Hi Lorenzo,
Thank you for joining the thread and sharing your experience.
But most of all, thank you for this book, which I find simply amazing.
Should you find some time to answer these follow-up questions, it would really help
Consider first a pure stoch vol model, calibrated either to the TS of VS vols, or ATMF vols, of vols of whichever moneyness you like.
Let's pick a short maturity (say 3m), a long maturity (say 5y) and say that our target levels are, for example:
- vol of the 3m ATMF vol = 100% + decay of the vol of ATMF vols with exponent 0.6
- correlation between 3m and 5Y ATMF vols = 50%
- correlation between the spot and the 3m ATMF vol = -80%
- correlation between the spot and the 5Y ATMF vol = -40%
Here at SG traders use a code of mine that generates in real time the corresponding model parameters. For the values I've just mentioned,with the Eurostoxx50 term structure of VS vols of 20 jan 17, I get sigma = 256%, theta = 17.8%, k1 = 8.64, k2 = 0.68, rho_XY = -35%, rho_SX = -70.3%, rho_SY = -11%. Sorry, I would have preferred to post a snapshot of the spreadsheet, but it's not possible to post images here.
The beauty of the 2-factor model is that it is not overparameterized with respect to our specs (2 implied vols of different maturities, their vols, correlations and correlations with the spot) and parameter generation is instantaneous.
I guess that when considering N > 2 implied vols of different maturities and their associated (co)-variances the model ends up being overparameterised - i.e. the optimisation problem to be solved becomes overconstrained. It would also be my guess that since the 2-factor forward variance model is a market model for a 1 dimensional set of instruments, it will have a hard time fitting the whole vanilla market (2 dimensional set of instruments), such that if I introduce target ATMF skew and curvature levels I will experience the same kind of overdetermination, which pushes us towards LSV as well.
Now let's move to the same 2-factor model in its LSV version. The constraint of calibration to the market smile places restrictions on the levels you're able to get for your physical levels, but you can still code the real-time optimization that generates them using the approximate formulae that I give in chapter XII of my book. I'm presently coding this up. What do you do once you have model parameters?
These are indeed the formulas I've used. Yet, here also I am willing to work with a full term structure of (i) (lognormal) vol of ATMF vol, (ii) spot/ATMF vol correlations (iii) skew-stickiness ratios -
this is to "correctly" embed the sizeable volga and vanna risks within my exotics prices - along with (iv) the whole vanilla volatility surface -
this is so that the prices of my hedge instruments are directly "built in" the model as well.
I extract the target levels for the physical quantities above from historical time series using the usual statistical estimators plus the one you describe in your book in the particular case of the skew-stickiness ratio. However it seems to lead to an overconstrained optimisation problem: it is straightforward to show that when fixing the vol of ATMF vols, their correlations with the spot price and the prevailing ATMF skew TS then the skew-stickiness ratio TS is unequivocally characterised, so this may partly explain why.
Maybe I was too optimistic and should get rid of some of these constraints and/or replace some by others to get to a better-determined optimisation problem hence a more robust calibration step. Alternatively, maybe I should work with 2 implied vols as you suggest and trust the model to generate the dynamics and consistently "fill the gaps" in between?
In the SV model, levels of vols/correlations of physical quantities have little dependence on the TS of vols that the model takes as input. Thus, you're set.
In the LSV model, on the other hand, with fixed model parameters, as the market smile changes your break-even levels for physical quantities change. So, keeping model parameters fixed, you have to feed the LSV model various smiles from past history and check that model-generated break-even levels hover around your desired target levels. Otherwise, generate model parameters using a different market smile.
When you say
different market smile do you mean some kind of synthetic smile which does not evolve through time? In that case, wouldn't that mean that the model is not fit to the vanilla market anymore?
Sticking with fixed model parameters to obtain a well-defined delta-vega hedging P&L over the instrument's life, what is your take on the frequency at which the SV parameters needs to be recalibrated (the local volatility component is recalibrated every day)?
I guess what I am naively looking for are how do you recommend to perform the calibration of a 2-factor mixed Bergomi model in practice.
Thanks a lot for your time and help and sorry for the profuse questions,
Kind regards