Hello All Does anyone know how to implement static hedging for exotic options i.e. barrier? Any good papers about this topic? Many thx!

google Peter Carr, he has a number of papers on the subject...

You need this...http://www.esnips.com/doc/89b431d2-e2a4 ... atelHere's the abstract:This project discusses two methods for obtaining static replicatingportfolios for barrier options. The first method discussed is themethod of (Carr & Chou 1997a) and the second is the methodof (Derman, Ergener & Kani 1995). The methods are tackledfrom both a theoretical point of view as well as from a practicalimplementation point of view. Hence, Matlab code has also beenprovided implementing these methods. The inputs and outputs ofthis code is also discussed. The type of barrier options dealt within this project are vanilla barrier options. That is, your basic up-and-(out/in) and down-and-(out/in), constant-barrier, standardEuropean put and call options.

I despair! The problem with the methods that are being advocated here is that they do not have a totally sound rationale behind static hedging. The result can be that you think you have hedged model risk but all you may have done is hide it, and therefore made it more dangerous. Look at these: http://ideas.repec.org/a/taf/apmtfi/v6y1999i1p1-18.html and http://www.wilmott.com/detail.cfm?articleID=245 as well.P

Paul, I agree with your point that static hedges make you think that you have hedged (some) model risk.A specific example would be the hedge for a sale of one touch on a forward price with a constant lower barrier and payment at expiry.Then in the Black model, the static hedge can be shown to be to buy 2 binary puts and sell one vanilla put with everything struck at the barrierand everything maturing with the one touch. Thus, the net payoff at the maturity date T from the static hedge if it were held to expiry would be 2 * 1(F_T<L) - (L-F_T)^+ where the lower barrier L<F_0. It can be shown that this same static hedge works perfectly in the model if the sigma parameter in the Black model is generalized into any stochastic process for the instantaneous volatility, so long as that process evolves independently of the Brownian motion that Black assumed drives forward prices. Importantly, the dynamics of this stochastic volatility process do not need to be known and this is why one has mitigated some model risk. In particular, one does not need to know the market price of vol risk. Other model risks remaineg. jumps in the forward price over the barrier make the static hedge imperfect. Could you pls clarify your point as to why putting this static hedge on makes you think that you may have hidden the risk of incorrectly specifying an independent stochastic process for instantaneous volatility?

Well put! But I've always said you should think of a contract as exotic only if you cannot perfectly statically hedge with vanillas! (That defines exotic to me...and that's my get-out-of-jail-free card in these arguments!) That said, if you can find a model-free perfect hedge then you must take that into account otherwise there is a trivial arbitrage. (As I say in my blog you must take trivial arbitrages seriously, but not necessarily the highly model-dependent arbitrages.)So I don't like the sort of static hedging that comes as an afterthought (when one realises one has some nasty model risk) and which doesn't have any guarantees about what risk, if any, you have got rid of. (It's like the Michael Moore film 'Sicko,' you think you are insured/hedged but it turns out you aren't!)P

> Could you pls clarify your point as to why putting this static hedge on makes you think that you may have hidden the risk of incorrectly specifying an independent stochastic process for instantaneous volatility?PutorCall, your static hedge rests on three assumptions:1) the barrier is imposed on the forward price, not on the spot price2) no price jumps3) no correlation between the spot and the volAs we know, none of these assumptions holds in practice, so if you use this static hedge for practical trading you misspecify the vol and gap risks

I agree that the static hedge follows from those 3 sufficient conditions, none of which hold in practice.When one or more sufficient conditions are violated, the static hedge made or may not work in theory. It does not necessarily follow that violating a sufficient condition leads to vol risk or gap risk - you have to both define these risks and show that they occur. But even if you can do that and I think we both can, what is superior to the hedge I proposed,either theoretically or in practice?

Paul, Thanks for your reply which I agree with completely. I know of institutionswho think they have hedged exotics because they put in place static hedges that neutralize all greeks out to second order. These kinds of static hedges sound like the kind of afterthought you are referring to.

> I agree that the static hedge follows from those 3 sufficient conditions, none of which hold in practice.Moreover, as I recall for this hedge you have to trade with bets struck at K=barrier and K=barrier^2. Since these strike are not necessarily traded on the market, to explain changes in your hedge portfolio p&l you have to map your position to the implied vol skew, this mapping is far from being "model-independent".> It does not necessarily follow that violating a sufficient condition leads to vol risk or gap risk - you have to both define these risks and show that they occur. Gap risk occurs when the barrier is hit by a price-jump so that you have a mismatch between your hedges and pay-off. Vol risk occurs if the skew flattens or steepens - so that MTM change of your hedges does not really explain MTM change of the barrier option.Both do occur.> But even if you can do that and I think we both can, what is superior to the hedge I proposed, either theoretically or in practice? Robust implementation of a model with price jumps and possibly stochastic vol.

As I wrote -- for the one touch hedge, everything is at the same strike, which is thebarrier of the one touch. With respect to my question on a superior hedge, you gave me alternative dynamics but no mention of a hedge, so I'm compelled to repeat my question.

Calibarate your model to the skew, price your barrier.Blip some points on the skew -> recalibrate your model -> reprice your barrierBy inverting the Jacobian of these sensitivities, you get how much vanillas to buy at these point on the skew - this is your hedging portfolio.Although theoretically it does not sound robust, I don't see any other way than that.

seppar,but that's the 'afterthought' (PC's word, I usually say 'fudge') that's inconsistent with the model and also misleading in that you (or let's say, some people!) will think they've removed risk. P

Paul,you can also hedge off the model. It depends how much you trust your model, how much liquid are hedge instruments, how well it explains the actual world, how much you are risk-averse etc. In the best case, the hedges implied by the model parameter are the same as the skew-consistent hedges (those that are obtained by inverting the Jacobian).

Seppar Thanks for answering how you would specify your hedging portfolio.I have three questions about it:1) Is it static i.e. would the number of vanillas initially obtained by this method also be the number of vanillas held afterwards?2) In what sense is it a hedge? I think your model value of the vanilla portfolio will immediately deviate from your model value of the barrier option if the initial movement of the skew is anything other than the blip you used to compute the portfolio weights, which I take to occur with probability one. Even if you rule out moves of any size other than your blip, then if your vanilla portfolio is static,its model value will change in a different way than your model value of the barrier option onall subsequent moves. 3) Do you know of any study using either data or simulations, that finds that your method works better than well known alternatives such as delta hedging in the Black model or the static hedge that I proposed?