Please, could you explain this point in a bit quantitative way? I don't think we need stochastic calculus, but at least the hedge ratio written as function of Vanna. Thank you
Not applicable. The vanna will increase as spot diverges from your neutral strike. You don't need to know change per se, but should model +/- on the vol-line at each hedge trigger. IOW you need to be familiar with vol-smile. You'll want to shave deltas as the index declines and vice-versa. You need to calc wing-sensitivity if you're trading a flat-smile (like most blue-chips).
How is it wrong? Whatever vol you hedge an option at, you should theoretically have the same expected value. This of course in theory which is based on the black-schole model assumptions around volatility being static, no transaction costs, and log normal distribution with drift = risk free rate, etc. If it were any other way then you could create synthetic options via hedging at different vols and make arbitrage.
Yes it does. But if you and I are trading the same option and we model it with different parameters we will get different hedge ratios. Our expected value is the same, but our pnls can be different because of the evolution of the stock price and the variance of our pnl will be drastically different. You are saying that someone who buys an OTM call and doesn't hedge it has a different expected value than someone who has hedged it? One guy is effectively saying the vol is zero and the other is not.
The midpoint of the OTM call market (vola) is representative of the expectancy under a perfect and continuous hedge under a flat surface. The transactional expectancy in aggregate should be zero-sum if your modeled vol = realized and your can hedge continuously and vol is static. BSM assumes static vol and therefore vanna is zero under the context of expectancy. It's not zero under a stoch-model or if you simply disassociate the concept of static vol and the current vol of the option in question when the position is hedged.
That is true, but how does your using a stochastic vol model change your expected value vs my using a static vol model? If you use a stochastic vol model you might get a better hedge and thus a lower variance of pnl; but it should not affect your expected value.
It doesn't. Use BSM and accept the new vol-figure that is modeled at the time of the hedge. It's (more so) representative of the true expectancy. Are you stating that the assumption of static vol does not alter the expectancy? Intermediate vols under BSM and stoch will be different even if they are equivalent at the inception of the trade. "Inception expectancy" is really a flat-Earth argument. We both know that static vol and continuous hedge assumptions are bullshit. Your expectancy argument assumes a perfect hedge and static vol. You cannot simply toss it out by stating expectancy is equivalent and model-independent when hedging is critical to the model. Expectancy and hedging size, freq, etc., are inextricably linked and BSM assumes zero-convexity.
The house is short gamma. They cannot lose as a group, because the house is still there and was there. Who is then the sucker (as a group)? If you get an edge on short gamma, then I would listen.