Price and Return's distribution

The normal distribution often occurrs in the nature. Although the prices are not normalyl distributed, the price return in many cases has a normal distribution. The return in the ordinary sense can be defined as

\( \left( \frac{S_{t+1}}{S_t} - 1 \right) \) where St is the market price in the moment t.

But let's assume an investor has shares, the price of which is $100. If, at first, the price of shares rises 10% (from $100 to $110), and then falls at 10% (from $110 to $99), investor loses 1$, while by summing returns he receives 10%-10%=0. The right result can be received not by summing but by multiplying the price ratios 110/100*99/110=0.99. Actually, the final price is 0.99 from the initial one.

The examination of the logarithm of price ratio instead of simply price ratio allows one to use addition instead of multiplication. Thus, it is more convenient to define a

return as \( \ln\left(\frac{S_{t+1}}{S_t}\right) \) In accordance with common sense, when

we define the return as a logarithm, the price remains the same after 10% increasing and 10% falling.

\( \ln \left( \frac{S_1}{S_0} \right) = 0.1 \Rightarrow S_1 = S_0 * e^{0.1} \)

\( \ln\left(\frac{S_2}{S_1}\right) = -0.1 \Rightarrow S_2 = S_1 * e^{-0.1} = S_0 * e^{0.1} * e^{-0.1} = S_0 \)

A random variable X is said to have a lognormal distribution if its natural logarithm, \( Y = \log{(X)} \)

has a normal distribution. We assume that the return has a normal distribution,

i.e. \( \ln\left(\frac{S_t}{S_0}\right) \sim N\left(\mu^*t, \sigma \sqrt{t}\right) \) where S0 - is the initial price, St price at the time instance t.

\( \ln(S_t) \sim \ln(S_0) + N\left(\mu^*t, \sigma \sqrt{t}\right) \)