## Wednesday, March 9, 2016

### Markets: Sucker Punch - Trading Events Based on Standard Vol Parameters

The high expectation before the Thursday ECB has made the smile in both rates and equities very acute (I have not checked the FX, but that should be no different). For example DAX and Euro Stoxx 50 is priced for a crash with skewed and convex smile favoring the puts. However, you still hear many market participants talking about how the vol of vol or skew is still cheaper - probably under the impression that these skews and vol of vol is still not enough to capture the fat tails that can result from such an event.

Which is a bit surprising, given the expected outcome. It is generally agreed that whatever ECB does, we will have a significant move in either side and then the level will settle down. If this is the case, what we are talking about is a classical case of bimodal outcome. And compared to that, the vol of vols and skews - i.e. in general the tails are quite over-priced. To see why, read on.

It is not very clear to everyone when someone talks about cheap vol of vol or skew (or whatever parameters), what exactly is being expressed by that view. Technically this should mean different things to different styles of trading. For a speculator (no delta hedging), this means the underlying distribution with the implied parameters is different than what she expects to realized. In particular, if one thinks vol of vol is cheap, the expected realized distribution has wider tails than the implied one. For a market maker (delta hedger), the relevant distribution is the PnL distribution after delta-hedge, which is quite different (and a bit more complex - a gamma weighted function of above) than the case of a speculator.

Nevertheless, let's examine the case of the speculator in a situation like Thursday - a bimodal outcome. One simple way to capture a bimodal distribution is what is known as skew bimodal normal (opens PDF, a bit technical). This distribution can be described as below

$$\Psi(x) = \Phi(x) - a(x)\phi(x)$$

Here $\Psi(x)$ is the cumulative distribution function (CDF) for the bimodal distribution, $\Phi(x)$ is the CDF of a normal distribution, and $\phi(x)$ is the PDF of the same. Here $a(x)$ is a linear function of x. Using a normal distribution with mean $\mu$ and variance $\sigma^2=1/\psi$, it is useful to express $a(x)$ as

$$a(x)= \frac{(x+\mu-2\beta)}{1+2\psi [\delta+(\beta-\mu)^2]}$$

This allows us to parameterize the bimodal distribution in terms of $\beta$ and $\mu$ as the location parameters (mean), $\psi$ as the scale parameter (inverse of variance approximately) and $\delta$ as the bimodality parameter.

With this framework, we pick-up a 1 month option with ATM forward at 100 and ATM vol at 25% (annualized), and tweak the $\delta$ parameters to generate a range of bimodal distribution of the underlying (matching the forward and variance to above values, i.e. the first and second moments). Compare these with the normal distribution.

Next step is to use these distributions to price the entire smile for each case, inverting the price to get BS vol. We get the following results:
Yes! This is what a typical smile under bimodal distribution looks like. This is counter-intuitive. The fair tail vols are actually lower than ATM for a bimodal outcome - typical of what could happen in ECB (or in June during Brexit). This is of course extreme and idealized version. But the point is measuring the vol of vol (or skew) in the conventional ways through fitted parameters or through price of a fly is not a very useful way to trade options around such events. We are trying to fit a log-normal like distribution to a one that is completely different. It is not a case of wrong pricing parameters, it is wrong a bit more fundamentally! Bimodal has thinner tails.

We are better off to try to pick the strikes near ATM (or biased towards one of the peaks depending on your view - but before the intersections)  - this will be cheaper than what is priced in if this distribution is realized  And cheapen that with a spread by selling the tail strikes (which is costlier according to our assumptions, even go 2x if you want). That is the correct way to play an event with bimodal outcome.

Note, this is the fair smile only for speculators. For market makers, even for a clear bimodal outcome, the smile will never be like this - as she cannot control some one picking her off to some tail strikes and not trading the the entire smile.