JEL Classification: G12, G13
Introduction
Since the global financial crisis of 2008, the VIX has become a central gauge of financial-market uncertainty and the reference index for a major market in volatility derivatives (Whaley [28 ] , Luo et al. [23 ] ). The index itself is not directly tradable: it is calculated from S&P 500 option prices and summarizes the market’s risk-neutral expectation of S&P 500 variation over the next 30 days (Luo and Zhang [22 ] ). Tradable exposure is instead provided by VIX futures and options, which allow investors to hedge volatility risk and take directional positions on changes in market uncertainty (Luo et al. [23 ] ). At the same time, evidence that volatility exhibits rough, non-Markovian dynamics has prompted growing interest in the implications of roughness for VIX futures and options (Gatheral et al. [14 ] , Jacquier et al. [18 ] , Rømer [27 ] ). The prominence of the VIX market, together with this evidence, creates a need for models that can capture the dynamics of the VIX while remaining sufficiently tractable for derivative pricing.1
Existing models for VIX derivatives can be organized into two broad approaches. The first derives VIX from the joint risk-neutral dynamics of the S&P 500 and its latent variance process, thereby maintaining a structural link between SPX and VIX derivatives, but often requiring a relatively rich specification and joint calibration to both markets [30 , 31 , 9 , 20 , 4 , 13 ] . The second approach models variance, volatility, VIX, or VIX futures directly [21 , 22 , 26 , 17 , 23 ] (Dupoyet et al. 2011). By avoiding an explicit model for the SPX spot price, direct specifications are generally more parsimonious and yield more tractable VIX derivative prices, although they do not impose a structural link between the equity and volatility markets.
This paper follows the instantaneous squared VIX (ISVIX) approach developed by Luo and Zhang [22 ] , Luo et al. [23 ] , which models directly the short-horizon limiting object whose conditional average determines squared VIX. The resulting Markovian specification is parsimonious and tractable, but it restricts the transmission of current volatility conditions across maturities to the pattern implied by exponential mean reversion. A natural way to relax this restriction is to introduce memory. Rough-volatility models use singular fractional kernels to capture the low local regularity and history dependence observed in volatility dynamics [5 , 14 , 12 ] . Most applications to VIX derivatives, however, introduce roughness into the latent variance or forward-variance process of the equity index and then derive VIX as a nonlinear functional of that process. This route is well suited to joint SPX–VIX modeling, but it makes the VIX term structure and VIX option prices depend on a comparatively rich latent-state system.
We instead propose a rough instantaneous squared VIX (RISVIX) model, by introducing a fractional Volterra kernel directly into the ISVIX framework of Luo et al. [23 ] . We derive an explicit VIX term structure governed by a Mittag–Leffler decay function and a semi-closed pricing formula for VIX options based on an affine Fourier–Laplace transform. These results lead to a two-step procedure in which the VIX term structure is used to estimate the memory and mean-reversion parameters, while VIX options are used to calibrate the remaining distributional parameters. Empirically, we estimate RISVIX using VIX term-structure indices and VIX option prices from 2018 to 2025. The gains from fractional memory vary substantially with market conditions. RISVIX improves the fit of the VIX term structure, with larger gains for its slope and curvature and during periods of elevated volatility. It also improves out-of-sample VIX option pricing, reducing MSE and MAE by 13.06% and 6.33%, respectively, with larger gains at shorter maturities and in high-VIX states.
Our contributions are threefold. First, we introduce fractional memory into an ISVIX model, yielding a parsimonious rough framework for modeling the VIX term structure and pricing VIX options. Second, we retain the tractability of direct VIX modeling by deriving an explicit term-structure formula and a semi-closed-form option-pricing formula, which together support a transparent two-step estimation and calibration procedure. Third, we show empirically that the gains from fractional memory vary substantially across volatility regimes: RISVIX improves the fit of the VIX term structure and the out-of-sample pricing of VIX options more strongly when volatility is elevated.
Related literature. Our paper is related to two strands of literature. First, it contributes to the literature on direct modeling of VIX term structures and derivatives. Lin [21 ] price VIX futures by integrating information under the physical and risk-neutral measures, while Dupoyet et al. (2011) develop a direct volatility-futures model and compare CEV and CIR specifications with and without jumps. Huskaj and Nossman [17 ] model the term structure of VIX futures directly rather than deriving it from an equity-variance process, and Mencía and Sentana [26 ] conduct an extensive empirical evaluation of VIX derivative valuation models across different market conditions. More closely related to our work, Luo and Zhang [22 ] extend VIX to arbitrary horizons and introduce instantaneous and long-term squared VIX as the short- and long-maturity limits of the VIX term structure. Building on this representation, Luo et al. [23 ] develop a unified Markovian ISVIX model with a stochastic mean level and jumps for pricing VIX derivatives across horizons. RISVIX extends their framework by replacing the propagation pattern implied by Markovian mean reversion with a fractional one.
Second, our paper contributes to the literature on rough volatility and VIX derivatives. Rough-volatility models use fractional kernels to capture the low local regularity and history dependence of volatility [5 , 14 , 12 ] . Existing applications to VIX derivatives generally specify rough dynamics for the latent variance or forward-variance curve of the equity index and then derive VIX from that process [18 , 16 , 27 , 8 , 19 ] . The closest study is Ye et al. (2026), who derive the VIX term structure from rough Heston using a Markovian approximation. Our RISVIX differs from their approach in two respects. First, it retains the fractional Volterra kernel directly in the ISVIX dynamics, thereby preserving the non-Markovian memory structure without relying on a Markovian approximation. Second, its affine Volterra structure yields a semi-closed pricing formula for VIX options, allowing the VIX term structure and option prices to be studied within the same framework.
The remainder of the paper is organized as follows. § 2 reviews the ISVIX framework, fractional Brownian motion and affine Volterra processes. § 3 introduces RISVIX, derives the VIX term structure and option-pricing formula, and presents the estimation procedure. § 4 reports the empirical results for the VIX term structure and VIX options. § 5 concludes. The Appendix contains the proofs and auxiliary results.
Preliminaries
This section reviews instantaneous squared VIX, fractional Brownian motion and affine Volterra models, which will be used throughout this paper. We work on a filtered probability space ( Ω , F , ( F t ) t ≥ 0 , Q ) (\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{Q}) ( Ω , F , ( F t ) t ≥ 0 , Q ) Q \mathbb{Q} Q E t Q [ ⋅ ] = E Q [ ⋅ ∣ F t ] . \mathbb{E}^{\mathbb{Q}}_t[\cdot] = \mathbb{E}^{\mathbb{Q}}[\cdot\mid\mathcal{F}_t]. E t Q [ ⋅ ] = E Q [ ⋅ ∣ F t ] .
Instantaneous Squared VIX
Let S t S_t S t τ > 0 \tau> 0 τ > 0 [21 , 22 ] ):
V I X t , τ 2 = 2 τ E t Q [ ∫ t t + τ d S s S s − − log ( S t + τ S t ) ] . (1) \mathrm{VIX}_{t,\tau}^{2} = \frac{2}{\tau}\mathbb{E}^{\mathbb{Q}}_t\left[\int_t^{t+\tau}\frac{\mathrm{d}S_s}{S_{s-}}-\log\left(\frac{S_{t+\tau}}{S_t}\right)\right]. \tag*{(1)} VIX t , τ 2 = τ 2 E t Q [ ∫ t t + τ S s − d S s − log ( S t S t + τ ) ] . (1) Suppose that the continuous component of the index return has instantaneous variance v t v_t v t λ t x \lambda_t^x λ t x x x x (1) admits the instantaneous squared VIX (ISVIX) representation of Luo and Zhang [22 ] :
V t = v t + 2 λ t x E Q [ e x − 1 − x ] , V I X t , τ 2 = 1 τ E t Q [ ∫ t t + τ V s d s ] . (2) V_t = v_t + 2\lambda_t^x\mathbb{E}^{\mathbb{Q}}\left[e^x-1-x\right], \qquad\mathrm{VIX}_{t,\tau}^{2} = \frac{1}{\tau}\mathbb{E}^{\mathbb{Q}}_t\left[\int_t^{t+\tau}V_s\,\mathrm{d}s\right]. \tag*{(2)} V t = v t + 2 λ t x E Q [ e x − 1 − x ] , VIX t , τ 2 = τ 1 E t Q [ ∫ t t + τ V s d s ] . (2) If V V V L 1 ( Q ) L^1(\mathbb{Q}) L 1 ( Q ) V I X t , τ 2 → V t \mathrm{VIX}_{t,\tau}^{2}\to V_t VIX t , τ 2 → V t L 1 ( Q ) L^1(\mathbb{Q}) L 1 ( Q ) τ ↓ 0 \tau\downarrow0 τ ↓ 0 [23 ] ).
Fractional Brownian Motion and Fractional Kernels
A standard fractional Brownian motion (fBM) with Hurst parameter H ∈ ( 0 , 1 ) H\in(0,1) H ∈ ( 0 , 1 ) B H B^H B H [25 ] )
E Q [ B s H B t H ] = 1 2 ( s 2 H + t 2 H − ∣ t − s ∣ 2 H ) , s , t ≥ 0. (3) \mathbb{E}^{\mathbb{Q}}\left[B_s^H B_t^H\right] = \frac{1}{2}\left(s^{2H}+t^{2H}-|t-s|^{2H}\right), \qquad s,t\ge0. \tag*{(3)} E Q [ B s H B t H ] = 2 1 ( s 2 H + t 2 H − ∣ t − s ∣ 2 H ) , s , t ≥ 0. (3) It has stationary increments and admits sample paths that are locally Hölder continuous of every order strictly less than H H H H = 1 / 2 H=1/2 H = 1/2 H < 1 / 2 H<1/2 H < 1/2
We restrict attention to H ∈ ( 0 , 1 / 2 ] H\in(0,1/2] H ∈ ( 0 , 1/2 ] α = H + 1 / 2 \alpha=H+1/2 α = H + 1/2 K ( t ) = t α − 1 / Γ ( α ) K(t)=t^{\alpha-1}/\Gamma(\alpha) K ( t ) = t α − 1 /Γ ( α ) t > 0 t>0 t > 0 Γ \Gamma Γ B B B
B ~ t H = ∫ 0 t K ( t − s ) d B s . (4) \widetilde{B}_t^H = \int_0^t K(t-s)\,\mathrm{d}B_s. \tag*{(4)} B t H = ∫ 0 t K ( t − s ) d B s . (4) For H < 1 / 2 H<1/2 H < 1/2 [5 ] ). Since K K K α > 1 / 2 \alpha>1/2 α > 1/2 α = 1 \alpha=1 α = 1 K ≡ 1 K\equiv1 K ≡ 1 B ~ 1 / 2 = B \widetilde{B}^{1/2}=B B 1/2 = B
Affine Volterra Models
We consider a class of affine Volterra models with diffusion and jump innovations ([3 , 6 ] ):
X t = X 0 + ∫ 0 t K ( t − s ) ∘ { μ ( X s ) d s + σ ( X s ) d B s + d J s } , (5) X_t = X_0 + \int_0^t \mathcal{K}(t-s) \circ\{\mu(X_s)\,\mathrm{d}s + \sigma(X_s)\,\mathrm{d}B_s + \mathrm{d}J_s\}, \tag*{(5)} X t = X 0 + ∫ 0 t K ( t − s ) ∘ { μ ( X s ) d s + σ ( X s ) d B s + d J s } , (5) where X X X d d d X 0 X_0 X 0 B B B m m m K = ( K 1 , … , K d ) ⊤ \mathcal{K} = (K_1,\ldots,K_d)^\top K = ( K 1 , … , K d ) ⊤ ∘ \circ ∘ J J J λ y ( X s ) \lambda^y(X_s) λ y ( X s ) y ∈ R d y \in\mathbb{R}^d y ∈ R d E Q [ ∥ y ∥ 2 ] < ∞ \mathbb{E}^{\mathbb{Q}}[\lVert y\rVert^2] < \infty E Q [∥ y ∥ 2 ] < ∞ λ y ( X s ) E Q [ y ] d s \lambda^y(X_s)\mathbb{E}^{\mathbb{Q}}[y]\,\mathrm{d}s λ y ( X s ) E Q [ y ] d s (5) is understood to hold Q ⊗ d t \mathbb{Q} \otimes\mathrm{d}t Q ⊗ d t [6 ] ).
The drift, diffusion covariance and jump intensity are affine on the state space:
μ ( x ) = b 0 + ∑ k = 1 d b k x k , σ ( x ) σ ( x ) ⊤ = H 0 + ∑ k = 1 d H k x k , λ y ( x ) = λ 0 y + ∑ k = 1 d λ k y x k , (6) \mu(x) = b_0 + \sum_{k=1}^{d} b_kx_k,\quad\sigma(x)\sigma(x)^\top= H_0 + \sum_{k=1}^{d} H_kx_k,\quad\lambda^y(x) = \lambda^y_0 + \sum_{k=1}^{d}\lambda^y_kx_k, \tag*{(6)} μ ( x ) = b 0 + k = 1 ∑ d b k x k , σ ( x ) σ ( x ) ⊤ = H 0 + k = 1 ∑ d H k x k , λ y ( x ) = λ 0 y + k = 1 ∑ d λ k y x k , (6) where b k ∈ R d b_k \in\mathbb{R}^d b k ∈ R d H k ∈ R d × d H_k \in\mathbb{R}^{d \times d} H k ∈ R d × d λ k y ∈ R \lambda^y_k \in\mathbb{R} λ k y ∈ R k = 0 , … , d k = 0,\ldots,d k = 0 , … , d K i ≡ 1 K_i \equiv1 K i ≡ 1 λ y ≡ 0 \lambda^y \equiv0 λ y ≡ 0 [3 ] .
Whenever the expectation is absolutely convergent, write ϱ ( z ) = E Q [ e z ⊤ y ] \varrho(z) = \mathbb{E}^{\mathbb{Q}}[e^{z^\top y}] ϱ ( z ) = E Q [ e z ⊤ y ]
R k ( z ) = b k ⊤ z + 1 2 z ⊤ H k z + λ k y [ ϱ ( z ) − 1 − z ⊤ E Q [ y ] ] , k = 0 , … , d . (7) R_k(z) = b_k^\top z + \frac{1}{2}z^\top H_kz + \lambda^y_k\left[\varrho(z) - 1 - z^\top\mathbb{E}^{\mathbb{Q}}[y]\right],\quad k = 0,\ldots,d. \tag*{(7)} R k ( z ) = b k ⊤ z + 2 1 z ⊤ H k z + λ k y [ ϱ ( z ) − 1 − z ⊤ E Q [ y ] ] , k = 0 , … , d . (7) For a horizon T > 0 T > 0 T > 0 u ∈ C d u \in\mathbb{C}^d u ∈ C d
Ψ ( t , u ) = u ∘ K ( t ) + ∫ 0 t K ( t − s ) ∘ R ( Ψ ( s , u ) ) d s , 0 < t ≤ T , (8) \Psi(t,u) = u \circ\mathcal{K}(t) + \int_0^t \mathcal{K}(t-s) \circ R(\Psi(s,u))\,\mathrm{d}s,\quad0 < t \le T, \tag*{(8)} Ψ ( t , u ) = u ∘ K ( t ) + ∫ 0 t K ( t − s ) ∘ R ( Ψ ( s , u )) d s , 0 < t ≤ T , (8) where R ( z ) = ( R 1 ( z ) , … , R d ( z ) ) ⊤ R(z) = (R_1(z),\ldots,R_d(z))^\top R ( z ) = ( R 1 ( z ) , … , R d ( z ) ) ⊤
Under suitable integrability and martingale conditions, solutions of Equation (8) yield initial-time exponential-affine transforms; see Abi Jaber et al. [3 ] and Bondi et al. [6 ] . The functions R 1 , … , R d R_1,\ldots,R_d R 1 , … , R d R 0 R_0 R 0 X t X_t X t [3 , 6 ] ).
Framework
In this section, we propose a rough ISVIX (RISVIX) model and derive the corresponding VIX term structure and VIX option pricing formulas. Throughout the theoretical analysis, time zero denotes the valuation origin.
Model Specification
We extend the ISVIX framework of Luo et al. [23 ] using the fractional kernel K ( t ) = t α − 1 / Γ ( α ) K(t) = t^{\alpha-1}/\Gamma(\alpha) K ( t ) = t α − 1 /Γ ( α ) T > 0 T > 0 T > 0 Q \mathbb{Q} Q
V t = V 0 + ∫ 0 t K ( t − s ) { κ ( θ s − V s ) d s + σ V V s d B 1 , s + d J s y } , θ t = θ 0 + σ θ ∫ 0 t θ s d B 2 , s , 0 ≤ t ≤ T . (9) \begin{aligned}
V_t &= V_0 + \int_0^t K(t-s)\left\{\kappa(\theta_s - V_s)\,\mathrm{d}s + \sigma_V\sqrt{V_s}\,\mathrm{d}B_{1,s} + \mathrm{d}J_s^y\right\},\\
\theta_t &= \theta_0 + \sigma_\theta\int_0^t \sqrt{\theta_s}\,\mathrm{d}B_{2,s}, \qquad0 \le t \le T. \tag*{(9)}
\end{aligned} V t θ t = V 0 + ∫ 0 t K ( t − s ) { κ ( θ s − V s ) d s + σ V V s d B 1 , s + d J s y } , = θ 0 + σ θ ∫ 0 t θ s d B 2 , s , 0 ≤ t ≤ T . (9) Here V 0 , θ 0 ≥ 0 V_0, \theta_0 \ge0 V 0 , θ 0 ≥ 0 F 0 \mathcal{F}_0 F 0 E Q [ V 0 2 + θ 0 2 ] < ∞ \mathbb{E}^{\mathbb{Q}}[V_0^2 + \theta_0^2] < \infty E Q [ V 0 2 + θ 0 2 ] < ∞ B 1 B_1 B 1 B 2 B_2 B 2 N y N^y N y λ y \lambda^y λ y { Y n } n ≥ 1 \{Y_n\}_{n\ge1} { Y n } n ≥ 1 μ y \mu_y μ y B 1 B_1 B 1 B 2 B_2 B 2 N y N^y N y { Y n } n ≥ 1 \{Y_n\}_{n\ge1} { Y n } n ≥ 1 F 0 \mathcal{F}_0 F 0 J t y = ∑ n = 1 N t y Y n − λ y μ y t J_t^y = \sum_{n=1}^{N_t^y}Y_n - \lambda^y\mu_y t J t y = ∑ n = 1 N t y Y n − λ y μ y t J y J^y J y ϱ ( z ) = ( 1 − μ y z ) − 1 \varrho(z) = (1-\mu_y z)^{-1} ϱ ( z ) = ( 1 − μ y z ) − 1 Re ( z ) < μ y − 1 \operatorname{Re}(z) < \mu_y^{-1} Re ( z ) < μ y − 1
The parameters satisfy κ , μ y > 0 \kappa, \mu_y > 0 κ , μ y > 0 σ V , σ θ , λ y ≥ 0 \sigma_V, \sigma_\theta, \lambda^y \ge0 σ V , σ θ , λ y ≥ 0 α ∈ ( 1 / 2 , 1 ) \alpha\in(1/2,1) α ∈ ( 1/2 , 1 ) κ \kappa κ θ \theta θ σ V \sigma_V σ V σ θ \sigma_\theta σ θ α = 1 \alpha= 1 α = 1 K ≡ 1 K \equiv1 K ≡ 1
On [ 0 , T ] [0,T] [ 0 , T ] (9) admits a nonnegative weak solution ( V , θ ) (V,\theta) ( V , θ ) θ \theta θ V V V ∫ 0 T V s d s < ∞ \int_0^T V_s\,\mathrm{d}s < \infty ∫ 0 T V s d s < ∞ V ≥ 0 V \ge0 V ≥ 0 (9) hold Q ⊗ d t \mathbb{Q} \otimes\mathrm{d}t Q ⊗ d t Lemma A.2 gives E Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] < ∞ \mathbb{E}^{\mathbb{Q}}\left[\int_0^T (V_t^2+\theta_t^2)\,\mathrm{d}t\right] < \infty E Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] < ∞ θ \theta θ [1 ] . The following proposition gives the conditional mean of ISVIX.
