# Why does Black Scholes formula give inconsistent dimensional analysis result?

For example, distance = speed * time, m = m/s * s.

But this technique gives wrong answer on the Black Scholes formula. The square root in the denominator gives wrong unit inside of the culumulative probability function.

Is this because some assumptions used in the equation fundamentally changed the dimension? What is the fundamental reason for the dimension to be inconsistant?

$$C= S_0 N(d_1) - K e^{-rT} N(d_2)$$

$$C$$, $$S_0$$ and $$K$$ have units of currency (e.g. USD).

$$N(d1)$$ and $$N(d_2)$$ are unit-less (dimensionless), the formula is dimensionally correct.

Considering,

$$d1 = \frac {ln{\frac {S_0} K} + r T + \frac {\sigma^2} {2} T} {\sigma \sqrt T }$$

$$r$$ and $$\sigma^2$$ have units of "per year", as they are stated on an annualized basis.

So, $$\sigma$$ has unit of "square root of "per year"".

Hence, $$d1$$ is also dimension-less.

• Thank you! I realized my interpretation of the unit of variation is wrong. This answer is very helpful! – aspire1inspire2 May 10 '19 at 12:51