Strategy 4: Convex Adaptive

Overview

The most sophisticated guaranteed-completion strategy. Uses exponential (convex) scaling to make proportionally larger bets on larger discounts, combined with time-based urgency acceleration.

Type: Active / Convex-adaptive
Completion: Always 100%
Performance: +25-40 bps typical

Algorithm

Convex Scaling

Instead of linear response to discounts, Strategy 4 uses exponential scaling:

\[ \text{convex\_factor} = e^{\beta \cdot \text{deviation}} - 1 \]

Where: - \(\beta\) = 50 (default sensitivity) - \(\text{deviation} = \frac{\text{TWAP} - P_t}{\text{TWAP}}\)

Urgency Acceleration

Time pressure increases as deadline approaches:

\[ \text{urgency\_factor} = \left(\frac{t}{\text{max\_duration}}\right)^{\gamma} \]

Where \(\gamma\) = 1.0 controls acceleration steepness.

Combined Multiplier

\[ \text{multiplier} = \text{base} + \text{convex\_factor} \times (1 + \text{urgency}) \]

Clamped between MIN_MULTIPLIER (0.1) and MAX_MULTIPLIER (8.0).

Parameters

Parameter	Default	Description
beta	50.0	Convex sensitivity
gamma	1.0	Urgency acceleration
max_multiplier	8.0	Maximum speed-up
min_multiplier	0.1	Minimum (slowdown)

Implementation

def strategy_4(prices, benchmarks, total_usd, min_dur, max_dur, target_dur,
               beta=50.0, gamma=1.0):
    """Multi-factor convex adaptive execution."""
    for t in range(max_dur):
        price = prices[t]
        benchmark = benchmarks[t]
        deviation = (benchmark - price) / benchmark

        # Convex scaling
        if deviation > 0:
            convex_factor = math.exp(beta * deviation) - 1
        else:
            convex_factor = -math.exp(-beta * deviation) + 1

        # Time urgency
        progress = t / max_dur
        urgency = progress ** gamma

        # Combined multiplier
        multiplier = 1.0 + convex_factor * (1 + urgency)
        multiplier = max(MIN_MULT, min(MAX_MULT, multiplier))

        daily_usd = base_daily * multiplier
        # ...

Why Convex?

Linear scaling: 5% discount = 5x response
Convex scaling: 5% discount = ~12x response (exponential)

This means: - Small discounts: modest response - Large discounts: aggressive response - The strategy "saves ammunition" for the best opportunities

Performance Profile

Metric	Typical Value
Mean Performance	+25 to +40 bps
Std Dev	~70-85 bps
Min	-100 to -180 bps
Max	+250 to +400 bps

Higher variance

The aggressive convex scaling produces higher variance. This strategy is best for those seeking maximum expected return over many executions.

Tuning Beta

Beta	Behavior
25	Conservative, closer to linear
50	Balanced (default)
75	Aggressive
100	Very aggressive

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(c) Joerg Osterrieder 2025