Economics and similar, for the sleep-deprived
A subtle change has been made to the comments links, so they no longer pop up. Does this in any way help with the problem about comments not appearing on permalinked posts, readers?
Update: seemingly not
Update: Oh yeah!
Friday, July 21, 2006
This is why I am not an academic
Apparently the "Journal of Conflict Resolution" is a proper journal and it printed this. (via the CiF comments section).
Meanwhile in the world of option pricing ...
Basically, consider a standard hedge fund charging structure: 2% annual fee and 20% of performance above the high-water mark. One thing you might want to know is what the present value of this stream of performance fees is, out to perpetuity. On the face of it, this looks like it is going to be quite difficult to do, as it is a path-dependent process and in general path-dependent processes are a bugger to value. However, with a bit of ingenuity it isn't. (I apologise for the fact that from here on in, it is going to get a bit incomprehensible for people without a background in finance. I also apologise for just giving cookbook style instructions here rather than showing it rigorously - life is too short to do equations in HTML).
Basically, if you take a lattice approach, it all becomes incredibly easy. The pricing structure is a contingent claim written on the value of the underlying fund assets, so you convert the volatility of those assets to a risk-neutral probability of an up-step and a down-step and draw out (in your mind) the risk-neutral lattice for the underlying.
At every node of that lattice, you either get paid a performance fee or you don't. If you do, then you get paid the value of the underlying at that node, times 2%.
So, you can write another lattice for the payments you get at each node, if you get them at all. You then pick out a copy of this fantastic book and look up the "hitting time theorem". This is an extension of the Bertrand ballot theorem, which just tells you the probability that a discrete random walk will reach some distance d after time t, for the first time. This is exactly what you need to know at every node, because if that value has been reached before for the value of the underlying, then it isn't a high-water mark. So you multiply each node by its first-hit probability and Bob's your uncle.
Except of course he isn't, because if you actually do this, you find that the valuation goes to infinity. This is because as you go out to perpetuity, every time slice contributes a tiny bit of expected value to the overall valuation, and although this gets smaller with every step, it doesn't converge.
This is because, of course, we've ignored the fact that there is an absorbing barrier for the value of the underlying - if the fund goes to zero, it is gone and there will be no more payments. So you have to use the ballot theorem again, to calculate for every note what the probability is of crossing zero before you reach that level, and multiply by one minus that. Obviously the probability of going to zero first gets bigger the further out you go, and this means that the whole thing solves.
This is probably amazingly obvious and has been covered in the literature a million years ago but I felt rather clever when I did it.
this item posted by the management 7/21/2006 07:43:00 AM