What causes skew in an option?

Though Black Scholes assumes constant volatility across strikes, in reality volatilities varies across strikes. Normally implied volatility is lowest near the spot (i.e. at the money) and rises as we move to the flanks there by forming a convex curve termed as skew. There are many theories that try to offer an explanation for the formation of the skew. The prominent among them is “Crashophobia” by Rubinstein based on Black Monday (October 19, 1987) when Dow Jones lost more than 20% in a day. In this article I will be arguing that skew is more due to lopsided pay off from Put compared to that of Call. This argument is based rather on fundamental concept than on empirical evidence.
An option value is sum of Intrinsic Value (IV) and Time Value (TV) ; TV depends on the volatility and time left to maturity and IV depends on the current spot and the strike i.e. moneyness. Hence broadly option value depends on volatility and moneyness. Following 2X2 matrix looks at the impact of change in price (IV) and change in volatility (TV) on the valuation of Call and Put.

For instance in Quadrant 1, when price as well as volatility increases moneyness of Call is up and that of Put is down while both gain in TV as volatility increases. Hence Call gains from IV and TV while Put’s TV gain is offset by loss in IV. The degree of offset will depend on the amount of change in volatility and price, and the sensitivities i.e Greeks. So are the changes in values of Call and Put in other quadrants.
Based on the above premise we look at the scatter plot of one day change in VIX and S&P 500, the latter in percentage. The data series are from 1990.
The chart above shows that the relationship between S&P 500 and VIX is inverted and strong i.e. as VIX rises S&P 500 declines. This fits well with the risk – return concept if we accept VIX to be a measure of risk premium but Black Scholes assumes that Price is independent of volatility.
Now out of 4,806 data points 12%, 41%, 11% and 36% lie in Quadrants 1, 2, 3 and 4, respectively. When we overlay this scatter plot on the 2X2 matrix we find that Put losing its value due to decline in IV and TV is 41% (Quadrant 2) of trading days since 1990 and Put gaining from both IV and TV is 36%. Hence due to the inverse relationship between price and volatility Put would generate extreme returns compared to that of Call which makes Put riskier than Call and hence sellers of options demand higher premium for Put than Call. Thereby create a steep skew for Put which seeps into the skew that market looks at.