Forward Black Implied Volatility For Within Risk Neutral European Option Pricing

Question

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!

Answer 1

The 'model free forward implied volatility' is pretty useless for your purposes. First of all, it doesn't say anything about the price of future IVs, which you need, and worse it's pretty much unhedgeable.

So you can do two things: Given your model, calibrated to vanillas, price the exotic numerically. Here a word of 'warning' if I may: local volatility tends to underprice forward vol risk.

Another possibility, which is numerically less intensive and which could in theory be hedged, is to price the fees in terms of forward variance swap prices. These can be determined from vanilla prices alone, and can be hedged as well. It still will not be a perfect hedge because you determine the fees based on future IVs, and the future varswap is an average over all future implied variances, but it's at least somewhat of a hedge, and it's in theory hedgeable by trading (forward) varswaps.