Chapter 23 - THe log normal distribution Flashcards
elaborate on x and z
we use x to indicate a normally distributed variable.
We use z to indicate a standard normal variable
Recall the cumulative distribution and its usefulness
Point estimates is no thing with continuous distributions. Therefore, we use the cumulative distribution to find intervals. It is easy to use it for arbitrary intervals.
P(k < x < t) is found by finding P(x<t) - P(x<k)
and so on
name an important feature of any normal distribution that disappear when considering the log normal distribuiton
Symmetry. Symmetry has important properties in regards to probabilities.
Namely:
N(-a) = 1 - N(a)
what is capital N used for?
Capital represent cumulative probability distribution
CASE:
compute P(x < b) when x is a normally distributed random variable.
How do we do this?
It is common to transform it into a standard normal variable first, and then make our life easier.
(x-u)/sigma < (b-u)/sigma
Now we basically have: P(z < (b-u)/sigma), which is easier to compute because of the standardization.
We can go one step further with the notation:
N((b-u)/sigma), where N is the cumulative standard normal distribtion.
Define a stable distribution
A distribution is said to be stable if the sum of independent random variables of this distribution is a new random variable with the same distribution.
we do not require the same parameters, only the same distribution type.
The normal distribution is stable, but generally, distributions are not stable.
elaborate on log normality preservation
A log distributed variable is e^(something). if we multiply two log distributed variables, we get_ e^x e^y = e^(x+y).
Since the requirement of the log distribution is that the exponent is normal, and we know that a sum of normal variables is also normal because it is stable, we know that log distributed variables are stable in multiplication.
what is the relationship between the stock price at 2 different time points, and the continuousll compounded returns?
we have the relaitonship:
S_t/S_0 = e^x
Solving for x find the continuously compounded return over a unit time period.
We can extend it like this:
S_t/S_0 = e^(rt), where t is the time diration in which r applies.
elaborate on the expression for stock price when we assume that continuously compounded returns are independent and normally distributed
The formula is:
ln(S_t/S_0) is N[(alpha - delta - 0.5 sigma^2)t, sigma^2 t]
The book just present this without going through it first.
Apparently, the formula above is standard normal. It is better if we allow for a regular one, because the understanding of mean and variance is easier.
There is a bunch of fuckery and stochastic calculus used, but the thing is that we eventually arrive at the fact that when we take the expected value of this expression for S_t, we get:
S_0 e^(alpha - delta)t
And if we take the ln of the expectation of the ratio, we get:
ln E[S_t/S_0] = (alpha - delta)t
So our expression allows us to use a normal distribution on the stock returns.
S_t = S_0 e^((alpha - delta - 1/2 sigma^2)t+sigma sqrt(t) Z)
elaborate on the overall idea of log normal model of stock prices
if S_t is log normally distributed, we can write:
S_t/S_0 = e^x, where x represent the continuously compounded returns.
However, our goal is to provide a representation of x that makes sense. We want a “useful” way of thinking about stock prices.
We assume that continuously compounded returns are normally distributed. Following this assumption, we have that the continuously compounded returns are given by ln(S_t/S_0).
We also assume that it is normally distributed with mean (alpha - delta - 0.5 sigma^2)t and variance sigma^2t.
We now have some sort of normally distributed variable.
If we take some standard normal variable, Z, we can transform it into a variable that has arbitrary mean and variance by multiplying by the “arbitrary” standard deviation, and adding the “arbitrary” mean:
Z –> sigmaZ+ mean
we take z, and multiply by the standard deviation that we want. And we want it to have the same standard devation as sigma sqrt(t), because this aligns with our model.
We also add the mean of (alpha - delta - 0.5 sigma^2)t. We end up with:
ln(S_t/S_0) = (a - d - 0.5 s^2)t + s sqrt(t) Z
At this point in time, ln(S_t/S_0) behaves like a normal variable with mean and variance as noted.
this also means that the ratio of prices, S_t/S_0 is now log-normally distrubuted. We also get an expression for the stock price:
S_t = S_0 e^((a - d - 0.5 s^2)t + s sqrt(t) Z)
The obvious question is: What the hell is this? where do these shit terms come from?
AS it turns out, when we take the expected value of this expression, we end up at:
E[S_t] = S_0e^(a-d)t
This is an expression we recognize very well.
If we take the natural logarithm of the expected value of the ratio S_t/S_0, we get the rate of appreciation of the stock, which would then be alpha-delta.
Recall that S_t/S_0 is the gross return. x, as in S_t/S_0 = e^x, represent the continuously compounded returns. We isolate x by taking ln on both sides.
IMPORTANT: If we did not subtract 0.5 s^2 from the “mean” above, we’d still get a working and valid result. But now, the rate of appreciation would longer be (a-d). It would be (a-d+0.5s^2). This is valid, but it makes alpha extremely difficult to interpret.
The outcome of all of this is that we have a model for stock price growth that follow log-normal distribution. we can find the distribution of returns by considering S_t/S_0, and utilizing the fact that it is log-normally distributed.
what are we actually doing with the model for the log normal distribution in regards to stock prices?
the overall idea is that if we assume that continuously compounded returns are normally distributed, we can build a model that reflect expected return equal to S_0e^(a-d)t, which align with our knowledge from forward pricing and CAPM returns etc, BUT it is now stochastic and random, so we can use it to compute probabilities. with just the CAPM return, we didn’t have a framework for computing likelihood of various returns. By assuming something about the continuously compounded returns, we move towards a better framework for modeling. Whether the continuously compounded returns are actually normally distributed or not, remains to be tested (by me), but it is widely researched and known to have many issues.
with the new model/Framework for stock returns, how do we use it for probability?
we start by the fact that continuously compounded returns, given by ln(S_t/S_0), is normally distributed.
We can shift the mean of the normal distribution by adding ln(S_0), which also makes our LHS simpler: ln(S_t).
Now we know that ln(S_t) follow a normal dist. We want stnadard normal, so we transform it like we always do.
So, we had ln(S_t) is normally distributed with the fucked up mean and fucked up variance. Then obviously ln(S_t) is the random variable, and when we standarize we subtract its mean, which gives:
ln(S_t) - ln(S_0) ….
And then divide on standard deviation.
This outcome is a a variable that doesnt represent anything of great interpretation, but allow us to use easy probability theory. It is actually plug and play. if we want P(S_t < K) we just plug in K in the formula, and since the formula is stnadard normal we can easily solve.
define kurtosis
what the tails look like
define the term used when the kurtosis is such that the mean is extremely sharp
extremely misleading topic. the book define kurtosis as peakness, but this is wrong. It is actually related to the tails.
the term leptokurtosis is used to indicate fatter tails.
discuss the common observed distribution of stock returns
higher kurtosis and fat tails.
Interpretation is that most of the time, jack shit happens. then when shit happens, a lot of shit happens.
Another interpretation is that of mixed normals. if we have drawn samples that actually represent varying volatilities, we end up recigvign weird results.
what is kurtosis for a normal distribution
3
formally, how do we define and find the kurtosis
It is the fourth central moment. It is found like any moment:
E[(x-u)^4], where u is the mean.
