November 26th, 2008, 4:03 pm
In practice, you don't want to go much further than Leland in closed-form or something similarly simple. Otherwise, you run the risk of thinking you know more about the price process than you do, which can be extremely dangerous. Any result about an optimal hedging strategy (and so transactions costs), even if you can find one, is likely to be quite sensitive to your payoff as well as the underlying process. "Keep it simple" is best.That's why, if you know roughly how your trader is hedging, you're probably best off simulating this, as it should give you a result which is close to the actual outcome. If you can find a better hedging strategy, of course, your trader will thank you! The problem with optimal hedging strategies (with imperfect replication or transactions costs) is that they will depend on your risk-aversion as well as things like your cost of capital and margin costs. These things are brutally difficult to model so, again, simple is best.In particular, worrying about whether your PL distribution is chi-squared or not is probably a bad idea: empirically, your price process probably doesn't even have a finite (or stable) second moment. This is one of those cases where overmodeling is probably not a good idea.