Investment models

1. Trend Score: how strong is the price direction?

The model looks at three horizons: roughly 1 month, 3 months and 1 year. Each price move is divided by the current daily volatility, so the same move does not mean the same thing in a calm market and a nervous one.

1. $$r_t = \ln\frac{P_t}{P_{t-1}}$$

   Daily log return. P is the adjusted close.

2. $$\sigma_t^2 = 0.94\,\sigma_{t-1}^2 + 0.06\,r_t^2$$

   EWMA daily variance. Recent moves weigh more, with λ = 0.94.

3. $$z(h) = \frac{\ln(P_t / P_{t-h})}{\sigma_t \sqrt{h}}, \qquad h \in \{21,\ 63,\ 252\}$$

   Momentum over horizon h, adjusted for risk and for horizon length. h is in trading days.

4. $$\text{Trend Score} = \frac{100\tanh z(21) + 100\tanh z(63) + 100\tanh z(252)}{3}$$

   tanh caps the extremes, so the result always lies between −100 and +100. Positive means a rising trend, negative a falling one, near zero no direction.

2. IVCalibrated: what does the market expect?

Option-implied volatility expresses the swings the market expects. On its own it is systematically biased (usually too high), so the model calibrates it historically against the volatility actually realised over the following 21 trading days.

1. $$\text{IVVar}_t = \frac{(IV_t / 100)^2}{252}$$

   Implied volatility in percent converted to a daily variance.

2. $$\ln(\text{FutureVar21}_t) = \alpha_t + \beta_t \ln(\text{IVVar}_t) + \varepsilon_t$$

   Expanding OLS: the model only uses observations whose full 21 following days are already known, and needs at least 504 of them. α and β are therefore re-estimated each day from the past only.

3. $$\text{ForecastVar}_t = e^{\alpha_t + \beta_t \ln(\text{IVVar}_t)} \cdot \overline{e^{\varepsilon}}$$

   Duan's smearing factor (the mean of e to the residuals) corrects the bias of transforming back from logs.

4. $$\text{Expected Vol} = 100\sqrt{252 \cdot \text{ForecastVar}_t}, \qquad \text{Current Vol} = 100\sqrt{252 \cdot \overline{r^2}_{21}}$$

   Both are annualised and in percent, so they compare directly.

3. Position rules

• CASH → LONG: Trend Score > 0 and risk falling.
• LONG → CASH: Trend Score < 0 and risk rising.
• If neither holds, nothing changes: LONG stays LONG, CASH stays CASH. This limits needless switching on mixed signals.

Cooldown: after every real state change the model ignores new signals for the next five trading days. On day 6 the state may change again. A signal from the close is executed on the next trading day.

4. The daily report

The model runs on this site every trading day after the US close: it downloads the prices of the underlyings and the implied-volatility indices, recomputes all four markets and stores the results.

A language model, Google Gemini, then writes a short report. It receives only the numbers from the calculation, nothing else, and may not write about events that are not in them. Every number in the finished text is checked automatically against the calculation; text that fails is not published and a template writes the report instead.

Example: S&P 500 expected volatility on 30 September 2026

The VIX closed at 16.34. On that day the model had estimated α = −0.07230, β = 1.05394 and a smearing factor of 1.31278 (from 8,451 observations).

1. $$\text{IVVar} = \frac{(16.34/100)^2}{252} = \frac{0.0266996}{252} = 0.000105951$$

   Implied volatility as a daily variance.

2. $$\alpha + \beta\ln(\text{IVVar}) = -0.07230 + 1.05394 \cdot (-9.15254) = -9.71856$$

   The linear part of the regression.

3. $$\text{ForecastVar} = e^{-9.71856} \cdot 1.31278 = 0.0000601566 \cdot 1.31278 = 0.0000789725$$

   Back from logs, with Duan's correction.

4. $$\text{Expected Vol} = 100\sqrt{252 \cdot 0.0000789725} = 100\sqrt{0.0199011} \approx 14.11\%$$

   Annualised expected volatility.

Realised volatility over the last 21 days was 10.52%. Since 14.11% > 10.52%, risk is rising. The Trend Score was +41, positive, but with rising risk neither the LONG nor the CASH condition holds, so the model keeps its previous state.

Assumptions and limits

• The link between implied and future volatility is estimated from the past and assumed to hold. It may not survive structural breaks.
• The model ignores transaction costs, taxes and euro/dollar currency risk.
• Oil is modelled through the USO fund, not WTI futures directly.
• The LONG/CASH state is computed from the history stored on this site. It starts in CASH, so in the first days of history it may differ from the model's long-run path.

Related concepts

• Volatility
• Implied volatility
• Momentum
• Risk
• Diversification