Abstract
The $-\sigma^{2}/2$ of Black-Scholes is sometimes read as a correction for discounting or risk aversion. This paper shows that it is Itô’s second-order term itself, measurable directly as the gap between mean and median—the ratio agrees with $\exp(\sigma^{2}T/2)$ to $12$ digits. No new theorem or law is claimed.
First, the mean is $\mathbf{110.517091808}$ and the median $\mathbf{108.328706767}$ (Section 2).
Second, this is the core of the paper. Their ratio $\mathbf{1.020201340}$ agrees with $\exp(\sigma^{2}T/2)$ to $12$ digits (Section 2).
Third, the growth of the mean logarithm is $\mathbf{0.080000000000}$, exactly $\mu-\sigma^{2}/2$ (Section 2).
Fourth, the gap is second order in $\sigma$—doubling $\sigma$ multiplies it by $\mathbf{4.122849}$ (Section 3).
Fifth, the mean itself does not depend on $\sigma$—it stays $110.517091808$ (Section 3).
Sixth, the separator is whether one is looking at $E[S]$ or $E[\ln S]$ (Section 4).
Keywords: Itô’s lemma, geometric Brownian motion, lognormal distribution, mean and median, Black-Scholes formula.
Scope (what is not claimed)
No new theorem or law is claimed—Itô’s lemma, the lognormal distribution of geometric Brownian motion and the Black-Scholes formula are all standard. No pricing formula is derived—option valuation, risk-neutral measures and replicating portfolios are not the subject; only the distribution of the underlying. Itô’s $2$ is not re-derived—its anatomy belongs to Paper 132; the result is borrowed and its financial consequence measured. Estimation is excluded—$\mu$ and $\sigma$ are taken as given, with no estimation error, time-varying volatility or jumps. No investment advice—which quantity one ought to look at is not the subject; only that the two differ, shown in numbers. Relation to earlier papers: Paper 132 showed that the $2$ in Itô’s lemma is not a dimension—this paper measures where that $2$ appears in the distribution of prices. Paper 292 showed that only the logarithm adds—which is why the median belongs to the logarithmic side. Paper 484 showed that maximising expectation goes bust—the quantity at work there is this same $\sigma^{2}/2$. Paper 481 showed that without separators a refutation has no destination—this paper writes one of them. What is added is giving mean and median as $110.517091808$ and $108.328706767$, confirming the ratio against $\exp(\sigma^{2}T/2)$ to $12$ digits, confirming the logarithmic growth of $0.080000000000$ by inversion, giving the factor $4.122849$ for a doubled $\sigma$, and confirming that the mean ignores $\sigma$.
1. Notation and the Question
Setting: Let a price $S_{t}$ follow geometric Brownian motion $dS=\mu S dt+\sigma S dB$. Applying Itô’s lemma to $\ln S$ gives
$$d(\ln S)=\left(\mu-\frac{\sigma^{2}}{2}\right)dt+\sigma\,dB$$where $-\sigma^{2}/2$ comes from the second derivative of $\ln$ and from $(dB)^{2}=dt$. Below $S_{0}=100$, $\mu=0.10$, $\sigma=0.20$ and $T=1$.
Observation 1.1 The $-\sigma^{2}/2$ is sometimes read as a correction. Discounting for an aversion to risk—a reading that makes it about preferences or discount rates.
Observation 1.2 But the term contains no preference at all. It comes only from the second derivative of $\ln$ and from $(dB)^{2}=dt$—as Paper 132 showed, the $2$ is not a dimension but the consequence of stopping at second order.
Question: Where does that term appear in the distribution of prices.
