Multiple Subordinated Modeling of Asset Returns
MMultiple Subordinated Modeling of AssetReturns
Abootaleb Shirvani, Svetlozar T. Rachev,Department of Mathematics and Statistics,Texas Tech UniversityandFrank J. FabozziEDHEC Business School, United StatesJuly 31, 2019
Abstract
Subordination is an often used stochastic process in modeling asset prices. Sub-ordinated L`evy price processes and local volatility price processes are now the maintools in modern dynamic asset pricing theory. In this paper, we introduce the the-ory of multiple internally embedded financial time-clocks motivated by behavioralfinance. To be consistent with dynamic asset pricing theory and option pricing, assuggested by behavioral finance, the investors view is considered by introducing anintrinsic time process which we refer to as a behavioral subordinator. The process issubordinated to the Brownian motion process in the well-known log-normal model,resulting in a new log-price process. The number of embedded subordinations resultsin a new parameter that must be estimated and this parameter is as important asthe mean and variance of asset returns. We describe new distributions, demonstrat-ing how they can be applied to modeling the tail behavior of stock market returns.We apply the proposed models to modeling S&P 500 returns, treating the CBOEVolatility Index as intrinsic time change and the CBOE Volatility-of-Volatility Indexas the volatility subordinator. We find that these volatility indexes are not propertime-change subordinators in modeling the returns of the S&P 500.
Keywords: behavioral finance; dynamic asset pricing models; L`evy-stable distribution;normal-compound inverse Gaussian distribution; variance-gamma-gamma distribution.1 a r X i v : . [ q -f i n . M F ] J u l Introduction
There is a vast literature that has sought to model the dynamics of asset returns. Theassumption typically made is that asset returns follow a normal distribution despite thepreponderance of empirical evidence that rejects this distribution (see Rachev et al., 2005).There are several stylized facts about asset returns that should be recognized in modelingthe dynamics of asset returns (see Cont, 2001). Specifically, asset returns exhibit asymmetryand heavy tails. Modeling and analyzing the tail properties of asset returns are crucial forasset managers and risk managers. Consequently, the usefulness of the results of modelsthat assume asset returns follow the normal law are questionable.To deal with non-normality, the method of subordination has been proposed in theliterature to include business time and allow the variance of the normal distribution tochange over time. The subordination process in finance, also called random time change, Y ( t ) = X ( T ( t )) , under the assumption of independence of X ( t ) and T ( t ) , is a techniqueemployed to introduce additional parameters to the return model to reflect the heavy tailphenomena present in most asset returns and to generalize the classical asset pricing model.The concept of random time change was first applied to Brownian motion to obtain morerealistic speculative prices by Clark (1973). Hurst et al. (1997) applied various subordinatedlog-return processes to model the leptokurtic characteristics of stock-index returns. Theycompared the classical log-normal model, the Mandelbrot and Fama log-stable model, theClark model, the log symmetric generalized hyperbolic model, the Barndorff–Nielsen model,the log hyperbolic model and the log variance gamma model in order to find the best three-parameter model that adequately takes into account leptokurtic characteristics for indices. In the option pricing literature, Carr and Wu (2004) used random-time change to derive amore realistic price process and to extend the approach in Carr et al. (2003) by providingan efficient way to include the correlation between the stock price process and random-timechange. Klingler et al. (2013) introduced two new six-parameter processes based on timechanging tempered stable distributions and developed an option pricing model based on See Bochner (1995), Sato and Katok (1999), and Schoutens (2003). For a further discussion of the use of subordinators in financial modeling, see the books by Sato andKatok (1999) and Schoutens (2003). { V ( t ) = T ( U ( t )) , t ≥ } as stock intrinsic time, and { U ( t ) , t ≥ } as stock-volatility intrinsictime or volatility subordinator.We will define and investigate the properties of various multiple subordinated log-returnprocesses that are applied to model the leptokurtic characteristics of asset returns. Thepossible multiple subordinated models that we consider for the distribution of changes inasset returns are the α -stable, gamma and inverse Gaussian subordinated models. Thesemodels differ by their intrinsic time processes that are subordinated to the standard Brow-nian motion for modeling asset returns. They are simple models that usually add someextra parameters and reflect several fundamental probabilistic relationships such as asymp-totic laws, self-similarity, and infinite divisibility. There are other classes of distributionalmodels that could be used, but they are more complex and do not emphasize character-istics that arise from fundamental relationships. Also, we generalize the multiple α -stablesubordination to the τ -subordination or continuous subordination, arguing that τ could bean additional parameter that allows for heavy-tailedness in the modeling of asset returns.We show that two popular stock market volatility indexes – the CBOE volatility index3VIX ) and the CBOE volatility of volatility index (VVIX ) – are not proper intrinsictime change subordinators for modeling the stock market as measured by the SPDR S&P500 (an exchange-traded fund).The remainder of this paper is organized as follows. In Section 2, we introduce thedouble subordinated model and present a multiple subordinated model using a continuous-time change process. In Section 3, we empirically estimate the return distribution of thestock market index by applying the double subordinated models that we presented inSection 2 and offer some concluding remarks in Section 4. Consider a stock price process { S t , t ≥ , S > } , with dynamics given by its log-priceprocess L t = lnS t , L t = L + µ t + γ U ( t ) + ρ T ( U ( t )) + σ B T ( U ( t )) , t ≥ , µ ∈ R , γ ∈ R , ρ ∈ R , σ > , (1)where the triplet ( B s , T ( s ) , U ( s ) , s ≥ ) are independent processes generating stochasticbasis ( Ω , F , F = (F t , t ≥ ) , P ) representing the natural world with { B s , s ≥ } being astandard Brownian motion. { T ( s ) , U ( s ) , s ≥ } and { T ( ) = , U ( ) = } are L´evysubordinators. A L´evy subordinator is a L´evy process with increasing sample path. B t , T ( t ) and U ( t ) are F t -adopted processes whose trajectories are right-continuous withleft limits. We view V ( t ) = T ( U ( t )) , t ≥ stock intrinsic time , and U ( t ) , t ≥ stock-volatility intrinsic time or the volatility subordinator . For example, in modelingthe SPDR S&P 500 by the triplet ( L t , V ( t ) , U ( t )) , t ≥ ( i ) L t , t ≥ ( ii ) V ( t ) , t ≥ See SPDR S&P 500 ETF Indices, https://us.sprdrs.com/. See Chapter 6 in Sato and Katok (1999). V ( t ) representing the cumulative value of VIX in [ , t ] ) and ( iii ) U ( t ) t ≥ U ( t ) , t ≥ [ , t ] ).The general framework of behavioral finance provides an alternative view of the doublysubordinated price process. In their seminal paper, Tversky and Kahneman (1992) intro-duced the Cumulative Prospect Theory (CPT). According to this theory, positive and neg-ative returns on financial assets are viewed differently due to the general “fear”dispositionof investors. To quantify an investor’s fear disposition, Tversky and Kahneman (1992) andPrelec (1998) introduced a probability weighting function (PWF), w (R , S) : [ , ] → [ , ] ,transforming the asset return distribution F R ( x ) = P (R ≤ x ) , x ∈ R according to the in-vestor’s views to a new one F S ( x ) = P (S ≤ x ) = w (R , S) ( F R ( x )) , x ∈ R . Tversky and Kahne-man (1992) introduced the following PWF w (R , S ; TK ) ( u ) = u γ [ u γ + ( − u ) γ ] γ , u ∈ ( , ) , γ ∈ [ , ] . (2)Unfortunately, this choice of the PWF is inconsistent with dynamic asset pricing theory(DAPT) because F S is not an infinitely divisible distribution function, leading to arbitrageopportunities in behavioral asset pricing models. Prelec (1998) introduced an alternativeWPF: w (R , S ; P ) ( u ) = exp (− δ lnu ) ρ , u ∈ [ , ] , δ > , ρ ∈ ( , ) . (3)Prelec’s w (R , S ; P ) is consistent with DAPT only in the case when R has a Gumbel distributiongiven by F R ( x ) = exp (cid:16) − e − x − µβ (cid:17) , x ∈ R , µ ∈ R , β > . (4)Rachev et al. (2017) studied the general form of PWF consistent with DAPT. Followingtheir arguments, we view the R as the return in unit time of a single subordinated log-priceprocess; that is, R = M , where M t = lnS + µ t + γ U ( t ) + σ B U ( t ) , t ≥ , µ ∈ R , γ ∈ R , σ > . (5)The log-price process M t , t ≥ T ( t ) , resulting in a new log-price process, See Barberis and Thaler (2005). t = lnS + µ t + γ U ( t ) + ρ T ( U ( t )) + σ B T ( U ( t )) , t ≥ , ρ ∈ R . (6)The distribution of S = L is characterized by heavier tails than R , representing thegeneral fear disposition of the investor. The corresponding WPF, w (R , S) : [ , ] → [ , ] ,is defined by w (R , S) ( u ) = F S (cid:16) F in v R ( u ) (cid:17) where F in v R ( u ) = min { x : F R ( x ) > u } is the inversefunction of F R ( x ) . In this setting, the log-price parameters for M t , t ≥ M t , t ≥ lnS t as observed by spot traders at the current time, t =
