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

1Introduction

Volatility is the central input of option pricing, risk measurement and portfolio choice. Since the Black-Scholes-Merton framework [1], [2], the price of a dividend-paying asset is commonly modelled as a geometric Brownian motion, with jump components added to capture sudden discontinuous moves [3], [4]. Geometric Brownian motion, however, lets the price grow without bound. Logistic Brownian motion replaces the constant growth rate with one that declines as the price approaches a carrying capacity, and a line of work has developed this model for option pricing. The author and co-workers introduced a logistic Brownian motion with jump diffusion and used it to motivate volatility estimation [5], derived the corresponding price process [6], extended it to dividend-yielding assets [7], derived a modified Black-Scholes equation with dividends and jumps [8], and priced European options under logistic dynamics with a continuous dividend yield [9]. These contributions are analytical. None of them estimates the model from market data, and [9] states explicitly that its proposed calibration is not applied.

This paper takes that step. Doing so on Kenyan data raises a problem that is absent from the theory: the observation scheme. Under the NSE equity trading rules, the reference (closing) price of a security is the volume-weighted average price of all transactions in the trading session [10, Rule 7.6.1], and prices move on a discrete tick. Averaging smooths the price path, so returns computed from averages have smaller variance and positive first-order autocorrelation [11]. Every standard volatility estimator that treats the closing price as a point-in-time price inherits this bias. Rounding to the tick adds measurement noise [12], [13]. Neither feature is specific to the logistic model, but both determine whether any volatility estimate for an NSE-listed stock can be trusted.

The contributions are as follows. First, for the logistic jump-diffusion with continuous dividend yield under the physical measure, the solution is shown to exist, to be unique and positive, and to have a closed-form representation between jumps; the estimators of diffusive volatility considered here are shown to be invariant to the poorly identified logistic drift. Second, the moments of daily returns under session averaging and rounding are derived, and they yield the identity ${\gamma }_0+2{\gamma }_1={\sigma }^2\Delta$. Third, a truncated autocovariance-corrected volatility estimator is proposed, its long-span consistency and asymptotic normality are established, and its finite-sample behaviour is documented by simulation alongside realised, bipower and threshold variation [14]-[17] and quasi-maximum likelihood. Fourth, the model is calibrated to Safaricom PLC, the largest NSE stock, using a dividend record verified against company filings, and the consequences for value-at-risk are measured.

Section 2 sets out the model and the observation scheme. Section 3 presents the estimators and their asymptotic properties. Section 4 reports the Monte Carlo evidence, Section 5 the empirical application, and Section 6 discusses limitations. Section 7 concludes.

2Model and observation scheme

2.1Logistic jump-diffusion with dividend yield

Let $S_t$ denote the ex-dividend price, $K>0$ the carrying capacity, $\mu$ the intrinsic growth rate, $q\ge 0$ the continuous dividend yield, $W_t$ a standard Brownian motion and ${\Pi }_t$ a Poisson process with intensity $\lambda$, independent of $W$. Jump sizes are $e^Y$ with $Y\sim N\left(\gamma ,{\delta }^2\right)$ independent and identically distributed, and $\kappa =E\left[e^Y-1\right]=e^{\gamma +{\delta }^2/2}-1$. With $a=\mu -q-\lambda \kappa$ and $b=\mu /K$, the price under the physical measure satisfies

so that the drift equals $\mu S_t\left(1-S_t/K\right)-q{S}_t-\lambda \kappa S_t$. The compensator $-\lambda \kappa S_t$ keeps the expected growth rate equal to the logistic drift net of the dividend yield. As $K\to \infty$, (1) reduces to Merton’s jump-diffusion with dividend yield [3], and with $\lambda =0$ further to geometric Brownian motion.

Lemma 1 (linearisation between jumps). Let $X_t=1/S_t$. Between jump times, Itô’s formula gives the linear equation $d{X}_t=\left(\left({\sigma }^2-a\right)X_t+b\right)dt-\sigma X_{t}d{W}_t$, whose solution is

At a jump time $X$ is multiplied by $e^{-Y}$.

