# 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.

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...
• For the forward rate, I am finding the rate $r$ that makes the forward look mostly constant over strikes, see quant.stackexchange.com/questions/48781/… . I then of course have some different forward rates for different strikes, I thus set the $F^{bid}(T) := \max_kF^{bid}$ and similarly minimum of ask forward for the ask, what do you think? I am trying to fit Gatheral’s SVI (which I still have trouble doing for short dated SPX). Do you sugest that paper because the method presented is industry standard? – raptor22 Sep 25 at 16:16
• What I have finally done is to estimate $r$ and borrowing + dividend yield $b + q$ through regression on the collar mid prices. I then use the derived rates to compute the mid forward rate $F^{mid} := S_0e^{(r-b-q)T}$ that will be consistent with the parameters that I use in BS. – raptor22 Oct 3 at 10:53