# Does shifting/scaling the IV surface relatively/absolutely introduce arbitrage?

I am fitting a volatility surface for vanilla call options. I do this by fitting low-degree polynomials (or cubic splines) along the strike dimension per maturity and then linearly interpolating implied variance along iso-moneyness lines. I would like to make a rough guess about how the surface might change if I assume that the volatility of the overall market as displayed by the VIX is going to decrease in the future.

Would it be a bad idea to assume that a given stock's IV surface would change the same way in terms of an absolute shift or a relative scaling of each option's IV?

As far as I understand, an absolute shift will introduce arbitrage, while relative scale of IV might not. What would be simple and not-so-bad approach to model the market shifting into a lower-volatility period or a sudden IV shock?

Long story short: yes both might introduce static arbitrage opportunities if performed blindly.

There are 3 types of static arbitrage to consider:

• Calendar arbitrage: total (implied) variance should be an increasing function of time for fixed (forward) moneyness.
• Vertical arbitrage (or call spread arbitrage): call spreads should have a positive price
• Butterfly arbitrage: butterflies should have a positive price

By writing the corresponding conditions under additive/multiplicative transformations of the original IV in (time to maturity, strike) axes:

• Cal: As long as the shift is positive, one cannot introduce cal arb. For negative shifts however, this needs to be checked (especially on short maturities).
• Vert: Assuming there is no fly arb, both additive and multiplicative spread will work.
• Fly: Can theoretically appear as well and should thus be checked. Typically happens for multiplicative spreads that are greater than 1 (does not happen when smaller than 1). For usual additive spreads (unless very negative), does not happen.

So I guess you have two options:

• Use additive/multiplicative spreads but make sure they do not introduce arbitrage in the first place (depending on the market/underlying you're looking at, you might even show that historically this type of shift never introduced arbitrage, so you can safely "overlook" them).
• Use an arbitrage-free parametrisation of your IV, which you can easily "bump" to reflect level changes (or term-structure changes). An example would be SSVI. This however adds to your original problem the complexity of fitting such a parametrisation.
• Thanks for the detailed answer. By "multiplicative/additive spread", you mean a constant (in K and T) mulitplier (or constant summand respectively) on every IV data point, correct? Could I start by, e.g., multiplying every IV with x and then de-arbing the surface by solving an optimization problem such as "minimize sum of squarde differences to this naively transformed surface while penalizing arbitrage opportunities"? – JMC Dec 22 '20 at 13:19
• Yes that's what I meant indeed. Yes you could, but note that the arbitrage conditions are quite bulky to evaluate (you would need to discretise your time x strike grid). Be aware that including these constraints through penalisation will not guarantee the absence of arbitrage (just penalise it). I would definitely first try to see if such a spread lead to arbitrage for the situation you consider: in case the answer is yes in most cases, then think of a smarter procedure, otherwise just go with it :) It really depends on what you plan on doing with your results. – Quantuple Dec 22 '20 at 13:25