In general, it is known that market volatility increases during bear markets while it decreases during bull markets. Why is this the case? It seems strange that volatility is linked to the the general direction of price movements. Volatility is a measure of how much fluctuation there is in stock prices. During the bear market, prices go down and during a bull market, prices go up. There seems to me no relationship between the general direction of price movements and market volatility.
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There seems to me no relationship between the general direction of price movements and market volatility.
Clearly the market disagrees with you. As the saying goes, "markets go up in an escalator and down in an elevator"---just take the recent mid-March correction and subsequent rebound of, say, the S&P500 index.
Price and its properties are outcomes of market participants' trades. Direction of price movement is determined by direction of trade and volatility by size of trade. It is a consistently observed fact that when traders perceive that their positions are in danger of being wiped out (the perception possibly spurred by some exogenous event), they head for the exits en masse and quickly. This can also be seen during a "short squeeze", where shorts scramble to close their positions, or a bank run (e.g. the margin calls faced by Lehman Brothers, and subsequent liquidation of its positions, during the precursor to the financial crisis). In comparison, a crowded trade usually becomes crowded gradually.
The economic explanation for this market behavior is risk-aversion.
Much of finance is built on the equilibrium relationship between the first and second moment of prices, i.e. expected price movement and volatility. What matters is not just expected return $\mu$, but risk-adjusted return---this is precisely the Sharpe ratio $\frac{\mu - r}{\sigma}$ that arises in CAPM, where $\sigma$ is volatility. This is because risk aversion leads to market participants putting more weight on the downside than upside. Therefore high risk (volatility) means market demands high risk premium.
The flip-side of this coin is that, in a bull market, it's the upside that is being priced and risk-premium demanded by the market is low. Therefore volatility is low and price escalates up gradually. On the other hand, when perception of market risk changes, traders reduce their holdings as prevailing expected excess return falls short of required risk-premium. As more and more market participants exit their long positions, this cascades and becomes a self-fulfilling phenomenon.
Price becomes more volatile on the down-leg because the market is expecting more volatility, and vice versa.
(To be slightly more technical, the Sharpe ratio $\frac{\mu - r}{\sigma}$ is proportional to the degree of risk-aversion. Assuming the degree of risk-aversion remains constant, then expected excess return $\mu - r$ is proportional to $\sigma$. Therefore, low volatility, low return---escalator. High volatility, high return---elevator. If people become more risk-tolerant during bull market and more risk-averse during bear market, this would exaggerate the escalator/elevator contrast even more.)
The change in market perception of risk---implied volatility---can be large compared to (already dramatic, during a crash) change in price. See, for example, the comovement of the S&P500 and VIX indices during the mid-March episode. The drop in S&P500 is tame compared to the increase in VIX.
Further Comments
In economic theory, the fact that agents "put more weight on the downside than upside" is reflected by the stochastic discount factor (SDF). This is a standard object in asset pricing. The SDF is basically agent's state-dependent marginal utility, up to a multiplicative factor. Because of risk-aversion, in bad states of the world where consumption is low, marginal utility is high. Therefore agents put more weight on bad states than good states. A closely related object in mathematical finance is the risk-neural measure. The risk-neutral density and SDF differs by a discount factor.
VIX is the volatility of S&P500 implied by the Black-Scholes formula (one example of risk-neutral pricing). High VIX means market expects high volatility, as reflected by options prices.
The typical "volatility smile" is skewed towards the put side---one such example here. Put options are more expensive relative to calls, i.e. it is more expensive to buy insurance against price falling than betting on price rising. This again points to the fact traders put more weight on the downside.
In a world where market participants are risk-neutral, what you say would be true. But, as discussed above, the market tells us otherwise.