2. It Appears as the Gap Between Mean and Median
Computation 2.1: Evaluate three quantities from the distribution at $T=1$.
| Quantity | Value |
|---|---|
| Mean $E[S_{T}]=S_{0}e^{\mu T}$ | $\mathbf{110.517091808}$ |
| Median $S_{0}e^{(\mu-\sigma^{2}/2)T}$ | $\mathbf{108.328706767}$ |
| Ratio | $\mathbf{1.020201340}$ |
| $\exp(\sigma^{2}T/2)$ | $\mathbf{1.020201340}$ |
Observation 2.2 The ratio agrees with $\exp(\sigma^{2}T/2)$. $1.020201340027$ against $1.020201340027$—to $12$ digits: not close but the same.
Observation 2.3 The growth of the mean logarithm equals $\mu-\sigma^{2}/2$. $0.080000000000$ against $0.080000000000$—returning on inversion, and equal to $\mu=0.10$ less $\sigma^{2}/2=0.02$.
Observation 2.4 So Itô’s term is exactly the gap between two summary values. The $-\sigma^{2}/2$ in the equation—on the distribution, it fixes how far the mean and the median stand apart.
Observation 2.5 The gap is $2.020134$ per cent. At $\sigma=0.20$ and $T=1$—small-looking, but a quantity that compounds.
3. The Gap Is Second Order, and the Mean Ignores $\sigma$
Computation 3.1: Vary $\sigma$.
| $\sigma$ | Gap | Mean |
|---|---|---|
| $0$ | $0$ | $110.517091808$ |
| $0.20$ | $\mathbf{2.020134}$ per cent | $110.517091808$ |
| $0.40$ | $\mathbf{4.122849}$ times | $110.517091808$ |
Observation 3.2 Doubling $\sigma$ roughly quadruples the gap. A factor of $4.122849$—the signature of a second-order dependence, exceeding $4$ because of higher terms in the exponential.
Observation 3.3 At $\sigma=0$ the gap vanishes. The other side of the boundary—without fluctuation, mean and median coincide.
Observation 3.4 The mean itself does not depend on $\sigma$. It stays $110.517091808$—what moves is the median.
Observation 3.5 So “more volatility lowers the expectation” is wrong. What falls is the median—the mean is constant while the distribution grows a long right tail.
4. The Separator
The separator: is one looking at $E[S]$, or at $E[\ln S]$.
Observation 4.1 Looking at $E[S]$, $\sigma$ does not enter. $110.517091808$—fixed by $\mu$ alone.
Observation 4.2 Looking at $E[\ln S]$, $\sigma^{2}/2$ is subtracted. $0.080000000000$—Itô’s term appearing directly.
Observation 4.3 The two are distinct roots by the criterion of Paper 300. Both may be called “the expected value”—one an average of prices, one of logarithms: what was taken before averaging differs.
Observation 4.4 It is the quantity at work in Paper 484. Where maximising expectation went bust—there too $\sigma^{2}/2$ separated the expectation side from the growth side.
Observation 4.5 As Paper 292 said, only the logarithm adds. Prices move by multiplication—so an additive average departs from the multiplicative world’s summary value.
Observation 4.6 This is one of the separators of Paper 481. When “the price did not reach the forecast” is reported—whether $\mu$ was misjudged or mean and median were confused sends the repair elsewhere.
Observation 4.7 To be honest—the distribution is assumed lognormal. As a consequence of geometric Brownian motion—real prices have jumps and time-varying volatility, and then the closed form $\exp(\sigma^{2}T/2)$ does not survive. What survives is only the direction: taking an additive average in a multiplicative world introduces a gap.
5. Roll Call and Closing
- Mean $110.517091808$ (Computation 2.1): fixed by $\mu$ alone.
- Median $108.328706767$ (Computation 2.1): $\sigma^{2}/2$ subtracted.
- Ratio matches to $12$ digits (Observation 2.2): the core.
- Logarithmic growth $0.080000000000$ (Observation 2.3): returning on inversion.
- A gap of $2.020134$ per cent (Observation 2.5): it compounds.
- $4.122849$ for a doubled $\sigma$ (Observation 3.2): second order.
- Vanishing at $\sigma=0$ (Observation 3.3): the other side.
