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I'm doing a simulation of a CRR model and I'm trying to find parameters in order for the successive $S_t$s (stock prices) to be martingale.

I'm assuming that if I'd create a function (and picked the right parameter values) that's equivalent to the martingale condition $\mathbb{E}(\frac{S_{t+1}}{S_t})=1$ and then plot it over $t=1,...,T$ (time), then I'd see "martingale"?

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  • $\begingroup$ Hi mavavilj, welcome to Quant.SE! Please not the faq about what we expect here and please make your posts conform to those rules. This question is a bit unclear to me. What do you mean by seeing a martingale? It's an abstract property of a stochastic process. $\endgroup$ – Bob Jansen Nov 30 '15 at 6:51
  • $\begingroup$ I was thinking that it could be "seen" by plotting the process? $\endgroup$ – mavavilj Nov 30 '15 at 7:31
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    $\begingroup$ As I recall in introductory stochastic calculus, a stochastic process is a martingale if its drift is zero. That probably had other assumptions I don't recall at the moment math.stackexchange.com/questions/1035461/… $\endgroup$ – BCLC Dec 6 '15 at 15:46
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When you plot a martingale you might see some random movements up and down, but no overall tendency to increase (or decrease) over time, in other words no clear upward or downward trend. On the other hand a strong increase over time (such as a graph of world population over the last century) is a sign of a submartingale and a tendency to go down a supermartingale. Of course the visual impression is not enough, these are abstract properties that have to be investigated mathematically. HTH

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  • $\begingroup$ This is related to drift? I seem to recall driftlessness implies martingale, but googling reveals local martingale but not necessarily martingale. $\endgroup$ – BCLC Dec 6 '15 at 15:48

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