# 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?

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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 Wolf Aug 23 '12 at 16:52

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:

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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.

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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 Wolf 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 Wolf 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 Wolf 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 Wolf 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://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