Comparing Volatility Forecasting Models: GARCH, Realized Vol, and HAR-RV

Worked Solution

How to Think About It: This is a classic desk question -- the PM wants a vol forecast for position sizing or options pricing, and you need to justify your modeling choice. The tension is between sophistication and practicality. GARCH models are easy to fit on daily closes, but they miss intraday information. Realized vol uses tick data and is more precise, but it is noisy and requires clean high-frequency feeds. HAR-RV sits in between: it uses realized vol as an input but models it parsimoniously with daily, weekly, and monthly components. The right answer depends on your data infrastructure, the asset, and the horizon.

Key Insight: The three model families are not really competing -- they operate at different points on the data-richness vs. parsimony spectrum. The deeper insight on SOFR vs. Treasury vol is that vol is not just a statistical artifact; it reflects market structure, and derivatives supply can compress realized vol in ways that purely statistical models will not capture.

The Method:

*Part 1 -- Model Comparison:*

• GARCH(1,1): Models conditional variance as $\sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$. Requires only daily returns. Strengths: parsimonious, well-understood, easy to estimate via MLE, captures volatility clustering. Limitations: symmetric -- treats positive and negative shocks identically, slow to adapt to regime changes, ignores intraday information.

• EGARCH: Models $\log(\sigma_t^2)$ as a function of standardized residuals, allowing the *sign* of the shock to matter (leverage effect). Avoids the non-negativity constraints of standard GARCH. Strengths: captures asymmetric vol response (negative returns increase vol more than positive returns). Limitations: harder to interpret parameters, still limited to daily data, leverage effect is less pronounced in rates than in equities.

• GJR-GARCH: Adds an indicator term: $\sigma_t^2 = \omega + (\alpha + \gamma \mathbf{1}_{\varepsilon_{t-1}<0}) \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2$, where $\gamma > 0$ means negative shocks have a larger impact. Strengths: directly parameterizes asymmetry, easy to test ($\gamma = 0$ means symmetric). Limitations: same daily-data limitation as GARCH.

• Realized Volatility (RV): Computed from intraday returns: $RV_t = \sum_{i=1}^{M} r_{t,i}^2$ where $M$ is the number of intraday intervals. Strengths: model-free, uses all available price information, converges to integrated variance as sampling frequency increases. Limitations: microstructure noise biases RV upward at very high frequencies (need kernel-based or subsampling corrections), requires reliable tick data, sensitive to jumps.

• HAR-RV: Regresses future RV on daily, weekly, and monthly RV components: $RV_{t+1} = \beta_0 + \beta_D RV_t + \beta_W RV_t^{(w)} + \beta_M RV_t^{(m)} + \varepsilon_{t+1}$, where $RV_t^{(w)} = \frac{1}{5}\sum_{j=0}^{4} RV_{t-j}$ and similarly for monthly. Strengths: captures the well-documented cascade of vol components at different horizons (short-term traders, medium-term funds, long-term investors), simple linear model, strong out-of-sample performance in empirical studies. Limitations: requires realized vol as input (so you still need intraday data), linear structure may miss regime shifts, no built-in mechanism for asymmetry (though you can add signed jump components).

*Part 2 -- Why SOFR Vol Can Be Lower Than Treasury Vol:*

1. Derivatives market vol supply: Large dealers are structurally short SOFR options (they sell caps, floors, and swaptions to hedgers). When they delta-hedge these positions, they systematically *dampen* realized moves in SOFR. This is the same mechanism by which SPX overwriting strategies compress equity vol -- the derivatives market itself becomes a volatility sink.

1. Anchoring by Fed policy: SOFR is an overnight rate closely tied to the Fed's target range. Between FOMC meetings, SOFR is nearly pinned. Treasury yields, especially beyond the front end, incorporate term premium, supply/demand for duration, and global macro flows -- all of which add volatility.

1. Microstructure differences: SOFR is derived from overnight repo transactions, a deep and highly standardized market. Treasury yields reflect trading across a curve of maturities with varying liquidity, dealer inventory effects, and auction cycles. The repo market's structural simplicity compresses SOFR vol.

1. Reserve abundance regime: Under ample reserves, the Fed's administered rates (IORB, ON RRP) create a corridor that mechanically bounds SOFR. Treasury yields face no such corridor.

*Part 3 -- Practical Recommendation:*

For a live trading system needing next-day forecasts:

• If you have reliable intraday data for both instruments: HAR-RV is the best default choice. It is simple to implement, robust out of sample, and naturally adapts to multi-horizon vol dynamics. You can augment it with a jump component for event days (FOMC, NFP).

• If you only have daily closes: GJR-GARCH is the pragmatic fallback. It captures the most important stylized fact (vol clustering) with minimal data requirements, and the asymmetry parameter is a useful diagnostic even in rates.

• Key practical trade-offs: HAR-RV wins on forecast accuracy but requires a clean tick-data pipeline and RV computation infrastructure. GARCH wins on simplicity and robustness to data gaps. For SOFR specifically, be aware that any statistical model will underestimate vol around FOMC meetings and overestimate it in quiet periods -- you need a regime-aware overlay or at least a calendar adjustment.

Answer: GARCH-class models use daily returns and capture vol clustering but miss intraday information and (in standard form) vol asymmetry. EGARCH and GJR-GARCH fix the asymmetry issue. Realized vol is model-free and precise but requires clean tick data and microstructure noise correction. HAR-RV combines realized vol inputs with a parsimonious multi-horizon regression and tends to produce the best next-day forecasts empirically. SOFR vol can be lower than Treasury vol due to derivatives-market dampening (dealer hedging compresses moves), Fed corridor anchoring, and the structural simplicity of the repo market. For a production system, HAR-RV with intraday data is the preferred choice; GJR-GARCH is the best daily-data fallback.