Proposition 3.1. Suppose that Equation (9) admits the finite-horizon solution specified in Appendix § 8 .§ 2 on [ 0 , T ] [0,T] [ 0 , T ]
E Q [ V s ] = θ 0 + ( V 0 − θ 0 ) E α , 1 ( − κ s α ) , (10) \mathbb{E}^{\mathbb{Q}}[V_s] = \theta_0 + (V_0-\theta_0)E_{\alpha,1}(-\kappa s^\alpha), \tag*{(10)} E Q [ V s ] = θ 0 + ( V 0 − θ 0 ) E α , 1 ( − κ s α ) , (10) where E a , b ( z ) = ∑ n = 0 ∞ z n / Γ ( a n + b ) E_{a,b}(z) = \sum_{n=0}^{\infty} z^n/\Gamma(an+b) E a , b ( z ) = ∑ n = 0 ∞ z n /Γ ( an + b ) [ 0 , T ] [0,T] [ 0 , T ]
Remark. The conditional mean in Equation (10) depends only on ( α , κ , V 0 , θ 0 ) (\alpha,\kappa,V_0,\theta_0) ( α , κ , V 0 , θ 0 ) E 0 Q [ θ s ] = θ 0 \mathbb{E}^{\mathbb{Q}}_0[\theta_s] = \theta_0 E 0 Q [ θ s ] = θ 0 ( σ V , σ θ , λ y , μ y ) (\sigma_V, \sigma_\theta, \lambda^y, \mu_y) ( σ V , σ θ , λ y , μ y )
VIX Term Structure
Recall from Equation (2) that, for t ≥ 0 t \ge0 t ≥ 0 τ > 0 \tau> 0 τ > 0 V I X t , τ 2 = τ − 1 E t Q [ ∫ t t + τ V s d s ] \mathrm{VIX}_{t,\tau}^2 = \tau^{-1}\mathbb{E}^{\mathbb{Q}}_t\left[\int_t^{t+\tau} V_s\,\mathrm{d}s\right] VIX t , τ 2 = τ − 1 E t Q [ ∫ t t + τ V s d s ]
Theorem 3.1. Suppose that Equation (9) admits the finite-horizon solution specified in Appendix § 8 .§ 2 on [ 0 , T ] [0,T] [ 0 , T ] t ∈ [ 0 , T ) t \in[0,T) t ∈ [ 0 , T ) τ ∈ ( 0 , T − t ] \tau\in(0,T-t] τ ∈ ( 0 , T − t ]
V I X t , τ 2 = w ( τ ) V 0 + [ 1 − w ( τ ) ] θ t + ∫ 0 t a τ ( t − s ) { κ ( θ s − V 0 ) d s + σ V V s d B 1 , s + d J s y } . (11) \mathrm{VIX}_{t,\tau}^2 = w(\tau)V_0 + [1-w(\tau)]\theta_t + \int_0^t a_\tau(t-s)\left\{\kappa(\theta_s-V_0)\,\mathrm{d}s+\sigma_V\sqrt{V_s}\,\mathrm{d}B_{1,s}+\mathrm{d}J_s^y\right\}. \tag*{(11)} VIX t , τ 2 = w ( τ ) V 0 + [ 1 − w ( τ )] θ t + ∫ 0 t a τ ( t − s ) { κ ( θ s − V 0 ) d s + σ V V s d B 1 , s + d J s y } . (11) where w ( τ ) = E α , 2 ( − κ τ α ) w(\tau)=E_{\alpha,2}(-\kappa\tau^\alpha) w ( τ ) = E α , 2 ( − κ τ α ) a τ ( s ) = [ E α , 1 ( − κ s α ) − E α , 1 ( − κ ( s + τ ) α ) ] / ( κ τ ) a_\tau(s)=[E_{\alpha,1}(-\kappa s^\alpha)-E_{\alpha,1}(-\kappa(s+\tau)^\alpha)]/(\kappa\tau) a τ ( s ) = [ E α , 1 ( − κ s α ) − E α , 1 ( − κ ( s + τ ) α )] / ( κ τ )
Remark. At the valuation origin, Equation (11) reduces to V I X 0 , τ 2 = V 0 w ( τ ) + θ 0 [ 1 − w ( τ ) ] \mathrm{VIX}_{0,\tau}^2=V_0w(\tau)+\theta_0[1-w(\tau)] VIX 0 , τ 2 = V 0 w ( τ ) + θ 0 [ 1 − w ( τ )] w ( τ ) w(\tau) w ( τ ) τ \tau τ V 0 V_0 V 0 θ 0 \theta_0 θ 0 V 0 < θ 0 V_0<\theta_0 V 0 < θ 0 V 0 > θ 0 V_0>\theta_0 V 0 > θ 0 ( α , κ , V 0 , θ 0 ) (\alpha,\kappa,V_0,\theta_0) ( α , κ , V 0 , θ 0 ) (11) . This separation motivates the two-stage calibration used below.
VIX Option Pricing
Consider a European VIX call valued at time zero, with expiry T > 0 T>0 T > 0 K > 0 K>0 K > 0 τ 0 > 0 \tau_0>0 τ 0 > 0 (11) evaluated at t = T t=T t = T τ = τ 0 \tau=\tau_0 τ = τ 0
f ( u ; 0 , T ) = E 0 Q [ exp { u V I X T , τ 0 2 } ] . (12) f(u;0,T)=\mathbb{E}^{\mathbb{Q}}_0\left[\exp\left\{u\mathrm{VIX}_{T,\tau_0}^2\right\}\right]. \tag*{(12)} f ( u ; 0 , T ) = E 0 Q [ exp { u VIX T , τ 0 2 } ] . (12) Let R κ R_\kappa R κ κ K \kappa K κ K R κ + κ K ∗ R κ = κ K R_\kappa+\kappa K*R_\kappa=\kappa K R κ + κ K ∗ R κ = κ K Lemma A.1 ,
R κ ( s ) = κ s α − 1 E α , α ( − κ s α ) , a τ 0 ( s ) = 1 κ τ 0 ∫ s s + τ 0 R κ ( v ) d v . (13) R_\kappa(s)=\kappa s^{\alpha-1}E_{\alpha,\alpha}(-\kappa s^\alpha),\qquad a_{\tau_0}(s)=\frac{1}{\kappa\tau_0}\int_s^{s+\tau_0}R_\kappa(v)\,\mathrm{d}v. \tag*{(13)} R κ ( s ) = κ s α − 1 E α , α ( − κ s α ) , a τ 0 ( s ) = κ τ 0 1 ∫ s s + τ 0 R κ ( v ) d v . (13) Suppose that Equation (9) admits the finite-horizon solution specified in Appendix § 8 .§ 2 on [ 0 , T + τ 0 ] [0,T+\tau_0] [ 0 , T + τ 0 ]
Theorem 3.2. *There exists η ˉ T , τ 0 > 0 \bar{\eta}_{T,\tau_0}>0 η ˉ T , τ 0 > 0 u ∈ C u\in\mathbb{C} u ∈ C 0 ≤ Re ( u ) < η ˉ T , τ 0 0\le\operatorname{Re}(u)<\bar{\eta}_{T,\tau_0} 0 ≤ Re ( u ) < η ˉ T , τ 0
transform satisfies almost surely
f ( u ; 0 , T ) = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } , (14) f(u;0,T)=\exp\{\phi(T)+\chi_1(T)V_0+\psi_2(T)\theta_0\}, \tag*{(14)} f ( u ; 0 , T ) = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } , (14) where the coefficients ψ 1 \psi_1 ψ 1 ψ 2 \psi_2 ψ 2 χ 1 \chi_1 χ 1 ϕ \phi ϕ [ 0 , T ] [0,T] [ 0 , T ]
ψ 1 ( s ) = u a τ 0 ( s ) + σ V 2 2 κ ∫ 0 s R κ ( s − v ) ψ 1 ( v ) 2 d v , ψ 2 ( s ) = u [ 1 − w ( τ 0 ) ] + ∫ 0 s [ κ ψ 1 ( v ) + σ θ 2 2 ψ 2 ( v ) 2 ] d v , χ 1 ( s ) = u w ( τ 0 ) + ∫ 0 s [ − κ ψ 1 ( v ) + σ V 2 2 ψ 1 ( v ) 2 ] d v , ϕ ( s ) = λ y ∫ 0 s [ ρ ( ψ 1 ( v ) ) − 1 − μ y ψ 1 ( v ) ] d v . (15) \begin{aligned}
\psi_1(s)&=ua_{\tau_0}(s)+\frac{\sigma_V^2}{2\kappa}\int_0^s R_\kappa(s-v)\psi_1(v)^2\,\mathrm{d}v,\\
\psi_2(s)&=u[1-w(\tau_0)]+\int_0^s\left[\kappa\psi_1(v)+\frac{\sigma_\theta^2}{2}\psi_2(v)^2\right]\,\mathrm{d}v,\\
\chi_1(s)&=uw(\tau_0)+\int_0^s\left[-\kappa\psi_1(v)+\frac{\sigma_V^2}{2}\psi_1(v)^2\right]\,\mathrm{d}v,\\
\phi(s)&=\lambda^y\int_0^s[\rho(\psi_1(v))-1-\mu_y\psi_1(v)]\,\mathrm{d}v. \tag*{(15)}
\end{aligned} ψ 1 ( s ) ψ 2 ( s ) χ 1 ( s ) ϕ ( s ) = u a τ 0 ( s ) + 2 κ σ V 2 ∫ 0 s R κ ( s − v ) ψ 1 ( v ) 2 d v , = u [ 1 − w ( τ 0 )] + ∫ 0 s [ κ ψ 1 ( v ) + 2 σ θ 2 ψ 2 ( v ) 2 ] d v , = u w ( τ 0 ) + ∫ 0 s [ − κ ψ 1 ( v ) + 2 σ V 2 ψ 1 ( v ) 2 ] d v , = λ y ∫ 0 s [ ρ ( ψ 1 ( v )) − 1 − μ y ψ 1 ( v )] d v . (15) The solution satisfies sup 0 ≤ s ≤ T Re ψ 1 ( s ) < μ y − 1 \sup_{0\le s\le T}\operatorname{Re}\psi_1(s)<\mu_y^{-1} sup 0 ≤ s ≤ T Re ψ 1 ( s ) < μ y − 1 ρ \rho ρ
Remark. For each transform argument u u u ψ 1 \psi_1 ψ 1 ψ 2 \psi_2 ψ 2 χ 1 \chi_1 χ 1 ϕ \phi ϕ a τ 0 a_{\tau_0} a τ 0 a τ 0 a_{\tau_0} a τ 0 R κ R_\kappa R κ α = 1 \alpha=1 α = 1 a τ 0 ( s ) = w ( τ 0 ) e − κ s a_{\tau_0}(s)=w(\tau_0)e^{-\kappa s} a τ 0 ( s ) = w ( τ 0 ) e − κ s χ 1 = ψ 1 \chi_1=\psi_1 χ 1 = ψ 1 (15) to the corresponding Markovian Riccati system.
To recover option prices from Equation (14) , we view the call payoff as a function of squared VIX, g ( x ) = ( x − K ) + g(x)=(\sqrt{x}-K)^+ g ( x ) = ( x − K ) + x ≥ 0 x\ge0 x ≥ 0 ∫ 0 ∞ e − z x g ( x ) d x = π erfc ( K z ) / ( 2 z 3 / 2 ) \int_0^\infty e^{-zx}g(x)\,\mathrm{d}x=\sqrt{\pi}\,\operatorname{erfc}(K\sqrt{z})/(2z^{3/2}) ∫ 0 ∞ e − z x g ( x ) d x = π erfc ( K z ) / ( 2 z 3/2 ) Re ( z ) > 0 \operatorname{Re}(z)>0 Re ( z ) > 0 Theorem 3.2 guarantees that the VIX transform is finite on every vertical contour u = η + i ξ u=\eta+i\xi u = η + i ξ η ∈ ( 0 , η ˉ T , τ 0 ) \eta\in(0,\bar{\eta}_{T,\tau_0}) η ∈ ( 0 , η ˉ T , τ 0 ) η \eta η
Corollary 3.1. Assume a constant risk-free rate r r r η ∈ ( 0 , η ˉ T , τ 0 ) \eta\in(0,\bar{\eta}_{T,\tau_0}) η ∈ ( 0 , η ˉ T , τ 0 ) T T T K K K
c ( 0 , T ; K ) = e − r T E Q [ ( V I X T , τ 0 − K ) + ] = e − r T 2 π ∫ 0 ∞ Re [ erfc ( K η + i ξ ) ( η + i ξ ) 3 / 2 f ( η + i ξ ; 0 , T ) ] d ξ . (16) \begin{aligned}
c(0,T;K)&=e^{-rT}\mathbb{E}_{\mathbb{Q}}\left[(\mathrm{VIX}_{T,\tau_0}-K)^+\right]\\
&=\frac{e^{-rT}}{2\sqrt{\pi}}\int_0^\infty\operatorname{Re}\left[\frac{\operatorname{erfc}(K\sqrt{\eta+i\xi})}{(\eta+i\xi)^{3/2}}f(\eta+i\xi;0,T)\right]\,\mathrm{d}\xi. \tag*{(16)}
\end{aligned} c ( 0 , T ; K ) = e − r T E Q [ ( VIX T , τ 0 − K ) + ] = 2 π e − r T ∫ 0 ∞ Re [ ( η + i ξ ) 3/2 erfc ( K η + i ξ ) f ( η + i ξ ; 0 , T ) ] d ξ . (16) Here erfc ( z ) = 1 − 2 π ∫ 0 z e − v 2 d v \operatorname{erfc}(z)=1-\frac{2}{\sqrt{\pi}}\int_0^z e^{-v^2}\,\mathrm{d}v erfc ( z ) = 1 − π 2 ∫ 0 z e − v 2 d v
Thus, VIX option pricing under RISVIX reduces to solving one nonlinear Volterra equation for each transform argument and evaluating a one-dimensional numerical integral.
Calibration Strategy
The parameter separation leads naturally to a two-step procedure, following the estimation–calibration structure of Luo et al. [23 ] . The first step estimates the common term-structure parameters and date-specific initial states from VIX indices, while the second step calibrates the remaining distributional parameters to VIX option prices.
