# Why is the price of a security after $n$ intervals of additional time modeled using a lognormal distribution?

I am reading a book about financial mathematics. There's a problem in the book that says that if $S(n)$ denotes the price of a certain security at the end of $n$ additional weeks, we can model the evolution of its price by assuming that the price ratios $S(n)/S(n-1)$ are independent and identically distributed (i.i.d) lognormal random variables.

Now I understand that choosing a model for a particular problem is up to us, but we usually use models that are close to reality. So, is there a theoretical reason, or statistical justification, for why the price ratios follow lognormal distributions?

There is no theoretical reason. However, under certain conditions the central limit theorem (CLT) can be applied, which leads to geometrical Brownian motion (GBM), and subsequently to lognormal prices. I'll outline the reasoning and the empirical evidence further.

The evidence is rather easy to observe. Go to any source of market price data, such as Yahoo!Finance or Fred. Look up the prices of actively traded securities, e.g. SPY ETF. Take the logarithm of prices, then get the differences. This is the series of so called log-returns: $r_t=\ln (p_t/p_{t-1})$. Draw a histogram, and observe that it looks bell shaped. If you take it that this is a normal distribution, then the distribution of the price ratio $p_t/p_{t-1}$ would be log normal.

You could run some normality tests on the distribution of log returns and see that it may not (and often doesn't) pass as normal distribution. For instance, weekly returns of Apple shares over past three years do not pass normality test, but the same for Exxon Mobil does pass the test as you can see in next two pictures:

So, the reality is that GBM is a simplification. I'd say it's simplest model that is still useful and is used today in many applications.

The theoretical part comes from the following reasoning. The prices of securities in markets reflect everything that is known by humankind about these securities. The prices may change only when new information arrives. The new information is truly knew, it can't be devised from what's known about the universe. In that it's truly random. Moreover, the new information and subsequently price changes are independent from the past. In this kind of a framework, you can posit that maybe for a infinitesimal time period we have: $$r_{t+\delta t}=\mu+\varepsilon_{t+\delta t},$$ where $\varepsilon_{t+\delta t}$ is from some distribution. We may not know what is the distribution, but we assume that the draws from this distribution are independent, i.e. the variance of updates $var[\varepsilon_{s},\varepsilon_{t}]=0$ for $s\ne t$. Under these conditions, we apply the central limit theorem and get that for a finite time step $\Delta t$ the distribution of the errors must be normal: $$r_{t+\Delta t}=\mu+\varepsilon_{t+\Delta t},$$ because $\varepsilon_{t+\Delta t}=\sum_{\delta t} \varepsilon_{t+\delta t}$ I'm being quite liberal with the notation here just to hint you to the reasoning.

If you agree with this approach, then you'd agree that the distribution of log returns must be normal for a small finite time interval $\Delta t$. Again, the reality is such that this assumption can be grossly incorrect, e.g. financial time series are known to have fat tails.

The GBM assumption is used a lot even in its simplest form today in practice. It's like Newtonian mechanics, a very useful approximation of reality, vs. special relativity theory. You wouldn't use Newtonian mechanics when designing a particle accelerator or GPS satellite system, but certainly can use it to launch a mortar in a battlefield.

• Thanks. What do you mean by $\mathrm{var}[\varepsilon_{s},\varepsilon_{t}]$? Jul 26, 2018 at 14:20
• It's the covariance of two innovation terms. If you have independent innovations, then the covariance of two innovations will be zero. Jul 26, 2018 at 14:25
• In particular, the log part of lognormal is to model multiplicative random effects (i.e. changes by percentages). From Wikipedia: "A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive." Jul 26, 2018 at 22:28

This question is better suited for the Quantitative Finance Stack Exchange, but I'll answer it anyway.

What you're talking here gives us the Geometric Brownian Motion for stock price. GBM is a very powerful tool as this is what we use for Black Scholes model for option pricing.

While powerful, every single QF practicioner knows this is not right. Anybody who believes lognormal is appropriate is crazy. So, why are we using it?

GBM is simple, it gives us a good analytic solution for option pricing. A simple approximating model is generally better than a super complicated model. After all, modelling is about approximation (close enough is sufficient). Most options models other than BS model don't have an analytical solution. Lastly, any "realistic" model for reality require something like time-dependent multidimensional stochastic process; they are very complicated. Trust me, you don't want to work with complex roots.

Another reason. Our financial market is actually driven by supply and demand. Nothing you're studying make any sense as traders don't care about anything like lognormal distribution. It's just about supply and demand. The BS model is used as a statistical tool to work out the volatility from market option price. Please pay attention to what @Matt wrote in the link. Very important for your understanding! Gold.

BS is merely a translation tool, nothing more, nothing less. What is really priced by the market is implied volatility. What is traded, however, is the option price. Hence, as long as the market agrees on one standardized model it does not matter what exact model we are talking about.

The lognormal model follows from the following assumptions:

(1) There are various events that can affect the price.

(2) These events are numerous enough, and uncorrelated enough, for the CLT to apply.

(3) The effects change the price by a certain percentage; the change in absolute price is independent of the current price.

The idea is that if prices changes scale with the current price, then we can write $p_{i+1}=p_i+p_ix=(1+x)p_i$ where $x$ is some random variable. Then

$log(p_{i+1})=log(1+x)+log(p_i)$.

And

$log(p_n)=log(p_0)+\sum_{k=1}^nlog(1+x_k)$

Thus, the logarithm of the price is equal to the logarithm of the original price plus the sum of random variables. Applying the CLT gives us that this sum is normally distributed.