Hi all. We are kicking off a new paper on volatility foreknowledge. We simulate a price series (of a forward contract) with relatively constant volatility throughout, except for a massive price spike at the end. Thus the volatility of the whole series is about 3x the pre-spike vol. Then we compare the series given by the price of a Black76 call option using the post-spike vol, with the series of a replication call, i.e. taking the B76 option delta at each point in time, then cumulating the P/L obtained using that delta, *also* using the post-spike vol. 

It turns out (and this is fairly well known) that the error resulting from using the pre-spike vol is massive, but that the error is considerably reduced when we use the post-spike vol. Riccardo mentions the effect in his book on Volatility and Correlation (I have lost the reference but it's in my notes somewhere).

Question: is there any research on an analytic estimate of the residual error with foreknowledge?  Riccardo estimates the error using simulation techniques, but, from memory, provides no analytic estimate. 

I don't have access to a library at the moment for Covid reasons, any thoughts welcome.

If the volatility is a known deterministic function of time, [$]\sigma_t[$], then the replication error should tend to zero with the rebalancing step size. That's because the associated Black-Scholes model (and hedging) is exact (in the continuous-time limit). Of course, you have to hedge using use the correct model effective vol at each instant [$]t[$], which off-hand is [$]\sigma^{eff}(t) = \sqrt{\int_t^T \sigma^2_s \, ds / (T-t)}[$], where [$]T[$] is the expiration time. 

If your 'price spike' can be generated by a deterministic [$]\sigma_t[$], then the above should work, even assuming a discontinuous [$]\sigma_t[$]. But, if [$]\sigma_t[$] is not square-integrable, all bets are off.   

We hedge using a flat volatility throughout, rather than using a different effective vol at each rebalancing. Which sounds different to what you say above, am I right?

[EDIT] I see your book was published in 2016 - does it include an up to date literature review? In which case I might get it.

Thanks. I found it in the library. I would buy it but it's nearly $100 and I am retired!

That's good -- what library? (I keep track of those). 

British Library!

British Library!

I see -- thanks. Likely my earlier (year 2000) book, but if they really have my 2016 one (ISBN-13: 978-0967637211), I will be very glad to hear it!

I am afraid it is the 2000 version.