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Interpolation: From Spirochete's Helical Body to Implied Volatility  Surfaces

In a typical option pricing problem, implied volatility varies with the contract strike price. This contradicts the Black-Scholes model and has led to the development of a variety of alternative models. These models produce fat-tail distributions and approximate the implied volatility "smiles" and "smirks" observed in actual markets. Numerical implementations of these models lead to partial differential equations (PDE) or integro-differential equations (IPDE) on infinite domains with possibly infinitely large initial conditions.


We review the typical numerical difficulties and interesting problems investigated by the academic community of numerical analysts.  The academic community seems to ignore a second category of problems typical to option pricing, namely interpolation techniques.


Interpolation techniques are used in a variety of applications. One such technique is B-spline interpolation, which allows construction of differentiable curves. Additional constraints can be imposed on first and second derivatives of these curves. Examples of B-spline interpolations are provided for construction of a computational grid to model the fluid dynamics of a helical swimmer (spirochete modeling). Similar interpolation techniques can be applied to noisy implied volatility curves in financial applications, which lead to quadratic programming formulations for B-spline control points.