Going to preface this question with an acknowledgement with how silly the ask is, but alas that is the working world; if anyone can share any ideas I'm all ears.
We're pricing an exotic option in risk neutral:
The option is any arbitrary structure that can be constructed from euro style options; I'm going to use a Call expiring at time T.
The holder of the option owns N notional of the option at inception.
Here is where the option takes a turn down wild lane:
At times $0 < t_{1}, t_{2}, .... t_{n} < T$ the holder will be assessed a 'fee' which will reduce the notional $N$ of the holder.
The fee at time t is $fee_{t} = constant \times \frac{Call(S_{t}, r, q, \sigma_{S_{t}/k ,T-t})}{Call(S_{0}, r, q, \sigma_{S_{0}/k ,T})}$.
The terminal payout to the holder is $(S_{T} - K)^{+} \times N(1-fee_{t_{1}})(1-fee_{t_2})...$
The option is clearly exotic because the terminal payout depends on the path of $S_{t}$ but also the path of $Call(S_{t}, r, q, \sigma_{S_{t}/k ,T-t})$.
The question I have is regarding valuation; because of the path dependent nature; I was going to use a local volatility model calibrated to the current volatility surface to generate paths of $S_{t}$.
To determine determine the fee assessment within risk neutral I was going to use the implied forward term structure of $q$ and $r$ from the implied yield curves.
The tricky component is $\sigma_{S_{t}/k ,T-t}$; at first blush I think I can derive risk neutral implied volatilities for a given strike ratio and term to maturity (in the future) using the 'model free' volatility equation provided by glasserman:
https://www0.gsb.columbia.edu/faculty/pglasserman/Other/ForwardFutureImpliedVol2011.pdf
Using the implied forward volatility to plug in to the Black-Scholes equation inside the risk neutral path to determine $fee_{t}$.
Is this a viable approach?
Thanks!