Why Return Variance Does Not Determine Price Variance

# Variance of the price from returns variance


# Variance of the price from returns variance












Let's say that we have the variance of the daily return at $t_0$: $$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}-S_{t_0-1}}{S_{t_0-1}}]$$ for price process $S_t$. Is there a way to derive the formula for variance of the price at moment $t_0$ ($\text{Var}[S_{t_0}]$)? I tried thinking of $S_{t_0-1}$ as a known value, then: $$\sigma_{r_{t_0}}^2=\text{Var}[r_{t_0}]=\text{Var}[\frac{S_{t_0}}{S_{t_0-1}}-1]=\text{Var}[\frac{S_{t_0}}{S_{t_0-1}}]=\frac{1}{S^2_{t_0-1}}\text{Var}[S_{t_0}].$$ But I'm not sure if it's correct. Wider context: I want to use GARCH(1,1) forecasted standard deviations in the Bollinger band instead of moving ones (sd-s from last $n$ days). But GARCH model gives me variances/sd-s for returns and in the Bollinger band I would need ones for prices.

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/69808

No, you cannot reverse engineer the variance of the prices from the variance of the returns because the variance of the returns must be infinite for any credible candidate distribution for prices. That does not imply that you cannot get information about the variance of prices, but not from the variance of returns.

First, we need to mentally distinguish the idea of the sample variance from the population variance. We should also remember that the formula that most use for variance, in one version or another, is an algorithm that is optimized under specific conditions.

First, we must arrive at a distribution for returns, so that $r_t=r_t(S_t,S_{t+1})$. Since returns are a function of prices and prices are the actual data, that means that returns are a statistic. They are a function of data. Therefore, it is possible to derive the distribution of $r$.

We will ignore dividends, bankruptcy, mergers, and liquidity costs. In the real world you could not, but this post will be too long if we do not.

I will show you why you cannot recover it.

To avoid having to write subscripted notation for the remainder of the post, I am going to define the more generic $$Z=\frac{Y}{X}$$ in lieu of $$r_t=\frac{S_{t+1}}{S_t}.$$

So let us assume the relationship $$Z=\frac{Y}{X}$$ holds. We want the variance of $Y$ and $X$ from information in $Z$. We won't be able to do that but there will be information that we can recover.

The general solution to solve a ratio distribution is to note that the distribution function of $Z$ is $$D(z)=P(Z\le{z}).$$ That becomes $$D(z)=P(Y\le{zX}|X>0)+P(Y\ge{zX}|X<0)$$

If we assume that $f(x,y)$ is the joint density function of the variables, then $$D(z)=\int_0^\infty\int_0^{zx}f(x,y)\mathrm{d}y\mathrm{d}x+\int_{-\infty}^0\int_{zx}^0f(x,y)\mathrm{d}y\mathrm{d}x.$$

Skipping forward quite a bit,it is well known in the field of probability that if $f$ is any elliptical distribution such as the normal, Student's, or the Cauchy distribution, then the ratio will be the Cauchy distribution if $(\mu_x,\mu_y)=(0,0)$. If it is not, then it will be the convex combination of a Cauchy distribution and a finite variance distribution.

In my work, I have shown that there is a trick that you can play to make it work out to a Cauchy distribution if the mean is located at the equilibrium. The trick comes from converting everything to polar coordinates. $\Re^2$ is not an ordered set, so you can find the distribution of the errors rather than returns, but since the equilibrium return is a constant, you can just add that back. There are a few other transforms in the discussion required, however.

Noting that $$\tan(\theta)=\frac{Y}{X}$$ it follows that $$\theta=\tan^{-1}(\frac{Y}{X}).$$ Why one might care about that is that the distribution function of the Cauchy distribution is the arctangent. One of the limits of the integral becomes the arctangent of the ratio, which means that since $\int_a^bf(t)\mathrm{d}t=F(B)-F(A)$ and now the distribution function is $B$, every ratio distribution of two stock prices must contain the Cauchy distribution, which has an infinite variance.

Now the nasty trick I played was assuming that prices were approximately in equilibrium, that is to say, the short-run equilibrium equals the long-run equilibrium. By the law of total probability, that assumption cannot hold water without adding in the effect of the cases where the short-run distribution does not equal the long run distribution.

Nonetheless, we did assume the real world out of existence by eliminating bankruptcy, mergers and so forth. The real solution must contain the finite variance components. However, that would make this prohibitively long and your goal was simple, can you reverse engineer price variance from return variance?

You cannot because the Cauchy distribution, and we have eliminated the assumption of limited liability so there is no left bound, has a density function of $$g(r|\mu;\gamma)=\frac{1}{\pi}\frac{\gamma}{\gamma^2+(r-\mu)^2},\gamma=\frac{\sigma_{S_{t+1}}}{\sigma_{S_t}}$$ in the univariate case.

The scale parameter, $\gamma$, of the Cauchy distribution is the ratio of the population standard deviations of the prices. The reason that you cannot reverse engineer the standard deviations has to do with the nature of ratios. It is simpler to see in polar coordinates, but easy to explain.