0. Thus,the parameters of M t , t ≥ L t , t ≥ lnS t as seen by option traders. The motivationfor this choice for the doubly subordinated process L t , t ≥ ρ ∈ R , and the parametersfor the distribution of T ( ) and U ( ) should be calibrated from the risk-neutral dynamics L risk − neutralt , preserving the double subordinated structure of L t . Mandelbrot and Taylor (1967) were the first to apply a subordinated Brownian motion tomodeling asset returns. In their model, the log-price process is modeled by L t = L + µ t + ρ T ( t ) + σ B T ( t ) , t ≥ , µ ∈ R , ρ ∈ R , σ > , (7)where L´evy subordinator T ( t ) , t ≥ α T -stable subordinator (see Samorodnitsky andTaqqu, 1994, Proposition 1.3.1) for α T ∈ ( , ) independent of the Brownian motion B t , t ≥
0. The unit increment of T ( t ) , t ≥ The corresponding WPF, w (R , S) : [ , ] → [ , ] , is defined by w (R , S) ( u ) = F S (cid:16) F inv R ( u ) (cid:17) where F inv R ( u ) = min { x : F R ( x ) > u } is the inverse function of F R ( x ) . F S ( x ) = w (R , S) ( F R ( x )) , with F R ( x ) = u , x = F invR ( u ) ,and from, F S ( x ) = w (R , S) ( F R ( x )) , we have F S (cid:0) F invR ( u ) (cid:1) = w (R , S) ( u ) . See change of measure theorem for L´evy processes in Chapter 6 in Sato and Katok (1999) and Chapter3 in Jacod and Shiryaev (2005). T ( ) ( s ) = E e − sT ( ) = exp (cid:16) −( δ T s ) α T (cid:17) , s > , δ T > , (8)where parameter δ T > α T is the tail index . The tail-probabilityfunction S T ( ) ( x ) = P ( T ( ) > x ) , x ≥ of order (cid:0) − α T (cid:1) > − S T ( ) ∈ T I (cid:0) α T (cid:1) . Theexplicit form of all moments – E ( T ( ) p ) < ∞ , < p < α T – is given in Samorodnitskyand Taqqu (1994, p.18). Tagliani and Vel´asques (2004) provide a numerical procedure toapproximate (in total variation distance) the density f T ( ) ( x ) , x > E ( T ( ) p j ) = p j Γ ( − p j ) ∫ ∞ − L T ( ) ( s ) s p j + ds , < p j < α T , j = , . . . , J , (9)are given. In other words, if the sample moments of order p j ∈ (cid:0) , α T (cid:1) , j = , . . . , J areavailable and J is sufficiently large, we can approximate the probability density function(pdf) f T ( ) in L -distance (total variation distance).The subordinated Brownian motion, denoted by B T ( t ) , is a α T -stable motion with unitincrement with B T ( ) having characteristic function (Ch.f.) given by ϕ B T ( ) ( u ) = E exp (cid:8) iuB T ( ) (cid:9) = exp (cid:32) − (cid:18) δ T (cid:19) α T u α T (cid:33) . (10)That is, B T ( t ) , t ≥ α T - stable motion with scale parameter (cid:16) δ T (cid:17) α T , and thus, S | B T ( t ) | ( x ) = P (cid:0) | B T ( t ) | > x (cid:1) , x ≥ RV ( α T ) . Consider now the price process model (1) with two stablesubordinators T ( t ) , t ≥ U ( t ) , t ≥ L U ( ) ( s ) = E e − sU ( ) = exp (cid:16) −( δ U s ) α U (cid:17) , s > , δ U > . (11) See Samorodnitsky and Taqqu (1994) for more information about stable random variables and stableprocesses that we use in this paper. Recall that a function f : ( , ∞) → ( , ∞) is called regularly varying (at infinity) of order r ∈ R , denoted f ∈ RV ( α ) , if f ( x ) = x r L ( x ) , where L : ( , ∞) → ( , ∞) is a slowly varying function (at infinity); thatis, lim x ↑∞ f ( bx ) f ( x ) = b >
0. For f ∈ RV ( r ) , we call (− r ) the tail index of f , and denote it by f ∈ T I (− r ) . The proof is provided in Appendix A.1 in the supplementary material. V ( t ) = T ( U ( t )) , t ≥ V ( ) with Laplacetransform L V ( ) ( s ) = ∫ ∞ exp (cid:16) − u ( δ T s ) α T (cid:17) f U ( ) ( u ) du . (12)The only one known explicit form for f U ( ) is when U ( t ) , t ≥ b U >
0. The pdf of U ( ) is given by f U ( ) ( x ) = (cid:114) b U π x − exp (cid:18) − b U x (cid:19) , x > . (13)In this case, the Laplace transform of V ( ) has the following representation L V ( ) ( s ) = exp (cid:16) − (cid:112) b u ( δ T s ) α T (cid:17) . (14)That is, V ( ) is α T -stable subordinator. Therefore, the subordinated Brownian motion B V ( t ) , t ≥ α T -stable motion. We refer to the L´evy subordinator V ( t ) = T ( U ( t )) , t ≥ L V ( ) given by (14) as the double-stable subordinator . We shall call L ( t ) , t ≥ normal-double-stable log-price process . The subordinated Brownianmotion B V ( t ) , t ≥ α T -stable motion.Now let’s look at the distribution of the normal-compound-stable log-price process L t = lnS t , t ≥ L t = L + µ t + γ U ( t ) + ρ T ( U ( t )) + σ B T ( U ( t )) , t ≥ , µ ∈ R , γ ∈ R , ρ ∈ R , σ > . The triplet ( B s , T ( s ) , U ( s ) , s ≥ ) , U ( ) ∼ L´evy-stable ( b U ) , T ( ) ∼ L´evy-stable ( b T ) areindependent processes generating stochastic basis ( Ω , F , F = (F t , t ≥ ) , P ) representingthe natural world. B s , s ≥ T ( s ) , U ( s ) , s ≥ We shall often have the probability distributions of U ( ) , T ( ) , V ( ) , B T ( t ) , B V ( t ) , and L in closed formin terms of their characteristic functions, Laplace transforms, or moment-generating functions. We will notdiscuss particular estimation procedures. The estimation procedures are well studied in the literature thatdeals with estimating distributional parameters, probability density function and cumulative distributions.The probability density function is recovered by using characteristic functions, Laplace transforms, andmoment-generating functions. See, for example, Abate and Whitt (1999), Glasserman and Liu (2010),Tsionas (2012), Mnatsakanov and Sarkisian (2013), Carrasco and Kotchoni (2017), and Kateregga et al.(2017). T ( ) = , U ( ) = ) are L´evy subordinators. DenoteΛ: = L − L = µ + γ U ( ) + ρ V ( ) + σ B V ( ) . (15)The pdf of Λ is given by f Λ ( x ) = ρ √ b T b U σ ( π ) ∫ ∞ e ( x − µ − γ u ) ρσ − bU u K (cid:18) ρσ (cid:113) ( x − µ − γ u ) + b T σ u (cid:19) √ u (cid:113) ( x − µ − γ u ) + b T σ u du , (16)where K n ( x ) is the modified Bessel function of the second kind. The Ch.f. ϕ Λ ( v ) = E e i v Λ , v > ϕ Λ ( v ) = E e i v Λ = e i v µ exp − (cid:118)(cid:117)(cid:117)(cid:116) − b U (cid:169)(cid:173)(cid:171) i v γ − (cid:115) − b T (cid:18) i v ρ − v σ (cid:19)(cid:170)(cid:174)(cid:172) . (17)We call Λ-distribution the normal-compound-L´evy-stable distribution , and L ( t ) , t ≥ normal-compound-stable log-price process. We note that the moments of Λ are undefined.Here is an example of multiple L´evy stable subordinations. Let U ( n ) ( t ) , t ≥ b n >
0; that is, U ( n ) ( ) ∼ L´evy-stable ( b n ) . Set V ( ) ( t ) = U ( ) ( t ) and V ( n ) ( t ) = U ( n ) (cid:16) V ( n − ) ( t ) (cid:17) for n = , , . . . Then, the Laplace exponentof V ( n ) ( t ) is given by Φ V ( n ) ( s ) = s − n n (cid:214) k = ( b k ) − k , s > , n ∈ N = { , , . . . . } . (18)Letting n ↑ ∞ and assuming that sup n ∈N (cid:206) nk = ( b k ) − k < ∞ , the distribution of V ( n ) ( ) degenerates as the distributional mass of V ( n ) ( ) escapes to infinity as n ↑ ∞ . Asthe tail-probability function S V ( n ) ( ) ∈ T I ( − n ) , then the random variable ξ ( n ,β ) = V ( n ) ( ) − n β , β > ( a ) in the domain of attraction of β -stable random variable if β <
2, and ( b ) The proof is provided in Appendix A.2 in the supplementary material. The proof is provided in Appendix A.3 in the supplementary material. The distributional tail of V ( n ) ( ) becomes heavier and heavier as n ↑ ∞ . In the limit, ifsup n ∈N (cid:206) nk = ( b k ) − k < ∞ then lim n ↑∞ L V ( n ) ( ) ( s ) = lim n ↑∞ exp (cid:16) − s − n (cid:206) nk = ( b k ) − k (cid:17) = exp (cid:0) − s (cid:1) = exp (− ) .
