We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. show that when underlying stock price martingale admits finite log-moments E[|log (S)|^q] for some positive q, arbitrage-free growth in left wing of implied volatility smile is less constrained than Lee's bound. The result rationalised a market trading discretely monitored variance swaps wherein payoff function sq...

We revisit the foundational Moment Formula proved by Roger Lee fifteen years ago. show that in absence of arbitrage, if underlying stock price at time T admits finite log-moments E [ | log S q ] $\mathbb {E}[|\log S_T|^q]$ for some positive q, arbitrage-free growth left wing implied volatility smile is less constrained than Lee's bound. The result rationalized a market trading discretely monito...

Machine learning has been a popular option implied volatility pricing approach. It brings good generalization in by avoiding building different models for options. However, it suffers from relatively low prediction accuracy besides model selection issue. In this study, we propose novel hierarchical approach to enhance machine pricing. is designed the ‘learning-hard’ problem and boosts models’ p...