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the payoff curve isA= sum(max(3%-r(i),0)) i = 1 to 12payoff= max(36%-A,0)any idea to price it? ( No Monte carlo)Thanks

Is this an interview question?A = sum of puts. payoff = put on A. Hence this is a vol-of-vol trade.The sum of puts is a portfolio. The components (puts) have some vols and correlations - they are linked to vol-of-vol & autocorrelation of the underlier, use insert-your-favourite-stoch-vol-model pricing formula for put prices conditional on stoch variance. With standard stochastic vol models, A is a sum of correlated ln variables with some vols, hence it is a ln variable with some vol. Your payoff is a put on that, BS formula. Correlation comes from the fact that put(i) is conditioned on r(i-1), + autocorrelation. Now do all the integrals. Hence the price is determined by vol-of-vol & autocorrelation of the underlier. If you are conservative and put correlation=1, you don't have to do the integrals, although presumably nobody will buy the product then unless you are very cheap on vol-of-vol. If you are careless and put correlation=0, people will buy it but you may not like the results.... Do I pass?

Only one undelying , let's assume it is S&P500. And for r(i), should be corrected by r(t) where t=1 2,...,12 12 months.thanks

That is what I wrote. A portfolio of puts of different expiries on the same underlier.

Correlation comes from the fact that put(i) is conditioned on r(i-1), + autocorrelation. Now do all the integrals. I do not get it. Can yo explain more in detail? or could you write a formula here?My understanding is:r(t) where t=1,2,...,12 are i.i.d N(a,b)does the autocorrelation come from the vol(t)?Thanks

Your understanding is wrong. Returns are iid, not prices.In BS, conditional on s(t_i) = s_i, s(t_i+1) is LN( s_i, vol * root(t_i+1 - t_i) )So price of put(t_i+1) is not independent of put(t_i). Integrate over all possibilities for s_i, calculating the prices of put(t_i). For i=1..n.If this was a cliquet of puts, the put payouts would be iid. As you wrote it, they are not.Autocorrelation comes on top of that, allowing for the fact that returns may not be iid.

r(t) is the return of underlying. Not the price.so r(t) where t=1,2,..,12 are i.i.dCould you write down the integral with autocorrelation.Many thanks

quantworm,You have got to work on your communication skills...Autocorrelation can mean many things to many people. Try r_i = a r_i-1 + root(1-a^2) W_i where a is the (constant) autocorrelation and W_i's are independent. You can play with time decay or state dependence if you feel like it.Normally you wouldn't bother with autocorrelation, but I think your structure is quite sensitive to it - ie. your portfolio vol can go from almost 0 to asset vol depending on how you mark it, and that is what your final payout depends on.You haven't told me, do I pass?