# Model based PnL explain for FX Options

In FX options the vol surface for a given maturity is usually described by three or five points, I.e. Atm, 25 delta risk reversal and butterfly and 10 delta risk reversal and butterfly. Then models like Vanna Volga, Sabr or stochastic (local) volatility models are calibrated to these points and then used to inter- and extrapolate the vol surface. I'm interested in how people can get a PnL explain when using these models. In the black Scholes world everything is simple, I.e. you look at changes in implied vol for a particular strike K and can compute your vanna and volga pnl based on this. However, this means that essentially you are using a different model for each strike, since vol spot correlation and vol of vol are not uniform across strikes in practice. I would prefer to have a PnL explain based on the observables, e.g. Atm, rr-25 and bf-25, but then I need to use a vol model. How is this usually done in practice? As a start I took a vanna volga model, calibrated to atm, rr-25 and bf-25 and then used finite differences to compute first and second order greeks with respect to spot and the three observables. This worked fairly well for the PnL explain of Atm, but for the 25 delta options my Pnl explain was far off what the pricer delivered and what I got though a simple black Scholes pnl explain. My pnl explain was drifting away from the one obtained by the pricer. I'm not sure why. My vanna volga model is only calibrated up to 25 delta and the pricer would use the 10 deltas as well, but actually just looking at the vol smile, the pricer and my model are very close in a fairly large interval around the 25 deltas. Would I need to go to even higher order greeks? The thing is 2nd order is a great approximation for black Scholes, so why would I need even higher order for my model? I suppose vanna volga is not a good model for this purpose. Maybe SABR would be better as it captures the dynamics better. Would my approach be valid for sabr too, I.e. calibrating the model to the market quotes and the bumping these quotes and recalibrating sabr to obtain finite difference greeks? I'm sure people are looking at this kind of pnl explain in banks or hedge funds, so I would appreciate any comments on how this is usually done. Thanks!

• How do you do your FD exactly? It can make a big difference if you use central difference or forward difference the further away from ATM as shown here (specifically the second figure for OTM). Commented Aug 9, 2022 at 15:48