Step 1: Estimation from the VIX term structure Let T \mathcal{T} T τ 1 , … , τ J \tau_1,\ldots,\tau_J τ 1 , … , τ J t 0 ∈ T t_0 \in\mathcal{T} t 0 ∈ T t 0 t_0 t 0 ( V I X t 0 , τ j ) 2 = V t 0 w ( τ j ) + θ t 0 [ 1 − w ( τ j ) ] (\mathrm{VIX}_{t_0,\tau_j})^2 = V_{t_0}w(\tau_j) + \theta_{t_0}[1-w(\tau_j)] ( VIX t 0 , τ j ) 2 = V t 0 w ( τ j ) + θ t 0 [ 1 − w ( τ j )] ( V t 0 , θ t 0 ) (V_{t_0},\theta_{t_0}) ( V t 0 , θ t 0 ) t 0 t_0 t 0 ( α , κ ) (\alpha,\kappa) ( α , κ )
( α ^ , κ ^ , { V ^ t 0 , θ ^ t 0 } t 0 ∈ T ) = arg min α ∈ ( 1 / 2 , 1 ] , κ > 0 V t 0 ≥ 0 , θ t 0 ≥ 0 , t 0 ∈ T ∑ t 0 ∈ T ∑ j = 1 J [ V I X t 0 , τ j m k t − V I X t 0 , τ j m o d e l ( α , κ , V t 0 , θ t 0 ) ] 2 . (17) \left(\widehat{\alpha},\widehat{\kappa},\{\widehat{V}_{t_0},\widehat{\theta}_{t_0}\}_{t_0\in\mathcal{T}}\right)
=
\underset{\substack{\alpha\in(1/2,1],\ \kappa>0\\ V_{t_0}\geq0,\ \theta_{t_0}\geq0,\ t_0\in\mathcal{T}}}{\operatorname{arg\,min}}
\sum_{t_0\in\mathcal{T}}\sum_{j=1}^{J}
\left[
\mathrm{VIX}^{\mathrm{mkt}}_{t_0,\tau_j}
-
\mathrm{VIX}^{\mathrm{model}}_{t_0,\tau_j}(\alpha,\kappa,V_{t_0},\theta_{t_0})
\right]^2 . \tag*{(17)} ( α , κ , { V t 0 , θ t 0 } t 0 ∈ T ) = α ∈ ( 1/2 , 1 ] , κ > 0 V t 0 ≥ 0 , θ t 0 ≥ 0 , t 0 ∈ T arg min t 0 ∈ T ∑ j = 1 ∑ J [ VIX t 0 , τ j mkt − VIX t 0 , τ j model ( α , κ , V t 0 , θ t 0 ) ] 2 . (17) For any fixed ( α , κ ) (\alpha,\kappa) ( α , κ ) α = 1 \alpha=1 α = 1
Step 2: Calibration to VIX option prices Conditional on the Step 1 estimates, we calibrate ζ = ( σ V , σ θ , λ y , μ y ) ⊤ \zeta=(\sigma_V,\sigma_\theta,\lambda^y,\mu_y)^\top ζ = ( σ V , σ θ , λ y , μ y ) ⊤ T t r a i n \mathcal{T}_{\mathrm{train}} T train t 0 ∈ T t r a i n t_0\in\mathcal{T}_{\mathrm{train}} t 0 ∈ T train M t 0 \mathcal{M}_{t_0} M t 0 K t 0 , T \mathcal{K}_{t_0,T} K t 0 , T T ∈ M t 0 T\in\mathcal{M}_{t_0} T ∈ M t 0 t 0 t_0 t 0 ( V ^ t 0 , θ ^ t 0 ) (\widehat{V}_{t_0},\widehat{\theta}_{t_0}) ( V t 0 , θ t 0 ) T T T t 0 t_0 t 0 τ 0 \tau_0 τ 0
ζ ^ = arg min ζ ∈ P ∑ t 0 ∈ T t r a i n ∑ T ∈ M t 0 ∑ K ∈ K t 0 , T [ C t 0 m k t ( T , K ) − C t 0 m o d e l ( T , K ; ζ , α ^ , κ ^ , V ^ t 0 , θ ^ t 0 ) ] 2 . (18) \widehat{\zeta}
=
\underset{\zeta\in\mathcal{P}}{\operatorname{arg\,min}}
\sum_{t_0\in\mathcal{T}_{\mathrm{train}}}
\sum_{T\in\mathcal{M}_{t_0}}
\sum_{K\in\mathcal{K}_{t_0,T}}
\left[
C^{\mathrm{mkt}}_{t_0}(T,K)
-
C^{\mathrm{model}}_{t_0}\left(T,K;\zeta,\widehat{\alpha},\widehat{\kappa},\widehat{V}_{t_0},\widehat{\theta}_{t_0}\right)
\right]^2 . \tag*{(18)} ζ = ζ ∈ P arg min t 0 ∈ T train ∑ T ∈ M t 0 ∑ K ∈ K t 0 , T ∑ [ C t 0 mkt ( T , K ) − C t 0 model ( T , K ; ζ , α , κ , V t 0 , θ t 0 ) ] 2 . (18) Here C t 0 m k t ( T , K ) C^{\mathrm{mkt}}_{t_0}(T,K) C t 0 mkt ( T , K ) C t 0 m o d e l ( T , K ; ⋅ ) C^{\mathrm{model}}_{t_0}(T,K;\cdot) C t 0 model ( T , K ; ⋅ ) c ( 0 , T ; K ) c(0,T;K) c ( 0 , T ; K ) (16) , evaluated using the Step 1 estimates and the candidate value of ζ \zeta ζ ζ \zeta ζ
For each valuation date t 0 t_0 t 0 T T T [ 0 , T + τ 0 ] [0,T+\tau_0] [ 0 , T + τ 0 ] [ t 0 , t 0 + T + τ 0 ] [t_0,t_0+T+\tau_0] [ t 0 , t 0 + T + τ 0 ] [11 , 16 ] ). In our empirical implementation, each t 0 t_0 t 0 ( V ^ t 0 , θ ^ t 0 ) (\widehat{V}_{t_0},\widehat{\theta}_{t_0}) ( V t 0 , θ t 0 )
Empirical Study
Data
We obtain daily VIX term-structure observations from CBOE and VIX option quotes from OptionMetrics. The term structure consists of the closing values of VIX9D, VIX, VIX3M, VIX6M and VIX1Y, corresponding to volatility horizons of nine days, 30 days, three months, six months and one year, respectively. The sample runs from January 2, 2018 to June 30, 2025. Restricting attention to dates on which all five indices are observed gives a balanced panel of 1,883 trading days.
The VIX option data also run from January 2, 2018 to June 30, 2025 and contain trading dates, expiration dates, strikes, option types, closing bid and ask quotes, and trading volume. We use the bid–ask midpoint as the observed option price, with option prices and strikes expressed in VIX index points. We retain call options with 30–126 calendar days to expiration and spot moneyness K / V I X t ∈ [ 0.7 , 1.3 ] K/\mathrm{VIX}_{t} \in[0.7, 1.3] K / VIX t ∈ [ 0.7 , 1.3 ] K K K V I X t \mathrm{VIX}_{t} VIX t
Table 1 reports summary statistics for the two datasets. Panel A shows an upward-sloping average term structure, with mean index levels increasing from 19.29 for VIX9D to 23.55 for VIX1Y, while the corresponding standard deviations decrease from 9.50 to 4.92. Panel B reports an average option midquote of 4.28 points. Longer-dated calls have higher average prices, whereas shorter-dated calls exhibit greater positive skewness and kurtosis.
Table 1. Summary statistics of VIX indices and options. Panel A reports daily closing index levels. Panel B reports VIX call-option midquotes, grouped by calendar days to expiration.
Figure 1 illustrates the historical evolution and term-structure variation of the VIX indices. Panel (a) shows substantial co-movement across horizons, with shorter-horizon indices exhibiting sharper fluctuations. Panel (b) shows that the term structure is typically upward sloping during low-volatility periods but can invert during volatility spikes, most notably in early 2020. Thus, the data exhibit substantial variation in both the level and shape of the VIX term structure.
Figure 1. Historical dynamics and term structure of VIX indices. Panel (a) plots the available daily closing values of VIX9D, VIX, VIX3M, VIX6M and VIX1Y on a common scale. Panel (b) displays the balanced term-structure panel used in the empirical analysis.
Term-Structure Calibration
We compare RISVIX with the benchmark of Luo et al. [23 ] , denoted LZZ (α = 1 \alpha= 1 α = 1 § 3 . The full-sample comparison also includes the rough Heston term-structure model of Ye et al. (2026), denoted YFK. The full calibration sample in this subsection covers January 2018–June 2025 and contains 1,883 trading days and 9,415 horizon–date observations. We also examine a separate calibration of RISVIX and LZZ over January–June 2020. For the subsequent option pricing, the first-step inputs are estimated separately using January 2018–December 2023, and January 2024–June 2025 is reserved as the test period.
For RISVIX, the common parameters (α , κ \alpha, \kappa α , κ V t , θ t V_t, \theta_t V t , θ t α , κ \alpha, \kappa α , κ V t , θ t V_t, \theta_t V t , θ t α , κ \alpha, \kappa α , κ α = 1 \alpha= 1 α = 1
Full-Sample Fit
Table 2 reports the parameter estimates and calibration errors for January 2018–June 2025. We recalibrate YFK (Ye et al. 2026) to the same VIX term-structure observations using a 20-factor
Table 2. Full-sample term-structure calibration. Panel A reports parameter estimates for January 2018–June 2025 (1,883 trading days). LZZ imposes α = 1 \alpha= 1 α = 1 κ \kappa κ λ \lambda λ
Markovian lift of the fractional volatility kernel.
In addition to VIX levels, we assess term-structure slope and curvature using divided differences. Let y t , j y_{t,j} y t , j t t t τ j \tau_j τ j τ 1 < ⋯ < τ 5 \tau_1 < \cdots< \tau_5 τ 1 < ⋯ < τ 5
s t , j = y t , j + 1 − y t , j τ j + 1 − τ j , j = 1 , … , 4 , c t , j = 2 ( s t , j + 1 − s t , j ) τ j + 2 − τ j , j = 1 , … , 3. (19) \begin{aligned}
s_{t,j} &= \frac{y_{t,j+1} - y_{t,j}}{\tau_{j+1} - \tau_j}, && j = 1,\ldots,4,\\
c_{t,j} &= \frac{2(s_{t,j+1} - s_{t,j})}{\tau_{j+2} - \tau_j}, && j = 1,\ldots,3. \tag*{(19)}
\end{aligned} s t , j c t , j = τ j + 1 − τ j y t , j + 1 − y t , j , = τ j + 2 − τ j 2 ( s t , j + 1 − s t , j ) , j = 1 , … , 4 , j = 1 , … , 3. (19) These definitions account for unequal horizon spacings and are applied to both observed and model-implied VIX levels.
For each measure, MSE is the average squared difference between observed and model-implied values over the relevant dates and horizons. Each model is evaluated on 9,415 level errors, 7,532 slope errors and 5,649 curvature errors. The percentage MSE reduction relative to benchmark B ∈ { L Z Z , Y F K } B \in\{\mathrm{LZZ},\mathrm{YFK}\} B ∈ { LZZ , YFK } 100 ( 1 − M S E R I S V I X / M S E B ) 100(1-\mathrm{MSE}_{\mathrm{RISVIX}}/\mathrm{MSE}_B) 100 ( 1 − MSE RISVIX / MSE B )
RISVIX yields α ^ = 0.8942 \hat{\alpha} = 0.8942 α ^ = 0.8942 H ^ = 0.3942 \hat{H} = 0.3942 H ^ = 0.3942 α ^ = 0.6997 \hat{\alpha} = 0.6997 α ^ = 0.6997 H ^ = 0.1997 \hat{H} = 0.1997 H ^ = 0.1997
State-Dependent Fit
The relative gains of RISVIX vary substantially across market conditions. We split the full calibration sample into dates with large VIX increases and the remaining dates, keeping the full-sample model parameters fixed. A shock date is defined as a daily VIX increase at or above the 90th percentile of the calibration-sample changes. Figure 2 reports percentage MSE reductions for levels, slopes and curvature in the full sample and in the two groups. On shock dates, the respective reductions are 28.73%, 26.34% and 20.60%, compared with 1.18%, 11.71% and 9.31% on the remaining dates. RISVIX therefore improves all three diagnostics in both groups, with larger relative improvements when VIX rises sharply.
Figure 2. Term-structure fit by VIX shocks. Bars report percentage MSE reductions for RISVIX relative to LZZ for each diagnostic and group. The sample runs from January 2, 2018 to June 30, 2025.
We next examine the initial COVID-19 period from January 2 to June 30, 2020, which contains 125 trading days and 625 horizon–date observations. Both models are separately estimated using this period.
Table 3 shows a larger departure from the Markovian restriction during the COVID-19 period, with α ^ = 0.7051 \hat{\alpha} = 0.7051 α ^ = 0.7051 H ^ = 0.2051 \hat{H} = 0.2051 H ^ = 0.2051
Table 3. Term-structure calibration during the COVID-19 period. Panel A reports parameter values for January 2–June 30, 2020 (125 trading days); LZZ imposes α = 1 \alpha= 1 α = 1
Fit across Horizons
Figure 3 decomposes the percentage reduction in level MSE across VIX horizons. Panel (a) uses the full January 2018–June 2025 sample. The largest reduction is 33.30% at nine days, followed by 8.26% at three months and 7.34% at six months. At 30 days and one year, MSE is higher than LZZ, with reductions of −1.05% and −18.83%, respectively.
Figure 3. Term-structure fit across horizons. Bars report horizon-specific percentage reductions in level MSE for RISVIX relative to LZZ. Panel (a) uses the full January 2018–June 2025 sample; Panel (b) uses models separately calibrated over January–June 2020. Each panel has one bar per horizon. Green bars indicate positive reductions and red bars indicate negative reductions. Dashed lines show the corresponding aggregate level-MSE reductions. The panels share the same vertical scale.
Panel (b) uses the separate COVID-19 calibrations and displays one bar per horizon. The largest reduction, 47.69%, occurs at the 30-day horizon, followed by 23.47% at nine days and 18.64% at six months. The one-year MSE declines by 3.20%, while the three-month MSE increases by 3.33%. The dashed lines mark the overall level-MSE reductions of 10.42% and 17.60% for the two calibration samples.
VIX Option Pricing
We next evaluate whether the term-structure flexibility of RISVIX is associated with more accurate out-of-sample VIX option prices. The training period is January 2018–December 2023, and the test period is January 2024–June 2025. The global parameter sets reported below are common to all dates, expirations and strikes and are held fixed throughout the test period. For each test date t 0 t_0 t 0 ( V t 0 , θ t 0 ) (V_{t_0}, \theta_{t_0}) ( V t 0 , θ t 0 ) ( α , κ ) (\alpha, \kappa) ( α , κ )
Parameter Estimates
The first-step inputs comprise the common term-structure parameters ( α , κ ) (\alpha, \kappa) ( α , κ ) ( V t 0 , θ t 0 ) (V_{t_0}, \theta_{t_0}) ( V t 0 , θ t 0 ) ( σ V , σ θ , λ y , μ y ) (\sigma_V, \sigma_\theta, \lambda_y, \mu_y) ( σ V , σ θ , λ y , μ y ) Table 4 reports the global parameter values used to generate the option prices in this subsection. RISVIX uses α ^ = 0.8937 \hat{\alpha}=0.8937 α ^ = 0.8937 H ^ = 0.3937 \hat{H}=0.3937 H ^ = 0.3937 κ ^ = 7.0357 \hat{\kappa}=7.0357 κ ^ = 7.0357 α = 1 \alpha=1 α = 1 κ ^ = 6.2450 \hat{\kappa}=6.2450 κ ^ = 6.2450 σ ^ V \hat{\sigma}_V σ ^ V σ ^ θ \hat{\sigma}_\theta σ ^ θ λ ^ y \hat{\lambda}^y λ ^ y μ ^ y \hat{\mu}_y μ ^ y
Table 4 . Option-pricing parameters.α \alpha α H H H κ \kappa κ H = α − 1 / 2 H=\alpha-1/2 H = α − 1/2
Table 4.
Pricing Performance
Table 5 reports mean squared error (MSE) and mean absolute error (MAE) for the complete option sample and for maturity and moneyness groups. Moneyness is measured as K / V I X t K/\mathrm{VIX}_t K / VIX t < 0.9 <0.9 < 0.9 ∈ [ 0.9 , 1.1 ] \in[0.9,1.1] ∈ [ 0.9 , 1.1 ] > 1.1 >1.1 > 1.1
Table 5. VIX option-pricing errors. Panel A uses 47,451 training-period quotes; Panel B uses 14,543 test-period quotes. MSE and MAE are computed from option prices in VIX index points. Red. is the percentage error reduction relative to LZZ; positive values favour RISVIX. The lower error in each model pair is in bold.
Table 5.
In sample, RISVIX lowers overall MSE from 0.5548 to 0.5015 and MAE from 0.5815 to 0.5528, corresponding to reductions of 9.61% and 4.93%. Both error measures are lower in every reported group, although the reductions are small for moneyness below 0.9. Out of sample, the overall MSE falls from 0.7238 to 0.6293 and MAE from 0.7537 to 0.7061, reductions of 13.06% and 6.33%. The shorter-maturity group has larger test-period gains, with MSE and MAE reductions of 15.10% and 6.83%, compared with 11.69% and 6.03% at 60–126 days. Across moneyness, RISVIX improves both measures in all three groups; the largest test-period reductions occur below 0.9, at 23.71% for MSE and 12.16% for MAE. The gains are smaller above 1.1, where MSE and MAE decline by 3.93% and 0.48%.
Figure 4 provides selected price cross sections for three training-period dates and one test-period date. Both models reproduce the decline in call prices across strikes. In these examples, LZZ tends to lie above market prices below spot VIX and below market prices at higher strikes. RISVIX reduces this tilt and has a lower mean squared error in each displayed cross section. The fitted curves remain close near the centre of the strike range, with more visible differences toward its edges. These selected examples illustrate cross-sectional shape; the complete-sample comparisons in Table 5 provide the aggregate model ranking.
Figure 4. Cross-sectional VIX option fits. Each panel compares market midquotes with RISVIX and LZZ prices across strikes for a fixed date and expiration. The vertical dotted line marks observed spot VIX. Panels (a)–(c) are in-sample; Panel (d) is out-of-sample. Model prices use the parameter values in Table 4 .
Pricing Performance across VIX States
Given the state dependence in term-structure fit, we next examine whether a similar pattern appears in VIX option prices. Figure 5 reports quote-weighted MSE reductions by the observed VIX level. Within each bin, MSE is computed by pooling all quote errors before taking the ratio between models.
Figure 5. Option-pricing fit across VIX states. Bars report quote-weighted percentage MSE reductions for RISVIX relative to LZZ across bins of observed VIX. Panel (a) is in-sample and Panel (b) is out-of-sample. Bins are left-closed and right-open, and the two panels share a vertical scale; positive values favour RISVIX.
In sample, the reductions are 4.4% below VIX 20 and 4.9% between 20 and 25, rising to 14.4% between 25 and 30, 16.1% between 30 and 40, and 17.9% at VIX 40 and above. Out of sample, RISVIX also improves every bin: MSE reductions are 10.0%, 21.5%, 37.4%, 45.9% and 39.6%, respectively. The largest test-period gain occurs in the 30–40 bin.
The broader split at VIX 25 supports the same pattern. For VIXt _t t ≥ \ge ≥
Implied-Volatility Fit
We complement the price-based evaluation with an implied-volatility representation for March 17, 2023. Figure 6 compares market implied volatilities with those obtained from the RISVIX prices for this date. For the Black conversion of model prices, each expiration uses the median forward inferred from its market midquotes and reported implied volatilities, with zero discounting; these conversion inputs are not observed futures quotes.
Figure 6. VIX option implied-volatility surface. Panel (a) compares market and RISVIX implied volatilities across moneyness for four maturities on March 17, 2023. Panel (b) shows the corresponding market points and the interpolated RISVIX surface.
Panel (a) reports slices at 33, 61, 96 and 124 days to expiration. Market and RISVIX implied volatilities both increase with K / V I X t K/\mathrm{VIX}_t K / VIX t
Panel (b) displays the same observations as market points and a piecewise-linear RISVIX surface across moneyness and maturity. The surface represents an interpolation of the model values at the observed contracts. The date is selected to illustrate four well-populated, upward-sloping maturity slices.
Comparison with Alternative VIX Option Pricing Models
We compare the following five models:
RISVIX: The proposed rough ISVIX model, which introduces a fractional Volterra kernel into the dynamics of instantaneous squared VIX.
LZZ: The Markovian ISVIX model of Luo et al. [23 ] , with α = 1 \alpha= 1 α = 1
HJT: The modulated Volterra stochastic volatility model of Horvath et al. [16 ] , which introduces an additional stochastic factor to capture the VIX implied-volatility smile.
AIL: The Gaussian polynomial volatility models of Abi Jaber et al. [2 ] , in which VIX is derived from conditional integrated variance.
BPS: The rough Hawkes–Heston model of Bondi et al. [7 ] , which combines fractional volatility dynamics with self-exciting jumps and uses affine transform methods for VIX option pricing.
RISVIX and LZZ model instantaneous squared VIX directly, whereas HJT, AIL and BPS derive VIX from the volatility dynamics of the underlying equity index. Table 6 reports their pricing errors on the same VIX call-option sample.
Table 6. Comparison with alternative VIX option pricing models. Calibration: January 2018–December 2023; evaluation: January 2024–June 2025.
