Time-Changed Lévy Process

Definition

A time-changed Lévy process is a log-price model of the form $Y_t = mt + Z_{\tau_t}$ where $Z$ is a Lévy process (independent and stationary increments, $\text{Var}(Z_1) < \infty$ , $Z_0 = 0$ ) and $\tau_t = \int_0^t \sigma_u\,du$ is a non-decreasing stochastic time-change, assumed independent of $Z$ ( $Z \perp \tau$ ). The time-change is driven by a non-negative spot variance process $\sigma_t$ . When $Z$ is Brownian motion this reduces to the standard diffusion SV model; non-Brownian $Z$ introduces infinite-activity or finite-activity jumps.

Key Ideas

Four canonical Lévy processes in financial economics: (i) Brownian motion (continuous paths); (ii) jump diffusion — Brownian motion plus compound Poisson (finite-activity jumps, Merton 1976); (iii) variance gamma (Madan-Seneta 1990, infinite-activity); (iv) normal inverse Gaussian, NIG (Barndorff-Nielsen 1998, infinite-activity). Only Brownian motion has continuous sample paths; equivalently, $k_4 = 0$ for Brownian motion.
Fourth cumulant characterization: For a symmetric Lévy process $Z$ , $k_4 = 0$ if and only if $Z$ is scaled Brownian motion plus drift. This follows from the Lévy-Khintchine representation: $k_4 = \int x^4 \,\nu(dx)$ where $\nu$ is the Lévy measure, so $k_4 = 0$ implies $\nu \equiv 0$ . Testing $k_4 = 0$ using high-frequency returns is therefore a test for absence of jumps.
Quadratic variation and inconsistency: $[Y]_t = k_2\tau_t + \sum_{s \leq t}(\Delta Z_{\tau_s})^2$ . In the non-Brownian case the jump-QV term $\sum (\Delta Z)^2$ does not vanish, so $[Y]_{d,i} \xrightarrow{p} [Y]_i \neq k_2\tau_i$ . The irreducible variance of the RV error is $k_4\Delta\xi$ (where $\xi = E[\sigma_t]$ , $\Delta$ = period length), which is $O(1)$ in $M$ — it does not shrink as sampling frequency increases.
Bias is small: $E([Y]_{d,i} - k_2\tau_i) = O(M^{-1})$ , and the squared bias $= O(M^{-2})$ is dominated by the $O(1)$ variance term in the MSE. The bias-variance decomposition shows that inconsistency matters primarily through increased estimation variance, not through bias.
ACF inequality: $\text{Cor}([Y]_{d,i}, [Y]_{d,i+s}) \leq \text{Cor}(\tau_i, \tau_{i+s})$ when $k_4 > 0$ , with equality only in the Brownian case. As $k_4 \to \infty$ , the RV autocorrelation $\to 0$ even if the true variance process is highly persistent. Volatility is more forecastable than RV-based ACFs suggest.
Subordination lineage: Bochner (1949) introduced time-changed Brownian motion; Clark (1973) applied it to speculative prices. Carr et al. (2003) and Carr-Wu (2004) extended to non-Brownian $Z$ for option pricing. Barndorff-Nielsen and Shephard (2001) derived second-order properties of $\tau_t$ for OU-driven $\sigma_t$ .

How It Works

Second-order properties of RV

Let $[Y]_{d,i}^{**} = [Y]_{d,i}$ be realized variance for period $i$ . Ignoring $O(M^{-2})$ terms: $E([Y]_{d,i}) \approx k_2\Delta\xi, \quad \text{Cov}([Y]_{d,i}, [Y]_{d,i+s}) \approx k_2^2\omega^2\tilde{r}^{**}_{\Delta s}$ $\text{Var}([Y]_{d,i}) \approx 2\omega^2 k_2^2 r_\Delta^{**} + k_4\Delta\xi\cdot\{(4k_1k_3 + 2k_2^2)(2\omega^2 Mr_d^{**} + M^{-1}\Delta^2\xi^2)\}$ where $r_s = \text{Cor}(\sigma_t, \sigma_{t+s})$ , $\omega^2 = \text{Var}(\sigma_t)$ , and $r_s^{**}$ , $\tilde{r}^{**}$ are integrated autocorrelation functions.