9n the domain of attraction of a normal law if β ≥ Next we define a continuous version of multiple L´evy stable subordinators. Let b n = B , n ∈ N = { , , . . . . } . We now extend (18) as follows: for every τ ≥
0, define V ( τ ) ( t ) , t ≥ Φ V ( τ ) ( s ) = s − τ ( B ) − − τ , s > . (19)Thus, V ( ) ( t ) = t , and for every τ > V ( τ ) ( t ) , t ≥ α -stable subordinator withstable index α = − τ . We call V ( τ ) ( t ) , t ≥ τ -compounded L´evy-stable subordinator withscale-intensity B >
0. Thus, every α -stable subordinator is a τ -compounded L´evy-stablesubordinator with τ = − ln ( α ) ln , α ∈ ( , ) .Consider next a log-price process L ( n ) t = lnS t , t ≥ , n = , , .. of the form L ( n ) t = L ( n ) + µ t + n (cid:213) k = γ k V ( k ) ( t ) + σ B T ( V ( n ) ( t )) , t ≥ , (20)where µ ∈ R , γ k ∈ R , k = , , .., and σ >
0. DenoteΛ ( n ) : = L ( n ) − L ( n ) = µ + n (cid:213) k = γ k V ( n ) ( ) + σ B T ( V ( n ) ( ) ) . Then, the Ch.f. of Λ ( n ) , n = , , . . . is given by ϕ Λ ( n ) ( v ) = E e iv Λ ( n ) = exp i v µ − (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) − b (cid:169)(cid:173)(cid:171) i v γ − (cid:118)(cid:117)(cid:116) − b (cid:32) . . . (cid:115) − b n − (cid:18) i v γ n − − (cid:113) − b n (cid:0) i v γ n − v σ (cid:1)(cid:19) . . . (cid:33)(cid:170)(cid:174)(cid:172) . (21) We call L ( n ) t , t ≥ normal-compound(n)-stablelog-price process . Suppose µ =
0, then the characteristic exponent of Λ ( n ) has the following Since the tail-probability function S V ( n ) ( ) ( x ) is RV ( − n ) , then ξ ( n ,β ) = V ( n ) ( ) − n β , β > β -stable random variable if β <
2, and in the domain of attraction of a normallaw if β ≥ P (cid:16) V ( n ) ( ) > x (cid:17) = x − − n L ( x ) , and P (cid:16) ξ ( n ,β ) = V ( n ) ( ) − n β > x (cid:17) = P (cid:18)(cid:16) V ( n ) ( ) − n β − (cid:17) n > x n (cid:19) = P (cid:18)(cid:16) V ( n ) (cid:17) β − > x n (cid:19) = P (cid:16) V ( n ) > x n β (cid:17) = (cid:0) x n β (cid:1) − − n L (cid:0) x n β (cid:1) = x − β L ( x ) . The proof is provided in Appendix A.4 in the supplementary material. The proof is provided in Appendix A.5 in the supplementary material. Λ ( n ) ( v ) = − ln ϕ Λ ( n ) ( v ) = (cid:113) − b n (cid:0) i v γ n − Ψ Λ ( n − ) ( v ) (cid:1) . Here we consider the case when the L´evy subordinators T ( t ) , t ≥ U ( t ) , t ≥ T ( ) ∼ Gamma ( α T , λ T ) , α T > λ T > f T ( ) ( x ) = λ α T T Γ ( α T ) x α T − e − λ T x , x ≥ , (22)and moment-generating function (MGF) M T ( ) ( v ) = (cid:18) − v λ T (cid:19) − α T , v < λ T , (23)and U ( ) ∼ Gamma ( α U , λ U ) . We refer to V ( t ) = T ( U ( t )) , t ≥ double-gammasubordinator .The pdf and MGF of V ( ) = T ( U ( )) are given by f V ( ) ( x ) = e − λ T x λ α U U Γ ( α U ) ∫ ∞ λ α T uT Γ ( α T u ) x α T u − u α U − e − λ U u du , (24)and M V ( ) ( v ) = (cid:18) + α T λ U ln (cid:18) − v λ T (cid:19) (cid:19) − α U , (25)for 0 < v < λ T (cid:16) − exp (cid:16) λ U α T (cid:17) (cid:17) . Thus, V ( ) has a finite exponential moment E e v V ( ) , forevery v ∈ (cid:16) , λ T (cid:16) − exp (cid:16) λ U α T (cid:17) (cid:17) (cid:17) . From the representation of the MGF, we determined allfour moments of V ( ) . The mean of V ( ) is given by E ( V ( )) = α T λ T α U λ U = E ( T ( )) E ( U ( )) . (26)For the variance of V ( ) we have v ar ( V ( )) = α T λ T α U λ U ( α T + λ U ) = v ar ( T ( )) v ar ( U ( )) ( α T + λ U ) , (27) See Schoutens (2003) and Applebaum (2009). Here, ∼ stands for equal in distribution between two random variables or two stochastic processes. The proof is provided in Appendix A.6 in the supplementary material. V ( ) is Sk e w ness [ V ( )] = E [ V ( )− E V ( )] [ v ar ( V ( )] = √ α U + λ U α T + (cid:16) λ U α T (cid:17) (cid:16) + λ U α T (cid:17) ≥ √ α U = Sk e w ness [ U ( )] . (28)The equality is reached for λ U α T ↓ implies λ U ↓
0. Finally, the excess kurtosis of T ( U ( )) is given by E xcessKurtosis ( V ( )) = E [ V ( )− E V ( )] [ v ar ( V ( )] − = α U + λ U α T + (cid:16) λ U α T (cid:17) + (cid:16) λ U α T (cid:17) (cid:16) + λ U α T (cid:17) ≥ α U = E xcessKurtosis ( U ( )) , (29)and the equality is reached for λ U α T ↓ variance-double-gamma process L t = L + µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) , t ≥ . (30)Because L t , t ≥ = L − L = µ + γ U ( ) + ρ V () + σ B V ( ) . We shall call Λ-distribution the variance-gamma-gamma distribution . The pdf of Λ is given by f Λ ( x ) = √ π λ α UU Γ ( α U ) ∫ ∞ (cid:18)∫ ∞ e − ( x − µ − γ u − ρ y ) σ y y α Tu − e − λ T y d y (cid:19) λ α T uT Γ ( α Tu ) u α U − e − λ U u du . (31)The expression for the pdf f Λ ( x ) , x ∈ R is computationally intractable in view of the twointegrals in the formula. We prefer to work with the MGF, M Λ ( v ) = E e v Λ , v > M Λ ( v ) = exp (cid:110) µ v − α U ln (cid:104) − γλ U v + α T λ U ln (cid:16) − ρλ T v − σ λ T v (cid:17) (cid:105) (cid:111) . (32)In (32) we require that 0 < v < √ ρ + λ T σ − ρσ and λ U + α T ln (cid:16) − λ T (cid:0) v ρ + v σ (cid:1) (cid:17) − v γ >
0, which should be fulfilled when v ↓ . Given the representation (32) we determinethe four moments of Λ . For the mean of Λ , we have the following representation E Λ = µ + α U λ U γ + α T λ T α U λ U ρ. (33) If together with λ U α T ↓
0, we also require that v ar ( V ( )) < ∞ , then λ U α T ↓ The proof is provided in Appendix A.7 in the supplementary material. v ar ( Λ ) = α U λ U (cid:18) α T λ T ρ + γ (cid:19) + α U λ U α T λ T (cid:18) σ + ρ λ T (cid:19) . (34)The skewness of Λ is Sk e w ness [ Λ ] = ( α T ρ + λ T γ ) { ( α T ρ + λ T γ ) + λ U α T λ T ( ρ + σ λ T )} + λ U α T ρ ( ρ + σ λ T ) √ α U ( ( α T ρ + λ T γ ) + α T λ U ( λ T σ + ρ )) . (35)For the excess kurtosis of Λ , we have ( E xcessKurtosis ( Λ )) = E [ Λ − E Λ ] [ v ar ( Λ )] − = (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) ( α T ρ + γλ T ) + λ U α T (cid:0) ρ + σ λ T (cid:1) ( α T ρ + γλ T ) ++ α T λ U ρ (cid:0) ρ + σ λ T (cid:1) ( α T ρ + γλ T ) ++ λ U α T (cid:0) ρ + σ λ T (cid:1) + λ U α T λ T (cid:0) ρ + σ (cid:1) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) α U [ ( α T ρ + γλ T ) + λ U α T ( σ λ T + ρ )] . (36)Note that from (36) it follows that the excess kurtosis of Λ can be negative if ρ and γ haveopposite signs. In this case the distribution of Λ can become platykurtic. However, in thecase where ρ and γ have the same sign, then the distribution of Λ is leptokurtic.Let us consider now the case of a compound subordination with multiple subordina-tors. Let U ( i ) ( t ) , t ≥ i = , . . . , n , n ∈ N = { , , . . . } be a sequence of indepen-dent gamma subordinators with U ( i ) ( ) ∼ Gamma ( α i , λ i ) , and define V ( ) ( t ) = U ( ) ( t ) , V ( i + ) ( t ) = V ( i ) (cid:16) U ( i + ) ( t ) (cid:17) for i = , , . . . , n −
1. We shall use the notation V ( n ) ( t ) = U ( ) ◦ U ( ) ◦ · · · ◦ U ( n ) ( t ) , t ≥ . Iteratively, we obtain the following representation forthe MGF of V ( n ) ( ) , n ∈ N : M V ( n ) ( ) ( v ) = (cid:16) + α n − λ n ln (cid:16) + α n − λ n − ln . . . ln (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) (cid:17) (cid:17) − α n , (37)where 0 < v < τ n < τ n − , and τ n : = λ (cid:18) − exp (cid:18) − λ α (cid:18) . . . (cid:18) − exp (cid:18) − λ n − α n − (cid:18) − ex p (cid:18) − λ n α n − (cid:19) (cid:19) (cid:19) (cid:19) ... (cid:19) (cid:19) (cid:19) . Note that for n = , , . . . we have the recursive formula: M V ( n ) ( ) ( v ) = (cid:18) − λ n lnM V ( n − ) ( ) ( v ) (cid:19) − α n , < v < τ n . (38) The proof is provided in Appendix A.8 in the supplementary material. V ( n ) ( ) (as n ↑ ∞) will either concentrate in 0, as τ n ↓