RISVIX achieves the lowest MSE and MAE among the five models in both periods. The relatively high calibration MSE of BPS mainly reflects larger pricing errors during the COVID-19 period in 2020. In the evaluation period, BPS is the closest competitor: its MSE is 0.6757, compared with 0.6293 for RISVIX, while their MAEs are nearly identical at 0.7067 and 0.7061, respectively.
Conclusion
This paper develops RISVIX, a fractional extension of the ISVIX framework of Luo et al. [23 ] for modeling the VIX term structure and pricing VIX options. We introduce a fractional Volterra kernel directly into instantaneous squared VIX and derive an explicit term structure governed by Mittag–Leffler decay, together with a semi-closed-form option-pricing formula based on an affine Fourier–Laplace transform. The separation between term-structure and distributional parameters supports a two-step procedure that estimates memory, mean reversion and date-specific initial states from VIX indices before calibrating the remaining parameters to VIX option prices.
Using daily VIX term-structure indices and option quotes from January 2018 to June 2025, we find that RISVIX improves term-structure fit, particularly during periods of market stress. Relative to LZZ, the full-sample MSE reductions are 10.42%, 17.21% and 13.18% for VIX levels, slopes and curvature, respectively. Over January–June 2020, these reductions increase to 17.60%, 41.14% and 46.59%, with an estimated fractional parameter of α ^ = 0.7051 \hat{\alpha} = 0.7051 α ^ = 0.7051
The option-pricing results show a similar pattern. With global parameters fixed at their training-sample estimates and date-specific initial states updated from the contemporaneous VIX term structure, RISVIX reduces out-of-sample MSE and MAE by 13.06% and 6.33%, respectively, relative to LZZ. The gains are larger for shorter-maturity options and high-VIX states, with an out-of-sample MSE reduction of 40.93% when V I X t ≥ 25 \mathrm{VIX}_t \ge25 VIX t ≥ 25
Future work could extend RISVIX to a joint SPX–VIX framework to impose consistency across equity and volatility markets ([9 , 27 ] ), or enrich the direct VIX specification with multiple memory factors and more general path-dependent feedback ([15 , 19 ] ).
Acknowledgements
Wen Su acknowledges the support from the EPSRC Centre for Doctoral Training in Mathematics of Random Systems: Analysis, Modelling and Simulation (EP/S023925/1).
Disclosure statement
No potential conflict of interest was reported by the author(s).
Technical Lemmas
Resolvent Identities
For scalar functions f f f g g g ( f ∗ g ) ( t ) = ∫ 0 t f ( t − s ) g ( s ) d s (f \ast g)(t) = \int_{0}^{t} f(t-s)g(s)\,\mathrm{d}s ( f ∗ g ) ( t ) = ∫ 0 t f ( t − s ) g ( s ) d s κ > 0 \kappa> 0 κ > 0 R κ R_{\kappa} R κ κ K \kappa K κ K
R κ + κ K ∗ R κ = κ K , K − R κ ∗ K = 1 κ R κ . (20) R_{\kappa}+\kappa K\ast R_{\kappa}=\kappa K,\qquad K-R_{\kappa}\ast K=\frac{1}{\kappa}R_{\kappa}. \tag*{(20)} R κ + κ K ∗ R κ = κ K , K − R κ ∗ K = κ 1 R κ . (20) Consequently, for locally integrable f f f g g g g = f − κ K ∗ g g=f-\kappa K\ast g g = f − κ K ∗ g g = f − R κ ∗ f g=f-R_{\kappa}\ast f g = f − R κ ∗ f [3 ] .
For a locally square-integrable martingale M M M ( K ∗ d M ) t = ∫ 0 t K ( t − s ) d M s (K\ast\mathrm{d}M)_t=\int_{0}^{t}K(t-s)\,\mathrm{d}M_s ( K ∗ d M ) t = ∫ 0 t K ( t − s ) d M s
g = f − κ K ∗ g + K ∗ d M ⟺ g = f − R κ ∗ f + 1 κ R κ ∗ d M . (21) g=f-\kappa K\ast g+K\ast\mathrm{d}M\quad\Longleftrightarrow\quad g=f-R_{\kappa}\ast f+\frac{1}{\kappa}R_{\kappa}\ast\mathrm{d}M. \tag*{(21)} g = f − κ K ∗ g + K ∗ d M ⟺ g = f − R κ ∗ f + κ 1 R κ ∗ d M . (21) Indeed, stochastic Fubini gives R κ ∗ ( K ∗ d M ) = ( R κ ∗ K ) ∗ d M R_{\kappa}\ast(K\ast\mathrm{d}M)=(R_{\kappa}\ast K)\ast\mathrm{d}M R κ ∗ ( K ∗ d M ) = ( R κ ∗ K ) ∗ d M M M M
Lemma A.1. Let K ( t ) = t α − 1 / Γ ( α ) K(t)=t^{\alpha-1}/\Gamma(\alpha) K ( t ) = t α − 1 /Γ ( α ) α ∈ ( 1 / 2 , 1 ] \alpha\in(1/2,1] α ∈ ( 1/2 , 1 ] κ > 0 \kappa>0 κ > 0 t > 0 t>0 t > 0
R κ ( t ) = κ t α − 1 E α , α ( − κ t α ) = − d d t E α , 1 ( − κ t α ) , R_{\kappa}(t)=\kappa t^{\alpha-1}E_{\alpha,\alpha}(-\kappa t^{\alpha})=-\frac{\mathrm{d}}{\mathrm{d}t}E_{\alpha,1}(-\kappa t^{\alpha}), R κ ( t ) = κ t α − 1 E α , α ( − κ t α ) = − d t d E α , 1 ( − κ t α ) , ∫ 0 t R κ ( s ) d s = 1 − E α , 1 ( − κ t α ) , (22) \int_{0}^{t}R_{\kappa}(s)\,\mathrm{d}s=1-E_{\alpha,1}(-\kappa t^{\alpha}), \tag*{(22)} ∫ 0 t R κ ( s ) d s = 1 − E α , 1 ( − κ t α ) , (22) ∫ 0 t E α , 1 ( − κ s α ) d s = t E α , 2 ( − κ t α ) = t [ 1 − κ t α E α , α + 2 ( − κ t α ) ] . \int_{0}^{t}E_{\alpha,1}(-\kappa s^{\alpha})\,\mathrm{d}s=tE_{\alpha,2}(-\kappa t^{\alpha})=t[1-\kappa t^{\alpha}E_{\alpha,\alpha+2}(-\kappa t^{\alpha})]. ∫ 0 t E α , 1 ( − κ s α ) d s = t E α , 2 ( − κ t α ) = t [ 1 − κ t α E α , α + 2 ( − κ t α )] . And the weight w w w (11) is strictly decreasing on ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) w ( 0 + ) = 1 w(0+)=1 w ( 0 + ) = 1 w ( ∞ ) = 0 w(\infty)=0 w ( ∞ ) = 0
Proof. The beta integral gives K ∗ n ( t ) = t n α − 1 / Γ ( n α ) K^{\ast n}(t)=t^{n\alpha-1}/\Gamma(n\alpha) K ∗ n ( t ) = t n α − 1 /Γ ( n α ) K ∗ n K^{\ast n} K ∗ n n n n K K K
R κ ( t ) = ∑ n = 1 ∞ ( − 1 ) n − 1 κ n K ∗ n ( t ) = κ t α − 1 E α , α ( − κ t α ) (23) R_{\kappa}(t)=\sum_{n=1}^{\infty}(-1)^{n-1}\kappa^{n}K^{\ast n}(t)=\kappa t^{\alpha-1}E_{\alpha,\alpha}(-\kappa t^{\alpha}) \tag*{(23)} R κ ( t ) = n = 1 ∑ ∞ ( − 1 ) n − 1 κ n K ∗ n ( t ) = κ t α − 1 E α , α ( − κ t α ) (23) converges in L 1 ( 0 , T ) L^{1}(0,T) L 1 ( 0 , T ) T T T ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) E α , 1 ( 0 ) = 1 E_{\alpha,1}(0)=1 E α , 1 ( 0 ) = 1 E α , 1 E_{\alpha,1} E α , 1 E α , 2 ( z ) = 1 + z E α , α + 2 ( z ) E_{\alpha,2}(z)=1+zE_{\alpha,\alpha+2}(z) E α , 2 ( z ) = 1 + z E α , α + 2 ( z )
For 0 < α < 1 0<\alpha<1 0 < α < 1 t ↦ E α , 1 ( − κ t α ) t\mapsto E_{\alpha,1}(-\kappa t^{\alpha}) t ↦ E α , 1 ( − κ t α ) [24 ] . For α = 1 \alpha=1 α = 1 e − κ t e^{-\kappa t} e − κ t w ( t ) = t − 1 ∫ 0 t E α , 1 ( − κ s α ) d s w(t)=t^{-1}\int_{0}^{t}E_{\alpha,1}(-\kappa s^{\alpha})\,\mathrm{d}s w ( t ) = t − 1 ∫ 0 t E α , 1 ( − κ s α ) d s w ′ ( t ) = [ E α , 1 ( − κ t α ) − w ( t ) ] / t < 0 w'(t) = [E_{\alpha,1}(-\kappa t^\alpha) - w(t)]/t < 0 w ′ ( t ) = [ E α , 1 ( − κ t α ) − w ( t )] / t < 0
Remark. Integrating the first identity in Equation A.3 over [ s , s + τ 0 ] [s, s+\tau_0] [ s , s + τ 0 ] (13) . This expression extends continuously to s = 0 s=0 s = 0 α = 1 \alpha=1 α = 1 a τ 0 ( s ) = w ( τ 0 ) e − κ s a_{\tau_0}(s) = w(\tau_0)e^{-\kappa s} a τ 0 ( s ) = w ( τ 0 ) e − κ s
Moment Conditions
Fix T > 0 T>0 T > 0 α ∈ ( 1 / 2 , 1 ) \alpha\in(1/2,1) α ∈ ( 1/2 , 1 ) κ , μ y > 0 \kappa,\mu_y>0 κ , μ y > 0 σ V , σ θ , λ y ≥ 0 \sigma_V,\sigma_\theta,\lambda^y\geq0 σ V , σ θ , λ y ≥ 0 ( V , θ ) (V,\theta) ( V , θ ) (9) on [ 0 , T ] [0,T] [ 0 , T ] F 0 \mathcal{F}_0 F 0 E Q [ V 0 2 + θ 0 2 ] < ∞ \mathbb{E}^{\mathbb{Q}}[V_0^2+\theta_0^2]<\infty E Q [ V 0 2 + θ 0 2 ] < ∞ θ \theta θ V V V ∫ 0 T V s d s < ∞ \int_0^T V_s\,\mathrm{d}s<\infty ∫ 0 T V s d s < ∞ V V V Q ⊗ d t \mathbb{Q}\otimes\mathrm{d}t Q ⊗ d t
Lemma A.2. Under the preceding assumptions, θ \theta θ 0 ≤ s ≤ t ≤ T 0\leq s\leq t\leq T 0 ≤ s ≤ t ≤ T E s Q [ θ t ] = θ s \mathbb{E}^{\mathbb{Q}}_s[\theta_t]=\theta_s E s Q [ θ t ] = θ s E s Q [ θ t 2 ] = θ s 2 + σ θ 2 ( t − s ) θ s \mathbb{E}^{\mathbb{Q}}_s[\theta_t^2]=\theta_s^2+\sigma_\theta^2(t-s)\theta_s E s Q [ θ t 2 ] = θ s 2 + σ θ 2 ( t − s ) θ s
E Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] ≤ C T ( 1 + E Q [ V 0 2 + θ 0 2 ] ) < ∞ , (24) \mathbb{E}^{\mathbb{Q}}\left[\int_0^T (V_t^2+\theta_t^2)\,\mathrm{d}t\right]\leq C_T\left(1+\mathbb{E}^{\mathbb{Q}}[V_0^2+\theta_0^2]\right)<\infty, \tag*{(24)} E Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] ≤ C T ( 1 + E Q [ V 0 2 + θ 0 2 ] ) < ∞ , (24) where C T C_T C T T , α , κ , σ V , σ θ , λ y , μ y T,\alpha,\kappa,\sigma_V,\sigma_\theta,\lambda^y,\mu_y T , α , κ , σ V , σ θ , λ y , μ y E 0 Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] < ∞ \mathbb{E}^{\mathbb{Q}}_0\left[\int_0^T(V_t^2+\theta_t^2)\,\mathrm{d}t\right]<\infty E 0 Q [ ∫ 0 T ( V t 2 + θ t 2 ) d t ] < ∞
Proof. Since θ \theta θ E Q [ ∫ 0 T θ s d s ] ≤ T E Q [ θ 0 ] < ∞ \mathbb{E}^{\mathbb{Q}}[\int_0^T\theta_s\,\mathrm{d}s]\leq T\mathbb{E}^{\mathbb{Q}}[\theta_0]<\infty E Q [ ∫ 0 T θ s d s ] ≤ T E Q [ θ 0 ] < ∞ θ \theta θ θ 0 ∈ L 2 ( Q ) \theta_0\in L^2(\mathbb{Q}) θ 0 ∈ L 2 ( Q ) E s Q [ ( θ t − θ s ) 2 ] = σ θ 2 ( t − s ) θ s \mathbb{E}^{\mathbb{Q}}_s[(\theta_t-\theta_s)^2]=\sigma_\theta^2(t-s)\theta_s E s Q [( θ t − θ s ) 2 ] = σ θ 2 ( t − s ) θ s
E Q [ ∫ 0 T θ t 2 d t ] = T E Q [ θ 0 2 ] + σ θ 2 T 2 2 E Q [ θ 0 ] . (25) \mathbb{E}^{\mathbb{Q}}\left[\int_0^T\theta_t^2\,\mathrm{d}t\right]
=T\mathbb{E}^{\mathbb{Q}}[\theta_0^2]+\frac{\sigma_\theta^2T^2}{2}\mathbb{E}^{\mathbb{Q}}[\theta_0]. \tag*{(25)} E Q [ ∫ 0 T θ t 2 d t ] = T E Q [ θ 0 2 ] + 2 σ θ 2 T 2 E Q [ θ 0 ] . (25) Let R κ R_\kappa R κ q κ ( t ) = E α , 1 ( − κ t α ) q_\kappa(t)=E_{\alpha,1}(-\kappa t^\alpha) q κ ( t ) = E α , 1 ( − κ t α ) Lemma A.1 , q κ q_\kappa q κ R κ ∈ L 1 ( 0 , T ) ∩ L 2 ( 0 , T ) R_\kappa\in L^1(0,T)\cap L^2(0,T) R κ ∈ L 1 ( 0 , T ) ∩ L 2 ( 0 , T ) K ∈ L 2 ( 0 , T ) K\in L^2(0,T) K ∈ L 2 ( 0 , T )
τ n = inf { t ∈ [ 0 , T ] : ∫ 0 t V s d s ≥ n } ∧ T , I s ( n ) = 1 { s ≤ τ n } . (26) \tau_n=\inf\left\{t\in[0,T]:\int_0^t V_s\,\mathrm{d}s\geq n\right\}\wedge T,\qquad I_s^{(n)}=\mathbf{1}_{\{s\leq\tau_n\}}. \tag*{(26)} τ n = inf { t ∈ [ 0 , T ] : ∫ 0 t V s d s ≥ n } ∧ T , I s ( n ) = 1 { s ≤ τ n } . (26) The cumulative integral is continuous and adapted, so τ n \tau_n τ n I ( n ) I^{(n)} I ( n ) ∫ 0 T I s ( n ) V s d s ≤ n \int_0^T I_s^{(n)}V_s\,\mathrm{d}s\leq n ∫ 0 T I s ( n ) V s d s ≤ n τ n ↑ T \tau_n\uparrow T τ n ↑ T h ∈ L 2 ( 0 , T ) h \in L^2(0,T) h ∈ L 2 ( 0 , T )
E Q [ ∫ 0 T ∣ ∫ 0 t h ( t − s ) I s ( n ) V s d B 1 , s ∣ 2 d t ] = E Q [ ∫ 0 T ∫ 0 t h ( t − s ) 2 I s ( n ) V s d s d t ] ≤ n ∥ h ∥ L 2 ( 0 , T ) 2 . (27) \mathbb{E}^{\mathbb{Q}}\left[\int_0^T\left|\int_0^t h(t-s)I_s^{(n)}\sqrt{V_s}\,\mathrm{d}B_{1,s}\right|^2\mathrm{d}t\right]
=
\mathbb{E}^{\mathbb{Q}}\left[\int_0^T\int_0^t h(t-s)^2I_s^{(n)}V_s\,\mathrm{d}s\,\mathrm{d}t\right]
