I didn’t think utility mattered at the start, but the numbers get so large so fast that it probably should. So the first issue is identifying if you are theoretically optimizing (max value in min time) or satisficing. Optimizing is simple—always invest when +EV exists/sell when -EV exists. Even if you ruin 99% of the time, you’ll average the highest returns (1 - note below on actual optimization).
The only reason this is complicated is because of the second issue; the risk of ruin. This issue is actually minor in a normal random walk situation.
I’ve simulated a few rules and assumptions to validate the argument. Since this leverages timing, even a negative average rate of return in the underlying shows a very high rate of return for the strategy. http://imgur.com/a/ZrH6g for a quick snapshot. There are a few issues that I didn’t bother to fix (namely with compound rules of holding onto stock), as the effect differs in only a minor way. For reference, actual mean should be about 1.0002, but I’ve only shown 0.9, 1, and 1.1.
The issue becomes more complicated if you stop assuming a normal model. The risk of ruin increases dramatically for any probability of major loss. To mitigate this risk, simply have a trading rule that prevents that risk from materializing, ie: don’t buy if this risk exists, sell if it does. There are lots of ways, both technical and not, to implement this rule. I ran it as a maximum of the sum of negative impact (probability x loss) using two distinct normal distributions (emulating a binary market choice, each with their own distributions). It works, which means the proper technical approach would be even better.
In general, the rule is simple; buy when the average return is positive, hold until it the average return is negative, sell when return is negative. To protect against the risk of ruin, reject any major possibility of loss; eg: do not trade when there is a high SD (or a lumpy negative distribution).
(1) Actual optimization is pretty complicated. All of the above assumes that you get the information and make the transaction immediately. However, true optimization would involve compensating for random walks away from the information you know to be true on the assumption that it would return to the mean. For instance, with the knowledge of the actual range you could break the “buy and hold” strategy below certain thresholds, such as if the stock random walked much higher/lower than would be expected (eg: bought expecting an average of 1.05 → random walked to 1.10; sell at 1.10 as it is probable to return to 1.05 by end of day).
I didn’t think utility mattered at the start, but the numbers get so large so fast that it probably should. So the first issue is identifying if you are theoretically optimizing (max value in min time) or satisficing. Optimizing is simple—always invest when +EV exists/sell when -EV exists. Even if you ruin 99% of the time, you’ll average the highest returns (1 - note below on actual optimization).
The only reason this is complicated is because of the second issue; the risk of ruin. This issue is actually minor in a normal random walk situation.
I’ve simulated a few rules and assumptions to validate the argument. Since this leverages timing, even a negative average rate of return in the underlying shows a very high rate of return for the strategy. http://imgur.com/a/ZrH6g for a quick snapshot. There are a few issues that I didn’t bother to fix (namely with compound rules of holding onto stock), as the effect differs in only a minor way. For reference, actual mean should be about 1.0002, but I’ve only shown 0.9, 1, and 1.1.
The issue becomes more complicated if you stop assuming a normal model. The risk of ruin increases dramatically for any probability of major loss. To mitigate this risk, simply have a trading rule that prevents that risk from materializing, ie: don’t buy if this risk exists, sell if it does. There are lots of ways, both technical and not, to implement this rule. I ran it as a maximum of the sum of negative impact (probability x loss) using two distinct normal distributions (emulating a binary market choice, each with their own distributions). It works, which means the proper technical approach would be even better.
In general, the rule is simple; buy when the average return is positive, hold until it the average return is negative, sell when return is negative. To protect against the risk of ruin, reject any major possibility of loss; eg: do not trade when there is a high SD (or a lumpy negative distribution).
(1) Actual optimization is pretty complicated. All of the above assumes that you get the information and make the transaction immediately. However, true optimization would involve compensating for random walks away from the information you know to be true on the assumption that it would return to the mean. For instance, with the knowledge of the actual range you could break the “buy and hold” strategy below certain thresholds, such as if the stock random walked much higher/lower than would be expected (eg: bought expecting an average of 1.05 → random walked to 1.10; sell at 1.10 as it is probable to return to 1.05 by end of day).