ) − K p ,together with Proposition 4. It is important to note that a single integral representation forthe call price is inaccessible due to inapplicability of the Fubini theorem when integrationacts over [ ˜ K , ∞ ) (cid:51) x . (cid:228) We simply use the bounded-ness and Hermitian property of φ t , T ( · ; ∆ ) in order to apply (cid:52) T to (41) inside the real part of the integral. For the call option we use the parity relation(42). (cid:228) First consider the symmetric power put option with the payoff decomposition (48), sothat we may write P ( p ,(s)) t = K p F t , T ( K ; ∆ ) + ∞ (cid:88) k = (cid:195) pk (cid:33) ( − k K p − k ¯ E k and for every k ∈ N ++ using the argument in Section 8.7 we have¯ E k = π (cid:90) ∞ Re (cid:104) φ t , T ( l ; ∆ ) (cid:90) K e − i l x x k /2 d x (cid:105) d l = π (cid:90) ∞ Re (cid:104) φ t , T ( l ; ∆ ) Γ ( k /2 + − Γ ( k /2 +
1, i K l )(i l ) k /2 + (cid:105) d l . eixuan Xia which obviously allows the series to be augmented to k = C ( p ,(s)) t = (cid:98) p (cid:99) (cid:88) k = (cid:195) pk (cid:33) ( − K ) k ˘ E p − k + Σ ( p ) t ,where all the summands with index k > p in conditional expectation are put into Σ ( p ) t . Forevery 0 ≤ k < p note that˘ E p − k = S ( p − k ) t − (cid:90) K x ( p − k )/2 f t , T ( x ; ∆ )( x )d x ,so that a parity argument can be employed where the integral on the left-hand side isevaluated in the same vein as in (71). If p ∈ N then Σ ( p ) t is clearly naught. On the otherhand, if p ∉ N , then we write Σ ( p ) t = ∞ (cid:88) k =(cid:98) p (cid:99)+ (cid:195) pk (cid:33) ( − K ) k ˘ E k ,where ˘ E k = (cid:90) ∞ K x ( p − k )/2 f t , T ( x ; ∆ )d x = π (cid:90) ∞ Re (cid:104) φ t , T ( l ; ∆ ) (cid:90) ∞ K e − i l x x − ( k − p )/2 d x (cid:105) d l .Here the Fubini theorem applies because k − p >
0. At this point it suffices to observe that (cid:90) ∞ K e − i l x x − ( k − p )/2 d x = Γ (1 − ( k − p )/2, i K l )(i l ) − ( k − p )/2 ,which is well-defined as p − k cannot be an even number. (cid:228) The proof is similar to that of Corollary 3, except that (cid:52) T acts on (50) and (51) termwise,where the interchange of integration and differentiation is permitted for the same reasonas in Section 8.8. (cid:228) By mimicking the steps in Section 8.2, it can be deduced from (57) that the characteristicfunction of T t for a fixed t > φ T t ( l ) : = (cid:69) (cid:163) e i l T t (cid:164) = exp (cid:90) t log φ Y (cid:179) l (cid:90) ts η ( v , s )d v (cid:180) d s , l ∈ R . ower-type volatility derivatives By assumption the process Y has its own filtration { σ (( Y s ) s ∈ [0, t ] )} t ≥ independent fromthat of X . Therefore, via subsequent conditioning we have ˜ φ t ( l ; ∆ ) : = (cid:69) (cid:163) e i u ˜ I t ( ∆ ) (cid:164) = (cid:69) (cid:163) (cid:69) (cid:163) e i uI T t ( ∆ ) (cid:175)(cid:175) σ (( Y s ) s ∈ [0, t ] ) (cid:164)(cid:164) = (cid:69) [ φ T t ( l ; ∆ )]. (72)Since the law of the time change is absolutely continuous, (72) can be written using in-verse Fourier transform as˜ φ t ( l ; ∆ ) = π (cid:90) ∞ φ s ( l ; ∆ ) (cid:90) ∞ Re[ e − i λ s φ T t ( λ )]d λ d s ,and this is exactly the same as (58). (cid:228) References [1] Abramowitz, M. & Stegun, I.A. (1972).
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