\le n\|h\|_{L^2(0,T)}^2. \tag*{(27)} E Q [ ∫ 0 T ∫ 0 t h ( t − s ) I s ( n ) V s d B 1 , s 2 d t ] = E Q [ ∫ 0 T ∫ 0 t h ( t − s ) 2 I s ( n ) V s d s d t ] ≤ n ∥ h ∥ L 2 ( 0 , T ) 2 . (27) In particular, r ↦ ( ∫ 0 T K ( r − s ) 2 I s ( n ) V s d s ) 1 / 2 r \mapsto\left(\int_0^T K(r-s)^2I_s^{(n)}V_s\,\mathrm{d}s\right)^{1/2} r ↦ ( ∫ 0 T K ( r − s ) 2 I s ( n ) V s d s ) 1/2 L 2 ( 0 , T ) L^2(0,T) L 2 ( 0 , T ) ∣ R κ ∣ |\mathcal{R}_{\kappa}| ∣ R κ ∣ t < τ n t<\tau_n t < τ n
V t ( n ) = q κ ( t ) V 0 + ∫ 0 t R κ ( t − s ) θ s d s + σ V κ ∫ 0 t R κ ( t − s ) I s ( n ) V s d B 1 , s + 1 κ ∫ 0 t R κ ( t − s ) d J s Y (28) V_t^{(n)}=q_{\kappa}(t)V_0+\int_0^t R_{\kappa}(t-s)\theta_s\,\mathrm{d}s
+\frac{\sigma_V}{\kappa}\int_0^t R_{\kappa}(t-s)I_s^{(n)}\sqrt{V_s}\,\mathrm{d}B_{1,s}
+\frac{1}{\kappa}\int_0^t R_{\kappa}(t-s)\,\mathrm{d}J_s^Y \tag*{(28)} V t ( n ) = q κ ( t ) V 0 + ∫ 0 t R κ ( t − s ) θ s d s + κ σ V ∫ 0 t R κ ( t − s ) I s ( n ) V s d B 1 , s + κ 1 ∫ 0 t R κ ( t − s ) d J s Y (28) agrees with V t V_t V t { t < τ n } \{t<\tau_n\} { t < τ n } Q ⊗ d t \mathbb{Q}\otimes\mathrm{d}t Q ⊗ d t F n = E Q [ ∫ 0 T ∣ V t ( n ) ∣ 2 d t ] F_n=\mathbb{E}^{\mathbb{Q}}\left[\int_0^T|V_t^{(n)}|^2\mathrm{d}t\right] F n = E Q [ ∫ 0 T ∣ V t ( n ) ∣ 2 d t ] h = R κ h=R_{\kappa} h = R κ E Q [ Y 1 2 ] = 2 μ y 2 \mathbb{E}^{\mathbb{Q}}[Y_1^2]=2\mu_y^2 E Q [ Y 1 2 ] = 2 μ y 2
F n ≤ 4 T ∥ q κ ∥ L ∞ ( 0 , T ) 2 E Q [ V 0 2 ] + 4 ∥ R κ ∥ L 1 ( 0 , T ) 2 E Q [ ∫ 0 T θ s 2 d s ] + 8 λ μ y 2 T κ 2 ∥ R κ ∥ L 2 ( 0 , T ) 2 + 4 σ V 2 κ 2 ∥ R κ ∥ L 2 ( 0 , T ) 2 E Q [ ∫ 0 T I s ( n ) V s d s ] . (29) \begin{aligned}
F_n \le{}&4T\|q_{\kappa}\|_{L^\infty(0,T)}^2\mathbb{E}^{\mathbb{Q}}[V_0^2]
+4\|R_{\kappa}\|_{L^1(0,T)}^2\mathbb{E}^{\mathbb{Q}}\left[\int_0^T\theta_s^2\,\mathrm{d}s\right]\\
&+\frac{8\lambda\mu_y^2T}{\kappa^2}\|R_{\kappa}\|_{L^2(0,T)}^2
+\frac{4\sigma_V^2}{\kappa^2}\|R_{\kappa}\|_{L^2(0,T)}^2\mathbb{E}^{\mathbb{Q}}\left[\int_0^T I_s^{(n)}V_s\,\mathrm{d}s\right]. \tag*{(29)}
\end{aligned} F n ≤ 4 T ∥ q κ ∥ L ∞ ( 0 , T ) 2 E Q [ V 0 2 ] + 4∥ R κ ∥ L 1 ( 0 , T ) 2 E Q [ ∫ 0 T θ s 2 d s ] + κ 2 8 λ μ y 2 T ∥ R κ ∥ L 2 ( 0 , T ) 2 + κ 2 4 σ V 2 ∥ R κ ∥ L 2 ( 0 , T ) 2 E Q [ ∫ 0 T I s ( n ) V s d s ] . (29) The right-hand side is finite for each n n n n n n F n < ∞ F_n<\infty F n < ∞ τ n \tau_n τ n V V V
E Q [ ∫ 0 T I s ( n ) V s d s ] ≤ E Q [ ∫ 0 T V s ( n ) d s ] ≤ T F n . (30) \mathbb{E}^{\mathbb{Q}}\left[\int_0^T I_s^{(n)}V_s\,\mathrm{d}s\right]
\le\mathbb{E}^{\mathbb{Q}}\left[\int_0^T V_s^{(n)}\,\mathrm{d}s\right]
\le\sqrt{TF_n}. \tag*{(30)} E Q [ ∫ 0 T I s ( n ) V s d s ] ≤ E Q [ ∫ 0 T V s ( n ) d s ] ≤ T F n . (30) Substituting this bound into Equation A.10, using the moment formula for θ \theta θ E Q [ θ 0 ] ≤ ( 1 + E Q [ θ 0 2 ] ) / 2 \mathbb{E}^{\mathbb{Q}}[\theta_0]\le(1+\mathbb{E}^{\mathbb{Q}}[\theta_0^2])/2 E Q [ θ 0 ] ≤ ( 1 + E Q [ θ 0 2 ]) /2 F n ≤ C T ( 1 + E Q [ V 0 2 + θ 0 2 ] ) + F n / 2 F_n\le C_T(1+\mathbb{E}^{\mathbb{Q}}[V_0^2+\theta_0^2])+F_n/2 F n ≤ C T ( 1 + E Q [ V 0 2 + θ 0 2 ]) + F n /2 C T C_T C T n n n τ n ↑ T \tau_n\uparrow T τ n ↑ T τ n \tau_n τ n V ( n ) → V V^{(n)}\to V V ( n ) → V Q ⊗ d t \mathbb{Q}\otimes\mathrm{d}t Q ⊗ d t θ \theta θ F 0 \mathcal{F}_0 F 0
Lemma A.3. Suppose that Equation (9) admits the solution specified above on [ 0 , T ] [0,T] [ 0 , T ] c T > 0 c_T>0 c T > 0 T T T
E 0 Q [ exp { c ∫ 0 T V t d t } ] < ∞ (31) \mathbb{E}^{\mathbb{Q}}_0\left[\exp\left\{c\int_0^T V_t\,\mathrm{d}t\right\}\right]<\infty \tag*{(31)} E 0 Q [ exp { c ∫ 0 T V t d t } ] < ∞ (31) almost surely for every c ∈ [ 0 , c T ) c \in[0,c_T) c ∈ [ 0 , c T )
Proof. Let X T = ∫ 0 T V t d t X_T=\int_0^T V_t\,\mathrm{d}t X T = ∫ 0 T V t d t q κ ( t ) = E α , 1 ( − κ t α ) q_\kappa(t)=E_{\alpha,1}(-\kappa t^\alpha) q κ ( t ) = E α , 1 ( − κ t α ) Lemma A.2 , followed by integration in time, gives
X T = V 0 ∫ 0 T q κ ( t ) d t + ∫ 0 T b T ( s ) θ s d s + σ V κ ∫ 0 T b T ( s ) V s d B 1 , s + 1 κ ∫ 0 T b T ( s ) d J s Y . (32) X_T=V_0\int_0^T q_\kappa(t)\,\mathrm{d}t+\int_0^T b_T(s)\theta_s\,\mathrm{d}s
+\frac{\sigma_V}{\kappa}\int_0^T b_T(s)\sqrt{V_s}\,\mathrm{d}B_{1,s}+\frac{1}{\kappa}\int_0^T b_T(s)\,\mathrm{d}J_s^Y. \tag*{(32)} X T = V 0 ∫ 0 T q κ ( t ) d t + ∫ 0 T b T ( s ) θ s d s + κ σ V ∫ 0 T b T ( s ) V s d B 1 , s + κ 1 ∫ 0 T b T ( s ) d J s Y . (32) where b T ( s ) = 1 − q κ ( T − s ) ∈ [ 0 , 1 ] b_T(s)=1-q_\kappa(T-s)\in[0,1] b T ( s ) = 1 − q κ ( T − s ) ∈ [ 0 , 1 ] Lemma A.1 . Here R κ ∈ L 2 ( 0 , T ) R_\kappa\in L^2(0,T) R κ ∈ L 2 ( 0 , T ) E Q [ X T ] < ∞ \mathbb{E}^{\mathbb{Q}}[X_T]<\infty E Q [ X T ] < ∞ C T C_T C T T T T
Choose a > 0 a>0 a > 0 a σ θ 2 T ≤ 1 a\sigma_\theta^2T\leq1 a σ θ 2 T ≤ 1 s ∈ [ 0 , T ] s\in[0,T] s ∈ [ 0 , T ] exp { a θ v / [ 1 − a σ θ 2 ( s − v ) / 2 ] } \exp\{a\theta_v/[1-a\sigma_\theta^2(s-v)/2]\} exp { a θ v / [ 1 − a σ θ 2 ( s − v ) /2 ]} 0 ≤ v ≤ s 0\leq v\leq s 0 ≤ v ≤ s { θ 0 ≤ m } \{\theta_0\leq m\} { θ 0 ≤ m } θ \theta θ m ↑ ∞ m\uparrow\infty m ↑ ∞
E 0 Q [ e a θ s ] ≤ exp { a θ 0 1 − a σ θ 2 s / 2 } ≤ e 2 a θ 0 . (33) \mathbb{E}^{\mathbb{Q}}_0[e^{a\theta_s}]
\leq\exp\left\{\frac{a\theta_0}{1-a\sigma_\theta^2s/2}\right\}
\leq e^{2a\theta_0}. \tag*{(33)} E 0 Q [ e a θ s ] ≤ exp { 1 − a σ θ 2 s /2 a θ 0 } ≤ e 2 a θ 0 . (33) Conditional Markov’s inequality gives Q ( θ s > 2 θ 0 + y ∣ F 0 ) ≤ e − a y \mathbb{Q}(\theta_s>2\theta_0+y\mid\mathcal{F}_0)\leq e^{-ay} Q ( θ s > 2 θ 0 + y ∣ F 0 ) ≤ e − a y y ≥ 0 y\geq0 y ≥ 0 ( E 0 Q [ θ s p ] ) 1 / p ≤ 2 θ 0 + ( p ! ) 1 / p / a ≤ 2 θ 0 + p / a (\mathbb{E}^{\mathbb{Q}}_0[\theta_s^p])^{1/p}\leq2\theta_0+(p!)^{1/p}/a\leq2\theta_0+p/a ( E 0 Q [ θ s p ] ) 1/ p ≤ 2 θ 0 + ( p ! ) 1/ p / a ≤ 2 θ 0 + p / a p ≥ 2 p\geq2 p ≥ 2 L p L^p L p ∫ 0 T b T ( s ) θ s d s \int_0^T b_T(s)\theta_s\,\mathrm{d}s ∫ 0 T b T ( s ) θ s d s C T ( θ 0 + p ) C_T(\theta_0+p) C T ( θ 0 + p )
For the jump term, ∣ ∫ 0 T b T ( s ) d J s Y ∣ ≤ ∑ j = 1 N T Y Y j + λ Y μ Y T \left|\int_0^T b_T(s)\,\mathrm{d}J_s^Y\right|\leq\sum_{j=1}^{N_T^Y}Y_j+\lambda^Y\mu_YT ∫ 0 T b T ( s ) d J s Y ≤ ∑ j = 1 N T Y Y j + λ Y μ Y T F 0 \mathcal{F}_0 F 0 E 0 Q [ exp { ( 2 μ Y ) − 1 ∑ j = 1 N T Y Y j } ] = e λ Y T \mathbb{E}^{\mathbb{Q}}_0[\exp\{(2\mu_Y)^{-1}\sum_{j=1}^{N_T^Y}Y_j\}]=e^{\lambda^YT} E 0 Q [ exp {( 2 μ Y ) − 1 ∑ j = 1 N T Y Y j }] = e λ Y T L p L^p L p C T p C_Tp C T p p ≥ 2 p\geq2 p ≥ 2
Set x p = ( E 0 Q [ X T p ] ) 1 / p x_p=(\mathbb{E}^{\mathbb{Q}}_0[X_T^p])^{1/p} x p = ( E 0 Q [ X T p ] ) 1/ p Lemma A.2 and Cauchy–Schwarz, x 2 < ∞ x_2<\infty x 2 < ∞ L p L^p L p 2 2 p 2\sqrt{2p} 2 2 p b T 2 V ≤ V b_T^2V\leq V b T 2 V ≤ V p ≥ 4 p\geq4 p ≥ 4 x p / 2 < ∞ x_{p/2}<\infty x p /2 < ∞
x p ≤ C T ( 1 + V 0 + θ 0 + p ) + C T p x p / 2 1 / 2 . (34) x_p\leq C_T(1+V_0+\theta_0+p)+C_T\sqrt{p}\,x_{p/2}^{1/2}. \tag*{(34)} x p ≤ C T ( 1 + V 0 + θ 0 + p ) + C T p x p /2 1/2 . (34) For the conditional inequality, first stop the Brownian integral and restrict to F 0 \mathcal{F}_0 F 0 x p / 2 x_{p/2} x p /2 x p < ∞ x_p<\infty x p < ∞ p ≥ 2 p\geq2 p ≥ 2
For dyadic p ≥ 4 p\geq4 p ≥ 4 x p / 2 ≤ x p x_{p/2}\leq x_p x p /2 ≤ x p x p x_p x p L p L^p L p x p ≤ A T + C T p x_p\leq A_T+C_Tp x p ≤ A T + C T p p ≥ 2 p\geq2 p ≥ 2 A T = C T ( 1 + V 0 + θ 0 ) < ∞ A_T=C_T(1+V_0+\theta_0)<\infty A T = C T ( 1 + V 0 + θ 0 ) < ∞ C T C_T C T E 0 Q [ X T n ] ≤ 2 n − 1 ( A T n + C T n n n ) \mathbb{E}^{\mathbb{Q}}_0[X_T^n]\leq2^{n-1}(A_T^n+C_T^nn^n) E 0 Q [ X T n ] ≤ 2 n − 1 ( A T n + C T n n n ) n ≥ 2 n\geq2 n ≥ 2 n ! ≥ ( n / e ) n n!\geq(n/e)^n n ! ≥ ( n / e ) n E 0 Q [ e c X T ] < ∞ \mathbb{E}^{\mathbb{Q}}_0[e^{cX_T}]<\infty E 0 Q [ e c X T ] < ∞ 2 e C T c < 1 2eC_Tc<1 2 e C T c < 1 c T = ( 2 e C T ) − 1 c_T=(2eC_T)^{-1} c T = ( 2 e C T ) − 1
Fix T , τ 0 > 0 T,\tau_0>0 T , τ 0 > 0 η ≥ 0 \eta\ge0 η ≥ 0
p η ( s ) = η a τ 0 ( s ) + σ V 2 2 κ ∫ 0 s R κ ( s − v ) p η ( v ) 2 d v , 0 ≤ s ≤ T , (35) p_\eta(s)=\eta a_{\tau_0}(s)+\frac{\sigma_V^2}{2\kappa}\int_0^s R_\kappa(s-v)p_\eta(v)^2\,\mathrm{d}v,\qquad0\le s\le T, \tag*{(35)} p η ( s ) = η a τ 0 ( s ) + 2 κ σ V 2 ∫ 0 s R κ ( s − v ) p η ( v ) 2 d v , 0 ≤ s ≤ T , (35) together with the Riccati equation
ψ 2 ′ ( s ; η ) = κ p η ( s ) + σ θ 2 2 ψ 2 ( s ; η ) 2 , ψ 2 ( 0 ; η ) = η [ 1 − w ( τ 0 ) ] . (36) \psi_2'(s;\eta)=\kappa p_\eta(s)+\frac{\sigma_\theta^2}{2}\psi_2(s;\eta)^2,\qquad\psi_2(0;\eta)=\eta[1-w(\tau_0)]. \tag*{(36)} ψ 2 ′ ( s ; η ) = κ p η ( s ) + 2 σ θ 2 ψ 2 ( s ; η ) 2 , ψ 2 ( 0 ; η ) = η [ 1 − w ( τ 0 )] . (36) Lemma A.4. There exists η ˉ T , τ 0 > 0 \bar{\eta}_{T,\tau_0}>0 η ˉ T , τ 0 > 0 η ∈ [ 0 , η ˉ T , τ 0 ) \eta\in[0,\bar{\eta}_{T,\tau_0}) η ∈ [ 0 , η ˉ T , τ 0 ) ( p η , ψ 2 ( ⋅ ; η ) ) (p_\eta,\psi_2(\cdot;\eta)) ( p η , ψ 2 ( ⋅ ; η )) [ 0 , T ] [0,T] [ 0 , T ]
sup 0 ≤ s ≤ T p η ( s ) ≤ 2 η ∥ a τ 0 ∥ L ∞ ( 0 , T ) < μ y − 1 , \sup_{0\le s\le T}p_\eta(s)\le2\eta\|a_{\tau_0}\|_{L^\infty(0,T)}<\mu_y^{-1}, 0 ≤ s ≤ T sup p η ( s ) ≤ 2 η ∥ a τ 0 ∥ L ∞ ( 0 , T ) < μ y − 1 , sup 0 ≤ s ≤ T ψ 2 ( s ; η ) ≤ 2 η [ 1 − w ( τ 0 ) + 2 κ T ∥ a τ 0 ∥ L ∞ ( 0 , T ) ] . (37) \sup_{0\le s\le T}\psi_2(s;\eta)\le2\eta\left[1-w(\tau_0)+2\kappa T\|a_{\tau_0}\|_{L^\infty(0,T)}\right]. \tag*{(37)} 0 ≤ s ≤ T sup ψ 2 ( s ; η ) ≤ 2 η [ 1 − w ( τ 0 ) + 2 κ T ∥ a τ 0 ∥ L ∞ ( 0 , T ) ] . (37) In particular, both functions converge uniformly to zero as η ↓ 0 \eta\downarrow0 η ↓ 0
Proof. By Lemma A.1, R κ R_\kappa R κ L 1 ( 0 , T ) L^1(0,T) L 1 ( 0 , T ) a τ 0 a_{\tau_0} a τ 0 [ 0 , T ] [0,T] [ 0 , T ] 0 < w ( τ 0 ) < 1 0<w(\tau_0)<1 0 < w ( τ 0 ) < 1 A = ∥ a τ 0 ∥ L ∞ ( 0 , T ) A=\|a_{\tau_0}\|_{L^\infty(0,T)} A = ∥ a τ 0 ∥ L ∞ ( 0 , T ) L = ∥ R κ ∥ L 1 ( 0 , T ) L=\|R_\kappa\|_{L^1(0,T)} L = ∥ R κ ∥ L 1 ( 0 , T ) η > 0 \eta>0 η > 0 Γ η p = η a τ 0 + ( σ V 2 / ( 2 κ ) ) R κ ∗ p 2 \Gamma_\eta p=\eta a_{\tau_0}+(\sigma_V^2/(2\kappa))R_\kappa*p^2 Γ η p = η a τ 0 + ( σ V 2 / ( 2 κ )) R κ ∗ p 2 { p ∈ C ( [ 0 , T ] ; R ) : ∥ p ∥ ∞ ≤ 2 η A } \{p\in C([0,T];\mathbb{R}):\|p\|_\infty\le2\eta A\} { p ∈ C ([ 0 , T ] ; R ) : ∥ p ∥ ∞ ≤ 2 η A }
