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### swaptions..

Posted: January 22nd, 2011, 7:26 pm
so black scholes is clearly the wrong dynamic - so we would use a stochastic vol model. for rates (atleast vanillas) I have heard SABR is the preferred choice.Now because the vol and forward rates are modelled with a correlation this means that as rates move by a basis point there will be an impact on vol etc.Assuming I calibrate my model successfully I can use it to risk manage my option book.So in the case of 'delta' from SABR we have that it will include black delta + a correction term because as the forward moves by 1 basis point so will vol. Does anyone know a formula or how to derive this SABR delta and how it relates to black delta...?From work I have heard that this is called normalised delta and is a much more preferred way to think about the sensitivity to forward swap moves than the classical black delta as it also includes the fact that vol will change when the forward changes. From what I know vol*Forward is the annualized break even movement in the forward rate to make the option worth its premium (i.e. so you can gamma scalp to break even). So another question I have is that normalised delta is the preferred choice this means that my delta also include a bit of black vega, then I consider my normalised vega as black vega - something.. what is something..I understand academically my question doesn't have a point my practically I have seen this concept used a lot in swaption trading etc..people are always talking about normalised deltas, normalised moves etc..I understand normalised moves but what is normalised delta how do i get this..?

### swaptions..

Posted: January 22nd, 2011, 7:43 pm

### swaptions..

Posted: January 22nd, 2011, 7:57 pm
Thanks - I have read this paper briefly in the past - he doesn't really ever talk about anything practical about using SABR he numerically uses it to hedge a portfolio (maybe I missed his point)..In practice if norm vol = black vol * F then my delta would be the change in PVs when F -> F + 1, therefore to keep normalised vol constant I have to assume that black vol new = black vol *F/(F+1).This means that due to the 1bp change in the forward I earn a delta PnL of, black delta * 1bp + (black vol new - black vol) * vega is this my normalised delta? (Further I understand that my delta is now parameterised on the forward Curve corresponding to F - so we would have to reproject this in some funky way onto the spot swap curve - lets ignore this for the moment).Therefore, so far delta PnL = black delta * 1bp + (black vol new - black vol)*vega of that forward point. this can be simplified to [black delta * 1bp -(black vol/(F+1)) * vega] is this correct or am I missing something here, if so, what?Assuming the above is all correct by some stroke of luck then my problem as I mentioned above would be to reproject my black delta with respect to the forward swap back onto my swap curve - any ideas how to do this...?Thanks,

### swaptions..

Posted: January 22nd, 2011, 9:17 pm
Hello, first of all, normal vol is F*black vol only for ATM strikes. For everything else you need to use a more exact approximation (e.g. the one by Hagan), but for close to ATM strikes, sqrt(F*K)*black vol is acceptable as well.Secondly, if you want to keep normal vol constant, as you state in your example, then you should use a normal model for hedging. Thirdly, if you insist on using a lognormal model which requires a smile to correctly price the market, then you should really use a stochastic volatility model, e.g. SABR, which then gives you a much better term for what you call "(black vol new - black vol)*vega".On a related note, I think you may be confusing the meaning of the smile for hedging. The smile roughly represents the average of the local vols up to expiry. It does not represent the local vols themselves. So there is no need for what you call "new black vol" to be equal to blackvol*F(F+1). In fact, if the market implied vol is correct (in the sense of mkt efficiency), then you can hedge keeping this vol constant until expiry.Also, I dont understand why you mention spot rates. They dont enter at any point (exept for building the curve that you need to interpolate the fwd rate). What do you mean by projecting back to the swap curve?

### swaptions..

Posted: January 22nd, 2011, 9:30 pm
QuoteOriginally posted by: DocTocAssuming the above is all correct by some stroke of luck then my problem as I mentioned above would be to reproject my black delta with respect to the forward swap back onto my swap curve - any ideas how to do this...?Thanks,Are you perchance talking about converting your analytical delta to bucketed par risk or something like that?

### swaptions..

Posted: January 23rd, 2011, 11:11 am
Yes, after what I said I'd have my 10y10y swaptions risk on a forward curve - I need to re-bucket this into "par risk" so my 10y10y payer should have some delta in the 10y and 20y buckets on my spot curve.

### swaptions..

Posted: January 23rd, 2011, 11:17 am
Hi GMike thanks for your reply.I understand what you say about using a stochastic vol model for the term : (black vol new -...), the SABR paper is well written but doesn't really go into any detail in terms of hedging etc. with it. Have you got a reference as to where this is explained in depth?Thanks,

### swaptions..

Posted: January 23rd, 2011, 11:54 am
QuoteOriginally posted by: DocTocYes, after what I said I'd have my 10y10y swaptions risk on a forward curve - I need to re-bucket this into "par risk" so my 10y10y payer should have some delta in the 10y and 20y buckets on my spot curve.Well, given you have your model, why don't you do this "in simulation", i.e. calculate your bucketed delta by actually bumping the curve? You'll have all sorts of choices to make about where to bucket various risks, but that's pretty much semantics.