# Are financial asset prices (Pt) log-normally distributed or arithmetic returns (Pt/P0)?

In finance, the price of an asset is given by the following formula.

$$P_t=P_0* e^r$$

r=returns
Pt = Price at time "t".
P0 = Actual Price.

Likewise, it is assumed that the yields (r) will follow a normal distribution with mean $$u$$ and variance $$\sigma^2$$.

so, the distribution functions $$F_p(p) = P(p_t\le p_t)=P(p_0 *e^r\le p_t)=P(e^r\le \frac{pt}{p_0}) =P(r<=\ln(\frac{P_t}{P_0})) = F_r(r=\ln(\frac{P_t}{P_0}))$$ To obtain the density function and using the chain rule, we derive the distribution functions,

$$f_p(p)=f_r(\ln(\frac{P_t}{P_0}))*\frac{1}{P_t}$$ By hypothesis we have that the density function of r follows a normal with zero mean and variance sigma squared, then

$$=>\frac{1}{P_t\sqrt{2\pi \sigma^2}}\exp\{-(\ln(\frac{P_t}{P_0})^2/(2\sigma^2)\}$$ Applying the logarithm property, we have

$$=>\frac{1}{P_t\sqrt{2\pi \sigma^2}}\exp-\{(\ln(P_t)-\ln(P_0))^2/(2\sigma^2)\},$$ Concluding that , $$P_t \sim \text{LogNormal}(\ln(P_0), \sigma^2)$$.

Is this derivation of price theory in finance correct? That is, prices are distributed lognormally with mean $$\ln(P_o)$$ and variance $$(\sigma^2)$$.

I ask the question because a log-normal distribution has mean = $$\exp^(u+\frac{\sigma^2}{2})$$ and not $$\ln(x_0)$$.

Please someone who can support me.

• @usuario357057 Yes, I know that the prices of financial assets are modeled with the GBM. But imagine I have a model $P_t = P_0 * e^x$ (also known as log-lin model ). So what distribution does Pt have if I am aware that "r" is normally distributed $(r \sim N (0,\sigma^2)$. According to the steps described above, Pt is lognormally distributed. $P_t \sim LogNorm(Ln(P_0),\sigma^2)$. I wanted to know if this deduction is correct. Thank you very much. Commented May 1, 2022 at 22:24