QuanTAlib/lib/volatility/rvi/Rvi.md

RVI: Relative Volatility Index

"Not all volatility is created equal—upward volatility feels like profit, downward volatility feels like loss. RVI separates these psychological experiences into a quantifiable measure."

The Relative Volatility Index (RVI) is a directional volatility oscillator that distinguishes between upward and downward price volatility. Originally developed by Donald Dorsey in 1993, RVI measures the standard deviation of closing prices and categorizes this volatility based on whether prices are rising or falling. The result is an oscillator bounded between 0 and 100, where values above 50 indicate upward volatility dominance and values below 50 indicate downward volatility dominance.

Historical Context

Donald Dorsey introduced the Relative Volatility Index in the June 1993 issue of Technical Analysis of Stocks & Commodities magazine. Dorsey designed RVI as a confirmation indicator rather than a standalone signal generator, intending it to be used alongside RSI to confirm trend strength and momentum.

The key innovation was separating volatility into directional components. Traditional volatility measures (standard deviation, ATR) treat upward and downward price movements identically. Dorsey recognized that traders experience these movements differently: upward volatility in a long position feels like opportunity, while downward volatility feels like risk.

The original 1993 formula used a 10-period standard deviation and 14-period Wilder's smoothing (RMA). This implementation follows the PineScript reference which uses bias-corrected RMA to ensure proper warmup behavior during the initial periods.

Architecture & Physics

1. Rolling Population Standard Deviation

First, compute the population standard deviation of closing prices over stdevLength periods:

\sigma_t = \sqrt{\frac{\sum_{i=0}^{n-1}(P_{t-i} - \bar{P})^2}{n}}

where:

• P_t = Closing price at time t
• \bar{P} = Mean of prices in the window
• n = stdevLength (default 10)

The computational form uses running sums for O(1) updates:

\sigma_t = \sqrt{\frac{\sum P_i^2}{n} - \left(\frac{\sum P_i}{n}\right)^2}

2. Directional Classification

Based on price change direction, assign the volatility to either upward or downward:

\text{upStd}_t = \begin{cases}
\sigma_t & \text{if } P_t > P_{t-1} \\
0 & \text{otherwise}
\end{cases}

\text{downStd}_t = \begin{cases}
\sigma_t & \text{if } P_t < P_{t-1} \\
0 & \text{otherwise}
\end{cases}

Note: When P_t = P_{t-1} (unchanged), both upStd and downStd are zero. The volatility is "orphaned" rather than assigned to either direction.

3. Bias-Corrected RMA Smoothing

Both directional volatilities are smoothed using Wilder's RMA (Exponential Moving Average with \alpha = 1/n) with bias correction for proper warmup:

Raw RMA update:

\text{raw}_t = \frac{\text{raw}_{t-1} \cdot (n-1) + x_t}{n}

Bias correction factor:

e_t = (1 - \alpha) \cdot e_{t-1}

Corrected output:

\text{avgStd}_t = \begin{cases}
\frac{\text{raw}_t}{1 - e_t} & \text{if } e_t > \epsilon \\
\text{raw}_t & \text{otherwise}
\end{cases}

where \alpha = 1/\text{rmaLength} and \epsilon = 10^{-10}.

This bias correction compensates for the zero initialization of raw RMA, preventing artificially low values during warmup.

4. Final RVI Calculation

\text{RVI}_t = \begin{cases}
100 \times \frac{\text{avgUpStd}_t}{\text{avgUpStd}_t + \text{avgDownStd}_t} & \text{if sum} > 0 \\
50 & \text{otherwise}
\end{cases}

Mathematical Foundation

Relationship to RSI

RVI shares structural similarity with RSI, but measures different quantities:

Aspect	RSI	RVI
Measures	Price changes	Price volatility
Up component	Positive price change	Stddev when price rises
Down component	Negative price change	Stddev when price falls
Formula	100 \times \frac{\text{avgGain}}{\text{avgGain} + \text{avgLoss}}	100 \times \frac{\text{avgUpStd}}{\text{avgUpStd} + \text{avgDownStd}}
Range	0-100	0-100

Both use RMA smoothing for the components.

Bias Correction Derivation

Standard RMA initialized to zero produces biased estimates during warmup. After t updates:

\text{bias} = (1 - \alpha)^t

The correction factor 1/(1-e_t) cancels this bias, ensuring unbiased estimates from the first bar.

