Interpolations Techniques in Simple Terms

Interpolation is a mathematical technique used to estimate unknown values that fall between two known values. In finance, this is crucial for constructing models that deal with variables such as prices, interest rates, or volatility. Let’s explore different interpolation methods and their applications in financial engineering.

1. Linear Interpolation

Imagine drawing a straight line between two points on a graph. If you want to find the value of a point that lies between these two points, you just check where it lands on the line. Linear interpolation connects these dots directly and assumes a linear relationship between the known points.

\[ f(x) = f(x_1) + \frac{x - x_1}{x_2 - x_1}(f(x_2) - f(x_1)) \]

Where:

• \\( x_1, x_2 \\): Known points
• \\( x \\): Point at which you’re estimating the value

Think of a graph where the known points are joined by straight lines—simple but may produce rough transitions between points.

2. Cubic (Bessel) Hermite Interpolation

Sometimes, it’s not enough to pass through a set of points; you also care about the angle or slope at which the curve hits those points. Cubic Hermite interpolation ensures that the curve passes through given points and respects specific slopes at those points.

\[ H(t) = (2t^3 - 3t^2 + 1)y_1 + (t^3 - 2t^2 + t)m_1 + (-2t^3 + 3t^2)y_2 + (t^3 - t^2)m_2 \]

Where:

• \\( H(t) \\): Interpolated value
• \\( y_1, y_2 \\): Function values at known points
• \\( m_1, m_2 \\): Slopes (derivatives) at those points
• \\( t \\): Normalized parameter between 0 and 1¹

Imagine trying to drive a car through specific checkpoints on a route, but ensuring that you hit each checkpoint at a particular angle or speed. The curve here is more “customized” based on direction at each point.

3. Cubic Spline Interpolation

Cubic spline interpolation aims for a smooth and natural curve that passes through all given points. Unlike linear or cubic Hermite interpolation, cubic splines ensure the smoothest transition across multiple points, maintaining continuous first and second derivatives.

\[ S_i(x) = a_i + b_i(x - x_i) + c_i(x - x_i)^2 + d_i(x - x_i)^3 \]

Where each segment \\( S_i(x) \\) is a cubic polynomial between points \\( x_i \\) and \\( x_{i+1} \\). Coefficients \\( a_i, b_i, c_i, d_i \\) ensure smoothness.

Picture using a flexible, bendy ruler to connect multiple dots on paper, creating a smooth path that naturally adjusts its curvature between points. The curve is seamless but not as “tight” to each point as the cubic Hermite.

Applications in Quantitative Finance

1. Volatility Smile & Surface

In options markets, implied volatility (IV) varies with strike prices and maturities, creating a volatility “smile” or surface. To price options that don’t directly match market quotes, interpolation is used to estimate IV for specific strikes and maturities.

\[ Vol(K) = Vol(K_1) + \frac{K - K_1}{K_2 - K_1}(Vol(K_2) - Vol(K_1)) \]

2. Yield Curves

Yield curves represent interest rates across different maturities. They are often constructed from discrete market data points (e.g., bond maturities). To create a smooth curve or estimate yields for non-observed maturities, interpolation is used.

3. Numerical Methods in Option Pricing

For complex derivatives like exotic options, pricing methods (e.g., finite difference methods, binomial trees) use discrete steps or nodes. Interpolation fills gaps in these models, enabling smoother calculations.

¹ \\( t \\) is a scaling parameter that moves smoothly between the two endpoints while controlling the interpolation in a way that respects both function values and slopes at those endpoints.