C. Statistical Arbitrage (C.10--C.13)

Market-neutral relative value within the precious metals complex. These strategies exploit mean-reverting spreads between metals, miners, and levered products using cointegration, Kalman filtering, and structural decay models.

C.10 Gold/Silver Ratio

Module: qgtm_strategies/gold_silver_ratio.py Class: GoldSilverRatioStrategy ID: gold_silver_ratio

Economic Rationale

The gold/silver ratio (GSR) has historically oscillated between ~40x and ~90x. Extreme readings tend to mean-revert because gold and silver share common macro drivers (real rates, dollar) but have divergent supply/demand structures (monetary vs. industrial). When the ratio is stretched, the lagging metal tends to catch up.

Signal Formula

Step 1 -- Ratio and Kalman-filtered hedge ratio:

\[ \text{GSR}_t = \frac{P_{\text{GLD},t}}{P_{\text{SLV},t}} \]

The hedge ratio $\beta_t$ is estimated via a Kalman filter:

\[ \beta_t = \beta_{t-1} + K_t \cdot (P_{\text{GLD},t} - \beta_{t-1} \cdot P_{\text{SLV},t}) \]

where $K_t$ is the Kalman gain.

Step 2 -- Z-score:

\[ z_t = \frac{\text{GSR}_t - \bar{\text{GSR}}_N}{\sigma_{\text{GSR}, N}} \]

Step 3 -- Half-life gate (Ornstein-Uhlenbeck estimation):

\[ d\text{GSR} = \theta \cdot (\mu - \text{GSR}) \, dt + \sigma \, dW \]

\[ t_{1/2} = \frac{\ln 2}{\theta} \]

Gate: proceed only if $10 \leq t_{1/2} \leq 120$ days.

Step 4 -- Entry/exit rules:

Condition	Position
$z_t > 2$ and $t_{1/2} \in [10, 120]$	Long SLV / Short GLD
$z_t < -2$ and $t_{1/2} \in [10, 120]$	Long GLD / Short SLV
$\\|z_t\\| < 0.5$	Close

Spread Dynamics

graph TD
    GSR[Gold/Silver Ratio] --> ZS{z-score?}
    ZS -- "> +2" --> HL{Half-life<br/>10-120d?}
    HL -- Yes --> LS[Long SLV<br/>Short GLD]
    HL -- No --> FLAT1[FLAT<br/>No mean reversion]
    ZS -- "< -2" --> HL2{Half-life<br/>10-120d?}
    HL2 -- Yes --> LG[Long GLD<br/>Short SLV]
    HL2 -- No --> FLAT2[FLAT]
    ZS -- "-2 to +2" --> HOLD[Hold / Close<br/>at |z| < 0.5]

Factor Exposures

Factor	Loading
Metals	+0.3
Value	+0.6
Momentum	-0.2

Capacity & Decay

Max capacity: $25M
Alpha half-life: 63 days
Kill condition: Half-life exceeds 120 days persistently (cointegration breakdown)
Expected Sharpe: 0.7--1.1

References

Gatev, Goetzmann, Rouwenhorst (2006) "Pairs Trading"
Vidyamurthy (2004) "Pairs Trading" -- cointegration approach
Hamilton (1994) "Time Series Analysis" -- OU half-life estimation

C.11 Gold/Platinum

Status: Planned

Economic Rationale

Gold/platinum ratio exploits the divergent supply dynamics: gold is monetarily driven while platinum depends on autocatalyst demand (diesel vehicles, hydrogen fuel cells). When the ratio is extreme, mean reversion is driven by industrial cycle shifts. Platinum's smaller market amplifies the snap-back.

Signal Formula (Proposed)

\[ \text{GPR}_t = \frac{P_{\text{GLD},t}}{P_{\text{PPLT},t}} \]

Same Kalman + OU framework as C.10, but with:

Z-score threshold: $\pm 2.5$ (wider due to structural trend post-2015)
Half-life gate: $[15, 180]$ days (slower mean reversion)
Regime gate: exclude when auto sector in recession (proxy: global PMI < 48)

Expected Properties

Max capacity: $15M (PPLT is less liquid)
Expected Sharpe: 0.4--0.7

References

Johnson Matthey PGM Market Report (annual)
WPIC (World Platinum Investment Council) reports

C.12 Miners vs Metal

Status: Planned

Economic Rationale

Gold mining equities (GDX, GDXJ) are a leveraged play on the gold price because miners have fixed costs (AISC ~ $1,200/oz). When the GDX/GLD ratio deviates from the historical beta, the relative value trade captures the reversion. Miners also embed equity market beta that can be hedged.

Signal Formula (Proposed)

Step 1 -- Beta-adjusted spread:

\[ \text{spread}_t = \ln(P_{\text{GDX}}) - \beta_{\text{KF}} \cdot \ln(P_{\text{GLD}}) - \gamma \cdot \ln(P_{\text{SPY}}) \]

where $\gamma$ hedges the equity beta component.

Step 2 -- Z-score with vol-adjusted threshold:

\[ z_t = \frac{\text{spread}_t - \bar{\text{spread}}_N}{\sigma_{\text{spread}, N}} \]

Entry at $|z| > 2$. Size inversely proportional to GDX implied vol (scale down in high-vol regimes to control risk).

Expected Properties

Max capacity: $20M
Expected Sharpe: 0.5--0.9
Key risk: Miners can decouple from gold due to company-specific events (M&A, mine closures)

References

Tufano (1998) "The Determinants of Stock Price Exposure: Financial Engineering and the Gold Mining Industry"
Baur & McDermott (2010)

C.13 Levered ETF Decay

Status: Planned

Economic Rationale

Leveraged ETFs (NUGT 2x bull gold miners, DUST 2x bear) suffer volatility decay from daily rebalancing. In a range-bound or mean-reverting market, the levered product bleeds value relative to the underlying. This strategy shorts the levered product and hedges with the underlying, harvesting the structural decay.

Signal Formula (Proposed)

Theoretical decay rate:

\[ \text{decay}_T = (1 + L \cdot r)^T - (1 + r)^{L \cdot T} \approx -\frac{L(L-1)}{2} \sigma^2 T \]

where $L$ is the leverage factor, $r$ is the daily return, and $\sigma$ is daily vol.

Practical entry:

\[ \text{RV}_{21d} > \text{percentile}_{75}(\text{RV},\; 252) \quad \wedge \quad |\text{trend}_{21d}| < 1\sigma \]

When realized vol is high AND the market is range-bound (no strong trend), decay is maximized. Short NUGT + long GDX delta-hedged.

Expected Properties

Max capacity: $10M (levered ETF AUM limits)
Expected Sharpe: 0.6--1.0
Key risk: Strong trending markets where levered products outperform (trend gate mitigates)

References

Cheng & Madhavan (2009) "The Dynamics of Leveraged and Inverse Exchange-Traded Funds"
Avellaneda & Zhang (2010) "Path-Dependence of Leveraged ETF Returns"

Condition	Position
\(z_t > 2\) and \(t_{1/2} \in [10, 120]\)	Long SLV / Short GLD
\(z_t < -2\) and \(t_{1/2} \in [10, 120]\)	Long GLD / Short SLV
\(\\|z_t\\| < 0.5\)	Close