Polynomial interpolation

What Is Polynomial Interpolation?

Polynomial interpolation is a numerical method used to construct a polynomial function that passes precisely through a given set of data points. In essence, it involves finding a single polynomial curve that connects all specified points. This technique falls under the broader umbrella of numerical methods within quantitative finance, where it is employed to estimate values between known observations, often when an underlying functional relationship is not explicitly known. Polynomial interpolation ensures that the resulting curve goes through every single input point, making it a powerful tool for curve fitting.

History and Origin

The concept of polynomial interpolation has roots in classical mathematics, with significant contributions from prominent figures such as Isaac Newton and Joseph-Louis Lagrange. Sir Isaac Newton's method of divided differences, developed in the 17th century, provided a systematic way to construct interpolating polynomials. This method generalizes the process of fitting polynomials to discrete data points, allowing for flexibility even when data points are not uniformly spaced.¹⁸, ¹⁹, ²⁰ Independently, Joseph-Louis Lagrange developed the Lagrange interpolation formula, published in 1795, which also constructs a unique polynomial passing through a given set of points.¹⁵, ¹⁶, ¹⁷ These foundational contributions paved the way for numerical analysis techniques widely used today, including various applications in financial modeling.

Key Takeaways

Polynomial interpolation constructs a single polynomial that passes exactly through a set of given data points.
It is a core technique in numerical methods used for estimating values between known observations.
While it ensures the curve precisely hits all data points, it can lead to undesired oscillations, especially with a large number of points or widely spaced data, a phenomenon known as Runge's phenomenon.
Key applications in finance include yield curve construction and estimating values for financial instruments.
Alternative methods, such as spline interpolation, often address some of the limitations of polynomial interpolation.

Formula and Calculation

Polynomial interpolation can be implemented using various formulas, with Lagrange and Newton forms being common.

Lagrange Interpolation Formula:
Given (n+1) data points ((x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)), the interpolating polynomial (P(x)) is given by:

P(x) = \sum_{j=0}^{n} y_j L_j(x)

where (L_j(x)) are the Lagrange basis polynomials, defined as:

L_j(x) = \prod_{k=0, k \neq j}^{n} \frac{x - x_k}{x_j - x_k}

In this formula, (y_j) represents the value of the function at the (j)-th data point (x_j). The Lagrange basis polynomial (L_j(x)) is designed to be 1 at (x_j) and 0 at all other (x_k) (where (k \neq j)). The sum then combines these basis polynomials, weighted by their respective (y_j) values, to create a single polynomial that passes through all given points.

Newton's Divided Differences Formula:
Another common approach is Newton's form, which uses divided differences. Given the same (n+1) data points, the polynomial (P(x)) is:

P(x) = f[x_0] + f[x_0, x_1](x - x_0) + f[x_0, x_1, x_2](x - x_0)(x - x_1) + \dots + f[x_0, \dots, x_n](x - x_0)(x - x_1)\dots(x - x_{n-1})

where (f[x_0, \dots, x_k]) denotes the (k)-th order divided difference. This iterative method builds the polynomial incrementally.

Interpreting the Polynomial Interpolation

When a polynomial interpolation is applied, the resulting polynomial provides a continuous curve that precisely connects all the original data points. The interpretation of the interpolated value depends on the context of the data. For instance, if interpolating interest rates along a yield curve, an interpolated point represents the estimated interest rate for a maturity not directly observed in the market.

However, interpreting the behavior of the polynomial between the points requires caution. While it provides a unique fit through the given points, the shape of the curve between these points, especially with higher-degree polynomials, may not always reflect the true underlying relationship or market dynamics. It's crucial to understand that polynomial interpolation aims for exact fit at known points, but its behavior elsewhere can be sensitive to the number and distribution of inputs.

Hypothetical Example

Consider a scenario where an analyst needs to estimate the yield of a bond with a maturity of 2.5 years, but only has observed yields for bonds maturing in 1 year, 3 years, and 5 years.

