Introduction
Interpolation is a mathematical technique for estimating unknown data points that lie between known observations. In finance and trading, interpolation is commonly used to fill gaps in price series, smooth noisy data, estimate intermediate yields on the Treasury curve, and create cleaner inputs for technical indicators and models.
Key takeaways
– Interpolation estimates values inside the range spanned by known data points; extrapolation estimates outside that range.
– Common interpolation methods include linear, polynomial, piecewise (splines), and weighted/exponential smoothing.
– Traders use interpolation to reconstruct missing data, smooth price series, estimate yields for nonstandard maturities, and help build indicators or models.
– Interpolation simplifies analysis but introduces assumptions and can produce misleading results when data are volatile or nonlinear.
Understanding interpolation (short)
Given two (or more) known data points (x0, y0) and (x1, y1), interpolation finds an estimated value y at some x between x0 and x1. The simplest form—linear interpolation—assumes the unknown value lies on the straight line connecting the two points.
Common types of interpolation
– Linear interpolation: Assumes a straight-line relationship between adjacent known points. Formula:
y = y0 + (x – x0) * (y1 – y0) / (x1 – x0)
Useful when changes between points are roughly linear or when simplicity and robustness are desired.
• Polynomial interpolation: Uses a polynomial that goes exactly through multiple known points. Can fit curvature but may oscillate (Runge’s phenomenon) for high-degree polynomials.
• Piecewise/spline interpolation (e.g., cubic spline): Fits low-degree polynomials between adjacent intervals with continuity constraints. Produces smooth curves without large oscillations; commonly used for yield curves.
• Piecewise constant interpolation: Holds the previous value until the next point (step function). Simple but discontinuous.
• Exponential/weighted interpolation (smoothing): Applies weights to nearby data (recent points often weighted more heavily). In technical analysis, this concept is used in exponential moving averages (EMA) and other smoothing techniques—sometimes referred to loosely as exponential interpolation.
Example — simple linear interpolation
Suppose you have a security price recorded at:
– August (x1) = 100
– October (x3) = 110
You want an estimate for September (x2), which lies between August and October.
Using linear interpolation:
y_sep = 100 + (2 – 1) * (110 – 100) / (3 – 1) = 100 + 1 * 10 / 2 = 105
Practical steps to perform interpolation (for traders and analysts)
1. Define the purpose and scope
• Are you filling missing intraday data, smoothing a noisy time series, or estimating yields for a nonstandard maturity? Purpose guides method selection.
2. Prepare and clean data
• Remove or flag obvious errors/outliers.
• Ensure x-values (time, maturities, etc.) are correctly scaled and ordered.
• Decide whether to work in levels, log-prices, or returns (logs can linearize multiplicative processes).
3. Choose an interpolation method
• Use linear for simplicity and robustness.
• Use splines for smooth curves (yield curves, term-structure fits).
• Use exponential weighting or EMA-style techniques for smoothing with greater emphasis on recent data.
• Use polynomial only for small numbers of points or when curvature is known and controlled.
4. Implement the computation
• Manual formula for linear interpolation (see above).
• Use numerical libraries for production use:
• Python: numpy.interp for 1-D linear interpolation; scipy.interpolate.interp1d or CubicSpline for splines.
• Excel: FORECAST/FORECAST.LINEAR or spline add-ins.
• Example Python (linear):
from numpy import interp
y_interp = interp(x_query, x_known, y_known)
• Example Python (cubic spline):
from scipy.interpolate import CubicSpline
cs = CubicSpline(x_known, y_known)
y_query = cs(x_query)
5. Validate and test
• Backtest: compare interpolated values against withheld actuals when possible.
• Sensitivity: check how estimates change when you vary nearby points or method.
• Visual inspection: plot known points vs. the interpolated curve.
6. Apply in workflow
• Use interpolated series as inputs to indicators, risk models, trade signals, or yield-curve analyses.
• Keep metadata: method used, parameters, date/time of interpolation, and assumptions.
7. Monitor and update
• Recompute interpolation as new actual data arrive.
• Reassess chosen method if market regime or data spacing changes.
How traders specifically use interpolation
– Technical analysis smoothing: Create smoother price curves or mid-price approximations (e.g., smoothing highs/lows, estimating intra-period low/high).
– Moving averages and regression lines: Interpolation helps fill missing ticks and estimate averaged trends.
– Yield curve construction: Bootstrapping and spline interpolation between issued Treasury maturities to estimate yields at any target maturity.
– Data reconstruction: Filling gaps in historical intraday data or aligning series sampled at different timestamps.
