Suppose that the Black Scholes model is correct but a hedger only rebalances discretely in time. More precisely he or she tradesonly at equally spaced time points and trades so as to hold N(d_1) shares at each rebalancing date. We know that there will be replication error due to the discrete rebalancing.I'm interested in how the standard deviation SD of this hedging error is related to the length of the rebalancing interval(call it h). Derman and Kamal have an article in Risk which uses simulations to indicate that the SD is O(sqrt(h)). Thus one must quadruple the trading frequency to halve the hedging error.This result seems very depressing. Could the result be incorrect? Does anyone know of a derivation relating the SD of the hedge error to the trading horizon h? Please don't refer me to offline papers.thx

Paul's book "Theory & Practice of Financial Engg.," Chp. 20, derives the results. Mentions that original work on the subject by Boyle & Emanuel. Oops....glossed over the part that you want online references.....

On first glance, it wouldn't be surprising to me that the error is O(sqrt(h)), by just thinking of the Ito Isometry.

PutOrCallYou need to set transaction costs against your rebalancing time. In other words, a longer re-balancing time will lead to an increased hedge error, but also to lower transaction costs. This has two implications:1. Balancing transaction costs against re-hedge interval leads to an optimal re-hedging strategy. There is an excellent discussion of this whole topic in Paul's book. I really strongly suggest you get hold of the relevant chapter ... or if you can't get it, read the seminal paper by Hodges & Neuberger.2. Any derivatives trader needs to learn to structure their trading book in such a way as to reduce the size of this problem. A trivial example: if you buy a call you can dynamically hedge it, but this leads to hedging error and transaction costs. It's better to sell a similar call against it and hedge the residual. This will lead to smaller hedging error AND smaller transaction costs. i.e. it's a strictly dominating strategy.The subject you raise is of real practical interest to traders. I really recommend you read the chapter in Paul's book if you can possibly beg, borrow or steal a copy.

I agree that transactions costs are realbut I'd like to understand the effects of discrete time rebalancing before introducing them. I'm embarrassed to admit that I don't have a copy of Paul's book handy, but I took a look at Boyle and Emmanuel which is online if you have Science Direct.Conditioning on the stock price at a future time t, they show that the LOCAL hedge error is a deterministic constant lambda times (squared std normal - 1) times h,where h is the length of the rebalancing interval.The constant lambda depends on the calendar time t and the stock price S that we conditioned on in a nontrivial way.I'm interested in summing these local errors over the life of the hedge, taking into account the randomness of stock prices we conditioned on and thendetermining how the standard deviation of this sum depends on h. For those who have read this far, here's the nontrivial relation between the constant lambda and the calendar time tand the stock price S conditioned on:lambda(S,t) = constant S N'(d1(S,t))/sqrt(T-t)Successive hedge errors are not independent,hence, I'll be impressed if Paul or anybody cananalytically relate the standard deviation of the sum (under zero tranasactions costs) to h, the length of the rebalancing interval.The simulations in Derman and Kamal suggest that the relationship between local hedging error and h is of the same order as the relationship between global hedging error and h i.e. error = O(sqrt(h))As I said earlier, if this is correct, I personally regard it as a damaging indictment on the efficacy of this form of hedging.

well I've done this and I find that the variance is like c \Delta t . which is the same as the sd being O(sqrt(\Delta) t). the interesting thing is that the constant is small. So whilst to get microsopic error is hard, to get small error is easy. And this is whyDelta hedging is popular. MJ

I found a calculation related to the one I was looking for in the published paper by Bertsimas Kogan and Lo available athttp://web.mit.edu/alo/www/Papers/contin.pdfOn page 183 Th2 they show that the RMSE is O(sqrt(h)) as everybody claimsThey also give a closed form expression for the constant on the leading error term inon page 185 Th3 eqn 3.2However their results are asymtptotic i.e as h ->0.For h bounded away from zero, they do Monte carlo simulations like everybody else.So MJ's calculation would be new to my knowledge.Can you make this public?

I did MC too....MJ

For plain vanilla calls the std.dev. of the P/L does indeed drecrease at 1/sqrt(#hedgeppoints).The convergence order depends critically on the smoothness of the pay-off funtion (so it'snot a "CLT-\sqrt"). Juding from the thread, this "well-known in the community".Proofs (of a hard-core probability-theory nature) are in "Discrete time hedging errors for options with irregular pay-offs." Finance and Stochastics, Vol.5(3), pp.357-367 - 2001 (you can download it from Gobet's homepage). This probably isn't the only article showing such results. Cheers,Rolf

PutOrCallIt sounds as if you've more or less got the answers you were looking for. To summarise, the hedge error has mean zero and standard deviation O(h^0.5). The distribution is chi squared with a single degree of freedom. The practical consequences of this are that, for a long gamma position, you would expect to lose money from hedge error 2 days out of 3, but to make enough money on the third day to compensate.However, it's a pointless abstraction to consider discrete hedging in a world with no transaction costs: there would be no reason to hedge discretely. This discussion is really only worthwhile in a world with transaction costs, or jumps, or some other reason for not hedging discretely.

Thanks to all of you for pointing out relevant references.Regarding why one would trade discretely in the absence of explicit transactions costs,let me just point out that equity markets close every night.In a market without other frictions, this can be regarded as equivalent to transactions costs depending on time and being zero when markets are open and an implicit transaction cost ofinfinity when markets are closed.As an approximation to this, we might consider zero transactions costs only on the integers and infinite transactions costs otherwise i.e. discrete rebalancing.bestp

That's as good a reason as any other!

So far the discussion involves the case where the assumed vol for discrete hedging and realized vol were the same. Does anyone know how the shape of the P&L is affected when the hedging vol does not match the realized vol? I suspect that the P&L should become skewned with positve skew if the realized vol is lower than the hedging vol, and negatively skewed if the realized vol is larger than the hedgin vol.- mayaro

when hedging with the wrong vol I got the variance to bea+ b \Delta twith a >0MJ

Thanks M.J., That answers the variance question, but I suspect that the disitrbution should also be skewed - do you have a parametric form for this as well? I am assuming that your results were based on simulations - or is there an analytical closed form for the finite hedging case in addition to the asymptotics as n->infinity?- mayaro