∥ Γ η p ∥ ∞ ≤ η A + 2 σ V 2 κ L η 2 A 2 , ∥ Γ η p − Γ η q ∥ ∞ ≤ 2 σ V 2 κ L η A ∥ p − q ∥ ∞ . (38) \|\Gamma_\eta p\|_\infty\le\eta A+\frac{2\sigma_V^2}{\kappa}L\eta^2A^2,\qquad\|\Gamma_\eta p-\Gamma_\eta q\|_\infty\le\frac{2\sigma_V^2}{\kappa}L\eta A\|p-q\|_\infty. \tag*{(38)} ∥ Γ η p ∥ ∞ ≤ η A + κ 2 σ V 2 L η 2 A 2 , ∥ Γ η p − Γ η q ∥ ∞ ≤ κ 2 σ V 2 L η A ∥ p − q ∥ ∞ . (38) For sufficiently small η \eta η p η p_\eta p η η ˉ T , τ 0 \bar{\eta}_{T,\tau_0} η ˉ T , τ 0 2 η A < μ y − 1 2\eta A<\mu_y^{-1} 2 η A < μ y − 1 p ~ η \widetilde p_\eta p η ( σ V 2 / ( 2 κ ) ) R κ ∗ [ ( p η + p ~ η ) ∣ p η − p ~ η ∣ ] (\sigma_V^2/(2\kappa))R_\kappa*[(p_\eta+\widetilde p_\eta)|p_\eta-\widetilde p_\eta|] ( σ V 2 / ( 2 κ )) R κ ∗ [( p η + p η ) ∣ p η − p η ∣ ] ∫ 0 δ R κ ( s ) d s → 0 \int_0^\delta R_\kappa(s)\,\mathrm{d}s\to0 ∫ 0 δ R κ ( s ) d s → 0 δ ↓ 0 \delta\downarrow0 δ ↓ 0 [ 0 , T ] [0,T] [ 0 , T ]
Given p η p_\eta p η ψ 2 \psi_2 ψ 2 ψ 2 ( s ; η ) ≥ 0 \psi_2(s;\eta)\ge0 ψ 2 ( s ; η ) ≥ 0 C = 1 − w ( τ 0 ) + 2 κ T A > 0 C=1-w(\tau_0)+2\kappa T A>0 C = 1 − w ( τ 0 ) + 2 κ T A > 0 η ˉ T , τ 0 \bar{\eta}_{T,\tau_0} η ˉ T , τ 0 2 σ θ 2 T η C < 1 2\sigma_\theta^2T\eta C<1 2 σ θ 2 T η C < 1 ψ 2 \psi_2 ψ 2 2 η C 2\eta C 2 η C s ≤ T s\le T s ≤ T ψ 2 ( s ; η ) ≤ η C + 2 σ θ 2 T η 2 C 2 < 2 η C \psi_2(s;\eta)\le\eta C+2\sigma_\theta^2T\eta^2C^2<2\eta C ψ 2 ( s ; η ) ≤ η C + 2 σ θ 2 T η 2 C 2 < 2 η C T T T T T T [ 0 , T ] [0,T] [ 0 , T ] η = 0 \eta=0 η = 0 η ↓ 0 \eta\downarrow0 η ↓ 0
For u ∈ C u\in\mathbb{C} u ∈ C
ψ 1 ( s ; u ) = u a τ 0 ( s ) + σ V 2 2 κ ∫ 0 s R κ ( s − v ) ψ 1 ( v ; u ) 2 d v . (39) \psi_1(s;u)=u a_{\tau_0}(s)+\frac{\sigma_V^2}{2\kappa}\int_0^s R_\kappa(s-v)\psi_1(v;u)^2\,\mathrm{d}v. \tag*{(39)} ψ 1 ( s ; u ) = u a τ 0 ( s ) + 2 κ σ V 2 ∫ 0 s R κ ( s − v ) ψ 1 ( v ; u ) 2 d v . (39) Let η ∈ [ 0 , η ˉ T , τ 0 ) \eta\in[0,\bar{\eta}_{T,\tau_0}) η ∈ [ 0 , η ˉ T , τ 0 ) ξ ∈ R \xi\in\mathbb{R} ξ ∈ R p η p_\eta p η u = η + i ξ u=\eta+i\xi u = η + i ξ ψ 1 ( ⋅ ; η + i ξ ) \psi_1(\mathord{\cdot};\eta+i\xi) ψ 1 ( ⋅ ; η + i ξ ) [ 0 , T ] [0,T] [ 0 , T ]
Lemma A.5. Under the preceding assumptions, this solution is unique and satisfies
Re ψ 1 ( s ; η + i ξ ) ≤ p η ( s ) < μ y − 1 , 0 ≤ s ≤ T . (40) \operatorname{Re}\psi_1(s;\eta+i\xi)\leq p_\eta(s)<\mu_y^{-1},\qquad0\leq s\leq T. \tag*{(40)} Re ψ 1 ( s ; η + i ξ ) ≤ p η ( s ) < μ y − 1 , 0 ≤ s ≤ T . (40) Proof. Write y = Im ψ 1 ( ⋅ ; η + i ξ ) y=\operatorname{Im}\psi_1(\mathord{\cdot};\eta+i\xi) y = Im ψ 1 ( ⋅ ; η + i ξ ) d = p η − Re ψ 1 ( ⋅ ; η + i ξ ) d=p_\eta-\operatorname{Re}\psi_1(\mathord{\cdot};\eta+i\xi) d = p η − Re ψ 1 ( ⋅ ; η + i ξ )
d = σ V 2 2 κ R κ ∗ [ ( 2 p η − d ) d + y 2 ] . (41) d=\frac{\sigma_V^2}{2\kappa}R_\kappa*\left[(2p_\eta-d)d+y^2\right]. \tag*{(41)} d = 2 κ σ V 2 R κ ∗ [ ( 2 p η − d ) d + y 2 ] . (41) Since p η p_\eta p η d d d [ 0 , T ] [0,T] [ 0 , T ] λ ≥ 0 \lambda\geq0 λ ≥ 0 λ + ( σ V 2 / ( 2 κ ) ) ( 2 p η − d ) ≥ 0 \lambda+(\sigma_V^2/(2\kappa))(2p_\eta-d)\geq0 λ + ( σ V 2 / ( 2 κ )) ( 2 p η − d ) ≥ 0 ( I + λ R κ ∗ ) − 1 R κ ∗ = ( 1 + λ ) − 1 R κ ( 1 + λ ) ∗ (I+\lambda R_\kappa*)^{-1}R_\kappa*=(1+\lambda)^{-1}R_{\kappa(1+\lambda)}* ( I + λ R κ ∗ ) − 1 R κ ∗ = ( 1 + λ ) − 1 R κ ( 1 + λ ) ∗ λ R κ ∗ d \lambda R_\kappa*d λ R κ ∗ d
d = 1 1 + λ R κ ( 1 + λ ) ∗ [ { λ + σ V 2 2 κ ( 2 p η − d ) } d + σ V 2 2 κ y 2 ] . (42) d=\frac{1}{1+\lambda}R_{\kappa(1+\lambda)}*\left[\left\{\lambda+\frac{\sigma_V^2}{2\kappa}(2p_\eta-d)\right\}d+\frac{\sigma_V^2}{2\kappa}y^2\right]. \tag*{(42)} d = 1 + λ 1 R κ ( 1 + λ ) ∗ [ { λ + 2 κ σ V 2 ( 2 p η − d ) } d + 2 κ σ V 2 y 2 ] . (42) Treat the coefficient in braces as the fixed continuous function determined by the given solution. The resulting linear Volterra equation has a nonnegative kernel and nonnegative forcing. Boundedness of this coefficient and ∫ 0 δ R κ ( 1 + λ ) ( s ) d s → 0 \int_0^\delta R_{\kappa(1+\lambda)}(s)\,\mathrm{d}s\to0 ∫ 0 δ R κ ( 1 + λ ) ( s ) d s → 0 δ ↓ 0 \delta\downarrow0 δ ↓ 0 d d d d ≥ 0 d\geq0 d ≥ 0 [ 0 , T ] [0,T] [ 0 , T ]
Finally, subtracting Equation A.20 for two continuous solutions factors the difference of their squares into their sum times their difference. Both solutions are bounded and R κ ∈ L 1 ( 0 , T ) R_\kappa\in L^1(0,T) R κ ∈ L 1 ( 0 , T ) [ 0 , T ] [0,T] [ 0 , T ]
Proofs
Proof of Proposition 3.1
Proof. Fix T > 0 T>0 T > 0 m ( t ) = E Q [ V t ] m(t)=\mathbb{E}^{\mathbb{Q}}[V_t] m ( t ) = E Q [ V t ] t ∈ [ 0 , T ] t\in[0,T] t ∈ [ 0 , T ] m ∈ L 2 ( 0 , T ) m\in L^2(0,T) m ∈ L 2 ( 0 , T ) K ∈ L 2 ( 0 , T ) K\in L^2(0,T) K ∈ L 2 ( 0 , T ) E Q [ ∫ 0 T V s d s ] < ∞ \mathbb{E}^{\mathbb{Q}}[\int_0^T V_s\,\mathrm{d}s]<\infty E Q [ ∫ 0 T V s d s ] < ∞ t t t E Q [ Y 1 2 ] = 2 μ y 2 \mathbb{E}^{\mathbb{Q}}[Y_1^2]=2\mu_y^2 E Q [ Y 1 2 ] = 2 μ y 2 F 0 \mathcal{F}_0 F 0 E 0 Q [ θ s ] = θ 0 \mathbb{E}^{\mathbb{Q}}_0[\theta_s]=\theta_0 E 0 Q [ θ s ] = θ 0 Q ⊗ d t \mathbb{Q}\otimes dt Q ⊗ d t
m ( t ) − θ 0 = V 0 − θ 0 − κ ∫ 0 t K ( t − s ) [ m ( s ) − θ 0 ] d s . (43) m(t)-\theta_0=V_0-\theta_0-\kappa\int_0^t K(t-s)[m(s)-\theta_0]\,ds. \tag*{(43)} m ( t ) − θ 0 = V 0 − θ 0 − κ ∫ 0 t K ( t − s ) [ m ( s ) − θ 0 ] d s . (43) Applying the resolvent identity in Equation A.1 to this pathwise L 1 ( 0 , T ) L^1(0,T) L 1 ( 0 , T )
m ( t ) − θ 0 = ( V 0 − θ 0 ) [ 1 − ∫ 0 t R κ ( s ) d s ] = ( V 0 − θ 0 ) E α , 1 ( − κ t α ) , (44) m(t)-\theta_0=(V_0-\theta_0)\left[1-\int_0^t R_\kappa(s)\,ds\right]=(V_0-\theta_0)E_{\alpha,1}(-\kappa t^\alpha), \tag*{(44)} m ( t ) − θ 0 = ( V 0 − θ 0 ) [ 1 − ∫ 0 t R κ ( s ) d s ] = ( V 0 − θ 0 ) E α , 1 ( − κ t α ) , (44) where the second equality follows from Lemma A.1 . Adding θ 0 \theta_0 θ 0 (10) , which is continuous on [ 0 , T ] [0,T] [ 0 , T ] m m m
Proof of Theorem 3.1
Proof. Fix t ∈ [ 0 , T ) t\in[0,T) t ∈ [ 0 , T ) τ ∈ ( 0 , T − t ] \tau\in(0,T-t] τ ∈ ( 0 , T − t ] q κ ( s ) = E α , 1 ( − κ s α ) q_\kappa(s)=E_{\alpha,1}(-\kappa s^\alpha) q κ ( s ) = E α , 1 ( − κ s α ) M u = ∫ 0 u σ V V s d B 1 , s + J u y M_u=\int_0^u \sigma_V\sqrt{V_s}\,dB_{1,s}+J_u^y M u = ∫ 0 u σ V V s d B 1 , s + J u y Lemma A.2 , both θ \theta θ M M M Q ⊗ d u \mathbb{Q}\otimes du Q ⊗ d u [ 0 , T ] [0,T] [ 0 , T ]
V u = q κ ( u ) V 0 + ∫ 0 u R κ ( u − s ) θ s d s + 1 κ ∫ 0 u R κ ( u − s ) d M s . (45) V_u=q_\kappa(u)V_0+\int_0^u R_\kappa(u-s)\theta_s\,ds+\frac{1}{\kappa}\int_0^u R_\kappa(u-s)\,dM_s. \tag*{(45)} V u = q κ ( u ) V 0 + ∫ 0 u R κ ( u − s ) θ s d s + κ 1 ∫ 0 u R κ ( u − s ) d M s . (45) For the Brownian term, Tonelli’s theorem and Cauchy–Schwarz give
∫ 0 τ ( E Q [ ∫ 0 t + r R κ ( t + r − s ) 2 V s d s ] ) 1 / 2 d r ≤ τ ∥ R κ ∥ L 2 ( 0 , T ) ( E Q [ ∫ 0 T V s d s ] ) 1 / 2 < ∞ . (46) \int_0^\tau\left(\mathbb{E}^{\mathbb{Q}}\left[\int_0^{t+r}R_\kappa(t+r-s)^2V_s\,ds\right]\right)^{1/2}dr
\leq\sqrt{\tau}\,\lVert R_\kappa\rVert_{L^2(0,T)}
\left(\mathbb{E}^{\mathbb{Q}}\left[\int_0^T V_s\,ds\right]\right)^{1/2}<\infty. \tag*{(46)} ∫ 0 τ ( E Q [ ∫ 0 t + r R κ ( t + r − s ) 2 V s d s ] ) 1/2 d r ≤ τ ∥ R κ ∥ L 2 ( 0 , T ) ( E Q [ ∫ 0 T V s d s ] ) 1/2 < ∞. (46) This verifies the stochastic-Fubini condition for the window average. The finite-activity jump term admits pathwise Fubini, while R κ ∈ L 1 ( 0 , T ) R_\kappa\in L^1(0,T) R κ ∈ L 1 ( 0 , T ) Lemma A.2 justify the deterministic and conditional Fubini interchanges. For Lebesgue-almost every r ∈ ( 0 , τ ) r\in(0,\tau) r ∈ ( 0 , τ ) u = t + r u=t+r u = t + r t t t E t Q [ θ s ] = θ t \mathbb{E}^{\mathbb{Q}}_t[\theta_s]=\theta_t E t Q [ θ s ] = θ t s ≥ t s\geq t s ≥ t
E t Q [ V t + r ] = q κ ( t + r ) V 0 + [ 1 − q κ ( r ) ] θ t + ∫ 0 t R κ ( t + r − s ) θ s d s + 1 κ ∫ 0 t R κ ( t + r − s ) d M s , (47) \mathbb{E}^{\mathbb{Q}}_t[V_{t+r}]=q_\kappa(t+r)V_0+[1-q_\kappa(r)]\theta_t+\int_0^tR_\kappa(t+r-s)\theta_s\,ds+\frac{1}{\kappa}\int_0^tR_\kappa(t+r-s)\,dM_s, \tag*{(47)} E t Q [ V t + r ] = q κ ( t + r ) V 0 + [ 1 − q κ ( r )] θ t + ∫ 0 t R κ ( t + r − s ) θ s d s + κ 1 ∫ 0 t R κ ( t + r − s ) d M s , (47) where Lemma A.1 gives ∫ 0 r R κ ( s ) d s = 1 − q κ ( r ) \int_0^rR_\kappa(s)\,ds=1-q_\kappa(r) ∫ 0 r R κ ( s ) d s = 1 − q κ ( r ) r ∈ ( 0 , τ ) r\in(0,\tau) r ∈ ( 0 , τ )
V I X t , τ 2 = V 0 τ ∫ t t + τ q κ ( s ) d s + [ 1 − w ( τ ) ] θ t + ∫ 0 t a τ ( t − s ) { κ θ s d s + d M s } , (48) \mathrm{VIX}_{t,\tau}^{2}=\frac{V_0}{\tau}\int_t^{t+\tau}q_\kappa(s)\,\mathrm{d}s+[1-w(\tau)]\theta_t+\int_0^t a_\tau(t-s)\{\kappa\theta_s\,\mathrm{d}s+\mathrm{d}M_s\}, \tag*{(48)} VIX t , τ 2 = τ V 0 ∫ t t + τ q κ ( s ) d s + [ 1 − w ( τ )] θ t + ∫ 0 t a τ ( t − s ) { κ θ s d s + d M s } , (48) because
w ( τ ) = 1 τ ∫ 0 τ q κ ( s ) d s = E α , 2 ( − κ τ α ) , w(\tau)=\frac{1}{\tau}\int_0^\tau q_\kappa(s)\,\mathrm{d}s=E_{\alpha,2}(-\kappa\tau^\alpha), w ( τ ) = τ 1 ∫ 0 τ q κ ( s ) d s = E α , 2 ( − κ τ α ) , a τ ( s ) = 1 κ τ ∫ s s + τ R κ ( v ) d v = q κ ( s ) − q κ ( s + τ ) κ τ . (49) a_\tau(s)=\frac{1}{\kappa\tau}\int_s^{s+\tau}R_\kappa(v)\,\mathrm{d}v=\frac{q_\kappa(s)-q_\kappa(s+\tau)}{\kappa\tau}. \tag*{(49)} a τ ( s ) = κ τ 1 ∫ s s + τ R κ ( v ) d v = κ τ q κ ( s ) − q κ ( s + τ ) . (49) Moreover, the second identity in Equation B.7 gives
1 τ ∫ t t + τ q κ ( s ) d s = w ( τ ) − κ ∫ 0 t a τ ( s ) d s . (50) \frac{1}{\tau}\int_t^{t+\tau}q_\kappa(s)\,\mathrm{d}s=w(\tau)-\kappa\int_0^t a_\tau(s)\,\mathrm{d}s. \tag*{(50)} τ 1 ∫ t t + τ q κ ( s ) d s = w ( τ ) − κ ∫ 0 t a τ ( s ) d s . (50) Since a τ a_\tau a τ [ 0 , t ] [0,t] [ 0 , t ] ( t , τ ) (t,\tau) ( t , τ ) M M M (11) almost surely.
Proof of Theorem 3.2
Proof. Fix a real parameter u = η ∈ [ 0 , η ˉ T , τ 0 ) u=\eta\in[0,\bar{\eta}_{T,\tau_0}) u = η ∈ [ 0 , η ˉ T , τ 0 ) ψ 1 = p η \psi_1=p_\eta ψ 1 = p η ψ 2 = ψ 2 ( ⋅ ; η ) \psi_2=\psi_2(\cdot;\eta) ψ 2 = ψ 2 ( ⋅ ; η ) Lemma A.4 . Write d Z s = κ ( θ s − V s ) d s + σ V V s d B 1 , s + d J s y \mathrm{d}Z_s=\kappa(\theta_s-V_s)\,\mathrm{d}s+\sigma_V\sqrt{V_s}\,\mathrm{d}B_{1,s}+\mathrm{d}J_s^y d Z s = κ ( θ s − V s ) d s + σ V V s d B 1 , s + d J s y V − V 0 = K ∗ d Z V-V_0=K*\mathrm{d}Z V − V 0 = K ∗ d Z (11) ,
V I X T , τ 0 2 = w ( τ 0 ) V 0 + [ 1 − w ( τ 0 ) ] θ T + ∫ 0 T ( a τ 0 + κ K ∗ a τ 0 ) ( T − s ) d Z s . (51) \mathrm{VIX}_{T,\tau_0}^{2}=w(\tau_0)V_0+[1-w(\tau_0)]\theta_T+\int_0^T\left(a_{\tau_0}+\kappa K*a_{\tau_0}\right)(T-s)\,\mathrm{d}Z_s. \tag*{(51)} VIX T , τ 0 2 = w ( τ 0 ) V 0 + [ 1 − w ( τ 0 )] θ T + ∫ 0 T ( a τ 0 + κ K ∗ a τ 0 ) ( T − s ) d Z s . (51) Here and below, boundedness of the deterministic coefficient functions, K ∈ L 1 ( 0 , T ) ∩ L 2 ( 0 , T ) K\in L^1(0,T)\cap L^2(0,T) K ∈ L 1 ( 0 , T ) ∩ L 2 ( 0 , T ) Lemma A.2 justify the drift and Brownian Fubini interchanges; the finite-activity jump terms admit pathwise Fubini.