Let the observed data points be:

(1 year, 2.0%)
(3 years, 3.5%)
(5 years, 4.0%)

Here, we have three data points, so we can fit a polynomial of degree 2 (a quadratic polynomial). Using the Lagrange interpolation method:

Let (x_0=1, y_0=2.0); (x_1=3, y_1=3.5); (x_2=5, y_2=4.0). We want to estimate the yield at (x=2.5).

First, calculate the Lagrange basis polynomials:

L_0(x) = \frac{(x - x_1)(x - x_2)}{(x_0 - x_1)(x_0 - x_2)} = \frac{(x - 3)(x - 5)}{(1 - 3)(1 - 5)} = \frac{(x - 3)(x - 5)}{(-2)(-4)} = \frac{(x^2 - 8x + 15)}{8}

L_1(x) = \frac{(x - x_0)(x - x_2)}{(x_1 - x_0)(x_1 - x_2)} = \frac{(x - 1)(x - 5)}{(3 - 1)(3 - 5)} = \frac{(x - 1)(x - 5)}{(2)(-2)} = \frac{(x^2 - 6x + 5)}{-4}

L_2(x) = \frac{(x - x_0)(x - x_1)}{(x_2 - x_0)(x_2 - x_1)} = \frac{(x - 1)(x - 3)}{(5 - 1)(5 - 3)} = \frac{(x - 1)(x - 3)}{(4)(2)} = \frac{(x^2 - 4x + 3)}{8}

Now, substitute (x=2.5) into each (L_j(x)):

L_0(2.5) = \frac{(2.5 - 3)(2.5 - 5)}{8} = \frac{(-0.5)(-2.5)}{8} = \frac{1.25}{8} = 0.15625

L_1(2.5) = \frac{(2.5 - 1)(2.5 - 5)}{-4} = \frac{(1.5)(-2.5)}{-4} = \frac{-3.75}{-4} = 0.9375

L_2(2.5) = \frac{(2.5 - 1)(2.5 - 3)}{8} = \frac{(1.5)(-0.5)}{8} = \frac{-0.75}{8} = -0.09375

Finally, calculate the interpolated yield (P(2.5)):

P(2.5) = y_0 L_0(2.5) + y_1 L_1(2.5) + y_2 L_2(2.5)

P(2.5) = (2.0)(0.15625) + (3.5)(0.9375) + (4.0)(-0.09375)

P(2.5) = 0.3125 + 3.28125 - 0.375 = 3.21875

Therefore, using polynomial interpolation, the estimated yield for a 2.5-year bond is approximately 3.21875%. This process allows for the estimation of missing values within a range of known time series data.

Practical Applications

Polynomial interpolation serves several practical purposes in financial modeling and analysis:

Yield Curve Construction: One of the most common applications is in building continuous yield curves from discrete market observations of bond yields. Financial institutions and central banks use these curves to price fixed-income securities, understand market expectations of future interest rates, and assess economic conditions. While other methods like splines are often preferred for smoothness, polynomial interpolation can be a foundational approach.¹⁴
Derivatives Pricing: Pricing derivatives, especially those with maturities between standard quoted points, often requires estimating underlying parameters (like volatility or interest rates) for non-standard tenors. Polynomial interpolation can be used to achieve this.
Missing Data Estimation: In situations where complete time series data is unavailable, polynomial interpolation can provide reasonable estimates for missing values, aiding in historical analysis or filling gaps in datasets used for mathematical models.
Valuation Models: For the valuation of various financial instruments, especially those with irregular cash flows, interpolated rates or factors might be necessary for accurate discounting. The Federal Reserve Bank of San Francisco, for example, has discussed various approaches to modeling the yield curve, highlighting the importance of interpolation techniques.

