What is the reason that Implied Volatility and Stock Price are inversely related?
Is it possible to understand this qualitatively without getting into the math of the Black-Scholes formula?
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people are willing to pay higher premiums for options when stocks go down. Obviously the time value and intrinsic value and interests rates of the option doesn't change because of this so the miscalculation remainder is priced into the implied volatility part of the formula.
Basically, anything that suggests the stock price will get volatile (sharp moves in either direction) will increase the implied volatility of the option. For instance, around earnings reports, the IV in both calls and puts in the nearest expiration dates are very high.
When stocks go down sharply, the volatility is high because some people are buying puts for protection and others are buying calls because they think there will be a rebound move in the other direction.
People (the "sleep-at-night" investors, not the derivatives traders ;) ) tend to be calm when stocks are going up, and fearful when they are going down. The psychology is important to understand and observe and profit from, not to quantitatively prove. The first paragraph should be your qualitative answer