jsl1


Total Posts: 127 
Joined: Sep 2005 


Hi
I am looking at the bounds for a variance swaption on equity. More specifically, I am looking for constraints imposed by option on realised variance.
Off the top of my head I can think of the following:
option on forward realised variance >= variance swaption
where the left hand side is the realised variance covering a period (T1,T2) while the variance swaption expiries at T1 and pays the max between the varswap starting at T1 and ending at T2. Both at the same strike.
Are there any other suggestions? Thanks. 




good morning
can you be a bit more clear in your wording ? I don't really understand what the payoff of your variance swaption is.
For this type of thing I would always have another look at the static replication work of Peter Carr and see if there is a super/sub hedge. 


jsl1


Total Posts: 127 
Joined: Sep 2005 


sorry for the confusion, please find below a formulaic description:
denote the realised variance from T to T+delta T as:
the option on realised variance is:
the variance swaption is:
where is the time T variance swap price expiring at T+delta T.
The option on realised var described above is an upper bound of the variance swaption. A lower bound is the intrinsic value of the variance swaption. I was wondering whether someone could think of something else.
Hope that clarifies my question. Thanks. 




it is an interesting question. thanks for the clarification, I'd have to think about it a little more. Intuitively I guess the strength of the relationship depends on the volvol / mean reversion of vol, and also on whether there is a two way market in variance. 



Please see Carr and Lee's paper: Hedging Variance Options on Continuous Semimartingales. The paper is a direct answer of your question. They have the formula for a forwardstart option of realized variance. Below I am copying the abstract:
We find robust modelfree hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forwardstarting variance options, by dynamically trading the underlying asset and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans. 


