Isn't using IV (implied volatility) for statistics the same as using Beta as a measure of risk for stocks? I.e., if the stock dropped sharply and it's beta increases, but not the risk, it's actually a better price now. Same here – if the market declines today, does it really mean that tomorrow will be an even riskier day, as told by IV? If not it eliminates the need to trade fewer contracts on high IV times.
Beta and volatility are not comparable. Yes, in a broad sense you can say they measure a stock's volatility and tendency to undergo large moves. But the differences are very significant.
Beta MEASURES the PAST volatility of a single stock when compared with the volatility of a group of stocks. IV is an ESTIMATE of FUTURE volatility for an individual stock (or group of stocks).
Beta is RELATIVE and depends on the volatility of it's comparative index (SPX or DAX) when we talk about volatility in the options world, it is an independent measure. In other words, beta not only depends on the volatility of the individual stock, but it also depends on the volatility of the group.
Not the Risk
You said that the stock price declines, beta rises, and 'not the risk.' Why do you believe that risk is less just because the stock is trading at a lower price? Okay, the total that can be lost is less because the sock price is nearer to zero. So in that sense, risk is less.
However, risk is most often measured in terms of probability of losing money on the trade and not only in terms of dollars lost. Many traders believe a declining stock is more likely to decline further than reverse direction. That's the basis of technical analysis. Once support is broken, the bottom cannot be known. Trend followers jump on the bandwagon when stocks make big moves – in either direction. I do NOT agree that the lower stock price suggests that owning the stock is less risky.
Remember Enron? As the price declined, people bought the now 'less risky' stock – only to discover that risk had increased, not decreased.
There is a psychological rationale for buying stocks that fall. Investors think about the fact that they were planning to buy at a price above the current level, so it must be a good, less risky purchase now. Unless the stock has not broken support, there is no evidence that this is true. There is always that feel good felling when you catch the bottom, but in my opinion, it's is too risky to make that attempt.
Getting back to beta
IV is an ESTIMATE of future volatility for an individual stocks or group of stocks. Whereas implied volatility is very likely to increase as the market falls, there is no reason for beta to change unless it independently becomes more volatile than it used to be. Beta could decline if the individual stock moved less that its customary percentage of the index against which it is being measured.
When IV rises on a market decline, it is a fact that market participants believe that the market will be riskier tomorrow. The evidence is clear and overwhelming. Traders pay higher prices for options – and those option prices are what determines the implied volatility. Why do they pay those higher prices? Because they are afraid that tomorrow will bring more downside. They may be wrong, but that is the expectation. And IV is a measure of expectations.
Traders buy options when they want to insure a position. They buy options when afraid. They buy options when speculating. Whatever the individual reason, the 'marketplace' buys options in anticipation of something bad happening. That makes IV higher.
Remember when markets fall, they sometimes fall hard. That's why people expect tomorrow to be riskier after a big decline. I see nothing wrong with that idea. Sure it's okay to fade the down move and sell a bunch of puts into a big decline. You are getting a pumped price, but you are selling to the buyers who are far more afraid than you. I have no reason to believe they are any smarter, but it does take courage to fade the crowd when selling into a falling market. That's why there is a higher reward for option sellers who are willing to take the risk.
One reason for trading fewer contracts (as a premium seller) in a falling market is fear. The prices are attractive, we may be hoping that the decline will end, but there is that nagging fear of the huge bear taking hold of the marketplace. I suspect it's not that higher IV per se that makes trades sell fewer options under such circumstances.