The Fourier estimator of spot volatility: Unbounded coefficients and jumps in the price process

Battle Count: 70

The paper addresses fundamental volatility estimation, which is critical for options pricing, risk management, and trading strategies. The handling of jumps is particularly relevant since asset returns often exhibit jump behavior (as noted by Carr et al. [10]). However, the paper is highly theoretical and does not provide directly implementable trading algorithms. The Fourier estimator's ability to handle unbounded volatility and jumps makes it practically relevant for high-frequency trading and risk assessment, but significant engineering work would be needed for implementation.

The theoretical framework requires deep knowledge of stochastic calculus, Fourier analysis, and martingale theory. Practical implementation would involve: (1) computing Fourier coefficients of the price process differential via pathwise Itô integrals, (2) performing Bohr convolutions, (3) constructing Fejér trigonometric polynomials, (4) handling discrete observation schemes, and (5) managing the rescaling for jump detection. The parameter calibration (exponents p, α, β, g, r) and the trade-off between M and N require careful numerical consideration. The jump detection via rescaled polynomials adds additional complexity.

The paper is primarily theoretical with complete proofs provided. Numerical simulations (Section 7.6) illustrate results for a compensated Poisson process with specified parameters (λ=2, σ=1, 105-point grid). However, no code or data repository is provided. The theoretical results are fully self-contained with all lemmas and theorems proven.