Free lesson · On-chain markets

Time-weighted reference prices

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Start with the idea

A time-weighted average gives longer-lived prices more influence than brief prices. It reduces sensitivity to a momentary observation while introducing lag.

Symbols, units & horizon

P_i: reference price USD/base during interval i
Δt_i: interval duration in seconds
P-bar_T: arithmetic time-weighted USD/base average over the full covered horizon
intervals nonoverlapping and exhaustive

When and why to use this

Reconstruct reference observations and compare lag, manipulation sensitivity and data availability.

A time-weighted average gives longer-lived prices more influence than brief prices. It reduces sensitivity to a momentary observation while introducing lag.

For a piecewise-constant arithmetic reference, weight each price by the duration it was in force. This is not the same as trade-volume weighting.

Protocols can average ticks/log prices, use cumulative observations or rely on off-chain aggregators. The arithmetic example should not be substituted for a protocol-specific geometric or median construction. Document update and availability times.

{\overline{P}}_{T} = \frac{\sum_{i} P_{i} Δ t_{i}}{\sum_{i} Δ t_{i}}

Model assumptions, derivation and arithmetic

Time-weighted reference prices

Multiply each price by seconds spent at that price.
Sum price-time areas and total covered seconds.
Divide area by duration and inspect missing intervals separately.

Work it by hand

Price $100 for 30 seconds and $110 for 10 seconds gives (3000+1100)/40=$102.50, not the unweighted average $105.

Apply it in a strategy

Reconstruct reference observations and compare lag, manipulation sensitivity and data availability.
Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
Stress this failure condition: Averaging cannot create reliable information if underlying liquidity is thin or input prices are systematically distorted.

Research deliverable

Build and explain a time-weighted reference prices worksheet. Reconstruct reference observations and compare lag, manipulation sensitivity and data availability.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def arithmetic_twap(prices,durations):
    if not prices or len(prices)!=len(durations) or any(p<=0 for p in prices) or any(t<=0 for t in durations): raise ValueError("Aligned positive intervals required")
    return sum(p*t for p,t in zip(prices,durations))/sum(durations)

print(arithmetic_twap([100,110],[30,10]))

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