## necessary or sufficient conditions for no-arbitrage

In work I have had to work with extending volatility surface interpolation and extrapolation models to fit market data. One question I kept asking myself is, what exactly is the significance of some sort of dynamics underlying the volatility movement in time? As someone from a geometric background, my first intuition for surface fitting is always to take a geometrically natural object such as a minimal surface, satisfying some obvious arbitrage free conditions such as positivity. This can be easily accomplished if one transform the surface into log space. More explicitly, we look at the set of pairs $(t_i, K_i, \log v_i)$ where $v_i$ is the implied volatility at strike $K_i$ at maturity $t_i$, and simply find a minimal surface passing through all these points, that is, ensuring that the mean curvature $H =$ trace of the curvature operator (or the sum of the principal curvatures for the 3-d hypersurface) is zero. Here using logarithm to transform the target vol values is arbitrary, any function that maps the positive axis to the entire real line would do. But surely practitioners have given thought to this highly academic approach. One immediate objection I can see is that as time moves forward, certain “continuity” behavior is not captured by this naive model: e.g., if we look at 1 yr implied vol now, versus 9 month implied vol 3 months later, the minimal surface approach might not yield any connection between the two. What’s still not clear to me is a. how seriously one should take this intuitive term structure correlation, and b. how well do existing models such as Sabr capture such correlation compared to a non-dynamical approach. This I will certainly spend time investigating in the future. But while searching for related topic, I ran into a paper by Peter Carr and D.P. Madan on equivalent conditions for no-arbitrage in a set of market quotes. Since their paper is quite short I read it pretty quickly:
Their main proposal is to discrete time and strike, and confine the set of arbitrage strategies to so-called static arbitrages. This means essentially that the strategies at any moment can only depend on the prices of assets at that moment, i.e., a Markovian control. While the equivalence between arbitrage-freeness and existence of equivalent Martingale measure holds for arbitrary adapted strategies with mild and practical regularity assumptions, the problem of identifying arbitrage is much more difficult (especially in the continuous setting) if the strategy is allowed to depend on the past history of the asset prices, even though it’s not clear to me how knowing the past in addition to the present can exploit additional arbitrages. But the main contribution of the paper, leveraging off an exotic ancient paper by Kellerer written in German, is that restricting to Markovian strategies, absence of arbitrage is equivalent to absence of three specific types of arbitrages: call spread, butterfly spread, and calendar spread arbitrages.

Knowing the absence of these three specific types of arbs, the authors argue that similar statements hold for all convex payoff instruments via standard static replication approach. This then implies via Kellerer that there is a discrete time Markovian Martingale $M_t$ with which every quoted call price can be expressed as the expected value of the standard call payoff $E_t (M_t - K)_+$. If we then price every other instrument with this Martingale, then arbitrage-freeness is guaranteed.

The significance of results in this paper is that since there is a finite (and not too large) number of quotes available in the market at any given time, checking that the three types of arbs are absent among those quotes is fairly straightforward.