Recognizing Martingale Behavior in Simulated Stock Prices

Summary

The document asks how a martingale should appear in a simulation of a Cox–Ross–Rubinstein stock-price model. It proposes checking whether the expected ratio of successive prices is one, then wonders whether plotting prices through time would reveal the property. The answer describes the visual pattern one might expect: random movements without a clear overall upward or downward tendency.

That visual cue is only an intuition, not a test. A persistent rise or fall may suggest submartingale or supermartingale behavior, respectively, but a plot alone cannot establish these mathematical properties. They must be assessed through the relevant conditional expectation rather than inferred from one simulated path. The discussion offers no parameter choices, simulation results, or formal testing procedure, so it serves as a conceptual distinction between a path’s appearance and a martingale condition.

Key ideas

• A martingale path may fluctuate while showing no clear overall upward or downward tendency.
• A persistent expected rise is associated with submartingale behavior, while a fall suggests supermartingale behavior.
• A visual inspection of a simulated path cannot establish the martingale property.
• The property must be checked mathematically using conditional expectations.

Tags

# What does martingale look like?


# What does martingale look like?












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"?

## Answer by Alex C (score 2, accepted)

https://quant.stackexchange.com/a/22122

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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