Arbitrage Freedom of the IV surface

In practice we can only observe European option IVs, of a fixed maturity, at a finite set of strikes, K1, K2, . . . , Km. It is also the case that we can only obtain these skews for a finite set of maturities,T1, T2 ,... ,Tn.

Even if the strikes or maturities happened to be very close, the following criteria must be met in order for the surface to be arbitrage free.

 

1. For all maturities, all call spreads must be positive. 

Ivs freedom 1

 

2. An additional restriction on such spreads is that if we were to divide by the difference in strikes, we must have:

Ivs freedom 2

 

3. All Calendar spreads must be positive 

Ivs freedom 3

 

4. All butterfly spreads must be positive

Ivs freedom 4

 

The set of European options will be arbitrage free if all these conditions are met. We should concern ourselves that any model we use  to capture skew observes these conditions. The failure of a model’s calibration to meet these conditions is a solid criterion to reject such calibration. Any interpolation between the IVs of two consecutive strikes in the above set must also observe these conditions to be arbitrage free.

 

 

 

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