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.
Formula:
Value at x = Value at x1 + [(x - x1) / (x2 - x1)] * (Value at x2 - Value at x1)
Where:
-
x1, x2 are the known points.
- x is the point at which you’re estimating the value.
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.
Formula:
H(t) = (2t^3 - 3t^2 + 1) * y1 + (t^3 - 2t^2 + t) * m1 + (-2t^3 + 3t^2) * y2 + (t^3 - t^2) * m2
Where:
- H(t) provides the interpolated value at any position t along the curve between those two points.
- y1, y2 are the function values at the known points.
- m1, m2 are the slopes (derivatives) at those points.
- t is the normalized parameter between 0 and 1 (*see the footnote).
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.
Formula:
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, and coefficients a_i, b_i, c_i, d_i are determined to 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.
Comparing Methods:
-
Linear Interpolation: Connects points with straight lines—simple, but less smooth.
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Cubic Hermite Interpolation: Creates curves with attention to slopes at each point, ensuring specific angles are respected.
- Cubic Spline Interpolation: Provides smooth curves with gentle transitions, ensuring continuity in both first and second derivatives.
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.
For example, If volatilities for strike prices K1 and K2 are known, you can estimate the IV for an intermediate strike K:
Vol(K) = Vol(K1) + [(K - K1) / (K2 - K1)] * (Vol(K2) - Vol(K1))
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.
For example, this method creates a yield curve that smoothly transitions across maturities, making it easier to discount cash flows accurately.
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.
When calculating values between nodes or estimating option prices for non-standard strikes or maturities, interpolation fills gaps in the model.
When calibrating a model to market data, such as building a volatility surface, you often don’t have a perfectly complete dataset. Interpolation helps to fill in missing volatilities between quoted strikes and maturities, allowing for a more accurate pricing of derivatives and risk management.
(*) t is a scaling parameter that moves smoothly between the two endpoints while controlling the interpolation in a way that respects both the function values and slopes at those endpoints. This allows to create a curve that not only passes through the points but also respects the desired angles or directions at those points.
Key Takeaways:
-
Linear Interpolation: Uses a straight line to estimate values between two points; simple but lacks
smoothness.
-
Cubic Hermite Interpolation: Creates curves between points considering both position and slope,
providing a more accurate fit.
- Cubic Spline Interpolation: Ensures a smooth curve through all points with seamless transitions; ideal for modeling continuous processes.
-
Finance Applications:
-
- Volatility Surface: Estimates implied volatility for unquoted strikes/maturities.
- Yield Curves: Smooths yield curves for accurate rate estimations.
- Numerical Methods: Enhances option pricing models by filling gaps between discrete points.
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