# How to shock the IV surface w.r.t VIX and keep AOA

I have to compute the sensitivity of a set of option prices on a single sotck (range of tenor is over the whole surface) to an increase of 100% in the VIX.. and I am trying to get to the most reasonable way to do so. (The risk model in question only considers one source of randomness for IV).

The only way I can think about is to compare the volatility of the ATM IV of my single stock to the vola of the VIX, and use this to scale the shock in the VIX to a shock in the ATM IV of my stock (and thus assuming a correlation of 1, which is the assumption in the model since the VIX is normally distributed in the model). For concretness, say that this corresponds to a shock of $$(1 + 0.75)$$ for the (30-day) ATM IV of my single stock.

However, once I have a shock in the (30-day) ATM IV, how can I make it into a sensible shock of the whole IV surface (while preserving that it is free of static arbitrage)?

So the two questions (I am more interested in the second as I can always read and do some empirical studies to understand better the first one):

1. How do I translate a shock in the 30 day ATM IV to a shock in the whole ATM slice of my surface?

2. How does this new set of ATM IVs influence the whole surface in a way that preserves Absence Of Static Arbitrage (AOSA)?

In Gatheral's SSVI model, it seems to be possible solve 2. by only changing the ATM variance (in our example, we would e.g. define $$\tilde{\theta}_t := \theta_t \times 1.75.$$ This would not change absence of static arbitrage conditions as long as the shocked ATM variance $$\tilde{\theta}_t$$ is still an increasing function of time.

Also, in general absence of calendar-spread arbitrage should be conserved by a vertical shift of the whole surface. However, it does not seem to necessarily keep absence of butterfly arbitrage.

I would be glad to have some hints and experience on this one. Particularly, is the shocked surface that I obtain meaningful? Do I have any chance with something else than SSVI? Does it make sense to shift the whole surface upwards?

An important note is that I am not interested into looking at the "local" (in time) dynamics of the surface as this sensitivity is used for longer term considerations. So the main requirement here is robustness and stability.

• Would you be happy with a relative shift rather than an absolute shift? If you do this then arbitrage concerns go away.
– will
Sep 27 '19 at 18:39
• Sorry maybe I wasn’t clear but the initial shock to the VIX is relative (multiply by 2). You mean that if I scale each slice (fixed tenor) by some scale then I don’t introduce butterfly arbitrage? Sep 27 '19 at 18:49