Hi everybody,Does anyone have a (rigourous) explanation of a stock skew of volatility( why is it decreasing?).thanks a lot for your help

There are any number of plausible explanations for downward sloping skew in implied vols for stocks - none of them are particularly rigorous, however, nor are they mutually exclusive.One is just supply and demand - those who are long stock and who wish to hedge are “natural” call sellers (for income) and put buyers (for protection.) All other things being equal, this would tend to drive OTM call implieds down, and puts up.Another explanation hinges on the well-known inverse relationship between ATM implied vol and the price of the underlying - as the underlying goes up, ATM vol tends to decline. The skew can be interpreted as the markets expectation of what ATM implied vol will be if the underlying rallies or declines to the level of the OTM call or put strike price.Finally, to a certain degree the skew (or smile) is an artifact of the way implieds are calculated. Implieds are commonly calculated using models which explicitly or implicitly assume a normal (or log-normal) distribution, despite the fact that underlying returns are not normally distributed. If you build an option-pricing model which can incorporate non-normal distributions, you may find the skew (or smile) is much less pronounced.I hope this is helpful.

Other possible explanations:Leverage: As a company's stock price falls, it's Debt/Equity ratio rises meaning higher volatility (which can be made worse if the firm decides to invest in more risky assets).Jumps: Market is more likely to jump down than up (e.g. due to restrictions on short-selling, bad news have more difficulty in being gradually assimilated by the market leading to more abrupt assimilations)

Thank you guys,I found another explanation based on Heston model and the negative correlation between option price and the stock price.