0, or will escape to infinity, de-pending on the choice of ( α n , λ n ) as n ↑ ∞ . There is no central limit theorem-type resultsfor V ( n ) ( ) , n ↑ ∞ , as there is no linear transformation of V ( n ) ( ) leading to a proper dis-tribution as a weak limit. It requires power-transformation of V ( n ) ( ) to obtain non-trivialweak limits. However, those types of limiting results, while of potential academic interest,are beyond the scope of this paper.Define the moment-generating exponent of V ( n ) ( t ) , t ≥ V ( n ) ( ) , K V ( n ) ( v ) = lnM V ( n ) ( ) ( v ) = − α n ln (cid:16) − λ n K V ( n − ) ( v ) (cid:17) . Then the cumu-lants κ j , n , j ∈ N , of V ( n ) ( ) are given by κ j , n = (cid:104) ∂ j ∂ v j K V ( n ) ( v ) (cid:105) v = , and E V ( n ) ( ) = κ , n , v ar (cid:16) V ( n ) ( ) (cid:17) = κ , n , Sk e w ness (cid:2) V ( n ) ( ) (cid:3) = E [ V ( n ) ( )− E V ( n ) ( ) ] [ v ar ( V ( n ) ( ) ] = κ , n ( κ , n ) , and E xcessKurtosis ( V ( )) = E [ V ( )− E V ( )] [ v ar ( V ( )] − = κ n , n ( κ , n ) .The following recursive formulas for κ j , n , j = , ,
3, and 4 hold: κ , n = α n λ n κ , n − , κ , n = α n λ n (cid:18) λ n κ , n + κ , n (cid:19) ,κ , n = α n λ n (cid:32) α n κ , n − λ n + α n κ , n − κ , n − λ n + κ , n − (cid:33) ,κ , n = α n λ n (cid:32) κ , n − λ n + κ , n − κ , n − λ n + κ , n − + κ , n − κ , n − λ n + κ , n − (cid:33) , n ≥ κ , = α λ , κ , = α λ , κ , = α λ , κ , = α λ .Let U ( i ) ( t ) , t ≥ i = , . . . , n , n ∈ N = { , , . . . } be a sequence of independentgamma subordinators with U ( i ) ( ) ∼ Gamma ( α i , λ i ) , and define V ( ) ( t ) = U ( ) ( t ) , V ( i + ) ( t ) = V ( i ) (cid:16) U ( i + ) ( t ) (cid:17) for i = , , . . . , n −
1. Consider next a log-price process L ( n ) t = lnS t , t ≥ , n = , , .. of the form L ( n ) t = L ( n ) + µ t + n (cid:213) k = γ k ˜ V ( k ) ( t ) + σ B V ( n ) ( t ) , t ≥ , (39)where µ ∈ R , γ k ∈ R , σ >
0, ˜ V ( k ) ( t ) = U ( k ) (cid:16) U ( k − ) (cid:16) . . . (cid:16) U ( ) ( t ) (cid:17) . . . (cid:17) (cid:17) , k = , . . . n , n ∈ N ,and U ( i ) ( t ) , t ≥ , i = , . . . , n is a sequence of independent gamma subordinators with14 ( i ) ( ) ∼ Gamma ( α i , λ i ) . Denote Λ ( n ) : = L ( n ) − L ( n ) = µ + (cid:205) nk = γ k V ( k ) ( ) + σ B T ( V ( n ) ( ) ) . Then,the Ch.f. of Λ ( n ) , n = , , . . . is given by ϕ Λ ( n ) ( v ) = e i v µ (cid:16) − i v γ λ + α λ ln (cid:16) − · · · − i v γ n − λ n − + α n λ n − ln (cid:16) − i v γ n λ n + v σ λ n (cid:17) . . . (cid:17) (cid:17) − α , v ∈ R . (40) Here we consider the case when the subordinators T ( t ) , t ≥ U ( t ) , t ≥ T ( ) ∼ IG ( λ T , µ T ) , λ T >
0, and µ T > f T ( ) ( x ) = (cid:114) λ T π x exp (cid:32) − λ T ( x − µ T ) µ T x (cid:33) , x ≥ , (41)and U ( ) ∼ IG ( λ U , µ U ) . We refer to V ( t ) = T ( U ( t )) , t ≥ double-inverse Gaussiansubordinator . We shall also consider the following two particular cases leading to single IGsubordinators: ( i ) µ T =
1, and λ T ↑ ∞ , and thus T ( ) → L and ( ii ) µ U =
1, and λ U ↑ ∞ , and thus, U ( ) → L .The compound subordinator V ( t ) = T ( U ( t ) , t ≥ f V ( ) ( x ) f V ( ) ( x ) = π (cid:114) λ T λ U x ∫ ∞ u − exp (cid:32) − λ T ( x − µ T u ) µ T x − λ U ( u − µ T ) µ U u (cid:33) du , x > . (42)The MGF, M V ( ) ( v ) = E e v T ( U ( )) , v > v ∈ (cid:18) , λ T µ T (cid:20) − (cid:16) − λ U µ T µ U λ T (cid:17) (cid:21) (cid:19) is M V ( ) ( v ) = exp (cid:169)(cid:173)(cid:173)(cid:171) λ U µ U (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ U λ U λ T µ T (cid:169)(cid:173)(cid:171) − (cid:115) − µ T λ T v (cid:170)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172) , (43)and the Ch.f of V ( ) is ϕ V ( ) ( v ) = exp (cid:169)(cid:173)(cid:173)(cid:171) λ U µ U (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ U λ U λ T µ T (cid:169)(cid:173)(cid:171) − (cid:115) − µ T λ T v i (cid:170)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172) . (44) The proof is provided in Appendix A.9 in the supplementary material. A sequence { f n } of periodic, square-integrable functions is said to converge in L to a function f if thesequence of numbers ∫ | f n ( x ) − f ( x )| dx converges to 0. The proof is provided in Appendix A.10 in the supplementary material. The proof is provided in Appendix A.10 in the supplementary material.
15o find the four central moments of V ( ) , we use the cumulant-generating functionK V ( ) ( v ) = ln M V ( ) ( v ) , and the cumulants κ n = (cid:2) ∂ n ∂ u n K V ( ) ( u ) (cid:3) u = , n = , , , V ( ) we have E V ( ) = κ = µ T µ U = E T ( ) E U ( ) , and V ar ( V ( )) = κ = µ U µ T λ U + µ T µ U λ T . As V ar ( T ( )) = µ T λ T , and V ar ( U ( )) = µ U λ U , we have V ar ( V ( )) = ( V ar ( U ( ))) ( E ( T ( )) + ( V ar ( T ( )) E ( U ( )) . Therefore, if E ( T ( )) =
1, then v ar ( V ( )) > v ar ( U ( )) , and if E ( U ( ) =
1, then v ar ( V ( )) > v ar ( T ( )) . If E ( T ( )) = µ T = λ T ↑ ∞ , then T ( ) → L -sense, and v ar ( V ( )) → µ U λ U = v ar ( U ( )) . Similarly, if E ( U ( )) = µ U = λ U ↑ ∞ , then U ( ) → L -senseand V ar ( V ( )) → = µ T λ T = V ar ( T ( )) .The skewness of V ( ) is given by Sk e w ness [ V ( )] = κ κ − = µ U λ U + µ U µ T λ U λ T + µ T λ T µ U (cid:16) µ U λ U + µ T λ T (cid:17) . (45)If E ( T ( )) = µ T = λ T ↑ ∞ , then T ( ) → L -sense, and furthermore, Sk e w ness [ V ( )] → (cid:113) µ U λ U = Sk e w ness [ U ( )] . Similarly, if E ( U ( )) = µ U =
1, and λ U ↑ ∞ ,then Sk e w ness [ V ( )] → (cid:113) µ T λ T = Sk e w ness [ T ( )] .Consider next the case when E T ( ) = E U ( ) = V arT ( ) = V arU (( ) =
1. Then
Sk e w ness [ T ( )] = Sk e w ness [ U ( )] =
3, while
Sk e w ness [ V ( )] = = . . . . .For the excess kurtosis of V ( ) , we have the following expression E xcessKurtosis ( V ( )) = κ κ = (cid:34) (cid:18) µ T λ U (cid:19) + (cid:18) µ U µ T λ U λ T (cid:19) + (cid:18) µ U µ T λ U λ T (cid:19) + (cid:18) µ T λ T (cid:19)(cid:35) µ T (cid:18) µ U λ U + µ T λ T (cid:19) . (46)If E ( T ( )) = µ T = λ T ↑ ∞ , then E xcessKurtosis ( V ( )) → µ U λ U = E xcessKurtosis ( U ( )) . E ( U ( )) = µ U =
1, and λ U ↑ ∞ , then E xcessKurtosis ( V ( )) → µ T λ T = E xcessKurtosis ( U ( )) . Now consider the case when E T ( ) = E U ( ) = V arT ( ) = V arU (( ) =
1. Then
E xcessKurtosis [ T ( )] = E xcessKurtosis [ U ( )] =
15, while
E xcessKurtosis [ V ( )] = . . Now let’s study the distribution of the normal compound inverse Gaussian log-priceprocess , L t = lnS t , t ≥
0, given by L t = L + µ t + γ U ( t ) + ρ T ( U ( t )) + σ B T ( U ( t )) , t ≥ , µ ∈ R , γ ∈ R , ρ ∈ R , σ > , where the triplet ( B s , T ( s ) , U ( s ) , s ≥ ) , U ( ) ∼ L´evy-inverse Gaussian ( µ U , λ U ) , T ( ) ∼ L´evy-inverse Gaussian ( µ T , λ T ) are independent processes generating stochastic basis ( Ω , F , F = (F t , t ≥ ) , P ) representing the natural world. B s , s ≥ T ( s ) , U ( s ) , s ≥ ( T ( ) = , U ( ) = ) are L´evy subordinators. DenoteΛ: = L − L = µ + γ U ( ) + ρ V ( ) + σ B V ( ) . The pdf of Λ is given by f Λ ( x ) = π √ λ T λ U ∫ ∞ ∫ ∞ t exp (cid:16) − x − µ − γ u − ρ t σ √ t − λ T ( t − u µ T ) ut µ T − λ U ( u − µ U ) u µ U (cid:17) , (47)and for the Ch.f., ϕ Λ ( v ) = E e i v Λ , we have the following expression ϕ Λ ( v ) = E e i v Λ = e i v µ + λ U µ U − (cid:118)(cid:116) − µ U λ U (cid:32) λ T µ T (cid:32) − (cid:114) − µ T λ T ( i v ρ − v σ ) (cid:33) + i v γ (cid:33) . (48)The MGF, M Λ ( u ) , is obtained by setting u = v i , and thus is omitted.Having the representation given by (48), we can determine the mean and the varianceof Λ as follows E Λ = µ + µ U γ + µ U µ T ρ. (49)and for the variance of Λ, we have the following expression V ar ( Λ ) = µ U ρ µ T λ T + σ µ U µ T + µ U ( γ + ρµ T ) λ U . (50)Finally, the skewness of Λ is given by Sk e w ness [ Λ ] = ρ µ T λ T + ρσ µ T λ T + µ T ( γ + µ T ) λ U + µ U λ U (cid:16) ρ µ T λ T + σ µ T (cid:17) ( γ + ρµ T ) (cid:16) µ U (cid:16) σ µ T + ρ µ T λ t + µ U λ U ( γ + ρµ T ) (cid:17) (cid:17) . (51) The proof is provided in Appendix A.11 in the supplementary material.
