Hi Mark

You claim that value of the cliquet (at least for the example you give) at time T_1 is (approx) 0.4\sigma \sqrt(T) where T = T_2 – T_1 and \sigma is the implied volatility observable at time T_1, for options expiring at time T_2.

1) More precisely, is \sigma not the implied volatility observed at T_1 for an option struck at S_{T_1} , expiring at time T_2 (i.e. the value of sigma is dependent on S_{T_1}?

2) Then there is the issue of pricing this, which you seem to suggest should be straightforward. My understanding is as follows: If we use a deterministic smile model, using the definition you give in chapter 10, we can deduce the price of an option (using the assumed model) struck at S_{T_1} expiring at time T_2. I think we then use this price to back out the implied volatility for said option; this will clearly depend on S_{T_1}, so I denote it by \sigma(S_{T_1}). We therefore know the value of the cliquet at time T_1 as a function of S_1,is 0.4\sigma(S_{T_1})\sqrt(T) and we can price this just by treating it as a European option with expiry time T_1 and payoff as given by the last formula?

You later say (still assuming deterministic smile model) that the price of the optional cliquet is max( C(1) - K, 0). I'm not really sure what C(1) is...presumably you meant C(S_{T_1}) (= 0.4\sigma(S_{T_1})\sqrt(T) as above) so that the value of the optional cliquet is the value of a European call option with payoff max( C(S_{T_1}) - K, 0)?

I realise that the point you’re trying to make is that using a deterministic smile model is incorrect, but until I understand exactly how to price using such a model, I’m not really understanding much of this final section. Clearly I have misunderstood something....where am I going wrong?

3) Finally, what exactly do you mean by the strike ratio (when you are looking at a portfolio of two cliquets with different strikes). I can’t seem to deduce that the portfolio consisting of the difference of the two options (as in the example you give) has 0 value…not sure how you get this?

Thanks!