# Why and how is Implied volatility directly related to stock price but inversely related to strike price?

I know that in equity markets there is a volatility smirk which results in higher IV for lower strike price options because of crashophobia and leverage related factors but I can't wrap my head around why there is a direct relationship between share price and IV... A graph to explain it all would be really helpful.

• Changing the strike price and changing the stock price is inherently connected. See for instance the put-call-symmetry (and recall that put and calls have the same implied volatility). So, you can plot the IV smirk for varying strikes, for varying stock prices or for vaying moneyness such as $\frac{S_t}{K}$ or $\ln\left(\frac{F_t}{K}\right)$. – KeSchn Sep 12 '19 at 10:22
• I think it's quite hard to give a one fits all explanation for the skew, but broadly speaking investors (retail, pension funds, asset managers etc) are usually long the market. Losing money sucks, so they buy put options, driving the prices of put options up (and hence steepening the put skew). Once / if the market does crash they will monetize their puts, i.e. sell them and you'll actually see put skew softening when the market does tank even though overall volatility is elevated. – ilovevolatility Sep 12 '19 at 12:09
• Hi: this is simplistic but think of the case of a call. For a given stock price, as the strike goes up, you have less of a chance of being in the money. = less value. OTOH, for a given strike, as price goes up, you have a greater chance of being in the money. = more value. – mark leeds Sep 12 '19 at 12:47

90% of the time, credit just gets paid.

5% of the time, credit still gets paid; but a booming economy means that rates rise, so the increased certainty of getting paid is worth less than the decline to NPVs from the coupon becoming worth less.

3% of the time, there will be worries about the coupon becoming less affordable; but it still gets paid. A 1% probability of default becomes a perceived 2%, so bad things happen to credit prices.

1% of the time, there will be genuine worries about the coupon being affordable, but it is paid. The 10% odds of default on the subsequent ones will do bad things...

And 1% of the time, there will be an actual default. Which spurs a whole new round of speculation about default rates under liquidation and litigation. Is the bond worth 15, 25, 35 or 45 cents on the dollar? Volatility then would make stocks, even NatGas, look like a wimp’s game.

The point here is that the vol IS (negatively) correlated with price. With credit, this is for solvency reasons.

With equities, it is more to do with equities being perpetual assets. Let’s say, for simplicity’s sake, my stock is a 2% yield expected to grow by 4%.

Good news happens... new product is a huge success. Earnings and payouts jump 20%; but I assume is a sustainable gain in market share; don’t assume this acceleration persists. Price up 20%.

Bad news happens... new product is a total disaster. Earnings fall; but payouts can be maintained. I assume the ability to grow falls significantly (say 4% to 3%). Meanwhile, the negative newsflow causes me to raise my discount factor/risk premium from 6% to 7%.

Then d/(k-g) goes from 1/(6%-4%) = 50 to 1/(7%-3%) = 25. Down 50%.

This is obviously a figurative example; but hopefully the broad thrust is clear. For any asset driven by growth expectations, where the majority of the NPV will be delivered decades in the future, there is an intuitive fundamental tendency for price downside to coincide with volatility upside.