Interpretation Zones

RVI Range	Interpretation
80-100	Strong upward volatility dominance
60-80	Moderate upward volatility
40-60	Neutral/balanced volatility
20-40	Moderate downward volatility
0-20	Strong downward volatility dominance

Warmup Period

RVI requires stdevLength bars for the standard deviation calculation plus additional bars for RMA convergence. Effective warmup:

\text{warmup} \approx \text{stdevLength} + 3 \times \text{rmaLength}

With defaults (10, 14): approximately 52 bars for 95% convergence.

Performance Profile

Operation Count (Streaming Mode, Scalar)

Per-bar operations after warmup:

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	8	1	8
MUL	6	3	18
DIV	5	15	75
SQRT	1	15	15
CMP	3	1	3
Total	—	—	~119 cycles

Dominant cost: five divisions (63%) for variance calculation and RMA updates.

Memory Profile

• State struct: ~88 bytes (stddev buffer, RMA states, counters)
• RingBuffer: 8 bytes × stdevLength (default 80 bytes)
• 100 instances @ defaults: ~16.8 KB

Quality Metrics

Metric	Score	Notes
Accuracy	8/10	Precise directional volatility measurement
Timeliness	7/10	RMA smoothing introduces lag
Responsiveness	7/10	Responds to volatility regime changes
Smoothness	8/10	Double smoothing (stddev + RMA)
Interpretability	9/10	Clear 0-100 scale with intuitive meaning

Validation

Library	Status	Notes
TA-Lib	N/A	Not implemented
Skender	N/A	Not implemented
Tulip	N/A	Not implemented
OoplesFinance	❔	Different algorithm (RSI-based)
PineScript	✅	Matches rvi.pine reference

Note: Some libraries implement "RVI" as a different indicator (often RSI applied to volatility). This implementation follows Dorsey's original design using directional standard deviation.

Common Pitfalls

1. Confusion with other RVI indicators: "RVI" name is used for at least three different indicators. Dorsey's original (this implementation) uses directional standard deviation. Others use RSI-like calculations on price or volume. Verify the algorithm before comparing values.

2. Unchanged price handling: When price doesn't change (P_t = P_{t-1}), the volatility is orphaned (neither up nor down). Extended flat periods push RVI toward 50 regardless of prior trend.

3. Warmup period: With defaults, RVI needs ~52 bars to converge. Early values may be misleading. The IsHot property indicates warmup completion.

4. Not a standalone signal: Dorsey designed RVI as a confirmation indicator. Use with RSI or trend indicators, not alone. High RVI confirms uptrend strength; low RVI confirms downtrend strength.

5. Volatility vs direction confusion: RVI measures which direction has MORE volatility, not which direction price is moving. A slow steady uptrend with occasional sharp drops can show low RVI despite rising prices.

6. Parameter sensitivity: Shorter stdevLength increases noise sensitivity. Longer rmaLength increases lag. Default 10/14 balances responsiveness and stability.

7. Zero denominator: When both avgUpStd and avgDownStd approach zero (flat market), RVI defaults to 50. This is mathematically correct but may mask the lack of volatility.

Trading Applications

Trend Confirmation (Dorsey's Original Use)

Combine with RSI for confirmation:

RSI > 50 AND RVI > 50: Confirmed uptrend
RSI < 50 AND RVI < 50: Confirmed downtrend
RSI > 50 AND RVI < 50: Divergence - uptrend weakening
RSI < 50 AND RVI > 50: Divergence - downtrend weakening

Volatility Regime Detection

Track RVI for directional volatility shifts:

RVI crossing above 60: Upward volatility expanding
RVI crossing below 40: Downward volatility expanding
RVI oscillating 40-60: Balanced/consolidating

Entry/Exit Filters

Use RVI as a filter for other signals:

Only take long entries when RVI > 50 (upward volatility dominance)
Only take short entries when RVI < 50 (downward volatility dominance)

Divergence Trading

RVI divergences from price can signal reversals:

Price making higher highs + RVI making lower highs: Bearish divergence
Price making lower lows + RVI making higher lows: Bullish divergence

References

• Dorsey, D. (1993). "The Relative Volatility Index." Technical Analysis of Stocks & Commodities, 11(6), 253-256.
• Dorsey, D. (1995). "Refining the Relative Volatility Index." Technical Analysis of Stocks & Commodities, 13(9).
• TradingView. (2024). "PineScript Reference Implementation." rvi.pine source file.