The reciprocal transformation is the one used for pricing in [9]; here it is applied under the physical measure and with jumps, so that $S_t=1/X_t$ is, between jumps, a geometric Brownian motion divided by an exponential functional of Brownian motion.

Theorem 1 (existence, uniqueness, positivity). For every $S_0>0$, (1) has a unique càdlàg solution that is strictly positive and does not explode in finite time.

Proof. The coefficients $aS-b{S}^2$ and $\sigma S$ are locally Lipschitz, so a unique solution exists up to an explosion time [18]. With $V\left(S\right)=S$ the generator satisfies $\mathrm{L}V=aS-b{S}^2+\lambda \kappa S\le c_1-c_2{S}^2$ for constants $c_1,c_2>0$, which bounds $E\left[S_{t\wedge {\tau }_m}\right]$ uniformly in the localising sequence ${\tau }_m$ and excludes explosion. On each inter-jump interval the solution is given by (2), and $X_t>0$ there because ${\Phi }_t>0$ and $b\ge 0$; jumps multiply $S$ by $e^Y>0$. Hence $S_t>0$ for all $t$.$▫$

The log-price $Z_t=\log S_t$ satisfies $d{Z}_t=\left(a-b{S}_t-{\sigma }^2/2\right)dt+\sigma d{W}_t+Yd{\Pi }_t$. Its drift is locally bounded because $S$ is càdlàg and positive, which is what makes the volatility estimators below insensitive to $\mu$, $K$ and $q$.

2.2The NSE observation scheme

Time is measured in years and a trading day has length $\Delta =1/252$. The NSE trading session runs from 09:00 to 15:00 [10, Rule 6.1], and the published daily price is the session VWAP [10, Rule 7.6.1]. Let $f$, with $0<f\le 1$, be the share of each day’s price variation that falls inside the session. Treating volume as evenly spread over the session, the log of the published price on day $t$ is approximated by the session average

and the recorded price is rounded to a tick. In log terms the observed price is $P_t=A_t+u_t$, where the rounding errors $u_t$ are taken to be independent, with mean zero and variance ${\eta }_t$; this is accurate when the tick is small relative to the daily price variation [12]. Daily returns are $r_t=P_t-P_{t-1}$.

Proposition 1 (moments under averaging and rounding). Ignoring the drift and jumps, and with constant $\eta$,

and ${\gamma }_l=0$ for $l\ge 2$. Consequently

whatever the values of $f$ and $\eta$.

The proof is given in the Appendix. With $\eta =0$, (3) and (4) give Working’s first-order autocorrelation ${\rho }_1=\left(f/6\right)/\left(1-f/3\right)$, which equals $1/4$ when $f=1$ [11]. A conventional estimate of daily variance therefore recovers ${\sigma }^2\Delta \left(1-f/3\right)$, about 30 per cent below ${\sigma }^2\Delta$ when $f\approx 0.9$, and scaling it by the horizon understates multi-day variance by the factor $\left(1-f/3\right)/\left(1-f/\left(3H\right)\right)$ at horizon $H$. Compound Poisson jumps have independent increments and enter through the same averaging kernel, so with jumps the right-hand side of (5) becomes the total variance $\left({\sigma }^2+\lambda E\left[Y^2\right]\right)\Delta$.

3Estimation and asymptotic properties

3.1Dividend adjustment

On an ex-dividend day the price falls by about $\log \left(1+D_j/S_j\right)$, where $D_j$ is the dividend and $S_j$ the cum-dividend price. Left in the series, such drops would be counted as jumps. Adjusted returns are

where $t_j$ is the ex-dividend day and $\widehat{q}=T^{-1}\sum _j\log \left(1+D_j/S_j\right)$ over a sample of length $T$ years. This replaces discrete payouts by the model’s continuous yield. All estimators below use ${\tilde{r}}_t$, written $r_t$ from here on.