– Risk and scenario analysis: Generating continuous curves from discrete observations for valuation and stress-testing.
Which type of interpolation is used in technical analysis?
– Mostly linear interpolation for simple fills and linear regression lines.
– Exponential/interpolated smoothing (like EMA) is widely used when recent data should have greater influence.
– Some traders use piecewise linear regression or smoothed splines to build custom indicators; choice depends on the purpose (smoothing vs. exact fit).
Interpolation vs. extrapolation — key difference
– Interpolation: estimates values inside the range of known data; generally safer because it relies on local behavior.
– Extrapolation: estimates beyond the known range; riskier because it assumes that observed trends continue unchanged outside observed data.
Criticism and limitations
– Assumption risk: Interpolation assumes a predictable relationship between points; in volatile markets that assumption may be weak.
– Loss of signal: Excessive smoothing can remove important short-term signals.
– Misleading precision: Interpolated values appear precise but are model-dependent estimates, not observed facts.
– Not a cure for bad data: If underlying data are poor or irregularly spaced, interpolation can still produce inaccurate estimates.
Practical tips and best practices
– Keep methods as simple as possible consistent with the task (Occam’s razor).
– Use log-transforms where percentage changes matter.
– Always backtest interpolated inputs in any automated strategy.
– Document interpolation choices and re-evaluate them as market conditions change.
– Prefer spline or piecewise methods for term-structure fitting (bonds), and linear or EMA-style smoothing for price-series work.
The bottom line
Interpolation is a useful, widely used tool in finance for filling gaps, smoothing data, and constructing continuous curves from discrete observations. It is most reliable when applied inside the data range and with an appropriate method for the data’s behavior. Traders and analysts should pair interpolation with validation, clear documentation, and prudent use in decision-making because interpolated values carry modeling assumptions and uncertainty.
Reference
– Investopedia. “Interpolation.”
(Continuing the article)
Interpolation in practice: additional sections
Types of interpolation — expanded
– Linear interpolation: simplest; connects two neighboring points with a straight line. Formula: for points (x1,y1) and (x2,y2), estimate at x:
y = y1 + (y2 − y1) * (x − x1) / (x2 − x1).
– Polynomial interpolation: fits a single polynomial through a set of points. Accurate for small sets but can oscillate (Runge’s phenomenon) for many points.
– Spline interpolation (piecewise polynomial): fits lower-degree polynomials between each pair (or small groups) of points, ensuring smoothness at joins. Cubic splines are common for yield curves.
– Piecewise constant (nearest-neighbor): assigns the value of the nearest known point; simple but discontinuous.
– Exponential interpolation / smoothing: weights nearby points exponentially (recent observations count more); common in time series (e.g., exponential moving average).
– Kernel and local regression methods (LOESS/LOWESS): flexible, useful when relationships change across the domain.
Practical steps: how to perform interpolation (general recipe)
1. Define the objective: decide what you need to estimate (price at a time, intermediate yield, missing data point). Be explicit whether you need a point estimate, a smoothed series, or a curve fit.
2. Inspect and clean data: remove outliers if they reflect errors, align time stamps, ensure units and maturities are comparable.
3. Choose an interpolation method: use linear for simplicity and transparency; splines for smooth curves (yield curve); exponential/EMA for smoothing price series; polynomial only for small sets.
4. Select the interpolation grid: decide the x-values where you need estimates (times, maturities, price ticks).
5. Compute interpolation with a reliable library/tool or formula: test on withheld points to validate.
6. Validate and backtest: compare interpolated values against held-out or subsequently observed values; measure errors (RMSE, MAE).
7. Document assumptions and limitations: record method choice, parameter values (e.g., spline knots, smoothing factor), and sensitivity.
8. Monitor and update: re-interpolate as more data arrives; re-evaluate model performance.
Practical tools and quick implementations
– Excel: FORECAST.LINEAR (or FORECAST) for linear interpolation/forecasting over two points; TREND and GROWTH for regression; you can also use chart trendlines or LINEST.
– Python: numpy.interp for simple linear; scipy.interpolate.interp1d for linear, cubic, and spline options; pandas.Series.interpolate for time-series modes.
Example: numpy.interp(x_new, x_known, y_known) returns linear interpolated values.
– R: approx() (linear) and spline() for splines; loess() for local regression.
– Charting platforms: most trading/charting packages apply smoothing or interpolation automatically for line charts; check settings for “missing data” handling.
