Hi. I have problems with time inhomogeneous models. E.g. Heston or SABR with time dependent parameters to capture the term structure or any short rate model with such parameters. To me that is overfitting and no dynamics is captured well. Can anybody recommend good paper where the hedging ratios derivation and some discussion about the hedging properties are well discussed? Many thanks!

I don't know of such a paper, but suspect if there was one it would begin like this:QuoteIn this paper, we show that the hedging performance of the Heston or SABR model with time-dependent parameters is worsethan both simple Black-Scholes model hedging and the same models with constant parameters...

Just googled that abstract, and nope, no survey paper like that. Still keep hope.

QuoteTo me that is overfitting and no dynamics is captured well.It is a sad fact that quants have banged on about this for ever, and been loudly ignored.It is a point of fact that people have been hedging with time dependent models for donkeys years (eg just within black-scholes,or local vol models)My conclusion is that for most products the dynamics are relatively small/unimportant. have a look at the first 8? pages of bergomi summer school.by vega hedging ( which basically requires a time dependent model to fit the option surface) you are only exposed to higher order termshow do you propose to both fit dynamics and option prices in Heston?

Yep, it's sad, and more sad is that, at least to my knowledge, I haven't seen this to be finally resolved, even in Bergomi series, aren't it? What confuses me exactly in the posed context of time inhomogeneous case is: OK, we need to find a good stoch. process for the dynamics of [spot(t), vol(t)] or [short rate(t)]. Fitting a vol surface (or an yield curve) gives the stoch. process marginal distributions at a discrete set of maturities. Fitting the increments gives the total law. The first fitting is calibration. Vanilas (bonds) are enough here. The second fitting is a calibration, if done to exotics, and estimation, if done to time series. In the first fitting, we have a nonlinear function of the stoch. process parameters made close to the market data. In the second one a second function of the stoch. process parameters is made close to the market data (or its history in estimation). It can happen that some parameters participate only in the first fitting others in the second (or have marginal weight in the first or second, e.g. Bergomi demonstrated the vol of vol does not in the first and does in the second, etc.). Now imagine that in this setting instead of clean parameters we put in the process time valued functions (parametrized or not). What happens is that we artificially make a brutal attempt to separate cleanly the parameters' control over marginals and increments. So the time valued function (or its parameters) will explain possibly only the marginals and any other parameters the increments. In such a context some people claim, e.g. Carmona and Nadtochiy, that we have a full overfitting and there are no other parameters left for a "free control" of increments. In general, I am fine with that. Small caveat: we have still some freedom both for adding other parameters or even in the ones in the time valued function, if parametrized. So they propose to have a forward setting by a good codebook process that starts at the fitted surface (yield curve) and then evolves dynamically. However, in my opinion that is an overfitting in the other direction, in the direction of putting all the parameters to govern the increments. Right? The truth should be in the middle but which is it?

QuoteOK, we need to find a good stoch. process for the dynamics of [spot(t), vol(t)] or [short rate(t)]I think bergomi's point is that you do NOT need a good process for the spot [so Time inhomogeneous parameters are even ok]... because you are gamma/vega hedged - what you care about is a good process for the implied volatility surface ( and even that is more for the more extreme exotics).That's why I was raising the issue that this has been done forever black scholes (timevarying vols)... This is clearly "totally wrong" from your viewpoint ( ie no one believes realised vol is going to go up and down according to the shape of implied vols)... yet the fact that it was being done forever should suggest to you that it is actually better than any other way ( and certainly better than trying to run all your exotics on a timehomogeneous vol)...The whole point of gamma(/vega hedging ) with vanilla option is precisely to remove sensitivity to the stock price process ( and replace wih sensitivity to a less volatile quantity the implied vols)maybe you want to try out some experiments to find out what the volatility and sensitivity of these quantities are ( ie what is typical stock price volatility, ATM vol and smile volatiltiy together with gamma/ vega/ vanna /volga of your chosen exotic ?so I don't really understand your last message .. but I believe you are not understanding that the true process we are approximating is [S(t), sigma_I(t;K,T)] - the dynamics of the stock process and the dynamics of implied vol - so QuoteFitting a vol surface gives the stoch. process marginal distributions at a discrete set of maturities.is wrong... it is simply giving us the the spot and implied vol at time t. Using say a heston model is just way of getting consistent dynamics for [S(t), sigma_I(t;K,T)] (because as you will have seen from Carmona and Nadotchiy- the direct way is intractable]