3.2Benchmark estimators for point-in-time prices

For point-in-time prices observed at interval $\Delta$ over $\left[0,T\right]$, with $n=T/\Delta$ returns, realised variance is $RV=\sum r_t^2$, bipower variation is $BV={\mu }_1^{-2}\sum \left|r_t\right|\left|r_{t-1}\right|$ with ${\mu }_1=\sqrt{2/\pi }$ [14], [15], and threshold variation is $TV=\sum r_t^{2}1\{\left|r_t\right|\le u_n\}$ with $u_n=c\widehat{\sigma }{\Delta }^{0.49}$ [16], [17]. As $\Delta \to 0$ with $T$ fixed, $BV/T$ and $TV/T$ converge to ${\sigma }^2$ under (1), with

where $\vartheta ={\pi }^2/4+\pi -3\approx 2.609$, and limiting variance $2{\sigma }^4$ for $TV$ [14]-[17]. Because the drift of $Z_t$ is locally bounded, its contribution to these statistics is $O_p\left(\Delta \right)$, so the limits do not depend on $\mu$, $K$ or $q$. The ratio test of Huang and Tauchen [19] compares $BV$ with $RV$ to test for jumps. These results presume point-in-time prices. Under (3) and (4) all three statistics instead estimate ${\sigma }^2\left(1-f/3\right)$ plus rounding noise, and the bias does not vanish as the sample grows.

3.3Truncated autocovariance-corrected estimator

Identity (5) suggests estimating ${\sigma }^2$ from the sum of the sample variance and twice the first sample autocovariance, an estimator of the type introduced by Zhou [20] for noisy high-frequency data. Let $x_t=r_t-\bar{r}$ and let $V_t$ be a local variance of daily returns computed from the lag-two bipower ${\mu }_1^{-2}\left|x_s\right|\left|x_{s-2}\right|$ over a centred window of 63 days that excludes day $t$; lag two is used because it is free of the first-order dependence in (4). Define the truncation indicator $1_t=1\{\left|x_t\right|\le c\sqrt{V_t}\}$ with $c=4$ and

The untruncated version, with every $1_t=1$, is denoted ${\tilde{\sigma }}_{AC}^2$.

Theorem 2 (long-span limit theory). Let $\sigma$ be constant, let the rounding errors be independent of $\left(W,\Pi \right)$ with constant variance, and let $\Delta$ and $f$ be fixed while $n\to \infty$.

(i) Without jumps, ${\tilde{\sigma }}_{AC}^2\to {\sigma }^2$ in probability and

where $\Omega =\sum _{\left|l\right|\le 2}\mathrm{Cov}\left(z_t,z_{t-l}\right)$ and $z_t=x_t^2+2x_t{x}_{t-1}$.

(ii) With compound Poisson jumps, ${\tilde{\sigma }}_{AC}^2\to {\sigma }^2+\lambda E\left[Y^2\right]$, the total variance per unit time.

(iii) The Newey-West estimator of $\Omega$ with bandwidth $L\ge 2$ [21] is consistent, so ${\widehat{\sigma }}_{AC}\pm 1.96\widehat{\mathrm{se}}$ with $\widehat{\mathrm{se}}={\widehat{\Omega }}^{1/2}/\left(2{\widehat{\sigma }}_{AC}\Delta \sqrt{n}\right)$ is an asymptotic 95 per cent interval for $\sigma$.

