Suppose you are a market maker with a model that is producing an implied volatility surface for you. Suppose you quote bid/ask prices (vols) around the prices given by your implied vol surface. In order to manage inventory and deal with asymmetric information risks it is necessary to adjust you bid/ask quotes as you are hit/lifted. For example if you quote 10/12 and get lifted at 12 then you are net short at 12 and would like to move your spread to cover your short position. For example you may move the spread to 11/13. That way you are more likely to buy and less to sell and you may cover your short at 12 by buying back at 11 and making 1.
My question is this: how does this dynamic carry over to an environment where you are trading across strikes and maturities and have a vol surface? Say I get lifted on the December 100 XYZ call. It is intuitive that all the XYZ options are correlated products so I should adjust my spread up across the whole vol surface. But my question is how is this actually done. How does the change in the spread depend on where you are on the surface? In other words: how do I change my vol surface in response to a trade?
How a market maker should adjust its quotes on a vol surface with respect to his inventory?
How to adjust the vol surface when a new trade is observed on the markets?
Let me focus on the market making question, and that for you need to be familiar with optimal trading and optimal market making literature:
A breakthrough has been made around the year 2010: first Avellaneda and Stoïkov wrote properly the market making problem using a stochastic control framework, then Guéant, L and Fernandez-Tapia solved it in closed form. This report by Stanford students summarizes these two papers. It tells you by how much to adjust your quotes when time lapses.
Stoikov and Saglam wrote one paper in 2009 explaining how to simultaneously make market (in theory) on one option and its underlying. This is not exactly what you have in mind since the option and its underlying are considered to be isolated: no vol surface.
Nevertheless few years later Abergel and Emilio Said tackled the problem a more generic way, opening the door very recently (2019) to a paper closer to your question: Market Impact: A Systematic Study of the High-Frequency Options Market. Their empirical study covers a 2-year period from June 2016 through June 2018 on the KOSPI 200 options. They show formally and empirically that
liquidity consumption near the money shifts up the volatility smile (Figure 1)
liquidity consumption off the money "rotates the smile around the strike" of the option (Figure 2)