Lognormal Distribution and Continuous Compounding

A random variable \(Y\) is lognormally distributed if its natural logarithm, ln \(Y\), is normally distributed. The opposite is true. If ln \(Y\) is normally distributed, then \(Y\) is lognormally distributed.

The lognormal distribution is positively skewed, meaning it’s skewed to the right and has a long right tail. In this distribution, values are bounded by 0. Typically, the mean is greater than the mode.

Consider the following graph of two probability density functions (pdfs) of two lognormal distributions.

Like the normal distribution, two parameters – the mean and variance of the associated normal distribution – fully describe the lognormal distribution.

Expressions for Mean and Variance of Lognormal Distribution

Assume that \(X\) is normally distributed with the mean \(\mu\) and variance \(\sigma^2\). Also, define the variable \(Y=e^X\).

Then \(\ln{Y}=\ln{\left(e^X\right)}=X\) is lognormally distributed with the following mean and variance expressions:

$$
\text{Mean}=\mu_L=e^{\left(\mu+\frac{1}{2}\sigma^2\right)} \\
\text{Variance}=\sigma_L^2=e^{2u+\sigma^2}\left(e^{\sigma^2}-1\right)
$$

Why the Lognormal Distribution is Used to Model Stock Prices

The lognormal distribution works well for modeling asset prices that cannot be negative because it has a lower bound at zero.

When the continuously compounded returns on a stock follow a normal distribution, the stock prices follow a lognormal distribution. Note that even if returns do not follow a normal distribution, the lognormal distribution is still the most appropriate for stock prices.

Continuously Compounded Rate of Return

Remember that given the investment horizon from time \(t=0\) to time \(t=T\), the continuously compounded return of a stock is given by:

$$ r_{0,T}=\ln{\left(\frac{P_T}{P_0}\right)} $$

If we apply the exponential function on both sides of the equation, we have the following:

$$ P_T=P_0e^{r_{0,T}} $$

Note that \(\frac{P_T}{P_0}\) can be written as:

$$ \frac{P_T}{P_0}=\left(\frac{P_T}{P_{T-1}}\right)\left(\frac{P_{T-1}}{P_{T-2}}\right)\ldots\left(\frac{P_1}{P_0}\right) $$

If we take natural logarithm on both sides of the above equation:

$$ \begin{align*}
ln \left(\frac{P_T}{P_0}\right) & =\ln{\left(\left(\frac{P_T}{P_{T-1}}\right)\left(\frac{P_{T-1}}{P_{T-2}}\right)\ldots\left(\frac{P_1}{P_0}\right)\right)} \\ \Rightarrow r_{0,T} & =r_{T-1,T}+r_{T-2,T-1}+\ldots+r_{0,1}
\end{align*} $$

Therefore, the continuously compounded return to time \(T\) equals the sum of one-period continuously compounded returns.

Remember that a linear combination of normal random variables is also normal. Therefore, if the shorter period returns \(r_{T-1,T}, r_{T-2,T-1},\ldots,r_{0,1}\) are normally distributed or approximately normal, the \(r_{0,T}\) is approximately normal.

As such, if we assume that the one-period continuously compounded returns \(r_{T-1, T}, r_{T-2, T-1},\ldots,r_{0,1}\) are independently and identically distributed (i.i.d) random variables, the mean of \(\mu\) and variance of \(\sigma^2\), then:

• The expected value of the continuously compounded return over a holding period of \(T\) periods is given by: $$ E\left(r_{0, T}\right)=E\left(r_{T-1, T}\right)+E\left(r_{T-2, T-1}\right)+\ldots+E\left(r_{0,1}\right)=\mu T $$
• The variance of the continuously compounded return over a holding period is given by: $$ \sigma^2\left(r_{0, T}\right)=\sigma^2T $$

The standard deviation of the continuously compounded returns, also known as volatility, is given by:

$$ \sigma(r_{0,T})=\sigma \sqrt{T} $$

In other words, if \(r_{T-1,T}, r_{T-2,T-1},\ldots,r_{0,1}\) are normally distributed with the mean of \(\mu\) and variance of \(\sigma^2\) then \(r_{0,T}\) is normally distributed with the mean of \(\mu T\) and variance of \(\sigma^2T\).

Let us go back to the formula:

$$ P_T=P_0e^{r_{0,T}} $$

If \(X\) is normally distributed with the mean \(\mu\) and variance \(\sigma^2\) and that \(Y=e^X\) then, \(\ln{Y}=\ln{\left(e^X\right)}=X\) is lognormally distributed. Assume we apply this intuition in the above formula. In that case, it is easy to see that we can model \(P_T\) as a lognormally distributed random variable since \(r_{0, T}\) is approximately normally distributed.