Proof. By the representation in the Appendix, each $r_t$ is a stochastic integral of the driving Lévy process over the interval of length $\left(1+f\right)\Delta$ that ends at $t\Delta$, plus $u_t-u_{t-1}$. These windows have length $\left(1+f\right)\Delta <2\Delta$, so $r_t$ and $r_{t-l}$ are independent for $l\ge 2$, and $z_t$ is a stationary 2-dependent sequence with finite variance. By (5), $E\left[z_t\right]={\sigma }^2\Delta$ without jumps and $\left({\sigma }^2+\lambda E\left[Y^2\right]\right)\Delta$ with them; demeaning by $\bar{r}$ changes the sum by $O_p\left(1\right)$. The law of large numbers and the central limit theorem for $m$-dependent sequences [22] give (i) and (ii), and (iii) follows from the consistency of kernel estimators of the long-run variance [21].$▫$

Three remarks complete the theory. First, the logistic drift adds at most $O\left(\Delta \right)$ to $E\left[z_t\right]/\Delta$, of order ${10}^{-4}$ for daily data, and does not affect (7) at the precision reported here. Second, truncation targets the diffusive part ${\sigma }^2$: jumps above $c\sqrt{V_t}$ are removed, so the remaining bias comes from jumps small enough to pass the threshold and from the small share of diffusive returns beyond four local standard deviations. Section 4 measures this bias, and Section 5 checks it by a parametric bootstrap at the fitted values. Third, (6) uses no estimate of $f$ or $\eta$, which is what makes it robust to the observation scheme.

3.4Jumps and drift

Jump days are detected with the test of Lee and Mykland [23]: the statistic $L_t=x_t/\sqrt{V_t}$ is compared with the Gumbel critical value for the maximum of $n$ standardised returns at level 5 per cent, which controls the probability of any false detection in the sample. The jump intensity is the number of detections per year, and the mean and dispersion of the jump size are taken from the detected returns after removing the local diffusive variance. The logistic drift is estimated by regressing $r_t$ on a constant and the lagged price, $r_t=\alpha -\beta S_{t-1}+e_t$, which identifies $\beta =\mu \Delta /K$ and $\alpha =\left(\mu -q-\lambda \kappa -{\sigma }^2/2\right)\Delta$, with Newey-West standard errors.

4Monte Carlo evidence

Paths of (1) were generated with 20 Euler steps per day on the log scale, which keeps the price positive, using $S_0=5$, $K=60$, $\mu =0.15$, $q=0.06$, $\sigma =0.30$, $\lambda =4$, $\gamma =-0.02$ and $\delta =0.06$, chosen to resemble Safaricom. Daily observation, $\Delta =1/252$, was used throughout, with spans of 1, 5 and 17 years and 100 to 1,000 replications per design and fixed random seeds.

With point-in-time prices and no jumps, the infill theory of Section 3.2 is accurate: all estimators are unbiased, the nominal 95 per cent intervals cover 94.4 to 96.1 per cent of the time, and the jump test rejects at 5.6 to 6.4 per cent at the 5 per cent level. With jumps, bipower variation overstates $\sigma$ by about 0.012 at the baseline because jumps leak into the products of neighbouring returns, and at daily frequency this does not shrink. Since the interval narrows as the span grows, coverage falls to 12.5 per cent over 17 years. Threshold variation is less biased, and both biases grow with jump intensity (Figure 1a). The estimates are identical for $K=20$, 60 and 1,000, as the invariance result predicts, and the dividend adjustment of Section 3.1 restores the baseline values to within simulation error.

The second set of designs reproduces the NSE scheme: the recorded price is the average of the simulated price over the last 90 per cent of each day ($f=0.9$), rounded to a relative tick of 0.3 or 1.4 per cent, which bracket the Safaricom sample. The simulated first-order autocorrelation is 0.21 against Working’s value of 0.214. Table 1 and Figure 1b show that every conventional estimator is biased downward, by 11 to 16 per cent without jumps, whereas the corrected estimator is essentially unbiased, with coverage of 92 to 94 per cent. With jumps at $\lambda =4$ per year the corrected estimator retains a bias of about 2.5 per cent; the untruncated version, which also counts jump variance, is 8 per cent too high, in line with Theorem 2(ii). With volatility changing every year between 0.15 and 0.45, the corrected estimator tracks the average integrated volatility with a bias of 2.2 per cent.

