Pricing a Crash Cliquet as a Down-and-Out Barrier Option
Summary
The document describes a one-year daily-reset payoff based on negative stock returns, with early termination if the stock’s daily ratio crosses a downside threshold. Its simplified assumptions make the problem a jump model: jump arrivals follow a Poisson distribution, jump days are uniformly distributed, and the jump-size distribution is constant and independent of the number of jumps. The question asks whether a proposed value based on the probability of at least one jump times the expected conditional payoff is sufficient.
The answer reframes the contract as a down-and-out barrier option on daily price ratios. It points to general barrier-option valuation methods and mentions sequential Monte Carlo as one possible numerical technique. No pricing derivation or numerical result is supplied, so the suggested classification is a starting point rather than a complete solution. Valuation still requires accounting for the path-dependent knock-out condition and the stated jump process; the answer does not establish that the proposed shortcut captures these effects.
Key ideas
- The daily ratio payoff and early termination rule create a path-dependent barrier feature.
- The contract can be viewed as a down-and-out barrier option on successive price ratios.
- A Poisson jump model alone does not make the proposed jump-probability-times-payoff shortcut a demonstrated valuation method.
- General barrier-option methods, including sequential Monte Carlo, are suggested as possible approaches.
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Full text
# Crash cliquet price
# 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.
## Answer by torbonde (score 1)
https://quant.stackexchange.com/a/41787
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.