Numeric example: linear interpolation (simple)
Known points:
– Date A (x1) = day 1, price y1 = 10
– Date B (x2) = day 3, price y2 = 16
Estimate price on day 2 (x = 2):
y = 10 + (16 − 10) * (2 − 1) / (3 − 1) = 10 + 6 * 0.5 = 13
Yield-curve example (illustrative)
Suppose you observe:
– 1-year Treasury yield = 1.50%
– 2-year Treasury yield = 2.00%
Estimate 18-month (1.5-year) yield by linear interpolation:
yield(1.5) = 1.50% + (2.00% − 1.50%) * (0.5 / 1.0) = 1.75%
In practice, fixed-income desks often prefer spline/bootstrapping methods to ensure smoothness and arbitrage-free properties.
Example: spline vs linear for a bond yield curve
– Linear interpolation: easy and transparent but can create kinked forward rates and poor hedging characteristics.
– Cubic spline: smoother forward curve; preferred when hedging or pricing interest-rate derivatives because it avoids unrealistic jumps.
Interpolation vs. smoothing — how traders use each
– Interpolation fills in missing points between known observations.
– Smoothing reduces noise by averaging across observations (moving average, EMA, Savitzky–Golay filters).
Traders use interpolation to reconstruct continuous curves from discrete quotes (yield curves, intraday price curves) and smoothing to discern trend vs noise (moving averages, regression lines).
Common pitfalls and criticisms (expanded)
– False precision: an interpolated point can look precise but may hide large uncertainty; the interpolated value is conditional on the chosen model.
– Model risk: different interpolation methods produce different estimates—choice matters.
– Overfitting (esp. high-degree polynomials): can create unrealistic oscillation between points.
– Misuse in volatile markets: financial markets are noisy and nonstationary; interpolated past behavior is not a guaranteed predictor of future prices.
– Extrapolation danger: applying an interpolation model beyond the known data range (extrapolation) increases error risk substantially.
Best practices and risk controls
– Use simpler methods first (linear, low-degree splines) and only increase complexity if justified by validation.
– Perform cross-validation or holdout testing—compare interpolated values to real observations.
– Quantify uncertainty: report confidence intervals or error statistics for interpolated values where feasible.
– Avoid overreliance: use interpolation as one tool among analyses (fundamentals, order flow, macro indicators).
– Beware of data frequency mismatches: interpolation can conceal gaps resulting from thin markets or stale quotes.
How interpolation appears in common trading and analysis workflows
– Technical analysis charts: line charts are often interpolated between closing prices; intraday charting may interpolate between ticks for smoother visuals.
– Order-book / depth-of-market smoothing: reconstruct continuous price impact curves from discrete order levels.
– Risk management & pricing: interpolation of discount factors or volatilities for pricing bonds, swaps, and options.
– Data cleaning: filling missing ticks/records to compute indicators that require a complete time series (e.g., rolling volatility).
Advanced topics (brief)
– Bootstrapping a zero curve: iterative method to construct zero rates from coupon-bearing bonds; interpolation is used between pillars.
– Arbitrage-free interpolation: methods (e.g., monotone convex, Hagan–West) that ensure no arbitrage in forward rates extracted from interpolated curves.
– Interpolation of implied volatilities: practitioners commonly use spline or SABR-based interpolation for option surface smoothing that respects arbitrage constraints.
Additional examples
1) Intraday price reconstruction
Scenario: A level-2 feed has quotes at irregular intervals. You need per-minute prices for an indicator.
Practical approach:
– Align timestamps to minute bins.
– Use last-known-price (forward-fill) or linear interpolate between ticks if estimating intra-minute price path.
– Validate by comparing to a higher-resolution feed when available.
2) Missing data in historical backtest
Scenario: A backtest requires a continuous daily return series but some days have missing prices.
Practical approach:
– If gaps are short, apply linear interpolation to fill prices.
– For longer gaps or illiquid assets, consider excluding periods or using conservative assumptions (e.g., carry-forward last price) and document the choice.
When to prefer extrapolation (and be cautious)
– Extrapolation is appropriate only when you have strong theoretical or empirical reason to extend a trend beyond observed data (e.g., deterministic curve asymptotes).
– Always quantify the increased uncertainty and avoid using extrapolated points for precise trading thresholds.
Concluding summary
Interpolation is a fundamental quantitative tool in finance: it lets analysts and traders estimate unknown values between known data points to construct continuous curves, fill missing observations, and smooth noisy series. The method and parameters you choose—linear, spline, exponential smoothing, or local regression—affect results and should match the objective (simplicity, smoothness, arbitrage constraints). Always clean and inspect data, validate interpolated values against held-out points, and report uncertainty. Interpolated values are estimates, not truths—use them wisely alongside other models and domain judgment.
Source
– Investopedia, “Interpolation,”