Design	Tick	Naive	BV	TV	AC
$$\lambda =0$$	0.3%	-16.0	-15.4	-16.0	-0.2
$$\lambda =0$$	1.4%	-11.0	-15.9	-11.1	-0.4
$$\lambda =4$$	0.3%	-9.1	-10.9	-13.4	2.5
$$\lambda =4$$	1.4%	-4.3	-11.4	-8.4	2.3
Yearly $\sigma$	0.3%	-9.8	-11.4	-13.7	2.2

Session VWAP with $f=0.9$, 17 years, 300 replications. Naive: standard deviation of daily returns. AC: estimator (6).

Quasi-maximum likelihood based on the Poisson-normal mixture density was also examined. With point-in-time prices, joint estimation of all parameters recovers $\sigma$ without bias (0.3000, coverage 94 per cent) and the jump parameters with coverage of 91 to 96 per cent, whereas a two-step version that plugs in bipower volatility underestimates $\lambda$ by about 40 per cent at $\lambda =12$. Rounding to a 1.4 per cent tick raises the joint estimate of $\sigma$ by 4 per cent. Under the NSE scheme, a version of the likelihood applied after inverting the first-order moving average recovers $\sigma$ within 2 per cent, but sampling weekly to escape the averaging destroys the identification of the jump law. These results matter for the application below.

5Application to Safaricom PLC

5.1Data

The data are the daily NSE prices and volumes of Safaricom PLC (ticker SCOM) from its listing on 9 June 2008 to 31 December 2024, taken from the published NSE all-stocks datasets of Wanjawa [24]-[28]. Cleaning re-dated one misdated record, removed one exact duplicate and resolved three conflicting duplicates by matching the next day’s reported previous price, leaving 4,121 daily returns over $T=16.35$ years. No day has zero volume, so stale prices are not an issue. The tick is KES 0.05, except from May 2017 to December 2018, when prices moved in steps of KES 0.25. Because the tick is large relative to the price early in the sample (about 1.4 per cent in 2009 to 2012), 23.7 per cent of raw daily returns are exactly zero.

The dividend record contains 20 cash distributions with book-closure dates between 2009 and 2024, including special dividends in 2016 and 2019 and interim dividends from 2021, compiled from Safaricom’s annual reports and AGM notices [29]. An event study fixes the ex-dividend day as the first trading day after book closure: across the 19 events with a confirmed date, the return on that day averages $-0.92$ times the dividend yield, and 95 per cent of these returns are negative ($t=-6.0$). The book-closure date of the FY2015 final dividend could not be confirmed, so that dividend enters $\widehat{q}$ but its price drop is not added back. The resulting dividend yield is $\widehat{q}=0.0475$ per year. Figure 2 shows the price path with ex-dividend days and the detected jumps.

5.2The VWAP signature

Adjusted daily returns have an annualised standard deviation of 0.274, excess kurtosis of 4.4, and strong dependence in squared returns (Ljung-Box $p<0.001$). Their first-order autocorrelation is 0.221 and stable across subperiods (0.229 in 2008 to 2012, 0.204 in 2013 to 2018 and 0.226 in 2019 to 2024), while autocorrelations at lags 2 to 10 lie between $-0.04$ and $0.00$ (Figure 3a). This is the pattern of Proposition 1: positive dependence at lag one only. Inverting Working’s formula gives $\widehat{f}=6{\widehat{\rho }}_1/\left(1+2{\widehat{\rho }}_1\right)=0.92$, so almost all price variation falls inside the session. Variance ratios rise from 1 to 1.28 within three days and broadly follow the curve implied by $\widehat{f}$ (Figure 3b). Daily returns from closing prices therefore understate the variance of the underlying price by roughly 30 per cent.