Applying I + κ K ∗ I+\kappa K* I + κ K ∗ R κ + κ K ∗ R κ = κ K R_\kappa+\kappa K*R_\kappa=\kappa K R κ + κ K ∗ R κ = κ K
ψ 1 = u ( a τ 0 + κ K ∗ a τ 0 ) + K ∗ ( − κ ψ 1 + σ V 2 2 ψ 1 2 ) . (52) \psi_1=u\left(a_{\tau_0}+\kappa K*a_{\tau_0}\right)+K*\left(-\kappa\psi_1+\frac{\sigma_V^2}{2}\psi_1^2\right). \tag*{(52)} ψ 1 = u ( a τ 0 + κ K ∗ a τ 0 ) + K ∗ ( − κ ψ 1 + 2 σ V 2 ψ 1 2 ) . (52) Define the remaining coefficients by
χ 1 ( s ) = u w ( τ 0 ) + ∫ 0 s [ − κ ψ 1 ( v ) + σ V 2 2 ψ 1 ( v ) 2 ] d v , \chi_1(s)=uw(\tau_0)+\int_0^s\left[-\kappa\psi_1(v)+\frac{\sigma_V^2}{2}\psi_1(v)^2\right]\,\mathrm{d}v, χ 1 ( s ) = u w ( τ 0 ) + ∫ 0 s [ − κ ψ 1 ( v ) + 2 σ V 2 ψ 1 ( v ) 2 ] d v , ϕ ( s ) = λ y ∫ 0 s [ ϱ ( ψ 1 ( v ) ) − 1 − μ y ψ 1 ( v ) ] d v . (53) \phi(s)=\lambda^y\int_0^s[\varrho(\psi_1(v))-1-\mu_y\psi_1(v)]\,\mathrm{d}v. \tag*{(53)} ϕ ( s ) = λ y ∫ 0 s [ ϱ ( ψ 1 ( v )) − 1 − μ y ψ 1 ( v )] d v . (53) Substituting Equation B.10 into Equation B.9 and using V − V 0 = K ∗ d Z V-V_0=K*\mathrm{d}Z V − V 0 = K ∗ d Z
u VIX T , τ 0 2 = χ 1 ( T ) V 0 + u [ 1 − w ( τ 0 ) ] θ T + κ ∫ 0 T ψ 1 ( T − s ) θ s d s + ∫ 0 T σ V ψ 1 ( T − s ) V s d B 1 , s + ∫ 0 T ψ 1 ( T − s ) d J s y − σ V 2 2 ∫ 0 T ψ 1 ( T − s ) 2 V s d s . (54) u\operatorname{VIX}_{T,\tau_0}^{2}=\chi_1(T)V_0+u[1-w(\tau_0)]\theta_T+\kappa\int_0^T\psi_1(T-s)\theta_s\,\mathrm{d}s
+\int_0^T\sigma_V\psi_1(T-s)\sqrt{V_s}\,\mathrm{d}B_{1,s}+\int_0^T\psi_1(T-s)\,\mathrm{d}J_s^y
-\frac{\sigma_V^2}{2}\int_0^T\psi_1(T-s)^2V_s\,\mathrm{d}s. \tag*{(54)} u VIX T , τ 0 2 = χ 1 ( T ) V 0 + u [ 1 − w ( τ 0 )] θ T + κ ∫ 0 T ψ 1 ( T − s ) θ s d s + ∫ 0 T σ V ψ 1 ( T − s ) V s d B 1 , s + ∫ 0 T ψ 1 ( T − s ) d J s y − 2 σ V 2 ∫ 0 T ψ 1 ( T − s ) 2 V s d s . (54) For the mean factor, apply Itô’s formula to ψ 2 ( T − s ) θ s \psi_2(T-s)\theta_s ψ 2 ( T − s ) θ s d θ s = σ θ θ s d B 2 , s \mathrm{d}\theta_s=\sigma_\theta\sqrt{\theta_s}\,\mathrm{d}B_{2,s} d θ s = σ θ θ s d B 2 , s ψ 2 \psi_2 ψ 2
u [ 1 − w ( τ 0 ) ] θ T + κ ∫ 0 T ψ 1 ( T − s ) θ s d s = ψ 2 ( T ) θ 0 + ∫ 0 T σ θ ψ 2 ( T − s ) θ s d B 2 , s − σ θ 2 2 ∫ 0 T ψ 2 ( T − s ) 2 θ s d s . (55) u[1-w(\tau_0)]\theta_T+\kappa\int_0^T\psi_1(T-s)\theta_s\,\mathrm{d}s
=\psi_2(T)\theta_0+\int_0^T\sigma_\theta\psi_2(T-s)\sqrt{\theta_s}\,\mathrm{d}B_{2,s}
-\frac{\sigma_\theta^2}{2}\int_0^T\psi_2(T-s)^2\theta_s\,\mathrm{d}s. \tag*{(55)} u [ 1 − w ( τ 0 )] θ T + κ ∫ 0 T ψ 1 ( T − s ) θ s d s = ψ 2 ( T ) θ 0 + ∫ 0 T σ θ ψ 2 ( T − s ) θ s d B 2 , s − 2 σ θ 2 ∫ 0 T ψ 2 ( T − s ) 2 θ s d s . (55) Combining these identities and exponentiating gives
exp { u VIX T , τ 0 2 } = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } E T , (56) \exp\left\{u\operatorname{VIX}_{T,\tau_0}^{2}\right\}
=\exp\{\phi(T)+\chi_1(T)V_0+\psi_2(T)\theta_0\}\mathcal{E}_T, \tag*{(56)} exp { u VIX T , τ 0 2 } = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } E T , (56) where, for 0 ≤ t ≤ T 0\le t\le T 0 ≤ t ≤ T
E t = exp { ∫ 0 t σ V ψ 1 ( T − s ) V s d B 1 , s + ∫ 0 t σ θ ψ 2 ( T − s ) θ s d B 2 , s − 1 2 ∫ 0 t [ σ V 2 ψ 1 ( T − s ) 2 V s + σ θ 2 ψ 2 ( T − s ) 2 θ s ] d s + ∫ 0 t ψ 1 ( T − s ) d J s y − λ y ∫ 0 t [ ϱ ( ψ 1 ( T − s ) ) − 1 − μ y ψ 1 ( T − s ) ] d s } . (57) \mathcal{E}_t=\exp\left\{
\int_0^t\sigma_V\psi_1(T-s)\sqrt{V_s}\,\mathrm{d}B_{1,s}
+\int_0^t\sigma_\theta\psi_2(T-s)\sqrt{\theta_s}\,\mathrm{d}B_{2,s}
-\frac{1}{2}\int_0^t\left[\sigma_V^2\psi_1(T-s)^2V_s+\sigma_\theta^2\psi_2(T-s)^2\theta_s\right]\mathrm{d}s
+\int_0^t\psi_1(T-s)\,\mathrm{d}J_s^y
-\lambda^y\int_0^t[\varrho(\psi_1(T-s))-1-\mu_y\psi_1(T-s)]\mathrm{d}s
\right\}. \tag*{(57)} E t = exp { ∫ 0 t σ V ψ 1 ( T − s ) V s d B 1 , s + ∫ 0 t σ θ ψ 2 ( T − s ) θ s d B 2 , s − 2 1 ∫ 0 t [ σ V 2 ψ 1 ( T − s ) 2 V s + σ θ 2 ψ 2 ( T − s ) 2 θ s ] d s + ∫ 0 t ψ 1 ( T − s ) d J s y − λ y ∫ 0 t [ ϱ ( ψ 1 ( T − s )) − 1 − μ y ψ 1 ( T − s )] d s } . (57) We first prove the martingale property for real u = η ∈ [ 0 , η ˉ T , τ 0 ) u=\eta\in[0,\bar{\eta}_{T,\tau_0}) u = η ∈ [ 0 , η ˉ T , τ 0 ) ψ 1 = p η \psi_1=p_\eta ψ 1 = p η ψ 2 = ψ 2 ( ⋅ ; η ) \psi_2=\psi_2(\mathord{\cdot};\eta) ψ 2 = ψ 2 ( ⋅ ; η ) Lemma A.4 . Write E t = D t exp { L t − ⟨ L ⟩ t / 2 } \mathcal{E}_t=D_t\exp\{L_t-\langle L\rangle_t/2\} E t = D t exp { L t − ⟨ L ⟩ t /2 } L L L D D D E \mathcal{E} E B 1 B_1 B 1 B 2 B_2 B 2
⟨ L ⟩ T = ∫ 0 T [ σ V 2 p η ( T − s ) 2 V s + σ θ 2 ψ 2 ( T − s ; η ) 2 θ s ] d s . \langle L\rangle_T=\int_0^T[\sigma_V^2p_\eta(T-s)^2V_s+\sigma_\theta^2\psi_2(T-s;\eta)^2\theta_s]\,\mathrm{d}s. ⟨ L ⟩ T = ∫ 0 T [ σ V 2 p η ( T − s ) 2 V s + σ θ 2 ψ 2 ( T − s ; η ) 2 θ s ] d s . Choose a > 0 a>0 a > 0 a σ θ 2 T ≤ 1 a\sigma_\theta^2T\le1 a σ θ 2 T ≤ 1 E 0 Q [ e a θ s ] \mathbb{E}_0^{\mathbb{Q}}[e^{a\theta_s}] E 0 Q [ e a θ s ] E 0 Q [ exp { ( a / T ) ∫ 0 T θ s d s } ] ≤ e 2 a θ 0 \mathbb{E}_0^{\mathbb{Q}}[\exp\{(a/T)\int_0^T\theta_s\,\mathrm{d}s\}]\le e^{2a\theta_0} E 0 Q [ exp {( a / T ) ∫ 0 T θ s d s }] ≤ e 2 a θ 0 η ↓ 0 \eta\downarrow0 η ↓ 0 η ˉ T , τ 0 \bar{\eta}_{T,\tau_0} η ˉ T , τ 0 2 σ V 2 ∥ p η ∥ ∞ 2 < c T 2\sigma_V^2\|p_\eta\|_\infty^2<c_T 2 σ V 2 ∥ p η ∥ ∞ 2 < c T 2 σ θ 2 ∥ ψ 2 ( ⋅ ; η ) ∥ ∞ 2 < a / T 2\sigma_\theta^2\|\psi_2(\mathord{\cdot};\eta)\|_\infty^2<a/T 2 σ θ 2 ∥ ψ 2 ( ⋅ ; η ) ∥ ∞ 2 < a / T 2 ∥ p η ∥ ∞ < μ y − 1 2\|p_\eta\|_\infty<\mu_y^{-1} 2∥ p η ∥ ∞ < μ y − 1 [ 0 , T ] [0,T] [ 0 , T ] E 0 Q [ e ⟨ L ⟩ T ] < ∞ \mathbb{E}_0^{\mathbb{Q}}[e^{\langle L\rangle_T}]<\infty E 0 Q [ e ⟨ L ⟩ T ] < ∞
The exponential formula for the marked Poisson process shows that D D D E 0 Q [ D T ] = 1 \mathbb{E}_0^{\mathbb{Q}}[D_T]=1 E 0 Q [ D T ] = 1
E 0 Q [ D T 2 ] = exp { λ y ∫ 0 T [ ϱ ( 2 p η ( s ) ) − 2 ϱ ( p η ( s ) ) + 1 ] d s } < ∞ . (58) \mathbb{E}_0^{\mathbb{Q}}[D_T^2]=\exp\left\{\lambda^y\int_0^T[\varrho(2p_\eta(s))-2\varrho(p_\eta(s))+1]\,\mathrm{d}s\right\}<\infty. \tag*{(58)} E 0 Q [ D T 2 ] = exp { λ y ∫ 0 T [ ϱ ( 2 p η ( s )) − 2 ϱ ( p η ( s )) + 1 ] d s } < ∞. (58) Define Q y \mathbb{Q}^{y} Q y F T \mathcal{F}_{T} F T d Q y / d Q = D T \mathrm{d}\mathbb{Q}^{y}/\mathrm{d}\mathbb{Q}=D_{T} d Q y / d Q = D T B 1 , B 2 B_{1}, B_{2} B 1 , B 2 Q y \mathbb{Q}^{y} Q y Q \mathbb{Q} Q F 0 \mathcal{F}_{0} F 0
E 0 Q y [ e ⟨ L ⟩ T / 2 ] = E 0 Q [ D T e ⟨ L ⟩ T / 2 ] ≤ ( E 0 Q [ D T 2 ] ) 1 / 2 ( E 0 Q [ e ⟨ L ⟩ T ] ) 1 / 2 < ∞ . (59) \mathbb{E}_{0}^{\mathbb{Q}^{y}}\left[e^{\langle L\rangle_{T}/2}\right]=\mathbb{E}_{0}^{\mathbb{Q}}\left[D_{T}e^{\langle L\rangle_{T}/2}\right]\leq\left(\mathbb{E}_{0}^{\mathbb{Q}}\left[D_{T}^{2}\right]\right)^{1/2}\left(\mathbb{E}_{0}^{\mathbb{Q}}\left[e^{\langle L\rangle_{T}}\right]\right)^{1/2}<\infty. \tag*{(59)} E 0 Q y [ e ⟨ L ⟩ T /2 ] = E 0 Q [ D T e ⟨ L ⟩ T /2 ] ≤ ( E 0 Q [ D T 2 ] ) 1/2 ( E 0 Q [ e ⟨ L ⟩ T ] ) 1/2 < ∞. (59) To apply Novikov’s criterion, first condition Q y \mathbb{Q}^{y} Q y F 0 \mathcal{F}_{0} F 0 { E 0 Q y [ e ⟨ L ⟩ T / 2 ] ≤ n } \{\mathbb{E}_{0}^{\mathbb{Q}^{y}}[e^{\langle L\rangle_{T}/2}]\leq n\} { E 0 Q y [ e ⟨ L ⟩ T /2 ] ≤ n } E 0 Q y [ exp { L T − ⟨ L ⟩ T / 2 } ] = 1 \mathbb{E}_{0}^{\mathbb{Q}^{y}}[\exp\{L_{T}-\langle L\rangle_{T}/2\}]=1 E 0 Q y [ exp { L T − ⟨ L ⟩ T /2 }] = 1 E 0 Q [ E T ] = 1 \mathbb{E}_{0}^{\mathbb{Q}}[\mathcal{E}_{T}]=1 E 0 Q [ E T ] = 1 E \mathcal{E} E [ 0 , T ] [0,T] [ 0 , T ] λ y = 0 \lambda^{y}=0 λ y = 0 D ≡ 1 D\equiv1 D ≡ 1 Q y = Q \mathbb{Q}^{y}=\mathbb{Q} Q y = Q f ( η ; 0 , T ) = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } < ∞ f(\eta;0,T)=\exp\{\phi(T)+\chi_{1}(T)V_{0}+\psi_{2}(T)\theta_{0}\}<\infty f ( η ; 0 , T ) = exp { ϕ ( T ) + χ 1 ( T ) V 0 + ψ 2 ( T ) θ 0 } < ∞ η ≥ 0 \eta\geq0 η ≥ 0
It remains to extend the identity to complex arguments. By the definition of squared VIX, conditional Jensen’s inequality, and Lemma A.3, after reducing η ˉ T , τ 0 \bar{\eta}_{T,\tau_{0}} η ˉ T , τ 0
E 0 Q [ e η VIX T , τ 0 2 ] ≤ E 0 Q [ exp { η τ 0 ∫ T T + τ 0 V s d s } ] ≤ E 0 Q [ exp { η τ 0 ∫ 0 T + τ 0 V s d s } ] < ∞ . (60) \begin{aligned}
\mathbb{E}_{0}^{\mathbb{Q}}\left[e^{\eta\operatorname{VIX}_{T,\tau_{0}}^{2}}\right]\leq\mathbb{E}_{0}^{\mathbb{Q}}\left[\exp\left\{\frac{\eta}{\tau_{0}}\int_{T}^{T+\tau_{0}}V_{s}\,\mathrm{d}s\right\}\right] \\
&\leq\mathbb{E}_{0}^{\mathbb{Q}}\left[\exp\left\{\frac{\eta}{\tau_{0}}\int_{0}^{T+\tau_{0}}V_{s}\,\mathrm{d}s\right\}\right]<\infty. \tag*{(60)}
\end{aligned} E 0 Q [ e η VIX T , τ 0 2 ] ≤ E 0 Q [ exp { τ 0 η ∫ T T + τ 0 V s d s } ] ≤ E 0 Q [ exp { τ 0 η ∫ 0 T + τ 0 V s d s } ] < ∞. (60) for every η ∈ ( 0 , η ˉ T , τ 0 ) \eta\in(0,\bar{\eta}_{T,\tau_{0}}) η ∈ ( 0 , η ˉ T , τ 0 ) u = η + i ξ u=\eta+i\xi u = η + i ξ 0 ≤ η < η ˉ T , τ 0 0\leq\eta<\bar{\eta}_{T,\tau_{0}} 0 ≤ η < η ˉ T , τ 0 ξ ∈ R \xi\in\mathbb{R} ξ ∈ R R κ ∈ L 1 ( 0 , T ) R_{\kappa}\in L^{1}(0,T) R κ ∈ L 1 ( 0 , T ) Re ψ 1 ( s ; u ) ≤ p η ( s ) \operatorname{Re}\psi_{1}(s;u)\leq p_{\eta}(s) Re ψ 1 ( s ; u ) ≤ p η ( s )
