The *lognormal distribution* is a continuous probability distribution of a [[Random Variable|r.v.]] whose natural logarithm follows as [[Gaussian Distribution|Normal Distribution]]. It is used to model variables that are positively skewed and cannot take negative values, such as stock prices, income distributions, and biological growth processes. ## Definition A r.v. $Y$ follows a lognormal distribution if there exists a r.v. $X$ that is normally distributed, and the relationship $Y=e^X$ holds. $ Y\sim\mathrm{Lognormal}(\mu, \sigma^2) \begin{cases} X\sim \mathcal N(\mu, \sigma^2) \\ Y=e^X \end{cases} $ This means that taking the natural logarithm of $Y$ transforms the lognormal r.v. into a Gaussian r.v.. $\ln(Y) =X$ ![[lognormal.png|center|500]] ## Properties - *Positivity*: The distribution only takes positive values $(Y>0)$. - *Skewness*: The distribution is positively skewed, with a long tail extending to the right. - *Moment-Generating Function*: The [[Moment Generating Function]] does not exist in closed-form for the lognormal distribution due to the exponential growth of moments. ## Probability Density Function The [[Probability Density Function|PDF]] of a lognormal random variable $Y$ is given by: $ f_Y(y; \mu, \sigma^2)=\frac{1}{y\sigma \sqrt{2\pi}}\exp\left(-\frac{(\ln y-\mu)^2}{2 \sigma^2} \right), \quad \text{when} \, y>0 $ where: - $\mu$ is the mean of the logarithm of $Y$ (not directly the mean of $Y$). - $\sigma^2$ is the variance of the logarithm of $Y$ (not directly the variance of $Y$). The derivation of the lognormal PDF uses the change of variables technique, as applied in [[Monotonic Functions of Random Variables]]. $ f_Y(y)=f_X(x)* \left \lvert \frac{dx}{dy}\right \rvert$ Given the relationship of $X = \ln(Y)$, the derivative of $x$ w.r.t. $y$ is as follows: $\frac{dx}{dy}\, \ln(y)= \frac{1}{y}$ Substituting the PDF of the normal distribution $f_X(x)$, we get: $ \begin{align} f_Y(y) &= f_X(x)*\frac{1}{y} \\ f_Y(y) &=\frac{1}{\sqrt{2\pi\sigma}}\exp\left(-\frac{(x-\mu)^2}{2 \sigma^2} \right) * \frac{1}{y}\\[2pt] f_Y(y) &=\frac{1}{y\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln y-\mu)^2}{2 \sigma^2} \right) \end{align} $ ## Expectation The expectation of $Y$ relies on the relationship $Y = e^X$. This implies: $ \mathbb E[Y]= \mathbb E[e^X]$ For the normal r.v. $X\sim \mathcal N(\mu, \sigma^2)$, we know the form of its [[Moment Generating Function]] $M_X(t)$. By setting $t=1$, we get the desired $\mathbb E[e^X]$. $ \begin{align} M_X(t) &= \mathbb E[e^{tX}]= \exp \left(\mu t+\frac{\sigma^2 t^2}{2} \right) \\ &=\mathbb E[e^X] = \exp \left (\mu+\frac{\sigma^2}{2} \right) \quad \text{when}\: t=1 \end{align} $ Note that when we talk about interest rate returns (i.e. the rate of change) in the normal r.v. $X$, then the relationship adjusts: $ E[Y]= \mathbb E[e^X-1] = \mathbb E[e^X]-1$ This leads to a slightly different definition of the expectation: $ \mathbb E[Y] = \exp \left (\mu+\frac{\sigma^2}{2} \right) -1$ ## Variance As we know the [[Variance]] can be defined as follows. $ \mathrm{Var}(Y)= \mathbb E[Y^2] - \mathbb E[Y]^2$ *Step 1: Compute $\mathbb E[Y]^2$* The above derived expectation is being squared. Taking powers of exponents means to multiply them. $ \mathbb E[Y]^2 = \exp \left (\mu+\frac{\sigma^2}{2} \right)^2 = \exp\left(2\mu+\sigma^2 \right)$ *Step 2: Compute $\mathbb E[Y^2]$* Using the relationship of $Y=e^X$, we can see that $Y^2=(e^X)^2$, which can also be written as $e^{2X}$. $ \mathbb E[Y^2] = \mathbb E[e^{2X}]$ Again, we can use the MGF of the normal r.v. $X$ and set $t=2$ to derive $\mathbb E[e^{2X}]$. $ \mathbb E[e^{2X}]= M_X(t=2)= \exp (2 \mu+2\sigma^2)$ *Step 3: Combine results* $ \begin{align} \mathrm{Var}(Y)&=\mathbb E[Y^2] - \mathbb E[Y]^2 \\[2pt] &=\exp(2\mu + 2\sigma^2)- \exp(2\mu + \sigma^2) \\[2pt] &=\exp(2\mu+\sigma^2) * (\exp(\sigma^2) -1) \end{align} $ >[!note:] >The variance is not impacted by the case where we talk about returns (we had to deduct $1$ in the expectation).