5.3Volatility

Table 2 compares the estimators over the full sample. The conventional estimators, which treat closing prices as point-in-time prices, give 0.256 to 0.274. The corrected estimator (6) gives ${\widehat{\sigma }}_{AC}=0.308$ with standard error 0.010; the untruncated version gives 0.329, and the difference, ${0.329}^2-{0.308}^2=0.013$, is the variance contributed by the 31 truncated days. A parametric bootstrap with 300 paths, simulated under the NSE scheme with year-specific volatilities, the detected jump law and $\widehat{f}$, finds a bias of $+0.0005$ (0.16 per cent), a sampling standard deviation of 0.0062 against a mean analytic standard error of 0.0068, and interval coverage of 96 per cent. The bias-corrected estimate is 0.308 with bootstrap 95 per cent interval 0.295 to 0.320. The estimate is stable to the window (0.308 to 0.314 for 21 to 126 days) and to adding autocovariances at lags 2 and 3 (0.314 and 0.310). It is more sensitive to the truncation constant (0.282 at $c=3$, 0.325 at $c=5$). At $c=3$ about 2.4 per cent of days are removed, which discards diffusive returns from volatile periods as well as jumps, so $c=4$, fixed before seeing the data and supported by the simulations, is retained.

Estimator	$$\widehat{\sigma }$$	s.e.
Naive s.d. of returns	0.274	-
Bipower variation	0.263	0.006
Threshold variation	0.257	0.005
AC, untruncated	0.329	-
AC, estimator (6)	0.308	0.010
AC, bootstrap	0.308	0.006

4,121 daily returns. The standard errors of bipower and threshold variation assume point-in-time prices and understate their error. Bootstrap: 300 parametric paths under the NSE scheme.

Volatility is not constant over time. By calendar year, the corrected estimate ranges from 0.199 in 2019 to 0.393 in 2023, with 0.575 in the partial year 2008 (Figure 4), and a test of equal variance across the full years 2009 to 2024 rejects decisively ($H=59.2$ on 15 degrees of freedom, $p<{10}^{-6}$). The conventional estimate lies below the corrected one in 15 of the 17 years.

5.4Jumps and the logistic drift

The Lee-Mykland test detects 15 jump days at the 5 per cent level (20 at 10 per cent, 10 at 1 per cent), 10 upward and 5 downward. The implied intensity is $\widehat{\lambda }=0.92$ per year (standard error 0.24), with mean jump $\widehat{\gamma }=0.024$ (standard error 0.017, not significantly different from zero) and dispersion $\widehat{\delta }=0.064$. Detection requires a move of about 4.4 local standard deviations, so this is the law of large moves rather than of all discontinuities. Two checks show that the jump law is only weakly identified in daily VWAP data. First, the joint likelihood fitted to the data assigns about 270 jumps per year of size 1.5 per cent, and the conventional likelihood degenerates entirely because a quarter of returns sit at a single value; in both cases the mixture is absorbing tick rounding and volatility clustering. Second, simulating the model with year-specific volatility and searching over $\left(\lambda ,\delta \right)$ to match the number and mean size of detected jumps [30] does not reproduce the data, which contain detections of moderate size during calm spells, a feature of within-year volatility clustering that the model does not include.

The drift regression gives a slope of $\widehat{\beta }=3\times {10}^{-5}$ with standard error $3\times {10}^{-5}$ ($p=0.31$), so the logistic term is not significant. The implied carrying capacity is $\widehat{K}=42$ KES, but its 95 per cent confidence set runs from 14 KES upward without bound and so includes the geometric limit $K=\infty$. The growth rate is $\widehat{\mu }=0.28$ with standard error 0.15. This confirms the identification problem noted in [9]: over 16 years the price never approaches a level at which the logistic damping can be measured. By the invariance result, this has no effect on the volatility estimate.

