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 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.


About aquazorcarson

math PhD at Stanford, studying probability
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