FunctionalScale

Research code for functional GARCH and functional GAS-GARCH models applied to intraday volatility surfaces. The models treat within-day return curves as functional observations, extending classical GARCH/GAS dynamics to function space ($L^2[0,1]$).

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Theory

Setting

Let ${r_t(u) : u \in [0,1]}_{t=1}^T$ denote a sequence of intraday log-return curves across $T$ trading days, where $u$ indexes the normalized time within a day ($u = 0$ at market open, $u = 1$ at market close).

The conditional variance curve on day $t$ given the past filtration $\mathcal{F}_{t-1}$ is defined as the functional object $\sigma_t^2(\cdot) \in L^2[0,1]$.

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Functional GARCH(p,q)

Following Cerovecki et al. (2018), the model specifies:

$$y_t = \sigma_t \eta_t, \qquad (\eta_t) \text{ is an iid innovation process on } L^2[0,1] \text{ with } \mathbb{E}[\eta_t(u)] = 0, \ \mathbb{E}[\eta_t^2(u)] = 1$$

$$\sigma_t^2 = \delta + \sum_{i=1}^{q} \alpha_i(y_{t-i}^2) + \sum_{j=1}^{p} \beta_j(\sigma_{t-j}^2)$$

where multiplication is pointwise, the intercept curve $\delta \in H^+_*$, and $\alpha_i, \beta_j$ are positive kernel operators defined by:

$$\alpha(x) = \int_0^1 K_\alpha(u, v), x(v), dv, \qquad K_\alpha(u,v) \geq 0$$

Stationarity

A unique strictly stationary, non-anticipative solution exists if the top Lyapunov exponent of the companion operator sequence satisfies $\gamma < 0$. For a GARCH(1,1) process, this reduces to:

$$\mathbb{E}\log\left|(\alpha\Upsilon_{t-1} + \beta)\cdots(\alpha\Upsilon_1 + \beta)\right| < 0$$

where $\Upsilon_t$ is the pointwise multiplication operator by $\eta_t^2$.

Bernstein Parametrisation

To estimate the infinite-dimensional operators, we expand them using a basis of non-negative Bernstein polynomials $\varphi_1,\ldots,\varphi_M$:

$$\delta = \sum_{k=1}^M d_k,\varphi_k, \qquad \alpha_i = \sum_{k,\ell=1}^M a_{k\ell}^{(i)},\varphi_k \otimes \varphi_\ell, \qquad \beta_j = \sum_{k,\ell=1}^M b_{k\ell}^{(j)},\varphi_k \otimes \varphi_\ell$$

where $(\varphi_k \otimes \varphi_\ell)(x) = \varphi_k\langle x, \varphi_\ell\rangle$. Non-negativity of the coefficients guarantees $\alpha_i, \beta_j \in K^+(H)$.

QMLE Estimation

The parameters are estimated via Quasi-Maximum Likelihood:

$$\hat{\theta}_n = \arg\min_{\theta \in \Theta}; \frac{1}{n}\sum_{t=1}^n \sum_{m=1}^M \left(\frac{\langle y_t^2,, \varphi_m \rangle}{\langle \tilde{\sigma}_t^2,, \varphi_m \rangle} + \log\langle \tilde{\sigma}_t^2,, \varphi_m \rangle\right)$$

Note: The current implementation (loss_func in garch.py) uses a Bernstein-projected MSE approximation rather than the formal QMLE criterion. The inner products are evaluated via sample means on a uniform grid of $N$ points, optimized using scipy.optimize (SLSQP).

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Functional GAS-GARCH

The Generalized Autoregressive Score (GAS) extension maps the infinite-dimensional update to a score-driven recursion on a low-dimensional cubic B-spline coefficient vector $b_t$.

Let $\Phi(u) = (\phi_1(u), \ldots, \phi_M(u))^\top$ be the B-spline basis. The log-volatility curve is parameterized as:

$$\log \sigma_t(u) = \Phi(u)^\top b_t$$

The coefficients evolve via the GAS recursion:

$$b_t = \omega + B,b_{t-1} + A,s_{t-1}$$

Likelihood & Score

Evaluated on a discrete grid $u_1, \dots, u_N$, the return vector $r_t \in \mathbb{R}^N$ is modeled as a multivariate Student-$t$ distribution with $\nu$ degrees of freedom and an Ornstein-Uhlenbeck covariance kernel $\Lambda_\delta$ ($[\Lambda_\delta]_{ij} = \exp(-|u_i - u_j|/\delta)$):

$$r_t \mid \mathcal{F}_{t-1} \sim t_\nu\left(0,\ S_t \Lambda_\delta S_t\right), \quad S_t = \mathrm{diag}(\exp(\sigma_t(u_i)/2))$$

Let $\tilde{r}_t = S_t^{-1} r_t$ be the standardized returns, and $A_1 = 1 + \tilde{r}t^\top \Lambda\delta^{-1} \tilde{r}t,/,\nu$. The analytic score $s_t = \nabla{b_t} \log p(r_t | b_t)$ evaluates to:

$$s_t = -\frac{1}{2},\Phi,\mathbf{1}_N + \frac{\nu + N}{2\nu A_1},\Phi\left(\tilde{r}_t \odot \Lambda_\delta^{-1}\tilde{r}_t\right)$$

where $\Phi$ is the $(M \times N)$ basis matrix and $\odot$ denotes the Hadamard product.

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