Willow tree algorithms for pricing VIX derivatives under stochastic volatility models
Bibliographic record
Abstract
VIX futures and options are the most popular contracts traded in the Chicago Board Options Exchange. The bid-ask spreads of traded VIX derivatives remain to be wide, possibly due to the lack of reliable pricing models. In this paper, we consider pricing VIX derivatives under the consistent model approach, which considers joint modeling of the dynamics of the S&P index and its instantaneous variance. Under the affine jump-diffusion formulation with stochastic volatility, analytic integral formulas can be derived to price VIX futures and options. However, these integral formulas invariably involve Fourier inversion integrals with cumbersome hyper-geometric functions, thus posing various challenges in numerical evaluation. We propose a unified numerical approach based on the willow tree algorithms to price VIX derivatives under various common types of joint process of the S&P index and its instantaneous variance. Given the analytic form of the characteristic function of the instantaneous variance of the S&P index process in the Fourier domain, we apply the fast Fourier transform algorithm to obtain the transition density function numerically in the real domain. We then construct the willow tree that approximates the dynamics of the instantaneous variance process up to the fourth order moment. Our comprehensive numerical tests performed on the willow tree algorithms demonstrate high level of numerical accuracy, runtime efficiency and reliability for pricing VIX futures and both European and American options under the affine model and 3/2-model. We also examine the implied volatility smirks and the term structures of the implied skewness of VIX options.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.001 |
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".