volatility smirk for equity option and leverage

[ajsa]

Hi David,

"Capital structure leverage: as a company’s equity value decreases, its leverage (i.e., debt-to-equity or debt-to-total capital ratio) increases. Higher leverage implies higher volatility."

I am confused because I think volatility smirk chart assumes equity value is constant and strike price changes.. could you pls clarify?

Thanks.

 

[David Harper CFA FRM]

Hi asja,

The volatility skew/smirk typically does have strike price on the x-axis. But it can also be the ratio of Strike/Stock price; e.g., the FRM handbook uses ratio of Strike/Stock (see http://forum.bionicturtle.com/viewthread/2031/)

So, the volatility smirk still tends to look like this: http://forum.bionicturtle.com/viewthread/1673/
...if on X-axis we replace strike with strike/stock ratio

Okay, but beyond that, I frankly *cannot* follow Hull's attempt to explain the volatility skew with leverage.
If we assume fixed strike price, if asset (firm) price = Debt + Equity and Strike price = Debt, then:
as stock price decreases, lower equity value implies higher leverage = high strike/asset ratio which should corresponds to lower volatility on the skew
and higher stock price implies lower leverage = lower strike/asset (i.e., in-the-money call) which should correspond to higher volatility on the skew

put another way, high leverage (high debt to equity), seems to me to correspond to an OTM (not ITM) call option...when Hull says "this argument shows that we can expect the volatility of equity to be a decreasing function of price and is consistent with Figures 18.3..."
...I am confused, b/c the skew seems to be a decreasing function of strike not stock price

David

 

[ajsa]

Hi David,

BTW does smirk mean decreasing monotonically? i am asking because it does not seem from here :
http://forum.bionicturtle.com/viewthread/1673/
(the implied vol of ATM is still the lowest like smile)

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

I would not apply the term "decreasing monotonically" simply because this business of volatility smile/smirk/skew is, really, just empirical observation. I am not aware of a function/model that claims to characterize the smile (it only "appeared" after black monday 1987). The point Hull makes, which i think is the essential point, is that the smile implies, and is implied by, a distribution (i.e., if perfectly lognormal, then no smile) ...

Because the smile is a function of observed prices, the pattern of the smile is empircal and to expect any function-like characteristics (decreasing monotonically) is equivalent to expecting a smooth distribution for *actual* returns (i.e., unrealistic)

...the Google chart i ran is a good example because i had to draw several samples to get one that looked like Hull's example, the actual smiles I got were all over the place

...so the only general statement I would make is something like: they've studied equities a lot and it seems to be "smirk like" (higher implied vol for ITM call/OTM put)

separately, I was just researching the "equity leverage" explanation b/c I can't agree with the Hull point. Peter James (Option Theory) offers the following as one interpretatoin of equity skew:
"Historic volatilities of stocks increase naturally when stock prices fall, because in these circumstances uncertainty and leverage increase for the company. This causes the out-of-the-money puts to be “overpriced” compared to out-of-the-money calls."
...that agrees with Hull but continues to confuse me (??): OTM calls associate with high leverage (i.e., high strike = high debt)....

David