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cook675
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Joined: April 18th, 2021, 7:27 pm

### Stock probability incorporating volatility

Hello, I'm trying to understand the best approach to model the probability of a stock move. My initial approach, however naive, was to create an X day rolling average of the stock prices over the trading history, then create a cumulative distribution function to determine what the probability was of a stock price being at or above a certain level X days from now

Ok, its im the ballpark probably somewhere. But recently I thought that, the probability must be somehow related to the current volatility estimate of the stock. If the stock's IV is very high right now, but for a large majority of the past it was very low, then the probability should be higher than what my rolling average analysis returns.

This lead me to discover the "Expected move" formula, which does seem to incorporate volatility. However, the classic expected move formula returns a range of prices for which the stock will be within 1 standard deviation over the time frame in question.

How could I rework this equation to answer my original question, or is there an alternate equation which incorporates volatility to get an estimate for the probability of a stock move?

Thanks for considering my questions. I am relatively new to this field but very eager to learn.

Alan
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### Re: Stock probability incorporating volatility

You're thinking correctly that you should use options. But I wouldn't use IV's directly. Instead, from option prices, using the Breeden-Litzenberger formula, one can infer (up to some issues with the tails) the forward-looking Q-distribution. Then, the issue becomes: how to best transform from that to the P-distribution?, which is the distribution you seek.

For the SPX, one answer is to use an exponential tilted distribution with tilt parameter $\gamma$. Historically, one can estimate, say from post WWI data, $\gamma \approx 3 \pm 0.8$. The interpretation is that $\gamma$ is the coefficient of relative risk aversion (CRRA) for a 'representative investor'.

For single-name stocks, the idea would be similar, but you use a stochastic discount factor transformation.
There's more detail in Appendix A of my paper on Option-based equity risk premiums. An associated book is forthcoming.
The single-name stock stuff needs fleshing out, but you could experiment with just the SPX case.

Note that the linked paper uses $\kappa$ for what I am now calling $\gamma$. (Have decided the latter is more conventional and adopted that for the book). Also, my best error estimate  for $\gamma$ is currently the one in this post.

Alan
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### Re: Stock probability incorporating volatility

Sorry, spotted a typo. That $\gamma$ estimate is roughly what you find from post WWII data. If you go back further and take, say 1926 to date, then you will estimate a point $\gamma$ that is somewhat lower. Generally, for any plausible data period, you'll estimate $\gamma \in (2,4)$.

cook675
Topic Author
Posts: 2
Joined: April 18th, 2021, 7:27 pm

### Re: Stock probability incorporating volatility

Thanks Alan im going to look into this heavily and report back with questions. I was also starting on SPX and had planned to eventually move onto single tickers. Very helpful, much appreciated

Alan
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Joined: December 19th, 2001, 4:01 am
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### Re: Stock probability incorporating volatility

You're welcome.

There's something worth adding. As you'll see, in the linked paper, I  fit option prices to a Gaussian mixture model. This computationally intensive fit (as I show in my book) can be avoided (with a small loss of accuracy) for the equity risk premium estimates. For your problem, estimating the P-density, I'm not sure if you can skip that fit -- it's an interesting issue.

Posts: 9
Joined: July 24th, 2020, 4:22 pm

### Re: Stock probability incorporating volatility

You're thinking correctly that you should use options. But I wouldn't use IV's directly. Instead, from option prices, using the Breeden-Litzenberger formula, one can infer (up to some issues with the tails) the forward-looking Q-distribution. Then, the issue becomes: how to best transform from that to the P-distribution?, which is the distribution you seek.

For the SPX, one answer is to use an exponential tilted distribution with tilt parameter $\gamma$. Historically, one can estimate, say from post WWI data, $\gamma \approx 3 \pm 0.8$. The interpretation is that $\gamma$ is the coefficient of relative risk aversion (CRRA) for a 'representative investor'.

For single-name stocks, the idea would be similar, but you use a stochastic discount factor transformation.
There's more detail in Appendix A of my paper on Option-based equity risk premiums. An associated book is forthcoming.
The single-name stock stuff needs fleshing out, but you could experiment with just the SPX case.

Note that the linked paper uses $\kappa$ for what I am now calling $\gamma$. (Have decided the latter is more conventional and adopted that for the book). Also, my best error estimate  for $\gamma$ is currently the one in this post.
Hi Alan,

Thanks for sharing your work on this topic (I remember reading your other ERP covid 19 article last year) congrats on the paper and looking fwd to the book!

I have two questions and a comment all of which might be too naïve.

1. This might be too textbook-like but when using the exponential tilting method you usually get a cumulant or moment term, I didn't catch any mention of it in the article and I'm wondering if somehow it can be shown to be not relevant, maybe its a common practice in the prior works you referenced (I haven't checked those yet)

2. Would you consider a RNP and RWP dist for the Risk Free term as a potential extension to this work?  I can see how it complicates things and might be be irrelevant in the time horizon, but curious if you've given it any though or have seen other authors work on this.

3. My comment is pretty simple, would you consider adding the P superscript in the Annex in formulas 24-32 where applicable, or I'm I wrong to assume that those those expectations are under Real World Porbabilities?

cheers,

M

Alan
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