I'm running through some delta hedging simulations of floating strike lookback call options (that is, I'm short the options) during a volatile (downside) period for the underlying and some very odd things have resulted.
First of all, these options have no Gamma, at least not by the standard understanding of gamma being $$\frac{\partial^2 C}{\partial S^2}$$
I've obtained Gamma values both by a finite difference approximation and by partially differentiating the analytical solution (messy!), and both come to 0. However, the delta values are very volatile, and the hedge is getting crushed. So, a few thoughts I had from drawing on analogies from vanilla calls.
1) If call deltas are known to underestimate the price increase in up moves and overestimate the loss in down moves, then being short delta in a down market should be good.
2) Again, there is no Gamma, at least by the "traditional" measure, or whatever you want to call it. However, in a volatile market, being short gamma (by being short either a call or put) is bad.
3) For vanillas, we know that Gamma is high for at-the-money options.
Now, this is where things get really weird in my head. Given that, as mentioned, I'm hedging short calls in a down market, the structure of the floating strike lookback option means that basically the strike is getting continually reset to the current value of the underlying. That is, the options are basically always at-the-money during this period. So, I have options with a volatile delta that are at-the-money. So in effect, the hedge is suffering for the same reason that a short call hedge might suffer by being short Gamma, yet I'm not short Gamma...!?
Can anyone clarify this for me, and perhaps lead me to some literature about hedging floating lookback options? It seems to be unfortunately sparse.
Thank you.