17e now consider the case of compound subordination with multiple subordinators. Let U ( i ) ( t ) , t ≥ i = , . . . , n , n ∈ N = { , , . . . } be a sequence of independent IG subordinatorswith U ( i ) ( ) ∼ IG ( µ i , λ i ) , and define V ( ) ( t ) = U ( ) ( t ) , V ( i + ) ( t ) = V ( i ) (cid:16) U ( i + ) ( t ) (cid:17) for i = , , . . . , n − . . We shall use the notation V ( n ) ( t ) = U ( ) ◦ U ( ) ◦ · · · ◦ U ( n ) ( t ) , t ≥ . Iteratively,we obtain the following representation for the Ch.f of V ( n ) ( ) , n ∈ N : ϕ V ( n ) ( ) ( v ) = e λ n µ n (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − µ n λ n − λ n µ n − (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ n − λ n − λ n − µ n − (cid:169)(cid:173)(cid:171) ..... (cid:118)(cid:116) − µ λ λ µ (cid:32) − (cid:114) − µ λ i v (cid:33)(cid:170)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:174)(cid:172) . (52)Note that for n = , , ... , we have the following recursive formula ϕ V ( n ) ( ) ( v ) = e λ n µ n (cid:32) − (cid:114) − µ n λ n ln ϕ V ( n − )( ) ( v ) (cid:33) . (53)Next consider a log-price process L ( n ) t = LnS t of the form (39) and again denote Λ ( n ) : = L ( n ) − L ( n ) = µ + n (cid:213) k = γ k V ( k ) ( ) + σ B T ( V ( n ) ( ) ) . Then the chf of Λ ( n ) , n = , , ... , will be obtained iteratively by the following representation: ϕ Λ ( n ) ( v ) = e − λ n µ n − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − µ n λ n − λ n µ n − (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ n − λ n − λ n − µ n − (cid:169)(cid:173)(cid:171) − ... (cid:118)(cid:116) − µ λ λ µ (cid:32) − (cid:114) − µ λ ( iv γ − v σ ) + iv γ (cid:33)(cid:170)(cid:174)(cid:172) ... (cid:170)(cid:174)(cid:174)(cid:172) + iv γ n + iva . (54) In this section, we apply the models we proposed in this paper to estimate the returns of abroad-based market index, the S&P 500 as measured by SPR S&P 500 which is an exchange-traded index. We use market indices by the triplet ( L t , T ( U ( t )) , U ( t )) , t ≥ ( i ) L t , t ≥ ( ii ) V ( t ) , t ≥ V ( t ) represents the cumulative value of VIX in [ , t ] ) (CBOE volatility index),and ( iii ) U ( t ) t ≥
0, as the cumulative VVIX (CBOE volatility of volatility index ) (i.e., U ( t ) , t ≥ [ , t ] ). The subordinator processes T ( t ) , t ≥ U ( t ) , t ≥ T ( ) ∼ IG ( λ T , µ T ) , λ T >
0, and µ T > The proof is provided in Appendix A.11 in the supplementary material.
18n the log-return model, Λ t = µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) , conditional on U ( t ) and V ( t ) ,the variance of Λ t is V ( t ) . Therefore, the conditional volatility of Λ t is (cid:112) V ( t ) . Since theVIX index is a measure of the stock market’s volatility, in modeling the variance of Λ t , V ( t ) , we use the squared value of the VIX index (VIX ).Similarly, in the V ( t ) log-return model conditional on U ( t ) , the variance of V ( t ) is U ( t ) ,and thus, conditional volatility is (cid:112) U ( t ) . Since the VVIX index measures the volatility ofthe price of the VIX index, to model the variance of V ( t ) , we apply the squared value ofVVIX index (VVIX ) as a representation of the variance.We then proceed as follows. First, we fit IG distribution to daily VVIX data andcompare the fitted density by the empirical kernel density. The kernel density estimatorˆ f n ( x ) , for estimating the density of f ( x ) at point x is defined asˆ f n ( x ) = nh n (cid:213) i = k (cid:16) x i − xn (cid:17) , (55)where k ( x ) = √ π e − x is the Gaussian kernel (see Epanechnikov, 1969). The mean andshape parameters of IG fitted on daily VVIX index data over the period from January2007 until the end of March 2019 using maximum likelihood methods are summarized inTable 1.Figure 1 shows the fitted IG distribution, corresponding to the empirical density for thedaily VVIX data. Our estimated model gives a good fit between the pdf and the empiricaldensity of the data. The Kolmogorov–Smirnov test for goodness of fit testing verifies thatthe fitted IG is a good fit. The Kolmogorov–Smirnov value for the p–value ((cid:39) ) fails toreject the null hypothesis that the IG distributions are sufficient to describe the data.In testing the double subordinated model, we view { U ( t ) , t ≥ } as the stock-volatilityintrinsic time (or volatility subordinator) and { V ( t ) = T ( U ( t )) , t ≥ } as stock intrinsic time.In our model, { V ( t ) = T ( U ( t )) , } is the variance of log-return process that is subordinatedby IG volatility subordinator. Thus, { V ( t ) = T ( U ( t ))} is a compound IG distribution (CIG)with four parameters. To estimate the model parameters, we fit the CIG distribution todaily VIX index data. The method of modeling fitting via the empirical characteristicfunction (ECF) is applied to estimate the model parameters because of the difficulty inmaximizing the likelihood function. We match the characteristic function derived from the19IG distribution to the ECF obtained from the daily VIX data. The ECF procedure wasfirst investigated by Paulson et al. (1975) and recently by Yu (2003). There is a one-to-one correspondence between the cumulative distribution function and the Ch.f because thepdf is the fast Fourier Transform (FFT) of the Ch.f. Therefore, inference and estimationthrough the ECF are as efficient as the likelihood methods (see Yu, 2003). To estimate themodel parameter, we minimized h ( r , x , θ ) = ∫ ∞−∞ (cid:32) n n (cid:213) i = e i θ x i − C ( r , θ ) (cid:33) dr (56)where C ( r , θ ) is the Ch.f of V ( t ) given by (44) . The daily VIX index data covering theperiod from January 1993 until the end of March 2019 consist of 6591 observations thatwe use to estimate the model’s parameters.Because the CIG distribution has four parameters, the optimization method is moresensitive to the input of initial values and can simply fail to converge or converge to a localoptimum. Here, the computational cost of estimating the four parameters model is high.The initial values are obtained from the method of moments estimation and additionally viainstructed guesses. For any initial value we estimated the model parameters and considerthe model as a good candidate to fit the data.To answer which model is the best in capturing the features of the data between thecandidate density forecasts models, we first focus on the probability integral transforms(PIT) of the data in the evaluation of density models. Diebold et al. (1998) showed thata PIT time series should be independent and identically distributed (iid) uniform if thesequence of densities is correct. They proposed testing the specification of a density modelby testing whether or not the transformed series is iid and uniform ( , ) . After evaluationof the density model, we selected the best model which was the one where the likelihoodvalue is the largest.We implemented the FFT to calculate the pdf and then computed the correspondinglikelihood values. The estimated parameters of the best model are reported in Table 2.The p-values ((cid:39) ) of the Kolmogorov (1933) and Kuiper (1960) uniformity tests do notlead to rejecting the uniformity of the PIT. Plotted in Figure 3 is the CIG density withestimated parameters, corresponding to the empirical density of the daily VIX index. The20gure reveals that our estimated model creates a good match between the pdf and theempirical density of the data.As noted earlier, V ( t ) is subordinated by an IG volatility subordinator, U ( t ) . FromTable 2 it can be seen that this volatility subordinator exhibits an IG distribution withmean µ U = . λ U = .
