To hedge/profit from a black swan move (index down 30%) jump, is there a way to quantify the strikes given the scenario above and VIX is trading >60, possibly at 100? I know that where delta = 25, that is the steepest part of the gamma and vega slope, peaking at ATM. Am I also right in saying this is where volga and speed is highest given the tangent line to the slope and concavity? If so, would buying a load of 1-5 puts vs 25 delta puts at the same dollar amount be a better P&L hedge/profit? I would like to see whether rate of change is higher for one vs other but I don't know what the IVs will be and how steep the skew gets when the move comes. Any way to extrapolate this? BTW, is there any retail software that will calculate 2nd and 3rd order derivatives and modify risk graphs for their inputs? Actually it doesn't have to be retail geared, price won't be a concern as a poor hedge will cost more. Search has yielded some results, but nothing conclusive.

Since you are invoking concavity , just give Nassim a call (no pun) , I'm sure he's worked all this boring stuff out long ago.

jj , all good questions ; hope atticus is around to answer,but...I would be more concern/planning of how to lock the profits if the above scenario happens. I think bid/ask spread will become huge and can erase half of theoretical/paper profit. Not so long ago GOOG collapsed 10-15% on intraday news and bid/ask spread was 5 (!) $.

try the eurex margin calculator, it is free http://www.eurexchange.com/clearing/risk/margin_calculator_en.html

You might try posting this question on www.riskdoctor.com and see what Charles has to say. The other option is to model this scenario and see which gives you the bigger bang for your buck. db

All right, I couldn't resist . Using option basics, I can tell you now that you're going to make a lot more money from your far otm (read delta 5 or less) puts than your 25 delta puts assuming you allocate the same dollar value to each position. Why? Because you will be long a lot more negative deltas with the far otm puts. Add to this a spike in iv with your black swan event and your profits will be very large. It doesn't matter that the slope is steepest at the 25 delta because your spot will move very fast past that point and your original 5 delta puts will be deep itm with deltas near 1. I don't think you need to worry about volga (aka vega gamma). db

there is no need to spend ours computing what are best best strikes for that setup because all calculations will have to be based on some assumption of where the price might go when the black swan scenario takes place and thus they are flawed or at least biased. in addition, the market makers have already made that calculation for you, which is shown in the iv skew. since the 87 episode that the black swan is pretty much priced in across the board. back in 87 many institutional option sellers went bust and the industry evolved to become extremely efficient when it comes to pricing options. the mm's have done good job pricing all anomalies such as interest rate surprise moves, potential take over announcements or sudden changes in dividend policies. to answer you question, i would say that any put that you decide to buy will have the same expectancy, that is, prob.success*potential_reward = prob.loss*potential_loss= zero. if you're just looking for the best bang for the buck and thus neglecting the fact that your expectancy is zero, then it is evident that you should go on and buy the cheapest puts available but dont expect more than with any lottery ticket.

Thanks guys for the responses. The reason I asked a general question is that it pertains to a more specific scenario in my trading. I've been trading put backspreads on the majors with good success, but to do it with any kind of credit I make the longs pretty far out as the shorts are already far out. Example would be ES -1 AUG 1350 put +2 1250 put. I do this with decent size for the %return I aim for. If the ES blows past my long strike I have no worries, but it's when it settles in the middle that I am concerned. Obviously with the choice of my strikes an instant move to the short strike alone would be considered a crash, and IV will explode. In this case I have determined that volga and speed(rate of change of gamma) is what will save me. However to input IV numbers into a model to calculate P&L I will need to know IV and skew steepness, something that I won't know until it happens. Hence the reason for asking the original question, would longs at the lowest part of the slope be a better hedge as they can climb the entire slope(most acceleration) or longs at the steepest part of the slope(fastest acceleration) for fastest gains be the better P&L hedge? I am aware that the projected move will be part of the answer, but since we don't know how big the black swan move is, how do we determine? Or is that the million dollar question?

If you're asking how to hedge or profit from a true Black Swan event, then don't bother with any type of credit spreads or any position involving shorts. They will get annihilated even in a 15% decline, nevermind 30%. Other than avoiding the market completely, the only way to protect yourself is with ultra-deep long otm puts. You can become an instant millionare if you own 1200 SPX puts or 1500 NDX puts when the event happens. ES doesn't go down to 1200 except on the quarterly futures options, but regardless, if you're long it's better to be long with index options because commissions are cheaper and the margin treatment is the same.