Thursday, June 03, 2010

Options on Options on Options

Equity is an option on the asset of a company, with the strike price being its total liabilities. Most 'options' are thus really options on options, and indeed some people (Geske) have made such explicit models (they aren't used much as a practical matter)

The VIX is a weighted blend of prices for a range of options on the S&P 500 index. In 2006 the index itself became directly tradeable via futures. Now we have the VXX and VXV, new ETFs that allow equity investors access to the VIX volatility index. The VXX trades a shload, about 30MM shares per day. This means, we now have options on the VIX. These aren't options on options, but options on implied vols of options on options (ack!). Looking at the implied volatilities by strike, and comparing them to the implieds on the SPY, we see they are quite different little beasts.

Note that for the SPY, volatilities increase as the strike goes down, because the market anticipates that downward movements in the index are correlated with higher volatility. The VXX, in contrast, has an upward sloping implied curve, because as the volatility increases, its volatility increases too.

The fact that volatility curve is not flat implies that the standard option models, such as Black-Scholes, are misspecified, as they assume a constant volatility. As Peter Carr has said, 'the implied volatility is the wrong volatility we use in the wrong model in order to get the right price'. Black-Scholes, and its American option counterparts, are still very useful, but this highlights that any model in practice is implemented within a kluge of ad hoc rules informed by reality. Having more price points to nail down how to adjust implied vols by strike/expiration is very useful, what Sandy Grossman called 'complete markets', allowing one to see more market derived probability estimates of various states of nature, where in this case the state is not just a price, but a volatility of a volatility of a price.


DaveinHackensack said...

This is part of the reason why we eschewed models in developing the algorithm for our hedging app Portfolio Armor. No models (and no parametric assumptions) = no chance of model error.

Question for you though related to volatility: Let's say you are bearish on an optionable stock and want to bet against it. Intuitively, it seems that an investor might want to avoid buying puts on it if volatility is high, and short it instead (assuming the investor has the temperament for shorting). Is there a rule of thumb, though, for when volatility might be considered too high in this context?

Anonymous said...

Eric, do you think there's any value in looking at different strikes ( C(x)-C(x+dx) ) to see what the price expectations are for the underlying?

Kinda makes sense to me but the market would have to be really efficient (no arb opportunities) for it to be relevant, right?.

Eric Falkenstein said...

This is pretty virgin territory, so I'm sure there's a lot of interesting stuff here.

David: if your have a correct belief that markets are declining, you basically maximize your return the higher the stock's Omega, but get worse returns the higher the implied vol. However, if you are correct, put options are the way to go regardless. I think a bigger issue is the bid-ask spread you encounter, which for options can be quite large.

I don't know, offhand, how C(x)-C(x+dx) would tell you anything.

DaveinHackensack said...

Thanks for the response, Eric. I'll have to start looking more closely at Omega. You raise a good point about the spreads. You may get a bigger percentage return on a put than the drop in the underlying, but you may need a big one to compensate for the spread.

One change I've made since reading your posts on risk is that now, when I'm buying puts to bet against a stock, I'll aim for ones at the money or close to it, rather than trying to be cheap and going for ones relatively far out of the money.

Anonymous said...

let P be the underlying.

then if P>x+dx, the value of C(x)=(P-x)*p0 and the value of C(x+dx)=(P-x-dx)*p1

for a small enough dx we can assume p0=p1 so the value of C(x)-C(x+dx)=(P-x)*p-(P-x-dx)*p=dx*p

so the implied probability of P>x+dx, p=(C(x)-C(x+dx))/dx

what thinks you? it could work as a prediction tool for P only if we assume there are no arb opportunities on that option chain, correct? so for heavy traded contracts?

Anonymous said...

This is really fascinating and a welcome relief from Eric's talk of genetic determinants of racial intelligence.
I really like hearing about the things Eric is an expert in, and wonder why such an otherwise smart guy, able to understand complexity, finds it necessary to simplify complex ethnic/religious/racial/ethnic/cultural histories with the belief that our genetics determine everything.
Eric ridicules the folks who find it easier to believe a creation myth than to use science to understand how the world was created. But he engages in the same absurd reductions.

Anonymous said...

Eric didn't come close to saying that genetics determined everything. His post implicitly ridiculed those who in shrill totalitarian insanity insist it determines NOTHING becuase that fits their world view. You are obviously such a fool, or just an idiot who cannot read. No offense.

I personally find the opinion of really smart people interesting, even if it's not about their chosen field.

Irish Pride said...

Penultimate Anon,

Mainstream scientists don't believe that genetics determines intelligence 100%. They believe it determines it by about 50%-80%. Genetics is the only logical explanation for the gaps in performance that occur on every sort of quasi-intelligence test, from SATs, military entrance tests, etc. Genetics (and reversion to the mean) is the only logical explanation for why the children of college educated, upper middle class black parents score worse on the SATs than the kids of poor, high school-educated white parents.

Anonymous said...

Irish Pride, you have to stop. Those are facts. He's dealing with politically correct wishes. Your facts are nothing to him.

More Facts said...

Irish Pride is a pretty recent phenomenon. Not that long ago, the Irish couldn't have any pride because they were a bunch of stupid drunk monkeys who couldn't even feed themselves. Apparently they managed to find a way to breed with some genetic superstars because their descendants seem to be a lot smarter and a lot more successful.

As recently as 20 years ago Ireland was a country with 20% unemployment and massive corruption. No wonder that most people who were born there left.

And then something happened, I think Bono had like 5 million kids and everything turned around.

Translation Services said...

This is really amazing to have option like this.