Dimensional analysis checks an equation by tracking the units and powers of units carried by its quantities. In quantitative finance, it can catch an annual volatility combined with a daily horizon, a missing price multiplier, or an attempt to add a return to a dollar amount before those mistakes become risk numbers.
The check asks whether the quantities fit together, not whether the arithmetic alone produces a number. It concerns measurement units; dimension in linear algebra counts independent directions.
A unit ledger for an option model
A unit is a chosen scale, such as a year or a session. A dimension records the underlying type, such as time; changing the unit changes the numerical value without changing the quantity. Ratios of comparable quantities can be dimensionless. These distinctions are developed in NIST's Guide to the SI, §§7.1, 7.10–7.11 and 7.14. We apply the same bookkeeping to currency and model time here; currency is not an SI base quantity.
For a non-dividend-paying share in the Black–Scholes model, take these stipulated inputs, with prices quoted for one share:
| Quantity | Value | Unit |
|---|---|---|
| Current price | 100 | USD |
| Strike | 100 | USD |
| Continuous interest-rate coefficient | 0.05 | |
| Volatility coefficient | 0.20 | |
| Remaining time | 0.25 | year |
Annualised volatility is often quoted as “20%”. In this continuous-time model, the coefficient has an inverse-square-root-time unit: multiplying it by gives the dimensionless standard deviation of the log return over that horizon. Brownian motion supplies the square-root scaling: its increment variance is proportional to elapsed time, as in Sigman's definition of standard Brownian motion, §1.2.
The ledger gives
Both are dimensionless. So are and . This is exactly what the formula needs:
The logarithm takes a ratio of prices in the same currency; the numerator adds dimensionless terms; the denominator is dimensionless. The discount factor also has a dimensionless exponent. Writing without specifying a reference price or numerical-unit convention would hide a dependence on whether prices were recorded in dollars or cents.
Change every input when changing the clock
For an illustrative clock with 252 equal model sessions per year, the same quarter-year horizon is 63 sessions. The rate coefficient becomes per session, while the volatility coefficient becomes per square-root session. These different conversion rules preserve the same and .
Original unit-conversion diagram for stipulated model inputs. The 252-session clock is a convention, not a claim about the actual calendar or observed returns.
Leaving volatility at 0.20 while replacing the horizon by 63 produces , instead of 0.10. The calculator cannot detect that its inputs refer to different clocks. Nor does choosing a session clock explain how overnight and weekend risk should be modelled.
A volatility number is not a dollar-risk number
Under constant-coefficient geometric Brownian motion, the one-session log-return standard deviation is
For a short interval, linearising the proportional price change gives the approximate dollar standard deviation
This is a local approximation, not the exact finite-horizon price standard deviation of a lognormal variable. Sigman's GBM notes, §§1–1.4, distinguish log-return variability from price moments. The price multiplier supplies the currency unit missing from the return volatility.
A successful unit audit is necessary, not sufficient. Replacing by preserves its units but changes the model. Units cannot establish a distribution, independence, a day-count convention or whether a volatility estimate remains useful tomorrow.
To apply this audit to a full pricing expression, see The Black-Scholes PDE, Formula and Implied Volatility, especially §4.3. Its public preview is available; the full lesson requires membership.
Sources
- NIST, Guide to the SI, Chapter 7, §§7.1, 7.10–7.11 and 7.14: quantities, numerical values, dimensionless ratios and dimensions.
- Karl Sigman, Introduction to Brownian Motion, Columbia University, §1.2, Definition 1.1: increment variance proportional to elapsed time.
- Karl Sigman, Geometric Brownian Motion, Columbia University (2006), §§1–1.4 and 1.7: log-return variance, exact price moments and Black–Scholes pricing.