Volatility and Continuously Compounded Returns

Volatility measures the standard deviation of the continuously compounded returns on the underlying asset. Conventionally, it is usually annualized.

We calculate volatility using the historical series of continuously compounded returns. Another method is converting daily holding returns into continuously compounded daily returns and then calculating annualized volatility.

We base annualizing volatility on 250 trading days in a year, which is an estimate of the business days the financial markets operate. The formula we use for annualizing volatility is:

$$ \sigma(r_{0,T})=\sigma \sqrt{T} $$

For example, if the daily volatility is 0.05, then the annual volatility is:

$$ \sigma(r_{0,T})=0.05\times \sqrt{250}=0.79 $$

Example: Lognormal Distribution and Continuous Compounding

Jess Kasuku is analyzing the stock of ABC Company, which is listed on the London Stock Exchange under the ABC ticker symbol. Kasuku wants to understand how the stock’s price changed during a particular week when significant developments in the global economy impacted the UK stock market. To do this, she calculates the stock’s volatility for that week using the closing prices shown in Table 1.

$$ \textbf{Table 1: ABC Company Daily Closing Prices} \\
\begin{array}{c|c}
\textbf{Day} & \textbf{Closing Price (GBP)} \\ \hline
\text{Monday} & 75 \\ \hline
\text{Tuesday} & 78 \\ \hline
\text{Wednesday} & 72 \\ \hline
\text{Thursday} & 70 \\ \hline
\text{Friday} & 68
\end{array} $$

Using the information in Table 1, calculate the annualized volatility of ABC Company’s stock for that week, assuming 250 trading days in a year.

Solution

Step 1: Calculate the continuously compounded daily returns for each day using the formula \(ln{\left(\frac{\text{Ending Price}}{\text{Beginning Price}}\right)}\):

$$ \begin{align*}
r_1 & =ln\left(\frac{78}{75}\right)=0.03922 \\
r_2 &=ln \left(\frac{72}{78}\right)=-0.08004 \\
r_3 & =ln \left(\frac{70}{72}\right)=-0.02817 \\
r_4 & =ln \left(\frac{68}{70}\right)=-0.02899
\end{align*} $$

Step 2: Calculate the mean of the continuously compounded daily returns:

$$ \begin{align*} \mu & =\frac{r_1+r_2+r_3+r_4}{4} \\
& =\frac{0.03922+\left(-0.08004\right)+\left(-0.02817\right)+\left(-0.02899\right)}{4} \\
& =-0.024495 \end{align*} $$

Step 3: Calculate the variance of the continuously compounded daily returns:

$$ \begin{align*}
\sigma^2 & =\frac{\left(r_1-\mu\right)^2+\left(r_2-\mu\right)^2+\left(r_3-\mu\right)^2+\left(r_4-\mu\right)^2}{4} \\
& = \frac {
{
\left[
\left(0.03922-\left(-0.24495\right)\right)^2 +
\left(-0.08004-\left(-0.024495\right)\right)^2 \\ +
\left(-0.02817-\left(-0.024495\right)\right)^2 +
\left(-0.02899-\left(-0.024495\right)\right)^2
\right]
}
}{4} \\
& =\frac{0.007179}{4}=0.001795 \end{align*} $$

Step 4: Calculate the standard deviation of the continuously compounded daily returns:

$$ \begin{align*}
\sigma & =\sqrt{\sigma^2} \\
& =\sqrt{0.001795}=0.042363 \end{align*} $$

Step 5: Annualize the volatility by multiplying the daily volatility by the square root of the number of trading days in a year.

We know that:

$$ \begin{align*}
\sigma(r_{0,T}) &=\sigma \sqrt T \\
\therefore\sigma_{\text{annualized}} & =\sigma_{\text{daily}}\times\sqrt{250} \\
& =0.042363\times\sqrt{250} \\
& =0.6698\approx67\% \end{align*} $$

So, the annualized volatility of ABC Company’s stock for that week was 67.23 percent.

Question

Which of the following is true about lognormal distributions compared to normal distributions?

1. They are skewed to the right.
2. They can take on negative values.
3. They are less suitable for describing asset prices than asset returns.

Solution

The correct answer is A.

Lognormal distributions are continuous probability distributions that only take positive values and are often skewed to the right.

B is incorrect because lognormal distributions only take on positive values.

C is incorrect because there is no evidence to suggest that lognormal distributions are less suitable for describing asset prices than asset returns.