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Random Walk - Market Invariants

Posted: May 27th, 2019, 7:23 pm
by TheCorpFinanceQuant
Reading through Atillio Meucci's questions for his Risk and Asset Allocation book. One of the questions 

Q3.1.1 - Generate a Merton jump-diffusion process Xat discrete times with arbitrary parameters. What are the invariants in this process?

Any ideas on what these invariants would be?

Re: Random Walk - Market Invariants

Posted: May 27th, 2019, 9:49 pm
by Alan
A brief google suggests his "invariants" just mean (a set of) IID variates. In that model, as in every exponential Levy process, the log-returns over (disjoint) fixed intervals are IID variates. So the answer is likely any set: [$]\{\log P(t_i)/P(t_{i-j})\}[$], with [$]P(t_i)[$] the model prices, and [$]j[$] some fixed integer. (Again, non-overlapping entries). The terminology seems idiosyncratic. I avoided using your [$]X_t[$] because I don't know if those are prices or log-prices.