Interpolation

Interpolation

What is interpolation?

Interpolation is a mathematical method for estimating unknown values that lie between known data points. In finance, it’s used to infer missing prices, yields, or other metrics by relying on the trend shown by surrounding observations. Interpolated values are commonly plotted on charts to help visualize price movement and support technical analysis.

How it works

• Identify known data points that bracket the missing value.
• Choose an interpolation method appropriate to the data and purpose.
• Calculate the estimated value that lies between the known points based on that method.

Common formal methods include:
– Linear interpolation: connects two adjacent points with a straight line and finds the in-between value along that line.
– Polynomial interpolation: fits a polynomial through a set of points and evaluates it between them.
– Piecewise methods: apply simple interpolations on short intervals to avoid large oscillations from high-degree polynomials.
– Exponential or weighted interpolation: gives more weight to certain points (e.g., recent prices or higher volume) when estimating values.

Example (linear interpolation)

If a security’s price is known for August and October but not September, linear interpolation estimates September’s price by drawing a straight line between the August and October prices and taking the point on that line corresponding to September. This is the simplest and most commonly used interpolation technique.

Uses in finance

• Yield curve construction: interpolated yield curves estimate yields for maturities where direct quotes are not available (e.g., between standard Treasury maturities).
• Charting and smoothing: traders and analysts interpolate missing price points to produce continuous lines, moving averages, or regression fits that summarize price behavior.
• Technical indicators: interpolation (smoothing) can be used to derive approximations of moving averages or the slope of price movement from high-low ranges.
• Data preparation: filling gaps in historical series to enable consistent backtesting or model estimation.

Interpolation vs. extrapolation

• Interpolation estimates values inside the range of known data points (filling the gaps).
• Extrapolation projects values outside the observed data range and generally carries higher risk of inaccuracy.

Limitations and criticism

• Precision: interpolation produces estimates, not actual observations; it can mask short-term volatility and sudden moves common in markets.
• Model risk: choosing an inappropriate interpolation method (e.g., high-degree polynomials) can distort inferred values.
• Overreliance: heavy interpolation may create a misleadingly smooth picture of price history, reducing sensitivity to real intraperiod variability.

Key takeaways

• Interpolation fills missing data by using nearby known values and is widely used in financial charting and yield-curve construction.
• Linear interpolation is the simplest and most common; weighted or exponential methods are used when some observations should influence the estimate more than others.
• While useful, interpolated values are approximations and should be treated cautiously, especially in volatile markets or when making forward-looking decisions.