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natkins
Topic Author
Posts: 13
Joined: August 18th, 2005, 9:24 am

### Bartlett hedging under SABR model confusion

HiI'm getting very confused with the paper by Bruce Bartlett about hedging under the SABR model and am looking for guidance.I'm not sure when in the paper alpha is meant to be the initial value (i.e. at t=0 and therefore fixed) and when it is meant to be a stochastic variable.Is alpha in the formula form the original Hagan paper for the 'backbone' fixed? Or should this also be moved with the 'expected' correlation to the underlying just as Bartlett does for his delta calculation? Because if alpha is expected to move then surely the 'expected' ATM path will not look anything like the backbones in the paper.Any thoughts much aprreciated.

prodiptag
Posts: 124
Joined: September 12th, 2008, 4:41 pm

### Bartlett hedging under SABR model confusion

I never heard of this paper before, and now that I look in to it, am doubtful of its utility!!this paper appears to treat alpha as the initial value in the beginning and then somewhat lost it later. You should look at the original paper by Hagan et al instead. and infact the delta hedge proposed by this paper is somewhat covered by the original paper (see eqn 3.9 and 3.10, corresponding to Bartlett's eqn 6 and 14). And besides, in practice getting so much nitty gritty in to the exact equations for delta might not be worth all the effort. SABR usually generates a good fit/interpolation, but the market most probably does not follow the SABR dynamics with so much precision (if it does follow SABR at all in the 1st place!!). Best way to find which hedge equation works best is to back-test and gain some confidence in that formula

natkins
Topic Author
Posts: 13
Joined: August 18th, 2005, 9:24 am

### Bartlett hedging under SABR model confusion

Thanks.That's what i thought but this later paper does seem to be the accepted delta - Rebanato's LMM-SABR book seems to say this is the one to use so i'm reluctant to go against him

largeN
Posts: 5
Joined: January 4th, 2011, 6:35 am

### Bartlett hedging under SABR model confusion

Bartlett delta is appropriate, if SABR is consistent with the actual market.As Rebonato's book shows, you can minimize the variance of P&L from delta hedging.Bartlett's delta is also recommended by original authorsBartlett's paper is written by Patrik Hagan and Andrew Lesniewski.

natkins
Topic Author
Posts: 13
Joined: August 18th, 2005, 9:24 am

### Bartlett hedging under SABR model confusion

Thanks, but i'm still confused that the 'backbone' mentioned earlier in the book (where alpha seems to be considered fixed) will be nothing remotely similar to the actual path the ATM vol will take if you do start moving alpha.I was under the impression that alpha is actually alpha^ at t=0 and that the only variables are f and K. Surely the pricing equations 'know' what the correlation is (rho is included in all calcs) and hence we only need alpha at t=0. And alpha must only be changed if the vol does not move as expected regardless of what the underlying does.I know i must be wrong but i don't know why.

largeN
Posts: 5
Joined: January 4th, 2011, 6:35 am

### Bartlett hedging under SABR model confusion

"back bone" is a naive concept for capturing the ATM dynamics.But, "back bone" does not capture the ATM dynamics. if beta equal to 1 or \rho \lambda is large. Here, \lambda is the strength of vol of vol compared to the local volatility, which is Eq.(3.1b) in the original Hagan paper.As Bartletts shows, change of underlying accompanines the following change of alpha:\Delta \alpha = \rho \nu / f^\beta \Delta f Then, approximated ATM vol ( \alpha / f^(1-\beta)) changes as:\Delta ATM = \alpha f^(\beta-2) ( (\beta -1) + \rho \lambda) \Delta fThe first term is the "back bone".The second term is due to the change of volatility.Using Barlett's delta, the change of ATM is also hedged by the underlying.If the SAMR model is right, Bartlet's delta is appropriate.

elan
Posts: 351
Joined: April 30th, 2003, 3:47 pm

### Bartlett hedging under SABR model confusion

There is actually a good deal of empirical evidence that Bartlett's delta is appropriate: market vol moves correlate significantly with vol moves consistent with SABR (formula (12) in Bartlett's paper).Additionally, Bartlett proposes a scenario to calculate the option vega in a way that is consistent with SABR (formulas (16) and (17)), which leads to a new expression for the vega risk, formula (19). The latter formula contains a mistake (a term missing) and should read:Since SABR has two independent risk factors, Bartlett's delta and vega is a complete hedge consistent with SABR's dynamics.