# How to normalize different instruments by volatility?

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?

• Please don't attach your signature to your posts. I notice other members of this community have had to correct your previous questions. – chrisaycock Aug 8 '12 at 12:55
• What are you trying to achieve? Portfolio creation? Stock comparison? Define some model? – SRKX Aug 13 '12 at 2:53
• @Freewind, could you possible be a bit more specific what you want to use the vol adjusted stock prices for? Is it used for pricing derivatives or other related products? Is it used for screening the adjusted prices, or to chart them? As you can see below it seems some of us (or all) may have completely misunderstood your question or the context of it. – Matt Aug 23 '12 at 16:52

## 3 Answers

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:

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.

• You are incorrectly suggesting that level (price) volatility measures are inferior to return volatility measures. Additionally, there are a number other volatility measures that you completely omitted. Finally I do not see how your digression into stochastic calculus 101 is at all related to volatility measures. Also care to explain your comments on PCA and how they relate to the very simple question of volatility normalization? If I could down vote multiple times I would, sorry but I dont see how you even tangent the question. – Matt Aug 18 '12 at 10:31
• @RockScience, this is exactly the academic arrogance of some quants I mentioned in my post (and later edited in my own answer). Lehalle presents a tiatribe of stochastic differential equations making tons of unproven assumptions without actually answering the question. But he took the luxury to downvote my answer without the courtasy of explaining his rational behind the downvote. It makes me chuckle because it's a perfect example how some quants try to shoot a very simple and almost trivial sparrow with a higher math bazooka. For reference I started as quant at an exotic rates desk and ... – Matt Aug 18 '12 at 12:17
• ...think I am qualified to comment on the nonsense posted above. The OP was simply asking how to adjust stock prices (or price returns if you insist Lehalle) for volatility. Simple task, no differential equations, no measure theory, no numeraires needed. Typical case of someone sitting very very far away from managing risk and actually generating pnl. Gee... – Matt Aug 18 '12 at 12:20
• @Freddy, please do not think that formalizing answers to try to be clear is academic arrogance. I try to be as concise as possible, we have the chance to share the formalism of stochastic calculus, I just use it. – lehalle Aug 23 '12 at 5:58
• sure if it answers the question, however, you are in no way even close to touching on the question. Instead you unnecessarily lift a simple normalization up to a science. I can only hope you are not wasting resources to such extent where you work at. – Matt Aug 23 '12 at 14:24

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.

• not sure what your discussion on quants really brings to Freewind's question. Sounds like a useless digression. – RockScience Aug 13 '12 at 6:41
• @RockScience, fair point, will edit as I think my answer stands on its own and sheds light on more different aspects of normalizing with vol measures than other answers so far. Thanks for the pointer. – Matt Aug 14 '12 at 1:47
• @Freddy volatility of the price is the multiplier of the normalized random part of the returns. What do you call "returns volatility"? – lehalle Aug 17 '12 at 21:05
• @Lehalle, that is you who downvoted? Price volatility is the variations of absolute price levels, return volatility is generally defined as the variation of log returns over specified time period. You made the incorrect assumption in your answer that level volatility measures are inferior to return volatility measures. Mathematically they are equivalently rigorous. Care to explain your down vote? – Matt Aug 18 '12 at 10:29
• @Freddy, the question was about normalization and not volatility definition. your generic remark about "there is no single answer" should be on renormalization and not volatility, don't you think? your answer is out of the scope of the question I think. – lehalle Aug 23 '12 at 5:47