6. Comparing these estimated parametersfor the fitted IG distribution to the estimated parameters for the VVIX index shown inTable 1, we see that there is a significant difference between the two models. This significantdifference in mean and shape parameters obtained for the models is an indication that theVVIX index cannot be a proper volatility subordinator for the VIX index. In Table 3, themean, variance, skewness, and excess kurtosis for the volatility subordinator model and theIG distribution fitted to the VVIX index are reported.In the case where we model the VIX index by using the VVIX index as the volatilitysubordinator, by comparing the skewness and excess kurtosis of the two models we cansee again there is a significant difference in skewness and kurtosis for both models. Thissuggests that using the VVIX index as a measure of time change cannot contain all the in-formation of stochastic volatility models; that is, the skewness and the fat-tail phenomenonof the VIX index are not properly captured by the VVIX index. Thus to have a propermodel for the VIX index, the VIX’s skewness and fat-tail phenomenon should be recoveredby specifying a different volatility subordinator.Next we investigate the distribution of Λ t = µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) as a stochasticmodel for the SPDR S&P 500 log-return index by fitting a normal compound inverseGaussian (NCIG) distribution to the data. Λ t is a stochastic process with eight parameters,four of the parameters of Λ t enter the model because of the intrinsic time change process.To estimate the parameters of the model, we use daily log-returns of the SPDR S&P 500index based on closing prices by implementing the Ch.f method. The database covers theperiod from January 1993 to March 2019 and includes 6591 observations collected fromYahoo Finance. As before, the optimization method is more sensitive to the input of initialvalues. The method of moments and instructed guess are used to obtain the initial values.We implemented the FFT to calculate the pdf and calculate the corresponding likelihoodvalues. The best model to fit and explain the observed data is chosen as the one with the21argest likelihood value. To evaluate the forecast density, we applied the PIT and inverse-normal-transform of the probability integral transform that should be iid standard normalas presented in Berkowitz (2001). In case of rejection of any tests, we changed the initialvalues and iterated the process. Finally, we calculated the likelihood value and selected thebest model by comparing their likelihoods. The estimated parameters of the best modelare summarized in Table 4. The p-values of the Kolmogorov–Smirnov (p-value =
1) andKuipers (p-value =
1) uniformity tests do not lead to rejecting the uniformity of PIT. Theperformed adjusted Jarque–Bera test (see Urzua, 1996) for the composite hypothesis ofnormality (p-value = . data exhibits a heavy tail in contrast to the subordinated model for the SPDR S&P500 log-return model. Also, we observe that the skewness of the CIG model fitted to VIX is more extreme than the time change subordinated model. Thus, it can be concluded thatthe VIX index is not a proper intrinsic time change for the SPDR S&P 500 index. Wesee that an index with a thinner tail and slight positive skewness than the VIX index canimprove the SPDR S&P 500 log-return model.Finally, we mention that for the SPDR S&P 500-log return model given by (15), thecoefficient of the volatility subordinator, γ , is zero. This finding suggests that the VVIXindex does not have too much influence directly in modeling the log-return of SPDR S&P500. This is because the VVIX is an indicator of the expected volatility of the VIX index,and VIX is not a proper time change subordinate in the model.22 Conclusion
In this paper, we generalized the classical asset pricing model by replacing physical timein the well-known return model with multiple stochastic intrinsic times subordinator. Thismodification to the return model takes into account tail effects, one of the stylized factsknown about stock return. We introduced the stock-volatility intrinsic time or volatilitysubordinator to the model to reflect the heavy-tail phenomena present in asset returns.This increased the number of parameters that are required to be estimated. The propertiesof the α -stable, gamma and inverse Gaussian multiple subordinator models are described.We defined the normal double stable, variance double gamma processes, and normal dou-ble inverse Gaussian processes for modeling asset returns. Our empirical results suggestthat the VIX and VVIX indexes are not the proper intrinsic time change and volatilitysubordinators for modeling the SPDR S&P 500 log-return, respectively. References
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Economet-ric Reviews 23 , 93–123. 25able 1: Estimated parameters of IG fitted to the daily VVIX data. µ U λ U Parameters λ U µ U λ T µ T Estimates 323.6 172.7 20.1 2.05Table 3: Mean, variance, skewness, and excess kurtosis of IG distribution, in VolatilitySubordinator model and IG model fitted to VVIX Model VVIX model Volatility Subordinator modelMean 8096.8 172.7Variance 5885600 15917Skewness 0.8989 2.1916Excess Kurtosis -1.6534 5.0053Table 4: The estimated parameters of NCIG distribution fitted to daily SPDR S&P 500log-returns λ U µ U λ T µ T µ γ ρ σ .Model CIG model fitted to VIX index SPDR time subordinator modelMean 354.03 0.2167Variance 66966 0.0025Skewness 2.191 0.6313Kurtosis 7.998 0.6707Figure 1: The IG fitted density via the kernel density of the daily VVIX data.27igure 2: The NCIG density of log-return SPDR S&P 500 via the kernel density.Figure 3: IG fitted CIG density via the kernel density of the daily VIX data.28 ppendix A.1: Characteristic function of α T -stable process If B T ( t ) is an α T -stable motion with unit increment B T ( ) , then the characteristic function(Ch.f.) of B T ( t ) is given by ϕ B T ( ) ( u ) = E T ( ) = v exp (cid:110) − v u (cid:111) = E exp (cid:110) − T ( ) u (cid:111) = L T ( ) (cid:16) u (cid:17) = exp (cid:18) − (cid:16) δ T (cid:17) α T u α T (cid:19) . A.2: Laplace exponent of double α T -stable subordinator The Laplace exponent of the compound subordinator V ( t ) = T ( U ( t )) , t ≥ U ( ) ∼ L´evy-stable ( b U ) and T ( ) ∼ L´evy-stable ( b T ) are independent processes, is given byΦ V ( s ) = − ln (cid:16) E U ( ) = u E e − sT ( u ) (cid:17) = − ln (cid:0) E U ( ) = u ( exp (− Φ T ( s )) ) u (cid:1) = − ln ( E exp (− U ( ) Φ T ( s ))) = Φ U ( Φ T ( s )) = ( δ U Φ T ( s )) α U = (cid:16) δ U ( δ T s ) α T (cid:17) α U = δ α U U ( δ T s ) α T α U . A.3: Ch.f. of normal-double-stable log-price process
Let L t be a normal-compound-stable log-price process L t = L + µ t + γ U ( t ) + ρ T ( U ( t )) + σ B T ( U ( t )) , t ≥ , µ ∈ R , γ ∈ R , ρ ∈ R , σ > , where the triplet ( B s , T ( s ) , U ( s ) , s ≥ ) , U ( ) ∼ L´evy-stable ( b U ) , T ( ) ∼ L´evy-stable ( b T ) are independent processes, and B s , s ≥ T ( s ) , U ( s ) , s ≥ ( T ( ) = , U ( ) = ) are L´evy subordinators, Denote Λ: = L − L = µ + γ U ( ) + ρ V ( ) + σ B V ( ) , then the Ch.f. of Λ is given by ϕ Λ ( v ) = e − Ψ Λ ( v ) = E e i v ( µ + γ U ( ) + ρ T ( U ( )) + σ B T ( U ( )) ) = e i v µ exp (cid:40) − (cid:115) − b U (cid:18) i v γ − (cid:113) − b T (cid:0) i v ρ − v σ (cid:1) (cid:19) (cid:41) . A.4: τ -compounded L´evy-stable subordinator Laplace exponent Let b n = B , n ∈ N = { , , . . . . } . Then for every τ > V ( τ ) ( t ) , t ≥ τ -compounded L´evy-stable subordinator with Laplace exponent denoted by Φ V ( τ ) ( s ) , s > . Therefore, for τ = n , the Laplace exponent of V ( n ) ( s ) is given byΦ V ( n ) ( s ) = s − n n (cid:214) k = ( b k ) − k = s − n n (cid:214) k = ( B ) − k = s − n ( B ) (cid:205) nk = − k = s − n ( B ) − − n . n ∈ N = { , , . . . } we find thatΦ V ( n ) ( s ) − Φ V ( n − ) ( s ) = s − n ex p (cid:32) n − (cid:213) k = ln (cid:16) ( b k ) − k (cid:17) (cid:33) (cid:16) ( b n ) − n − s − n − − − n (cid:17) . Thus, for every τ >
0, we haveΦ V ( τ + dt ) ( s ) − Φ V ( τ ) ( s ) = s − τ exp (cid:16)∫ τ − dt ln (cid:16) (cid:0) b y (cid:1) − y (cid:17) d y (cid:17) (cid:16) ( b τ ) − τ − − τ − dt − (cid:17) dt = s − τ exp (cid:16)∫ τ ln (cid:16) (cid:0) b y (cid:1) − y (cid:17) d y (cid:17) dt .Since Φ V ( τ ) ( s ) = Φ V ( n ) ( s ) Φ V ( n − ) ( s ) = s − n − − n − b n , a simple calculation shows thatΦ V ( n ) ( s ) − Φ V ( n − ) ( s ) Φ V ( n − ) ( s ) = s − n ( − ) b n − . Thus, we have Φ V ( τ + dt ) ( s )− Φ V ( τ ) ( s ) Φ V ( τ ) ( s ) = (cid:0) s − τ b τ − (cid:1) dt . Therefore, we find ∂∂τ ln Φ V ( τ ) ( s ) = (cid:0) s − τ b τ − (cid:1) and, ln Φ V ( τ ) ( s ) − ln Φ V ( ) ( s ) = ∫ τ s − − y b y d y .Setting V ( ) ( t ) = t , then Φ V ( ) ( s ) = − ln E e − sV ( ) ( ) = − ln E e − s = s .Finally, we find ln Φ V ( τ ) ( s ) = lns + ln (cid:16) ex p ∫ τ (cid:0) s − − y b y − (cid:1) d y (cid:17) = ln Φ V ( τ ) ( s ) = ln (cid:16) se − τ + ∫ τ s − − y b y d y (cid:17) ,Or, Φ V ( τ ) ( s ) = s − τ ( B ) − − τ , s > . A.5: Ch.f. of normal-compound(n)-stable log price process
