Chapter 18 - cliquets

[MattT]

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!

[mj]

1 yes

2 in this case C(t_1,S_{t_1}) does not depend on S_{t_1} so we are writing C(1) to denote the price at t_1

3 if we assume zero interest rates and dividends then the put at 1/1.1 and call at 1.1 are worth approximately the same cf theorem 10.1

[MattT]

Thanks for your reply Mark.

I'm still a bit puzzled by question 2. Let me write \mu(S, K, t, T) for the value of implied volatility at time t of a call option struck at K, when spot is S, and expiry time is T. This function is well defined for any model with future deterministic smiles.

Then, I think what you are saying is the value of the cliquet (at time t_1, when stock price is S_{t_1}) is C(t_1, S_{t_1}) = (approx) some (linear) function f( \mu(S_{t_1}), S_{t_1}, t_1, t_2) ). In fact, we have approx C(t_1, S_{t_1}) = 0.4 \mu(S_{t_1}, S_{t_1}, t_1, t_2) sqrt(t_2 - t_1).

Now as S_{t_1} changes, the RHS of this last expression changes (unless the model also has floating smiles). So the conclusion I get is that C(t_1, S_{t_1}) will depend on S_{t_1} for any model which has future deterministic smiles which stick to strike. Is this right?

If so, I think I see your point; you said that equities markets display floating smiles, so one would assume if working with a deterministic model, we would be working with a model with a floating smile, and then, as you say there is no S_{t_1} dependency....

[mj]

the text could have been better phrased. We are implicitly assuming that the AS_{T_1} implied vol is known in advance.