Over the years I have observed so many carelessly written academic papers that deserve some serious spanking that here I must yell out a few to vindicate the endless time spent chasing them down. Just an hour ago, I have been browsing through the following paper http://www2.imperial.ac.uk/~ajacquie/index_files/GJ%20-%20NoArbSVI.pdf on arbitrage freeness conditions for the so-called SVI model. Due to the mounting pressure at work to figure out a volatility extrapolation scheme, I can’t help googling academic papers on arbitrage free construction of volatility surface these days. This is one of the more promising articles I found. Without going into detail on what the main picture the paper is trying to present, I got bogged down by some detail in one of its lemmas: 2.1. Here it claims that the implied volatility surface w(K,t) is arbitrage free if and only if the implied variance is non-decreasing in t. The result should be true, since decreasing variance would imply negative volatility at some point in time, which would result in negative option price, a clear arbitrage. I am not able to formalize this into a proof however, since the relation between forward implied volatility and spot implied volatility is not yet clear to me. The proof in the paper however seems plainly false(of course I haven’t contacted the authors, why should I? It is courteous enough to bring their paper to public spotlight):
The tricky part is that the authors allow nonzero dividend rate. This could result in a forward being a supermartingale, instead of a submartingale as is the usual case in textbooks. The authors claim is a Martingale which was not very obvious to me at first but then I realized he is simply taking the dynamics to be
, where is the implied volatility for call option maturing at with strike . With this explicit parametric form the Martingale property is easy to verify.
So assuming this is true, and let be two strike levels at maturity respectively such that the moneynesses of the options are the same, i.e., . Now the authors invoke the following consequence of Jensen’s inequality:
for . This implies , where is the price of the call option struck at maturing at . This is all good but I don't see how this implies (which the authors then use to imply the time monotonicity of the implied variance by exploiting the monotonicity of the Black Scholes formula with respect to variance. In fact the implication holds only when , that is, when the dividend rate is superseded by the interest rate . However the claimed monotonicity must hold; just the proof is wrong.