Limitations and Criticisms

Despite its utility, polynomial interpolation has significant limitations, particularly when applied to large datasets or for extrapolation beyond the observed range:

Runge's Phenomenon: A major drawback is Runge's phenomenon. This occurs when using high-degree polynomials with equally spaced interpolation points, leading to large oscillations and inaccuracies, especially near the edges of the interpolation interval. The interpolating polynomial may exhibit wild swings between data points, which often does not reflect the true underlying function.¹¹, ¹², ¹³ This phenomenon demonstrates that simply increasing the degree of the polynomial does not guarantee improved accuracy and can, in fact, worsen the approximation.¹⁰
Computational Instability: Constructing high-degree interpolating polynomials can be numerically unstable, meaning small changes in input data points can lead to large changes in the polynomial coefficients and the resulting curve.⁹
Lack of Smoothness Control: Unlike some other curve fitting methods, basic polynomial interpolation does not offer direct control over the smoothness of the curve or its derivatives between the interpolation points. This can be problematic in finance where smooth transitions (e.g., in a yield curve) are often desirable to prevent arbitrage opportunities.
Extrapolation Risk: Using a polynomial interpolation for extrapolation (estimating values outside the range of known data) is highly unreliable. The polynomial's behavior outside the observed interval can become erratic and quickly diverge from any sensible trend.⁸

For these reasons, in many practical risk management and financial modeling applications, methods like spline interpolation are often preferred as they mitigate Runge's phenomenon by using lower-degree piecewise polynomials.

Polynomial Interpolation vs. Spline Interpolation

Polynomial interpolation and spline interpolation are both methods for curve fitting, but they differ significantly in their approach and resulting properties.

Feature	Polynomial Interpolation	Spline Interpolation
Nature of Curve	A single polynomial equation passes through all data points.	A series of piecewise polynomials connected at specific points (knots).
Degree of Polynomial	Can be of high degree, equal to (number of points - 1).	Typically uses lower-degree polynomials (e.g., cubic splines use degree 3) for each segment.
Smoothness	Exact fit at data points, but can exhibit wild oscillations between points (Runge's phenomenon).	Achieves continuity of the curve and its derivatives at the knots, resulting in a smoother overall curve.
Local vs. Global Impact	Changing one data point affects the entire polynomial.	Changing one data point only affects the local segment of the curve.
Use Cases	Simpler datasets, smaller number of points; sometimes used as a foundational concept.	Widely used in finance (e.g., yield curve construction) and engineering where smoothness and avoiding oscillations are critical.⁵, ⁶, ⁷

While polynomial interpolation creates a unique polynomial that passes through all given points, this global fit can lead to issues like Runge's phenomenon, especially with a large number of data points.³, ⁴ Spline interpolation addresses this by dividing the overall interval into smaller sub-intervals and fitting a separate, low-degree polynomial to each segment, ensuring that the segments join smoothly. This piecewise approach provides better control over the curve's behavior and is generally more stable and practical for many real-world applications in financial modeling. For example, the IMF has published working papers that employ cubic spline interpolation for estimating and forecasting the yield curve, often preferring it over single high-degree polynomial approaches due to its superior smoothness and stability.¹, ²

FAQs

What is the primary goal of polynomial interpolation?

The primary goal of polynomial interpolation is to find a single polynomial function that passes precisely through a given set of data points, allowing for the estimation of values at intermediate points.

Can polynomial interpolation be used for forecasting?

While polynomial interpolation estimates values between known data points (interpolation), its use for forecasting (estimating values outside the range of known data, also known as extrapolation) is generally not recommended due to its tendency to produce unreliable and wildly oscillating results.

What is Runge's phenomenon in polynomial interpolation?

Runge's phenomenon is a problem where, for certain functions and equally spaced data points, increasing the degree of the interpolating polynomial leads to large oscillations and significant errors, especially near the edges of the interval.

Is polynomial interpolation commonly used in finance today?

While polynomial interpolation is a foundational concept in numerical methods and some basic applications, more sophisticated curve fitting techniques like spline interpolation are often preferred in modern finance due to their better control over smoothness and stability, particularly for complex data like yield curves.