If $r=2$ then both $$\frac{100}{50}$$ and $$\frac{200}{100}$$ produce the same result but presumably, if they are both equilibria, then they have differing variances at different price levels. The standard deviation of the first ratio might be ten and the second twenty. The scale parameter, $\gamma$, describes the nature of the heteroskedasticity of prices.

That also assumes two more things. First, it assumes we are roughly in equilibrium. Second, it assumes that prices are stationary over the interval of time. If there is a structural break in the interim, we would have a very different discussion. Likewise, if you started with a very undervalued or overvalued security, that relationship would not hold. It is a different, more complicated, construction.

Now, we do not have to assume a normal distribution. It is possible to derive the distribution of prices for different asset classes. For example, since Old Masters sold at Sotheby's are in an English style auction, the winner's curse would obtain. The distribution of winning bids would be the Gumbel distribution. If you bought and sold a Rembrandt at Sotheby's, then $f$ would be the a joint Gumbel distribution. For equity securities, because stocks are sold in a double auction, the joint distribution of prices would be normal in equilibrium, if you scale the prices around the equilibrium price and normalize them by dividing by their standard deviations.

Prices can have a standard deviation, but only under some rather odd assumptions would returns have a finite variance.

One last technical note, if you were not using returns, but instead your dependent variable were something like the probability of bankruptcy, then the distribution would have a finite variance because it would be bound in the interval of zero to one. Taking a ratio, by itself, is not enough to determine the distribution of a model because you can transform any distribution into any other distribution with the right transformation. You could map the quantile function of the Cauchy distribution onto the quantile function of the Normal distribution, and vice versa. The only thing that would matter is that you really understood that you were doing a transform of the data.

However, if $S$ is a stock price, you cannot recover the variance from the sample variance of $R$ because $R$ has an infinite population variance. How would you estimate infinity from a finite sample?

EDIT

I thought I should comment on the comments.

The first one has to do with assuming returns as being normal. One could argue that such a decision is problematic because returns are a statistic and not data. Returns are a function of prices in your construction. Arguably, they are a function of prices and volume since a larger purchase and sale should result in a larger haircut. But as you are ignoring quantities, I will ignore it here. Please don’t in the real world.

So, $R_t=R(p_t,p_{t+1})$. $R$ is a function. Prices are data. To assume that $R$ follows a normal distribution is no different than performing the hypothesis test $H_0:\mu=0$ with a known variance $\sigma^2$ on normally distributed data and assuming the test distribution was Snecdor’s F distribution. If you solved the sampling distribution, you would find that $\bar{x}$ is normally distributed. Economics has always assumed the distribution of returns into existence because it did not know that the solution is a solved problem in the field of probability.

For the general case, I would argue that equity securities under pretty mild conditions, assuming away bankruptcy, mergers, and dividends, would follow a truncated Cauchy distribution if some convenient transformations would happen. There is an inconvenient solution. It makes the empirical work more difficult because of how financial data has been recorded, but it is the most general solution.

You can find it at:

> Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association, 60(309):193-204.

I would argue that consols, perpetuities, and rentes should follow truncated normal distributions if bankruptcy, interest renegotiation and delays are excluded.

The second issue has to do with GARCH. If you read the article where GARCH is discovered, the authors note that they tested GARCH on equity securities and found that they so severely violated the necessary assumptions for GARCH that its use with equity securities is precluded. The field chose to use it because it had no other tool.

The problem with the conditional probability, in my opinion, is that if you have a series such as $S_1,S_2\dots,S_\tau$ and you treat $S_1$ as known and find the conditional volatility for $S_2$ given $S_1$, how do you then turn around and say $S_2$ is now being treated as known.

There is a more significant issue.

For a series $$S_{t+1}=RS_t+\epsilon_{t+1}$$, where $R$ is unknown, and $\epsilon$ is drawn from a distribution centered on zero with finite variance, then we will have a problem in finance. In finance, if $R\le{1}$, then nobody would invest that is rational. If $R>1$, then the series has explosive roots.

White showed that the sampling distribution of $\hat{R}$ is the Cauchy distribution. It forms the basis of the Dickey-Fuller test. Intuitively, that is sensible.

If $S_1=1$ and $R=1.1$ and $S_0=0$, then the random shock was 1 unit. If there were no more random shocks, then $S_2=1.1$ and $S_3=1.21$. The shock will tend to infinity as time goes to infinity. The variance over the life of the series will go to infinity.

The actual proof is exceedingly subtle.

You can find it at:

> White, J. S. (1958). The limiting distribution of the serial correlation coefficient in the explosive case. The Annals of Mathematical Statistics, 29(4):1188-1197.

The population distribution, for sample size $n$, for $\hat{R}$ is the Cauchy distribution that lacks a mean and has infinite variance. The same problem persists.

However, if your goal was to work with Bollinger Bands, you could instead work with either Koenker’s quantile regression or Theil’s median-based method of polynomial regression. A median exists for any distribution. You would still want to seek a way to catch structural breaks. The regression wouldn’t care, of course, but you might.

If you were gambling actual money, you should seek a Bayesian solution. Structural breaks are also easier to handle and Bayesian methods generate coherent probabilities in the de Finetti sense of the word.