QuanTAlib/lib/statistics/convexity/Convexity.md

Convexity: Beta Convexity

The asymmetry between upside and downside beta reveals whether an asset delivers convex payoffs — the holy grail of portfolio construction.

Property	Value
Category	Statistic
Inputs	Dual (asset price, market price)
Parameters	period
Outputs	5 series (BetaStd, BetaUp, BetaDown, Ratio, Convexity)
Output range	Convexity ≥ 0; BetaStd/Up/Down unbounded
Warmup	period + 1 bars

• Convexity measures how asymmetrically an asset responds to market up-moves vs. down-moves.
• Similar: Beta, Correl | Trading note: Convexity > 0 signals favorable payoff asymmetry. Ratio > 1 = asset amplifies gains more than losses.
• Based on Skender.Stock.Indicators GetBeta(BetaType.All) implementation.

Beta Convexity decomposes the standard beta coefficient into its upside and downside components, then measures their squared difference. An asset with positive convexity captures more upside than downside — the ideal characteristic for portfolio construction. Harry Markowitz's Modern Portfolio Theory shows that investors should seek assets that maximise (β⁺ - β⁻)².

Historical Context

The concept of separating upside and downside beta was pioneered by Bawa and Lindenberg (1977) in their work on lower partial moments. It gained mainstream traction through Ang, Chen, and Xing's landmark 2006 paper "Downside Risk," which demonstrated that stocks with high downside beta earn higher returns — the so-called "downside risk premium." Skender's .NET implementation packages this as BetaType.All, computing standard, upside, downside, ratio, and convexity in a single pass.

Architecture & Physics

Convexity is built on the same dual-input pattern as Beta, but adds a conditional filtering step:

1. Standard Beta uses O(1) Kahan-compensated running sums for Cov(Ra, Rm) / Var(Rm)
2. Filtered Betas perform an O(period) scan of the ring buffer, partitioning returns by market direction:
   • Rm > 0 → contributes to BetaUp sums
   • Rm < 0 → contributes to BetaDown sums
   • Rm = 0 → excluded (following Skender's convention)

The Bar Correction Pattern

For streaming bar corrections (isNew = false), Convexity follows the proven Beta.cs pattern: only compensation values are saved/restored (not full sums), and a Kahan delta swaps the old return contribution for the new one. This ensures numerical stability across long-running sessions.

Mathematical Foundation

Standard Beta (all bars)

\beta = \frac{N \sum R_a R_m - \sum R_a \sum R_m}{N \sum R_m^2 - (\sum R_m)^2}

Upside Beta (market up bars only, R_m > 0)

\beta^+ = \frac{N^+ \sum_{R_m > 0} R_a R_m - \sum_{R_m > 0} R_a \sum_{R_m > 0} R_m}{N^+ \sum_{R_m > 0} R_m^2 - (\sum_{R_m > 0} R_m)^2}

Downside Beta (market down bars only, R_m < 0)

\beta^- = \frac{N^- \sum_{R_m < 0} R_a R_m - \sum_{R_m < 0} R_a \sum_{R_m < 0} R_m}{N^- \sum_{R_m < 0} R_m^2 - (\sum_{R_m < 0} R_m)^2}

Derived Outputs

\text{Ratio} = \frac{\beta^+}{\beta^-} \text{Convexity} = (\beta^+ - \beta^-)^2

Performance Profile

Operation Count (Streaming Mode)

Operation	Count	Cost (cycles)	Subtotal
Ring buffer add/evict (2 inputs)	2	3 cy	~6 cy
Compute asset + market returns	2	3 cy	~6 cy
Update 4 Kahan running sums	4	4 cy	~16 cy
Compute standard beta (FMA)	1	5 cy	~5 cy
O(period) scan for Up/Down beta	period	4 cy	~80 cy*
Compute ratio + convexity	2	3 cy	~6 cy
Total	O(period)	—	~119 cy

*Assuming period = 20. The O(period) scan is a simple branch-free iteration with no allocations.

Metric	Score	Notes
Throughput	~40 ns/bar	O(period) scan dominates.
Allocations	0	Zero-allocation hot path.
Complexity	O(period)	Linear scan for up/down filtering.
Accuracy	9	Kahan compensation prevents drift.
Timeliness	Lagged	Depends on the lookback period.
Smoothness	Low	Sensitive to period and market regime.

Validation

Library	Status	Notes
QuanTAlib	✅	Reference implementation.
Skender	✅	Matches GetBeta(BetaType.All) — 5-output bundle.

Common Pitfalls

1. Zero Variance in Up/Down Subsets: If all market up-day returns are identical, Var(Rm|Rm>0) = 0 and BetaUp is undefined (returns 0). This occurs with synthetic data; real market data always has variance.
2. Period Sensitivity: Short periods (< 10) may have too few up/down bars for meaningful beta decomposition. Typical institutional use: period = 60 (3-month daily).
3. Interpretation: Convexity = 0 does NOT mean beta = 0. It means upside and downside betas are equal (symmetric risk profile).

C# Usage

// Initialize with period 20
var conv = new Convexity(20);

// Update with Asset and Market prices
conv.Update(assetPrice, marketPrice);

Console.WriteLine($"BetaStd: {conv.BetaStd:F4}");
Console.WriteLine($"BetaUp:  {conv.BetaUp:F4}");
Console.WriteLine($"BetaDown: {conv.BetaDown:F4}");
Console.WriteLine($"Ratio:   {conv.Ratio:F4}");
Console.WriteLine($"Convexity: {conv.ConvexityValue:F4}");

Batch Mode

var (betaStd, betaUp, betaDown, ratio, convexity) =
    Convexity.Batch(assetSeries, marketSeries, period: 20);

Bar Correction

// New bar
conv.Update(assetPrice, marketPrice, isNew: true);

// Update same bar (price correction)
conv.Update(correctedAsset, correctedMarket, isNew: false);

Resources

• Skender GetBeta — BetaType.All returns all 5 outputs.
• Ang, Chen, Xing (2006). "Downside Risk" — Empirical evidence for downside risk premium.
• Bawa, Lindenberg (1977). "Capital Market Equilibrium in a Mean-Lower Partial Moment Framework" — Original lower partial moment theory.