Fitting an Arbitrage-Free Implied Volatility Surface with Bid-Ask Spreads

# Arbitrage-free IV surface definition vs. real arbitrage process


# Arbitrage-free IV surface definition vs. real arbitrage process












In the context of BS implied volatility surface fitting.

In the literature, it seems that conditions for arbitrage are defined in a way that assumes that options can be traded at the same price for buying and selling (i.e. no bid-ask spread).

In reality, we are sure to buy a certain quantity at an ask and sure to sell another one at bid.

My intuition would be aligned to that of the put-call parity re-expressed to account for bid-ask spreads:

$$F^{ask}(T) := k + e^{r_T T}(C^{bid}(k,T) - P^{ask}(k,T)),$$ $$F^{bid}(T) := k + e^{r_T T}(C^{ask}(k,T) - P^{bid}(k,T)),$$

so to have two volatility surfaces, one built with call bids, put asks and forward ask and another one built with call asks, puts bids and forward bid.

Some arbitrages then depend on both surfaces (which I guess might be a complete nightmare to define and fit) and the arbitrage constraint and fitting would have to be done on both surfaces at the same time.

I am missing something here? How is it done in practice, I am thinking in particular about hedging where I guess that we cannot just ignore bid-ask spreads.

## Answer by jherek (score 1)

https://quant.stackexchange.com/a/48908

In general, you don't want to build one arbitrage-free bid surface and one arbitrage-free ask surface, since such surfaces would have no practical use. As you mention, any hedging will involve both bid and ask.

In practice, people often use the mid call/put prices to build a single arbitrage-free surface. This is what you fit against, and then you make sure that the resulting prices are within the bid-ask spread, either by using appropriate weights in your least-squares fit, or by adding an explicit constraint. How to construct an arbitrage-free surface is detailed in Model-Free Stochastic Collocation for an Arbitrage-Free Implied Volatility, Part II.

Now what should be exactly the arbitrage-free forward? This is a little bit what you seem to wonder. It is a good question, which does not have any exact answer. If you fit on the mid, you would use the mid at-the-money implied forward. Note that in practice your $F^{bid}$ and $F^{ask}$ will depend on the strike as well...