Crash cliquet price

Question

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.

Answer 1

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.

I wrote my dissertation on barrier options a couple of years ago. You might be able to find some inspiration there. You can find it on github along with the matlab code I wrote for the project. (I think I accidentally pushed a non-final version of the actual dissertation, but it should still be fine to read).

If what you described is the model you will be working under, I'm afraid only the method described in Chapter 2 may be used, since it's very general. I'm not exactly sure, though, so maybe read through the other chapters as well? (If nothing else, then just to witness the beauty of the Sequential Monte Carlo method described in Chapter 4.)

I'm not sure this answers your question, but maybe I have given you a good starting point.