Consider a log-price process L ( n ) t = lnS t , t ≥ , n = , , .. of the form L ( n ) t = L ( n ) + µ t + n (cid:213) k = γ k V ( k ) ( t ) + σ B T ( V ( n ) ( t )) , t ≥ , (57)where the V k ( ) ∼ L´evy-compound-stable, and T ( ) ∼ L´evy-stable ( b T ) are independentprocesses, and B s , s ≥ ( n ) : = L ( n ) − L ( n ) = µ + n (cid:213) k = γ k V ( n ) ( ) + σ B T ( V ( n ) ( ) ) . Then, the Ch.f. of Λ ( n ) , n = , , . . . is given by30 Λ ( n ) ( v ) = E e i v Λ ( n ) = exp i v µ − (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) − b (cid:169)(cid:173)(cid:171) i v γ − (cid:118)(cid:117)(cid:116) − b (cid:32) . . . (cid:115) − b n − (cid:18) i v γ n − − (cid:113) − b n (cid:0) i v γ n − v σ (cid:1)(cid:19) . . . (cid:33)(cid:170)(cid:174)(cid:172) . For simplicity, we consider when n =
3, we have:Λ ( ) : = L ( ) − L ( ) = µ + γ V ( ) ( ) + γ V ( ) ( ) + γ V ( ) ( ) + σ B T ( V ( ) ( ) ) = µ + γ U ( ) ( ) + γ U ( ) (cid:16) U ( ) ( ) (cid:17) + γ U ( ) (cid:16) U ( ) (cid:16) U ( ) ( ) (cid:17) (cid:17) + σ B T ( U ( ) ( U ( ) ( U ( ) ( ) ))) . Thus the Ch.f. of Λ ( ) is given by ϕ Λ ( ) ( v ) = E e i v Λ ( ) = e i v µ E U ( ) ( ) = u E exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i v (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) γ + γ U ( ) ( ) ++ γ U ( ) (cid:16) U ( ) ( ) (cid:17) ++ σ B T ( U ( ) ( U ( ) ( ) )) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) u From U ( k ) ( ) ∼ L ´e v y stable ( b k ) , k = , , . . . , and ϕ Λ ( v ) = E e i v Λ = e i v µ exp (cid:40) − (cid:115) − b U (cid:18) i v γ − (cid:113) − b T (cid:0) i v ρ − v σ (cid:1) (cid:19) (cid:41) withΛ: = L − L = µ + γ U ( ) + ρ T ( U ( )) + σ B T ( U ( )) , it follows that E exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i v (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) γ + γ U ( ) ( ) ++ γ U ( ) (cid:16) U ( ) ( ) (cid:17) ++ σ B T ( U ( ) ( U ( ) ( ) )) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) = e i v γ exp (cid:40) − (cid:115) − b (cid:18) i v γ − (cid:113) − b (cid:0) i v γ − v σ (cid:1) (cid:19) (cid:41) . Thus, ϕ Λ ( ) ( v ) = E e i v Λ ( ) = e i v µ E U ( ) ( ) = u E exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) i v (cid:169)(cid:173)(cid:173)(cid:173)(cid:173)(cid:171) γ + γ U ( ) ( ) ++ γ U ( ) (cid:16) U ( ) ( ) (cid:17) ++ σ B T ( U ( ) ( U ( ) ( ) )) (cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:174)(cid:174)(cid:172) u = e i v µ E (cid:34) exp (cid:32)(cid:40) i v γ − (cid:115) − b (cid:18) i v γ − (cid:113) − b (cid:0) i v γ − v σ (cid:1) (cid:19) (cid:41) U ( ) ( ) (cid:33) (cid:35) = e i v µ E exp (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) i (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) i v γ − (cid:115) − b (cid:18) i v γ − (cid:113) − b ( i v γ − v σ ) (cid:19) i (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) U ( ) ( ) (cid:170)(cid:174)(cid:174)(cid:174)(cid:172) .We know the Ch.f of U ∼ L ´e v y stable ( b ) is ϕ U ( u ) = E e iuU = exp (cid:110) −√− ibu (cid:111) , u ∈ R . Thisleads to ϕ Λ ( ) ( v ) = e i v µ exp − (cid:118)(cid:116) − ib i v γ − (cid:115) − b (cid:18) i v γ − (cid:113) − b ( i v γ − v σ ) (cid:19) i exp i v µ − (cid:118)(cid:117)(cid:116) − b (cid:32) i v γ − (cid:115) − b (cid:18) i v γ − (cid:113) − b (cid:0) i v γ − v σ (cid:1) (cid:19) (cid:33) Consequently, for n > ϕ Λ ( n ) ( v ) = E e i v Λ ( n ) = exp i v µ − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − b (cid:169)(cid:173)(cid:173)(cid:173)(cid:171) i v γ − (cid:118)(cid:117)(cid:117)(cid:117)(cid:117)(cid:116) − b (cid:169)(cid:173)(cid:173)(cid:171) . . . (cid:118)(cid:117)(cid:117)(cid:116) − b n − (cid:169)(cid:173)(cid:171) i v γ n − − (cid:115) − b n (cid:18) i v γ n − v σ (cid:19)(cid:170)(cid:174)(cid:172) . . . (cid:170)(cid:174)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:174)(cid:172) . A.6: Double-gamma subordinator moment-generating function If T ( ) ∼ Gamma ( α T , λ T ) , α T , > , λ T >
0, and U ( ) ∼ Gamma ( α U , λ U ) , then we have thefollowing representation for the MGF for the double gamma subordinator T ( U ( t )) : M T ( U ( )) ( v ) = E U ( ) = u (cid:16) E e v T ( ) (cid:17) u = E U ( ) = u (cid:16) (cid:16) − v λ T (cid:17) − α T (cid:17) u = λ α UU Γ ( α U ) ∫ ∞ e (cid:16) − α T ln (cid:16) − v λ T (cid:17) (cid:17) u u α U − e − λ U u du = λ α UU (cid:16) λ U + α T ln (cid:16) − v λ T (cid:17) (cid:17) α U , < v < λ T . Thus, M T ( U ( t )) ( v ) = (cid:0) M T ( U ( )) ( v ) (cid:1) t = (cid:18) + α T λ U ln (cid:18) − v λ T (cid:19) (cid:19) − α U t . Note that we must have ( i ) − v λ T > ( ii ) + α T λ U ln (cid:16) − v λ T (cid:17) >
0. Therefore, thedomain of M T ( U ( t )) ( v ) is, 0 < v < min (cid:16) λ T , λ T (cid:16) − exp (cid:16) − λ U α T (cid:17) (cid:17) (cid:17) = λ T (cid:16) − exp (cid:16) − λ U α T (cid:17) (cid:17) . A.7: Variance gamma-gamma L´evy process density and characteristic func-tion
Let L t = L + µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) , t ≥ = L − L = µ + γ U ( ) + ρ V () + σ B V ( ) . The pdf of Λ is given by f Λ ( x ) = ∂∂ x P ( Λ ≤ x ) = ∂∂ x ∫ ∞ P (cid:16) µ + γ u + ρ T ( u ) + σ (cid:112) T ( u ) N ( , )≤ x (cid:17) f U ( ) ( u ) du = ∂∂ x ∫ ∞ (cid:16)∫ ∞ P (cid:16) N ( , ) ≤ x − µ − γ u − ρ y σ √ y (cid:17) f T ( u ) ( y ) d y (cid:17) f U ( ) ( u ) du = ∫ ∞ (cid:16)∫ ∞ f N ( , ) (cid:16) x − µ − γ u − ρ y σ √ y (cid:17) f T ( u ) ( y ) d y (cid:17) f U ( ) ( u ) du ∫ ∞ (cid:18)∫ ∞ √ π e − ( x − µ − γ u − ρ y ) σ y f T ( u ) ( y ) d y (cid:19) f U ( ) ( u ) du Next, because T ( u ) ∼ Gamma ( α T u , λ T ) and U ( ) ∼ Gamma ( α U , λ U ) , it follows that f Λ ( x ) = ∫ ∞ (cid:18)∫ ∞ √ π e − ( x − µ − γ u − ρ y ) σ y λ α T uT Γ ( α Tu ) y α Tu − e − λ T y d y (cid:19) λ α UU Γ ( α U ) u α U − e − λ U u du = √ π λ α UU Γ ( α U ) ∫ ∞ (cid:18)∫ ∞ e − ( x − µ − γ u − ρ y ) σ y y α Tu − e − λ T y d y (cid:19) λ α T uT Γ ( α Tu ) u α U − e − λ U u du . The expression for the pdf f Λ ( x ) , x ∈ R is computationally intractable in view of thetwo integrals in the formula. The Ch.f. of Λ, ϕ Λ ( v ) = E e i v Λ , v ∈ R , has the form ϕ Λ ( v ) = E U ( ) = u e i v ( µ + γ u ) E e i v ( ρ T ( u ) + σ B T ( u ) ) = E U ( ) = u e i v ( µ + γ u ) (cid:16) E e i v ( ρ T ( ) + σ B T ( ) ) (cid:17) u . Note that E e i v ( ρ T ( ) + σ B T ( ) ) = E T ( ) = y e i v ρ y e − v σ y = ∫ ∞ e − ( − i v ρ + v σ ) y λ α TT Γ ( α T ) y α T − e − λ T y d y = ∫ ∞ λ α TT Γ ( α T ) y α T − e − ( − i v ρ + v σ + λ T ) y d y = (cid:16) − i v ρλ T + v σ λ T (cid:17) − α T . Conditional on U ( ) in ϕ Λ ( v ) we have ϕ Λ ( v ) = e i v µ E U ( ) = u e i v γ u (cid:16) − i v ρλ T + v σ λ T (cid:17) − u α T = e i v µ ∫ ∞ e i v γ u (cid:16) − i v ρλ T + v σ λ T (cid:17) − u α T f U ( ) ( u ) du = e i v µ (cid:16) − i v γλ U + α T λ U ln (cid:16) − i v ρλ T + v σ λ T (cid:17) (cid:17) − α U ,and therefore ϕ Λ ( v ) = E e i v Λ = e i v µ (cid:18) − i v γλ U + α T λ U ln (cid:18) − i v ρλ T + v σ λ T (cid:19) (cid:19) − α U , v ∈ R . By setting u = v i we find the following form for the MGF M Λ ( u ) = exp (cid:26) µ u − α U ln (cid:20) − γλ U u + α T λ U ln (cid:18) − ρλ T u − σ λ T u (cid:19) (cid:21) (cid:27) , for u >
0, such that 1 − γλ U u + α T λ U ln (cid:16) − ρλ T u − σ λ T u (cid:17) > u > . A.8: Compound-(n) gamma subordinator moment-generating function
Let U ( i ) ( t ) , t ≥ , i = , . . . , n , n ∈ N = { , , . . . } be a sequence of independent gammasubordinators with U ( i ) ( ) ∼ gamma ( α i , λ i ) , and define V ( ) ( t ) = U ( ) ( t ) , V ( i + ) ( t ) = ( i ) (cid:16) U ( i + ) ( t ) (cid:17) for i = , , . . . , n −
1. We shall use the notation V ( n ) ( t ) = U ( ) ◦ U ( ) ◦· · · ◦ U ( n ) ( t ) , t ≥ n =
