# Crash cliquet price

Denote by $n$ the n-th trading day in a year and by $S_n$ the stock price on that day. An instrument expirying in 1 year pays $\max(0,1-\frac{S_n}{S_{n-1}})$ and early terminates if $\frac{S_n}{S_{n-1}}<0.8$ on any day $n$ before and including the expiry. Let's assume that number of jumps in one year follow Poisson distribution with $\lambda>0$. Also assume that days on which jumps occur are distributed uniformly and that on days when no jump occurs the stock price stays constant (i.e. jumps are the only driver of the moves in the stock price). Let us also assume that jump distribution is time-invariant, I.e. distribution of $\frac{S_n}{S_{n-1}}$ on a day when jump occured is the same for each $n$. Also assume that the jump size is independent of the number of jumps. If I know the expected value of $\max(0,1-\frac{S_n}{S_{n-1}})$ conditional on jump occuring on day $n$, how can I calculate the value of such instrument in this simplified model? I thought about this:

$$V=DF(0,0.5y)\cdot P(\mbox{at least one jump occurs before the expiry}) \cdot \phi$$

But I'm not quite sure this gives a correct answer.

Defining $\tilde{S}_n = S_n/S_{n-1}$ (which is well defined, assuming $S_n > 0$ for all $n$), the problem becomes that of barrier option pricing. In particular, you're looking to price a down-and-out barrier option.