Wavelet Optimized Finite-Difference Approach to Solve Jump-Diffusion type Partial Differential Equation for Option Pricing
Bibliographic record
Abstract
The sine and cosine functions used as the bases in Fourier analysis are very smooth (infinitely differentiable) and very broad (nonzero almost everywhere on the real line), and hence they are not effective for representing functions that change abruptly (jumps) or have highly localized support (diffusive). In response to this shortcoming, there has been intense interest in recent years in a new type of basis functions called wavelets. A given wavelet basis is generated from a single function, called a mother wavelet or scaling function, by dilation and translation. By replicating the mother wavelet at many different scales, it is possible to mimic the behavior of any function; this property of wavelets is called multiresolution. Wavelet is a powerful integral transform technique for studying many problems including financial derivatives such as options. Moreover, the approximation error is much smaller than that of the truncated Fourier expansion. Therefore, one can get better approximation of a function at jump discontinuity with the use of wavelet expansion rather than Fourier expansion. In the current study, we employ wavelet analysis to option pricing problem manifested as partial differential equation (PDE) with jump characteristics. We have used wavelets to develop an optimum finite differencing of the differential equations manifested by complex financial models. In particular, we apply wavelet optimized finite-difference (WOFD) technique on the partial differential equation. We describe how Lagrangian polynomial is used to approximate the partial derivatives on an irregular grid. We then describe how to determine sparse and dense grid with wavelets. Further work on implementation is going on.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".