Digital Option Volatility Arbitrage Done Wrong
From the views/likes on the previous post, I have decided to extend the previous discussion;; do let me know if you enjoy this series, I can dive even further.
In the previous post, we took a simple sane approach to making markets on digital options. One looked at the role of the market maker (to provide liquidity around market implied fair) and another from the perspective of an arbitrageur.
In the latter, the trader has view of integrated volatility over the tenor of the option, and believes that he is ‘on-average’ better than the market. Supposing this is the case, given this superior volatility estimator, a naive way to perform arbitrage is to input this into the BSM for digitals and quote around this fair.
Then the difference between the trader’s fair and the market’s fair is the trader’s directional view on volatility, and he expects to make money through the market implied converging to his estimator through the option vega. When he is wrong, he might attribute this to factors including path dependency, estimator standard errors and execution errors. Critically, no matter how much the execution is improved, or a more efficient estimator is found, this naive approach is doomed to slowly bleeding out.
An important extra error term he did not consider is model miscalibration.
Nuances
Successful market making asks that everything from technological demand to model construction be well aligned. If any “important issue” is lacking, we are immediately on the high way to losing hell.
Suppose the technological capacities are taken care of but not modelling, in this case your P(adverse | fill) is high, because sure, you can get execution as you desire, but your fills are primarily aggressive takers coming only when your model is significantly, and (importantly) poorly aligned with the market.
On the other hand, if the theoretical model were sufficient but not your technological capacity, your P(fill | adverse) blows up. When the market shifts and conditions become adverse, your cancels are not hasty enough and markouts are terrible.
Back to why the naive approach of digital option pricing would land you in the ‘lacking’ category. We all know that the BSM has a series of poor satisfied assumptions, lest I need repeat.
Of the rather concerning one is that market’s exhibit frequent jumps, and a pure diffusion model completely misprices short tenor digitals. Diffusion plays out and scales risk proportional to the square root of time, while jumps scale linearly. Jumps can dominate the realized volatility over shorter periods. Simply inflating a volatility estimator to input into the BSM is too obtuse of a solution due to jump processes being fundamentally distinct from volatility processes. So how do we jump-aware price a digital option?