5.5Value-at-risk

Each trading day from mid-2009, a one-year trailing window of returns gives two forecasts for a long position over horizons of 1, 5 and 10 days. The conventional forecast multiplies the sample variance of daily returns by the horizon. The corrected forecast uses (3) with $H$ in place of 1, that is ${\widehat{\sigma }}_{AC}^2\Delta \left(H-\widehat{f}/3\right)+h_t^2/6$, where $h_t$ is the relative tick and ${\widehat{\sigma }}_{AC}^2$ and $\widehat{f}$ are computed in the window. Forecasts use only past data, and the evaluation uses non-overlapping returns. Table 3 and Figure 5 report the results. The ratio of mean squared realised returns to mean forecast variance is 1.23 for the conventional rule at ten days, an understatement of 23 per cent, against 1.04 for the corrected forecast. At the 95 per cent level the corrected forecast passes the Kupiec test [31] at all horizons, with 4.7 per cent violations at ten days, while the conventional rule has 6.6 per cent violations at five days. Both Gaussian forecasts have too many violations at 99 per cent and fail the Christoffersen independence test [32] at one day, which reflects fat tails and volatility clustering rather than the averaging correction.

$$H$$	Forecast	Var. ratio	95% viol.	99% viol.
1	Naive	0.98	4.9	1.8*
1	Corrected	1.08	5.4	2.2*
5	Naive	1.28	6.6	2.2*
5	Corrected	1.10	5.6	2.1*
10	Naive	1.23	6.0	2.8*
10	Corrected	1.04	4.7	2.3*

Violations in per cent; 3,869, 773 and 386 forecasts at H = 1, 5, 10. Var. ratio: mean squared return over mean forecast variance. * Kupiec test rejects at 5 per cent.

The same correction matters for derivative pricing. At the money, an option price is close to proportional to volatility, so pricing with the conventional estimate of 0.274 instead of 0.308 understates the value of a Safaricom option by about 11 per cent, and by more if the untruncated total volatility is the relevant input.

6Discussion

Three findings stand out. First, the observation scheme matters more than the choice among conventional estimators: on NSE data every point-in-time estimator understates volatility, and the correction depends only on an identity that holds whatever the averaging fraction and rounding error. Second, the logistic structure is not identified from Safaricom prices, and the drift, including $\mu$ and $K$, is estimated with little precision. The invariance of the volatility estimator to the drift is therefore a practical result, not only a theoretical one. Third, the data reject constant volatility, and with daily VWAP data the jump law cannot be separated cleanly from volatility clustering.

The analysis has limits. VWAP is approximated by an equal-volume time average; intraday volume patterns change the weights but not the first-order moving-average structure on which (6) relies. Rounding errors are treated as independent, which is accurate for small ticks but only approximate early in the sample. The ±10 per cent daily price band [10, Rule 5.10.1] was reached on five days and is not modelled. Theorem 2 assumes constant volatility, whereas the data show time variation; under year-specific volatility the estimator targets average integrated volatility, as the simulations confirm. The dividend record omits the ex-date of one dividend. Finally, the study covers one stock. Extending it to other NSE stocks, to models with stochastic volatility, and to option pricing under the logistic dynamics of [9] with calibrated volatility are natural next steps.

7Conclusion

This paper has calibrated a logistic jump-diffusion with dividend yield to Nairobi Securities Exchange data. The model has a unique positive solution with a closed-form representation between jumps. Because NSE closing prices are session VWAPs on a discrete tick, daily returns follow a first-order moving average whose variance and first autocovariance combine to the true diffusive variance. A truncated autocovariance-corrected estimator built on this identity is consistent and asymptotically normal over a long span, robust to averaging and rounding, and invariant to the logistic drift. For Safaricom it gives an annualised volatility of 0.308, about 12 per cent above the conventional estimate, and it removes most of the understatement of multi-day value-at-risk produced by the square-root-of-time rule. Practitioners using NSE closing prices should correct for the averaging before using volatility estimates in pricing or risk management.

Acknowledgments

The author thanks Barack Wanjawa for making the NSE daily price datasets publicly available. The cleaned Safaricom series, the dividend record and the code that reproduces every table and figure are available from the author on request.

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