Volatility Estimation for a Logistic Jump-Diffusion with Dividend Yield under VWAP Closing Prices: Theory and Evidence from the Nairobi Securities Exchange

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@article{1724119,
      author = {Andanje Mulambula},
      title = {Volatility Estimation for a Logistic Jump-Diffusion with Dividend Yield under VWAP Closing Prices: Theory and Evidence from the Nairobi Securities Exchange},
      journal = {Iconic Research And Engineering Journals},
      year = {2026},
      volume = {10},
      number = {4},
      pages = {2707-2720},
      issn = {2456-8880},
      url = {https://www.irejournals.com/formatedpaper/1724119.pdf},
      abstract = {Daily closing prices on the Nairobi Securities Exchange (NSE) are session volume-weighted average prices (VWAP), quoted on a discrete tick. This paper studies what that observation scheme does to volatility estimation for a dividend-paying logistic jump-diffusion and proposes an estimator that is robust to it. For the logistic jump-diffusion with continuous dividend yield, existence, uniqueness and positivity of the solution are established, together with a closed-form representation between jumps. Under session averaging and rounding, daily returns are shown to follow a first-order moving average whose variance and first autocovariance satisfy γ_0+2γ_1=σ^2 Δ, independently of the averaging fraction and the rounding error. A truncated autocovariance-corrected estimator built on this identity is consistent and asymptotically normal over a long span, whereas realised, bipower and threshold variation computed from closing prices are biased downward by 11 to 16 per cent. Monte Carlo experiments confirm both results. Applied to Safaricom PLC (4,121 daily returns, June 2008 to December 2024, with a verified dividend record), the estimator gives an annualised diffusive volatility of 0.308 (95% interval 0.295 to 0.320) against 0.274 from the conventional estimator; volatility is not constant across years, the logistic carrying capacity is not identified, and about 0.9 large jumps occur per year. In a value-at-risk backtest, the conventional square-root-of-time rule understates ten-day variance by 23 per cent, while the corrected forecast is close to calibrated.},
      keywords = {jump-diffusion; logistic Brownian motion; Nairobi Securities Exchange; volatility estimation; volume-weighted average price},
      month = {October},
      doi = {https://doi.org/10.64388/IREV10I4-1724119}
  }