Quasi-likelihood estimation

Build a Gaussian QL from the above second-order properties; constrain $k_2 = 1$ (otherwise $k_2$ and $\xi$ are not separately identified from $\text{Var}([Y]_{d,i})$ ). Sandwich standard errors $I^{-1}JI^{-1}$ .

Computation: (i) Durbin algorithm ( $O(n^2)$ ): exploits Toeplitz structure of $\text{Cov}([Y]_{d}^{**})$ when $\tau$ is stationary; works for any parameterised autocorrelation function. (ii) Kalman filter ( $O(n)$ ): applies when $\tau_i - \Delta\xi = x'\$_i$ and $\$$ satisfies a linear state-space recursion; ~20× faster than Durbin for $n=3000$ and 2-dimensional state.

Model hierarchy

Model	$r_s$	State dim	Algorithm
Single OU	$e^{-\lambda s}$	2	Kalman
$J$ -factor superposition	$\sum_j w_j e^{-\lambda_j s}$	$2J$	Kalman
Log-normal OU (LNOU)	$[e^{\sigma^2_{\log}e^{-\lambda\|s\|}}-1]/[e^{\sigma^2_{\log}}-1]$	approx. $2\times 10$	Kalman (truncated Poisson approx.)
Gamma long-memory	$(1 + s/a)^{-2H}$ , $H < 1/2$	$\infty$	Durbin

For single OU: $\text{Cov}(\tau_i, \tau_{i+s})$ gives ARMA(1,1) for $\tau_i$ with AR root $\phi = e^{-\lambda\Delta}$ and analytic MA root from the ratio $r_1 = \phi(\text{Cor}(\tau_i, \tau_{i+1}) - \phi)/[(1+\phi^2) - 2\phi\,\text{Cor}(\tau_i, \tau_{i+1})]$ .

Empirical results (USD/DEM, Olsen data, 1986–1996, $M=288$ )

Single OU: poor ACF fit in both Brownian and Lévy cases (too little memory)
$J=2$ superposition: two well-separated components (persistent + rapidly reverting, roughly equal weight); Lévy logL −1195 vs. Brownian −1222; RV ACF tracked to lag 50
Log-normal OU (Lévy): logL −1219; most stable as $M$ decreases; $J=3$ Lévy ≈ $J=2$ LNOU
Gamma long-memory: $\hat{H} \approx 0.17$ (Lévy, pure gamma); somewhat less stable across $M$ ; consistent with FIGARCH estimates

Why It Matters

Time-changed Lévy processes provide a unified framework for non-Gaussian volatility models that encompasses both continuous (diffusion) and jump components. The RV inconsistency result quantifies the cost of jumps for realized variance estimation: the signal that RV recovers is the quadratic variation (continuous + jump), not the integrated variance (continuous only). The QL estimation framework connects high-frequency second-order properties directly to continuous-time model parameters without requiring distributional assumptions beyond moments, making estimation computationally tractable even for complex memory structures. The ACF inequality provides a formal justification for the claim that "true" volatility predictability exceeds what realized variance series reveal.

Open Questions

Leverage. The model assumes $Z \perp \tau$ , ruling out the leverage effect (negative correlation between returns and future volatility). Extending the framework to correlated $Z$ and $\tau$ is technically involved and remains an open area.
Microstructure noise. The theory assumes clean high-frequency prices; microstructure contamination biases RV upward and complicates inference on $k_4$ and the time-change.
Bipower variation. Under the Lévy framework, $BV_t = \frac{\pi}{2}\sum_j|r_j||r_{j-1}|$ consistently estimates the continuous component of QV even with finite-activity jumps (Barndorff-Nielsen-Shephard 2004), and $RV - BV$ consistently estimates jump variation. Whether BV-based estimation of $\tau_t$ outperforms RV-based in finite samples for infinite-activity Lévy processes is unresolved.