To exclude explosion, write D C α g ( s ) = Γ ( 1 − α ) − 1 ∫ 0 s ( s − v ) − α g ′ ( v ) d v D_{C}^{\alpha}g(s)=\Gamma(1-\alpha)^{-1}\int_{0}^{s}(s-v)^{-\alpha}g'(v)\,\mathrm{d}v D C α g ( s ) = Γ ( 1 − α ) − 1 ∫ 0 s ( s − v ) − α g ′ ( v ) d v ψ 1 = u a τ 0 + K ∗ [ − κ ψ 1 + κ u a τ 0 + ( σ V 2 / 2 ) ψ 1 2 ] \psi_{1}=ua_{\tau_{0}}+K\ast[-\kappa\psi_{1}+\kappa ua_{\tau_{0}}+(\sigma_{V}^{2}/2)\psi_{1}^{2}] ψ 1 = u a τ 0 + K ∗ [ − κ ψ 1 + κ u a τ 0 + ( σ V 2 /2 ) ψ 1 2 ] α \alpha α ψ 1 \psi_{1} ψ 1 2 α > 1 2\alpha>1 2 α > 1 K K K C ( s α − 1 + s 2 α − 1 ) C(s^{\alpha-1}+s^{2\alpha-1}) C ( s α − 1 + s 2 α − 1 ) a τ 0 a_{\tau_{0}} a τ 0 ψ 1 \psi_{1} ψ 1
D C α ψ 1 ( s ; u ) = u h ( s ) − κ ψ 1 ( s ; u ) + σ V 2 2 ψ 1 ( s ; u ) 2 , ψ 1 ( 0 ; u ) = u a τ 0 ( 0 ) , D_{C}^{\alpha}\psi_{1}(s;u)=uh(s)-\kappa\psi_{1}(s;u)+\frac{\sigma_{V}^{2}}{2}\psi_{1}(s;u)^{2},\qquad\psi_{1}(0;u)=ua_{\tau_{0}}(0), D C α ψ 1 ( s ; u ) = u h ( s ) − κ ψ 1 ( s ; u ) + 2 σ V 2 ψ 1 ( s ; u ) 2 , ψ 1 ( 0 ; u ) = u a τ 0 ( 0 ) , h ( s ) = − E α , 1 ( − κ ( s + τ 0 ) α ) τ 0 + 1 κ τ 0 Γ ( 1 − α ) ∫ 0 s ( s − v ) − α R κ ( v + τ 0 ) d v . (61) h(s)=-\frac{E_{\alpha,1}(-\kappa(s+\tau_{0})^{\alpha})}{\tau_{0}}+\frac{1}{\kappa\tau_{0}\Gamma(1-\alpha)}\int_{0}^{s}(s-v)^{-\alpha}R_{\kappa}(v+\tau_{0})\,\mathrm{d}v. \tag*{(61)} h ( s ) = − τ 0 E α , 1 ( − κ ( s + τ 0 ) α ) + κ τ 0 Γ ( 1 − α ) 1 ∫ 0 s ( s − v ) − α R κ ( v + τ 0 ) d v . (61) Here h = D C α a τ 0 + κ a τ 0 h=D_{C}^{\alpha}a_{\tau_{0}}+\kappa a_{\tau_{0}} h = D C α a τ 0 + κ a τ 0 Lemma A.1 . Since τ 0 > 0 \tau_{0}>0 τ 0 > 0 R κ ( ⋅ + τ 0 ) R_{\kappa}(\cdot+\tau_{0}) R κ ( ⋅ + τ 0 ) [ 0 , T ] [0,T] [ 0 , T ] h h h
Fractional integration by parts gives D C α ∣ ψ 1 ∣ 2 ≤ 2 Re ( ψ 1 ‾ D C α ψ 1 ) D^\alpha_C|\psi_1|^2 \le2\operatorname{Re}(\overline{\psi_1}D^\alpha_C\psi_1) D C α ∣ ψ 1 ∣ 2 ≤ 2 Re ( ψ 1 D C α ψ 1 )
1 Γ ( 1 − α ) [ ∣ ψ 1 ( s ; u ) − ψ 1 ( 0 ; u ) ∣ 2 s α + α ∫ 0 s ∣ ψ 1 ( s ; u ) − ψ 1 ( v ; u ) ∣ 2 ( s − v ) 1 + α d v ] ≥ 0. (62) \frac{1}{\Gamma(1-\alpha)}\left[\frac{|\psi_1(s;u)-\psi_1(0;u)|^2}{s^\alpha}+\alpha\int_0^s\frac{|\psi_1(s;u)-\psi_1(v;u)|^2}{(s-v)^{1+\alpha}}\,\mathrm{d}v\right]\ge0. \tag*{(62)} Γ ( 1 − α ) 1 [ s α ∣ ψ 1 ( s ; u ) − ψ 1 ( 0 ; u ) ∣ 2 + α ∫ 0 s ( s − v ) 1 + α ∣ ψ 1 ( s ; u ) − ψ 1 ( v ; u ) ∣ 2 d v ] ≥ 0. (62) Consequently, the real-part comparison yields
D C α ∣ ψ 1 ( s ; u ) ∣ 2 ≤ 2 Re ( ψ 1 ( s ; u ) ‾ u h ( s ) ) + [ − 2 κ + σ V 2 Re ψ 1 ( s ; u ) ] ∣ ψ 1 ( s ; u ) ∣ 2 ≤ ∣ u ∣ 2 ∥ h ∥ ∞ 2 + ( 1 + σ V 2 ∥ p η ∥ ∞ ) ∣ ψ 1 ( s ; u ) ∣ 2 . (63) \begin{aligned}
D^\alpha_C|\psi_1(s;u)|^2 &\le2\operatorname{Re}\left(\overline{\psi_1(s;u)}uh(s)\right)+\left[-2\kappa+\sigma_V^2\operatorname{Re}\psi_1(s;u)\right]|\psi_1(s;u)|^2\\
&\le|u|^2\|h\|_\infty^2+\left(1+\sigma_V^2\|p_\eta\|_\infty\right)|\psi_1(s;u)|^2. \tag*{(63)}
\end{aligned} D C α ∣ ψ 1 ( s ; u ) ∣ 2 ≤ 2 Re ( ψ 1 ( s ; u ) u h ( s ) ) + [ − 2 κ + σ V 2 Re ψ 1 ( s ; u ) ] ∣ ψ 1 ( s ; u ) ∣ 2 ≤ ∣ u ∣ 2 ∥ h ∥ ∞ 2 + ( 1 + σ V 2 ∥ p η ∥ ∞ ) ∣ ψ 1 ( s ; u ) ∣ 2 . (63) Applying K ∗ K* K ∗
∣ ψ 1 ( s ; u ) ∣ 2 ≤ ∣ u ∣ 2 [ a τ 0 ( 0 ) 2 + ∥ h ∥ ∞ 2 T α Γ ( 1 + α ) ] E α , 1 ( [ 1 + σ V 2 ∥ p η ∥ ∞ ] T α ) (64) |\psi_1(s;u)|^2\le|u|^2\left[a_{\tau_0}(0)^2+\frac{\|h\|_\infty^2T^\alpha}{\Gamma(1+\alpha)}\right]E_{\alpha,1}\left(\left[1+\sigma_V^2\|p_\eta\|_\infty\right]T^\alpha\right) \tag*{(64)} ∣ ψ 1 ( s ; u ) ∣ 2 ≤ ∣ u ∣ 2 [ a τ 0 ( 0 ) 2 + Γ ( 1 + α ) ∥ h ∥ ∞ 2 T α ] E α , 1 ( [ 1 + σ V 2 ∥ p η ∥ ∞ ] T α ) (64) throughout its interval of existence before T T T ψ 1 2 \psi_1^2 ψ 1 2 R κ R_\kappa R κ L 1 L^1 L 1 ψ 1 \psi_1 ψ 1 [ 0 , T ] [0,T] [ 0 , T ] Re ψ 1 ( s ; u ) ≤ p η ( s ) < μ y − 1 \operatorname{Re}\psi_1(s;u)\le p_\eta(s)<\mu_y^{-1} Re ψ 1 ( s ; u ) ≤ p η ( s ) < μ y − 1
Given this solution, consider the complex Riccati equation
ψ 2 ′ ( s ; u ) = κ ψ 1 ( s ; u ) + σ θ 2 2 ψ 2 ( s ; u ) 2 , ψ 2 ( 0 ; u ) = u [ 1 − w ( τ 0 ) ] . (65) \psi_2'(s;u)=\kappa\psi_1(s;u)+\frac{\sigma_\theta^2}{2}\psi_2(s;u)^2,\qquad\psi_2(0;u)=u[1-w(\tau_0)]. \tag*{(65)} ψ 2 ′ ( s ; u ) = κ ψ 1 ( s ; u ) + 2 σ θ 2 ψ 2 ( s ; u ) 2 , ψ 2 ( 0 ; u ) = u [ 1 − w ( τ 0 )] . (65) Its right-hand side is locally Lipschitz, so it has a unique maximal solution. Subtracting its real part from Equation A.17 gives a linear equation for ψ 2 ( s ; η ) − Re ψ 2 ( s ; u ) \psi_2(s;\eta)-\operatorname{Re}\psi_2(s;u) ψ 2 ( s ; η ) − Re ψ 2 ( s ; u ) κ [ p η ( s ) − Re ψ 1 ( s ; u ) ] + ( σ θ 2 / 2 ) [ Im ψ 2 ( s ; u ) ] 2 \kappa[p_\eta(s)-\operatorname{Re}\psi_1(s;u)]+(\sigma_\theta^2/2)[\operatorname{Im}\psi_2(s;u)]^2 κ [ p η ( s ) − Re ψ 1 ( s ; u )] + ( σ θ 2 /2 ) [ Im ψ 2 ( s ; u ) ] 2 Re ψ 2 ( s ; u ) ≤ ψ 2 ( s ; η ) \operatorname{Re}\psi_2(s;u)\le\psi_2(s;\eta) Re ψ 2 ( s ; u ) ≤ ψ 2 ( s ; η )
d d s ∣ ψ 2 ( s ; u ) ∣ 2 = 2 κ Re ( ψ 2 ( s ; u ) ‾ ψ 1 ( s ; u ) ) + σ θ 2 Re ψ 2 ( s ; u ) ∣ ψ 2 ( s ; u ) ∣ 2 ≤ κ ∣ ψ 1 ( s ; u ) ∣ 2 + [ κ + σ θ 2 ∥ ψ 2 ( ⋅ ; η ) ∥ ∞ ] ∣ ψ 2 ( s ; u ) ∣ 2 . (66) \begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}s}|\psi_2(s;u)|^2&=2\kappa\operatorname{Re}\left(\overline{\psi_2(s;u)}\psi_1(s;u)\right)+\sigma_\theta^2\operatorname{Re}\psi_2(s;u)|\psi_2(s;u)|^2\\
&\le\kappa|\psi_1(s;u)|^2+\left[\kappa+\sigma_\theta^2\|\psi_2(\cdot;\eta)\|_\infty\right]|\psi_2(s;u)|^2. \tag*{(66)}
\end{aligned} d s d ∣ ψ 2 ( s ; u ) ∣ 2 = 2 κ Re ( ψ 2 ( s ; u ) ψ 1 ( s ; u ) ) + σ θ 2 Re ψ 2 ( s ; u ) ∣ ψ 2 ( s ; u ) ∣ 2 ≤ κ ∣ ψ 1 ( s ; u ) ∣ 2 + [ κ + σ θ 2 ∥ ψ 2 ( ⋅ ; η ) ∥ ∞ ] ∣ ψ 2 ( s ; u ) ∣ 2 . (66) The preceding bound for ψ 1 \psi_1 ψ 1 ψ 2 \psi_2 ψ 2 T T T [ 0 , T ] [0,T] [ 0 , T ]
These bounds are uniform when u u u u u u u u u u ↦ ψ 1 ( ⋅ ; u ) u\mapsto\psi_1(\cdot;u) u ↦ ψ 1 ( ⋅ ; u ) u ↦ ψ 2 ( ⋅ ; u ) u\mapsto\psi_2(\cdot;u) u ↦ ψ 2 ( ⋅ ; u ) C ( [ 0 , T ] ) C([0,T]) C ([ 0 , T ]) 0 < Re ( u ) < η ˉ T , τ 0 0<\operatorname{Re}(u)<\bar\eta_{T,\tau_0} 0 < Re ( u ) < η ˉ T , τ 0 Re ( u ) = 0 \operatorname{Re}(u)=0 Re ( u ) = 0 χ 1 \chi_1 χ 1 ϕ \phi ϕ ψ 1 \psi_1 ψ 1 Re ψ 1 < μ y − 1 \operatorname{Re}\psi_1<\mu_y^{-1} Re ψ 1 < μ y − 1
Finally, take the conditional moment bounds in Equation B.18 and the real-parameter transform identities simultaneously on a countable dense subset of ( 0 , η ˉ T , τ 0 ) (0,\bar{\eta}_{T,\tau_0}) ( 0 , η ˉ T , τ 0 ) V I X T , τ 0 2 \mathrm{VIX}^{2}_{T,\tau_0} VIX T , τ 0 2 f ( u ; 0 , T ) f(u;0,T) f ( u ; 0 , T ) (14) throughout the open strip, with the coefficients specified above. Conditional dominated convergence and continuity of the coefficient functions extend the identity to Re ( u ) = 0 \operatorname{Re}(u)=0 Re ( u ) = 0
Proof of Corollary 3.1
Proof. For Re ( z ) > 0 \operatorname{Re}(z)>0 Re ( z ) > 0 x ↦ ( x − K ) + x\mapsto(\sqrt{x}-K)^{+} x ↦ ( x − K ) +
∫ 0 ∞ e − z x ( x − K ) + d x = π 2 z 3 / 2 erfc ( K z ) . (67) \int_{0}^{\infty}e^{-zx}(\sqrt{x}-K)^{+}\,\mathrm{d}x=\frac{\sqrt{\pi}}{2z^{3/2}}\operatorname{erfc}(K\sqrt{z}). \tag*{(67)} ∫ 0 ∞ e − z x ( x − K ) + d x = 2 z 3/2 π erfc ( K z ) . (67) Indeed, this follows by substituting x = y 2 x=y^{2} x = y 2
Fix η ∈ ( 0 , η ˉ T , τ 0 ) \eta\in(0,\bar{\eta}_{T,\tau_0}) η ∈ ( 0 , η ˉ T , τ 0 ) Theorem 3.2 , the entire contour u = η + i ξ u=\eta+i\xi u = η + i ξ ξ ∈ R \xi\in\mathbb{R} ξ ∈ R f ( η ; 0 , T ) < ∞ f(\eta;0,T)<\infty f ( η ; 0 , T ) < ∞ ∣ f ( η + i ξ ; 0 , T ) ∣ ≤ f ( η ; 0 , T ) \lvert f(\eta+i\xi;0,T)\rvert\le f(\eta;0,T) ∣ f ( η + i ξ ; 0 , T )∣ ≤ f ( η ; 0 , T ) z = η + i ξ z=\eta+i\xi z = η + i ξ
π 2 z 3 / 2 erfc ( K z ) = e − z K 2 2 K z 2 − 1 4 z 2 ∫ K 2 ∞ e − z x x − 3 / 2 d x , (68) \frac{\sqrt{\pi}}{2z^{3/2}}\operatorname{erfc}(K\sqrt{z})=\frac{e^{-zK^{2}}}{2Kz^{2}}-\frac{1}{4z^{2}}\int_{K^{2}}^{\infty}e^{-zx}x^{-3/2}\,\mathrm{d}x, \tag*{(68)} 2 z 3/2 π erfc ( K z ) = 2 K z 2 e − z K 2 − 4 z 2 1 ∫ K 2 ∞ e − z x x − 3/2 d x , (68) ∣ π 2 z 3 / 2 erfc ( K z ) ∣ ≤ e − η K 2 K ( η 2 + ξ 2 ) . \left\lvert\frac{\sqrt{\pi}}{2z^{3/2}}\operatorname{erfc}(K\sqrt{z})\right\rvert\le\frac{e^{-\eta K^{2}}}{K(\eta^{2}+\xi^{2})}. 2 z 3/2 π erfc ( K z ) ≤ K ( η 2 + ξ 2 ) e − η K 2 . The bound uses K > 0 K>0 K > 0 ∫ K 2 ∞ x − 3 / 2 d x = 2 / K \int_{K^{2}}^{\infty}x^{-3/2}\,\mathrm{d}x=2/K ∫ K 2 ∞ x − 3/2 d x = 2/ K
The function x ↦ e − η x ( x − K ) + x\mapsto e^{-\eta x}(\sqrt{x}-K)^{+} x ↦ e − η x ( x − K ) + [ 0 , ∞ ) [0,\infty) [ 0 , ∞ ) x x x x x x e η x e^{\eta x} e η x F 0 \mathcal{F}_{0} F 0 { f ( η ; 0 , T ) ≤ n } \{f(\eta;0,T)\le n\} { f ( η ; 0 , T ) ≤ n } n π e − η K 2 / ( K η ) < ∞ n\pi e^{-\eta K^{2}}/(K\eta)<\infty nπ e − η K 2 / ( K η ) < ∞
E 0 Q [ ( V I X T , τ 0 − K ) + ] = 1 4 π ∫ − ∞ ∞ erfc ( K η + i ξ ) ( η + i ξ ) 3 / 2 f ( η + i ξ ; 0 , T ) d ξ = 1 2 π ∫ 0 ∞ Re [ erfc ( K η + i ξ ) ( η + i ξ ) 3 / 2 f ( η + i ξ ; 0 , T ) ] d ξ . (69) \begin{aligned}
\mathbb{E}_{0}^{\mathbb{Q}}\left[(\mathrm{VIX}_{T,\tau_0}-K)^{+}\right]=\frac{1}{4\sqrt{\pi}}\int_{-\infty}^{\infty}\frac{\operatorname{erfc}(K\sqrt{\eta+i\xi})}{(\eta+i\xi)^{3/2}}f(\eta+i\xi;0,T)\,\mathrm{d}\xi\\
&=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{\operatorname{erfc}(K\sqrt{\eta+i\xi})}{(\eta+i\xi)^{3/2}}f(\eta+i\xi;0,T)\right]\mathrm{d}\xi. \tag*{(69)}
\end{aligned} E 0 Q [ ( VIX T , τ 0 − K ) + ] = 4 π 1 ∫ − ∞ ∞ ( η + i ξ ) 3/2 erfc ( K η + i ξ ) f ( η + i ξ ; 0 , T ) d ξ = 2 π 1 ∫ 0 ∞ Re [ ( η + i ξ ) 3/2 erfc ( K η + i ξ ) f ( η + i ξ ; 0 , T ) ] d ξ . (69) The second equality follows from conjugate symmetry. Multiplying by e − r T e^{-rT} e − r T (16) . ◻️
