## mathematical errata and BS detector

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 $F_t = E(S_t | \mathcal{F}_0)$ being a supermartingale, instead of a submartingale as is the usual case in textbooks. The authors claim $S_t / F_t$ is a Martingale which was not very obvious to me at first but then I realized he is simply taking the dynamics to be

$dS_t = S_t(\delta_t dt + \sigma dW_t)$, where $\sigma =\sigma(K,t)$ is the implied volatility for call option maturing at $t$ with strike $K$. With this explicit parametric form the Martingale property is easy to verify.
So assuming this is true, and let $K_1, K_2$ be two strike levels at maturity $t_1, t_2$ respectively such that the moneynesses of the options are the same, i.e., $K_1 / F_{t_1} = K_2 / F_{t_2}$. Now the authors invoke the following consequence of Jensen’s inequality:
$E((S_{t_2}/F_{t_2} - K_2/F_{t_2})^+) \ge E((S_{t_1}/F_{t_1} - K_1 / F_{t_1})^+)$
for $t_1 < t_2$. This implies $F_{t_1}^{-1} C_1 \le F_{t_2}^{-1} C_2$, where $C_1 := E(S_{t_1} - K_1)^+$ is the price of the call option struck at $K_1$ maturing at $t_1$. This is all good but I don't see how this implies $C_2 \ge C_1$ (which the authors then use to imply the time monotonicity of the implied variance $v(K,t):=\sigma(K,t)^2 t$ by exploiting the monotonicity of the Black Scholes formula with respect to variance. In fact the implication holds only when $F_{t_2} \ge F_{t_1}$, that is, when the dividend rate $\delta_t$ is superseded by the interest rate $r_t$. However the claimed monotonicity must hold; just the proof is wrong.