How to normalize different instruments by volatility?

Question

I'm trying to think on a way to normalize stocks to be on the same scale depending on their recent volatility.

Is there some theoretical reference on the subject or and experience you can share?

Answer 1

One more answer from my side in case you are interested in risk management.

In historical simulation (for details please see the references below) past returns are sometimes scaled by (i.e. devided by) some local volatility measure (this can e.g. be GARCH or EWM) such that the resulting scaled returns are theoretically stationary (with respect to volatility). This procedure is sometimes calles filtering.

Then at a later stage, when one considers scenarios, the filtered returns are multiplied by the most recent volatility measure. This gives (historically) simulated returns on the present volatility level that preserve historically seen correltions.

I would be happy to give you more details if an application to risk management is your aim.

References are:

1. INCORPORATING VOLATILITY UPDATING INTO THE HISTORICAL SIMULATION METHOD FOR VALUE AT RISK, John Hull and Alan White

2. For an SDE approach: A new approach for scenario generation in Risk management, Juan-Pablo Ortega, Rainer Pullirsch, Josef Teichmann, Julian Wergieluk

Answer 2

With a simple diffusion model (i.e. $dX_i=X_i \cdot (r_i \,dt+\sigma_i dW_i)$ for $i\in\{1,2\}$), you would probably want to normalize the returns (i.e. $dX_i$) and not the levels (i.e. $X_i=\int_t dX_i(t)$).

The most natural way to do it is to assume that the trends are structurally nulls (i.e. $r_i=0$ for all $i$) and just divide each return by an empirical estimate of $\sigma_i$, replacing $dX_i/X_i$ by $d{\tilde X}_i=dX_i/(X_i \sigma_i)$.

Renormalization can be seen as a rescaling on each variable you consider (as I proposed), but also a multi-dimensional way. You can operate sophisticated changes of measure or of coordinates, to obtain two stochastic processes $Y_1$ and $Y_2$ that are more homogeneous and related to the original $X_1$ and $X_2$. It is a way to renormalize in the sense that your $Y$s will contain essential components of the $X$s that are easier to compare. But what would mean observe some relationships between these two $Y$ and your original $X$?

For instance, just imagine that you try a PCA (Principal Component Analysis) on the de-trended parts of the returns (i.e. in the $(dX_1/X_1-r_1\,dt,dX_2/X_2-r_2\,dt)$ space). You will find a change of coordinate in the space of $(\sigma_1\, dW_1,\sigma_2\, dW_2)$ so that in this new space, the two processes are more orthogonal (in the L2 statistical sense in the increment space, i.e. independents in the space of the returns). It will be one step further than dividing each $dX_i/X_i-r_i\,dt$ by $\sigma_i$: the new $d{\tilde W}_i$ will now be independent. Of course it is an interesting property, but each time you will observe them, you will also have to go back in the original space and understand what it means. Namely you will have:

• to understand the meaning of the new components, analyzing the contribution of each $X_i$ (via its returns) to each of them;
• to monitor how the original $X_i$ are decomposed in a linear combination of ${\tilde W}_i$ through time.

For only two original instrument; is it worthwhile? Of course if you have 100 of them, it would be interesting.

Answer 3

May I recommend you first think what you try to achieve. As with almost anything in life there is no single answer. So, let me go ahead with an assumption and attempt to answer your question under given assumption.

The assumption is that you attempt to rank stocks and their price levels/return levels in comparison to other stocks. But just to be safe I consider you also want to just look at a single time series and observe how the volatility price levels/ returns evolved:

With that in mind I disagree with Lehalle in that it is preferable to scale price returns by return volatility. Quite a number quants scale price levels by price volatility. Mathematically such approach is in no way sub optimal.

What is also very important is whether you look at daily volatility measures or intraday volatility measures. Depending on such you will need to look into completely different ways to compute and scale volatility. Computational models for intraday measures include Garman Klass, DU, Symmetry measures, open/close, among others. Here are couple references I personally like regarding intraday volatility measures:

http://www.hedgeworld.com/research/download/Efficient_Estimation.pdf

http://erasmus-mundus.univ-paris1.fr/fichiers_etudiants/3963_dissertation.pdf

Here are some (in my humble opinion) excellent theoretical references for volatility measures in general, as requested:

http://arxiv.org/pdf/cond-mat/0202527.pdf

http://polymer.bu.edu/hes/articles/ymhbs05.pdf

http://www.nobid.com/mta/newsletter.pdf

And here my personal experience:

First of all, with so many different approaches to measuring volatility out there I do not believe that one measure is always better than the other. My personal experience as market practitioner is that its much more important to look at the dynamics of a single measure over time rather than comparing absolute levels between different vol measures. For example, I could not care less whether the historical (or even implied for this particular matter) vol levels of a given stock are annualized 30% or 80%. What I care much more about is how they compare relative to levels at other times in the history of this particular stock's time series. So, imho, it is completely irrelevant whether Garman Klass intrday vol gives you a daily vol of 2.00% and another mesure 1.80%. I highly recommend you do some reading of posted references (or whatever readings you find helpful), settle on a measure and go with it and start digging into the dynamics of how such volatility measures evolved during different market cycles. I do not want to digress too much into what particular measures I am using and what exactly I am looking at but I hope this helps to get you started and to not get bogged down by the sheer number of different approaches to measuring volatility. Even on the intraday volatility modeling side of whole implied vol surfaces there are at least 5-6 different core approaches used out there and I know some excellent index vol traders who use completely different approaches and each of those buddies generate excellent risk adjusted returns.

As an aside, I do admit that Garman Klass, for instance, highlights slightly different aspects of intraday volatility than looking at a different measure. Just look at the formulae and it should become apparent whether the open, high, low, or close is weighted more heavily vs. others. Again it comes down to what you exactly want to achieve. However, its a small aside, and generally I re-iterate that the different levels are less important than the dynamics over time.

Answer 4

You can just scale each stock volatility to the values from 0 to 1. Then you can compare them.

I hope I understood your question.