standardization of greeks for a portfolio

[shashishekhar]

Hello everyone,



Suppose I have a barrier book where I use plain vanilla options for hedging part of the gamma and vega risk arising out of the barriers. My barriers would are be valued (and correspondingly greeks computed) using the vanna volga model. My vanillas would be valued (and corresponding greeks computed) using the Black-Scholes model.



My query is, shouldn't the greeks that are computed arise out of the same model? As long as I am pricing plain vanillas using B-S, shouldn't the greek computation happen using the same model that is being used to compute the greeks of the barrier options to remove the model bias? Correspondingly, wouldn't it be correct to price all products in a barrier book using the same model?

[aaron]

Yes and no.



Let me give a simpler example of this question. You hold a bond portfolio with EUR and USD bonds. You want to compute a duration. You use an EUR yield curve to compute the duration of your EUR bonds, and a USD yield curve to compute the duration of your USD bonds. How do you define the duration of your portfolio?



One idea is just to combine the durations, ignoring the currency difference. This answers the question, what will my portfolio do if USD and EUR yield curves both rise by one basis point in all maturities? Is that a useful thing to know? I would say yes, if the two bonds in both currencies have smiilar distributions of maturities and credit qualities. But if the USD portfolio is short a lot of T-Bills and the EUR portfolio consists of long-term high-yield bonds, the combined duration is worse than useless.



Another idea is to define one curve as your reference for the duration, say the EUR curve. Then you estimate a Beta, how much you expect the USD yield curve to move if the EUR curve moves up one basis point. In this case you take the EUR duration plus Beta times the USD duration.



Getting back to the original question, it's generally a mistake to insist on one model for all positions, for the same reason you wouldn't put EUR and USD bonds on the same yield curve. When you have different models, you have to decide whether to treat all Greeks equally, or to pick one model as a reference and adjust the Greeks from other models. For general purposes like setting limits, adding inconsistent Greeks together can work, as long as there isn't a lot of offset. For specific purposes like hedging, you'll have to either adjust or hedge separately.

[NeroTulip]

Book a trade between your barrier book and your vanilla book. Stick the best vanilla static hedge possible in the barrier book to reduce model risk. This transfers the vanilla risk to your vanilla book, where you can manage it with BS.