- The mean ignores $\sigma$ (Observation 3.4): the median moves.
- The separator is $E[S]$ or $E[\ln S]$ (Observation 4.3): distinct roots by Paper 300.
- Lognormality is assumed (Observation 4.7): jumps break the form.
Closing: The $-\sigma^{2}/2$ of Black-Scholes is sometimes read as discounting for an aversion to risk—a reading that makes it about preferences. But the term contains no preference at all, coming only from the second derivative of $\ln$ and from $(dB)^{2}=dt$; as Paper 132 showed, the $2$ is not a dimension but the consequence of stopping at second order. It appears as the gap between mean and median—at $S_{0}=100$, $\mu=0.10$, $\sigma=0.20$ and $T=1$, the mean is $\mathbf{110.517091808}$, the median $\mathbf{108.328706767}$ and their ratio $\mathbf{1.020201340}$. That ratio agrees with $\exp(\sigma^{2}T/2)$ to $12$ digits: not close but the same. The growth of the mean logarithm is $\mathbf{0.080000000000}$, exactly $\mu-\sigma^{2}/2$, and it returns on inversion. So the $-\sigma^{2}/2$ in the equation fixes, on the distribution, how far the mean and the median stand apart. The gap is $2.020134$ per cent: small-looking, but a quantity that compounds. The gap is second order, and the mean ignores $\sigma$—doubling $\sigma$ multiplies the gap by $\mathbf{4.122849}$, the signature of a second-order dependence, exceeding $4$ because of higher terms in the exponential. At $\sigma=0$ the gap vanishes: without fluctuation, mean and median coincide. The mean itself stays $110.517091808$ throughout, and what moves is the median. So “more volatility lowers the expectation” is wrong—what falls is the median, while the mean is constant and the distribution grows a long right tail. One thing separates them—is one looking at $E[S]$, or at $E[\ln S]$. Looking at $E[S]$, $\sigma$ does not enter and $\mu$ alone fixes it. Looking at $E[\ln S]$, $\sigma^{2}/2$ is subtracted and Itô’s term appears directly. The two are distinct roots by the criterion of Paper 300: both may be called “the expected value”, but one averages prices and the other logarithms. It is the quantity at work in Paper 484, where maximising expectation went bust and $\sigma^{2}/2$ separated the expectation side from the growth side. As Paper 292 said, only the logarithm adds: prices move by multiplication, so an additive average departs from the multiplicative world’s summary value. This is one of the separators of Paper 481: when a price does not reach its forecast, whether $\mu$ was misjudged or mean and median were confused sends the repair elsewhere. To be honest—the distribution is assumed lognormal, as a consequence of geometric Brownian motion. Real prices have jumps and time-varying volatility, and then the closed form $\exp(\sigma^{2}T/2)$ does not survive. What survives is only the direction: taking an additive average in a multiplicative world introduces a gap.
References
- Itô, K. (1951). On Stochastic Differential Equations. Memoirs of the American Mathematical Society.
- Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy.
- Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science.
- Yamagishi, Y. (2026). The $2$ in Itô’s Lemma Is Not a Dimension [Paper 132]. Zenodo. 10.5281/zenodo.22080358
- Yamagishi, Y. (2026). Only the Logarithm Adds [Paper 292]. Zenodo. 10.5281/zenodo.22171519
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). A Broken Law Can Be Blamed on Any of the Assumptions [Paper 481]. Zenodo. 10.5281/zenodo.22733927
- Yamagishi, Y. (2026). The Bet That Maximises Expectation Goes Bust with Probability One [Paper 484]. Zenodo. 10.5281/zenodo.22759898
Reference
Yamagishi, Y. (2026). What Separates the Mean from the Median Is Itô’s Second-Order Term Itself [Paper 495]. YOI: https://www.yamagishiyuuki.com/yoi/495
DOI: 10.5281/zenodo.22760156
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 495. This paper is a preprint and has not been peer reviewed.