3. From A.6 we have M U ( ) ( U ( ) ( U ( ) ( ) )) ( v ) = E U ( ) ( ) = u (cid:16) (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) − α (cid:17) u = E U ( ) ( ) = u (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) − α u = λ α γ ( α ) ∫ ∞ u α − e − (cid:16) λ + α ln (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) (cid:17) u du = (cid:16) + α λ ln (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) (cid:17) − α Note that the domain of M U ( ) ( t ) ( v ) is, 0 < v < λ (cid:16) − exp (cid:16) λ α (cid:16) exp (cid:16) − λ α (cid:17) − (cid:17) (cid:17) (cid:17) . Now, let V ( n ) ( t ) = U ( ) ◦ U ( ) ◦ · · · ◦ U ( n ) ( t ) = V ( n − ) ( U ( t )) , t ≥
0. Therefore, for any n ∈ N , we find M V ( n ) ( ) ( v ) = (cid:18) + α n − λ n ln (cid:18) + α n − λ n − ln . . . ln (cid:18) + α λ ln (cid:18) − v λ (cid:19) (cid:19) (cid:19) (cid:19) − α n , < v < τ n , where τ n = λ (cid:16) − exp (cid:16) − λ α (cid:16) . . . (cid:16) − exp (cid:16) − λ n − α n − (cid:16) − ex p (cid:16) − λ n α n − (cid:17) (cid:17) (cid:17) (cid:17) ... (cid:17) (cid:17) (cid:17) . Then by the MGF we have (cid:16) M V ( n ) ( ) ( v ) (cid:17) − α n = + α n − λ n ln (cid:16) + α n − λ n − ln . . . ln (cid:16) + α λ ln (cid:16) − v λ (cid:17) (cid:17) (cid:17) = + α n − λ n (cid:16) − α n − (cid:17) lnM V ( n − ) ( ) ( v ) = − λ n lnM V ( n − ) ( ) ( v ) , Or, M V ( n ) ( ) ( v ) = (cid:18) − λ n lnM V ( n − ) ( ) ( v ) (cid:19) − α n . A.9: Ch.f of normal-compound variance gamma
Let U ( i ) ( t ) , t ≥ , i = , . . . , n , n ∈ N = { , , . . . } be a sequence of independent gammasubordinators with U ( i ) ( ) ∼ Gamma ( α i , λ i ) , and define V ( ) ( t ) = U ( ) ( t ) , V ( i + ) ( t ) = V ( i ) (cid:16) U ( i + ) ( t ) (cid:17) for i = , , . . . , n − . Consider next a log-price process L ( n ) t = lnS t , t ≥ , n = , , .. of the form L ( n ) t = L ( n ) + µ t + n (cid:213) k = γ k ˜ V ( k ) ( t ) + σ B V ( n ) ( t ) , t ≥ , where µ ∈ R , γ k ∈ R , k = , , .., σ > , ˜ V ( k ) ( t ) = U ( k ) (cid:16) U ( k − ) (cid:16) . . . (cid:16) U ( ) ( t ) (cid:17) . . . (cid:17) (cid:17) , k = , . . . , n , n ∈ N , and U ( i ) ( t ) , t ≥ , i = , . . . , n is a sequence of independent gamma subor-dinators with U ( i ) ( ) ∼ Gamma ( α i , λ i ) . Denote Λ ( n ) : = L ( n ) − L ( n ) = µ + (cid:205) nk = γ k V ( k ) ( ) + σ B T ( V ( n ) ( ) ) . Then, we find Ch.f. of Λ ( n ) , n = , , . . . . n = ϕ Λ ( ) ( v ) = E e i v Λ ( ) = E e i v (cid:16) µ + (cid:205) k = γ k V ( k ) ( ) + σ B V ( )( ) (cid:17) = E exp (cid:16) i v (cid:16) µ + (cid:205) k = γ k V ( k ) ( ) + σ B V ( ) ( ) (cid:17) (cid:17) = e i v µ E U ( ) = u E exp (cid:16) i v (cid:16) γ u + γ U ( ) ( u ) + γ U ( ) (cid:16) U ( ) ( u ) (cid:17) + σ B U ( ) ( U ( ) ( u ) ) (cid:17) (cid:17) = e i v µ E U ( ) = u (cid:110) E exp (cid:16) i v (cid:16) γ + γ U ( ) ( ) + γ U ( ) (cid:16) U ( ) ( ) (cid:17) + σ B U ( ) ( U ( ) ( ) ) (cid:17) (cid:17) (cid:111) u .We know that if L t = L + µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) , t ≥
0, then ϕ Λ ( v ) = E e i v Λ = e i v µ (cid:16) − i v γλ U + α T λ U ln (cid:16) − i v ρλ T + v σ λ T (cid:17) (cid:17) − α U , v ∈ R .Thus, we find ϕ Λ ( ) ( v ) = e i v µ E U ( ) = u exp (cid:104) i v γ u − α u ln (cid:16) − i v γ λ + α λ ln (cid:16) − i v γ λ + v σ λ (cid:17) (cid:17) (cid:105) Then, by having the distribution of U ( ) , f U ( ) ( x ) = λ α Γ ( α ) x α − e − λ x , x ≥ , it follows that ϕ Λ ( ) ( v ) = e i v µ E U ( ) = u exp (cid:104) i v γ u − α u ln (cid:16) − i v γ λ + α λ ln (cid:16) − i v γ λ + v σ λ (cid:17) (cid:17) (cid:105) = e i v µ (cid:104) − i v γ λ + α λ ln (cid:16) − i v γ λ + α λ ln (cid:16) − i v γ λ + v σ λ (cid:17) (cid:17) (cid:105) − α .Consequently, for any n ∈ N , we find ϕ Λ ( n ) ( v ) = e iv µ (cid:18) − i v γ λ + α λ ln (cid:18) − · · · − i v γ n − λ n − + α n λ n − ln (cid:18) − i v γ n λ n + v σ λ n (cid:19) . . . (cid:19) (cid:19) − α . A.10: Double-inverse Gaussian subordinator density and moment-generatingfunction.
Let’s consider the case when the subordinators, T ( t ) , t ≥ U ( t ) , t ≥ T ( ) ∼ IG ( λ T , µ T ) , λ T > , µ T > , and U ( ) ∼ IG ( λ U , µ U ) .Note that the MGF of T ( u ) , M T ( u ) ( v ) = (cid:0) M T ( ) ( v ) (cid:1) u = exp (cid:0) λ T u (cid:1) ( µ T u ) (cid:169)(cid:173)(cid:171) − (cid:115) − ( µ T u ) v λ T u (cid:170)(cid:174)(cid:172) , that is, T ( u ) ∼ IG (cid:0) λ T u , µ T u (cid:1) .Then the pdf f V ( ) ( x ) , x > f V ( ) ( x ) = ∂∂ x ∫ ∞ P ( T ( u ) ≤ x ) f U ( ) ( u ) du = ∫ ∞ f T ( u ) ( x ) f U ( ) ( u ) du = ∫ ∞ (cid:113) λ T u π x e − λ T u ( x − µ T u ) µ T u x (cid:113) λ U π u e − λ U ( u − µ T ) µ U u du = ∫ ∞ π (cid:113) λ T λ U ux exp (cid:16) − λ T ( x − µ T u ) µ T x − λ U ( u − µ T ) µ U u (cid:17) du . V ( ) has the form f V ( ) ( x ) = π (cid:114) λ T λ U x ∫ ∞ u − exp (cid:32) − λ T ( x − µ T u ) µ T x − λ U ( u − µ T ) µ U u (cid:33) du , x > . Next, the MGF of V ( ) is given by M V ( ) ( v ) = E U ( ) = u E e v T ( u ) = E U ( ) = u (cid:16) E e v T ( ) (cid:17) u = E U ( ) = u exp (cid:34) λ T u µ T (cid:32) − (cid:114) − µ T v λ T (cid:33) (cid:35) = E exp (cid:34) λ T µ T (cid:32) − (cid:114) − µ T v λ T (cid:33) U ( ) (cid:35) = exp λ U µ U (cid:169)(cid:173)(cid:171) − (cid:118)(cid:116) − µ U λ U λ T µ T (cid:32) − (cid:114) − µ T v λ T (cid:33)(cid:170)(cid:174)(cid:172) . Therefore, M V ( ) ( v ) = exp λ U µ U (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ U λ U λ T µ T (cid:169)(cid:173)(cid:171) − (cid:115) − µ T v λ T (cid:170)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172) . where v > , − µ T v λ T > , − µ U λ U λ T µ T (cid:32) − (cid:114) − µ T v λ T (cid:33) > . That is, v < = λ U µ U (cid:32) µ T − λ U µ U λ T (cid:33) . As a result we have M V ( ) ( v ) = exp λ U µ U (cid:169)(cid:173)(cid:173)(cid:171) − (cid:118)(cid:117)(cid:117)(cid:116) − µ U λ U λ T µ T (cid:169)(cid:173)(cid:171) − (cid:115) − µ T v λ T (cid:170)(cid:174)(cid:172)(cid:170)(cid:174)(cid:174)(cid:172) . with the restrictions0 < v < λ T µ T , if λ U µ T µ U λ T ≥
1, and 0 < v < min (cid:16) λ T µ T , λ U µ U (cid:16) µ T − λ U µ U λ T (cid:17) (cid:17) , if λ U µ T µ U λ T < A.11: Normal-compound inverse Gaussian L´evy process density and char-acteristic function
Let L t = L + µ t + γ U ( t ) + ρ V ( t ) + σ B V ( t ) , t ≥ , be a normal-compound inverse GaussianL´evy process, then its distribution is determined by the unit increment Λ = L − L = µ + γ U ( ) + ρ V () + σ B V ( ) . The pdf of Λ is given by f Λ ( x ) = ∂∂ x P ( Λ ≤ x ) = ∂∂ x ∫ ∞ P (cid:16) µ + γ u + ρ T ( u ) + σ (cid:112) T ( u ) N ( , )≤ x (cid:17) f U ( ) ( u ) du = ∂∂ x ∫ ∞ (cid:16)∫ ∞ P (cid:16) N ( , ) ≤ x − µ − γ u − ρ y σ √ y (cid:17) f T ( u ) ( y ) d y (cid:17) f U ( ) ( u ) du ∫ ∞ (cid:16)∫ ∞ f N ( , ) (cid:16) x − µ − γ u − ρ y σ √ y (cid:17) f T ( u ) ( y ) d y (cid:17) f U ( ) ( u ) du = ∫ ∞ (cid:18)∫ ∞ √ π e − ( x − µ − γ u − ρ y ) σ y f T ( u ) ( y ) d y (cid:19) f U ( ) ( u ) du Next, because T ( u ) ∼ IG ( µ T u , λ T u ) and U ( ) ∼ Gamma ( α U , λ U ) , we find f Λ ( x ) = π √ λ T λ U ∫ ∞ ∫ ∞ t exp (cid:16) − x − µ − γ u − ρ t σ √ t − λ T ( t − u µ T ) ut µ T − λ U ( u − µ U ) u µ U (cid:17) . The expression for the pdf f Λ ( x ) , x ∈ R , is computationally intractable in view of the twointegrals in the formula. The Ch.f. of Λ, ϕ Λ ( v ) = E e i v Λ , v ∈ R has the form ϕ Λ ( v ) = E U ( ) = u e i v ( µ + γ u ) E e i v ( ρ T ( u ) + σ B T ( u ) ) = E U ( ) = u e i v ( µ + γ u ) (cid:16) E e i v ( ρ T ( ) + σ B T ( ) ) (cid:17) u . Note that E e i v ( ρ T ( ) + σ B T ( ) ) = E T ( ) = y e i v ρ y e − v σ y = ∫ ∞ e − ( − i v ρ + v σ ) y (cid:113) λ T π y exp (cid:16) − λ T ( y − µ T ) µ T y (cid:17) d y = exp (cid:32) λ T µ T (cid:32) − (cid:114) + µ T λ T (cid:0) i v ρ − v σ (cid:1) (cid:33) (cid:33) . Conditional on U ( ) in ϕ Λ ( v ) we have ϕ Λ ( v ) = e i v µ E U ( ) = u e i v u γ u e λ T µ T (cid:32) − (cid:114) + µ T λ T ( i v ρ − v σ ) (cid:33) = e i v µ ∫ ∞ e i v γ u e u λ T µ T (cid:32) − (cid:114) + µ T λ T ( i v ρ − v σ ) (cid:33) f U ( ) ( u ) du . And, therefore, ϕ Λ ( v ) = E e i v Λ = e i v µ + λ U µ U − (cid:118)(cid:116) − µ U λ U (cid:32) λ T µ T (cid:32) − (cid:114) − µ T λ T ( i v ρ − v σ ) (cid:33) + i v γ (cid:33) , v ∈ R . The proof of the MGF, M Λ ( u ) , follows by setting u = v ii